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Forbidden Induced Subgraphs and the Łoś-Tarski Theorem

Yijia Chen
School of Computer Science
Fudan University
yijiachen@fudan.edu.cn
   Jörg Flum
Mathematisches Institut
Universität Freiburg
joerg.flum@math.uni-freiburg.de
Abstract

Let 𝒞\mathscr{C} be a class of finite and infinite graphs that is closed under induced subgraphs. The well-known Łoś-Tarski Theorem from classical model theory implies that 𝒞\mathscr{C} is definable in first-order logic (FO) by a sentence φ\varphi if and only if 𝒞\mathscr{C} has a finite set of forbidden induced finite subgraphs. It provides a powerful tool to show nontrivial characterizations of graphs of small vertex cover, of bounded tree-depth, of bounded shrub-depth, etc. in terms of forbidden induced finite subgraphs. Furthermore, by the Completeness Theorem, we can compute from φ\varphi the corresponding forbidden induced subgraphs. We show that this machinery fails on finite graphs.

  • There is a class 𝒞\mathscr{C} of finite graphs which is definable in FO and closed under induced subgraphs but has no finite set of forbidden induced subgraphs.

  • Even if we only consider classes 𝒞\mathscr{C} of finite graphs which can be characterized by a finite set of forbidden induced subgraphs, such a characterization cannot be computed from an FO-sentence φ\varphi, which defines 𝒞\mathscr{C}, and the size of the characterization cannot be bounded by f(|φ|)f(|\varphi|) for any computable function ff.

Besides their importance in graph theory, the above results also significantly strengthen similar known results for arbitrary structures.

1 Introduction

Many classes of graphs can be defined by a finite set of forbidden induced finite subgraphs. One of the simplest examples is the class of graphs of bounded degree. Let d1d\geq 1 and d\mathscr{F}_{d} consist of all graphs with vertex set {1,,d+2}\{1,\ldots,d+2\} and maximum degree exactly d+1d+1. Then a graph GG has degree at most dd if and only if no graph in d\mathscr{F}_{d} is isomorphic to an induced subgraph of GG. Less trivial examples include graphs of small vertex cover (attributed to Lovász [9]), of bounded tree-depth [5], and of bounded shrub-depth [13]. As a matter of fact, understanding forbidden induced subgraphs for those graph classes is an important question in structural graph theory [7, 21, 12, 11]. However, a straightforward adaptation of a result in [10] shows that it is in general impossible to compute the forbidden induced subgraphs from a description of classes of graphs by Turing machines.

It is folklore [1, 17] that characterization by finitely many forbidden induced finite subgraphs is equivalent to definability by a universal sentence of first-order logic (FO). But only very recently, it was realized [2] that such a characterization can be further understood by the Łoś-Tarski theorem. Łoś [15] and Tarski [19] proved the first so-called preservation theorem of classical model theory. In its simplest form it says that the class Graph(φ)\textsc{Graph}(\varphi) of finite and infinite graphs that are models of a sentence φ\varphi of first-order logic is closed under induced subgraphs (or, that φ\varphi is preserved under induced subgraphs) if and only if there is a universal FO-sentence μ\mu with Graph(φ)=Graph(μ)\textsc{Graph}(\varphi)=\textsc{Graph}(\mu). Recall that a universal sentence μ\mu is a sentence of the form x1xkμ0\forall x_{1}\ldots\forall x_{k}\,\mu_{0}, where μ0\mu_{0} is quantifier-free.

Such a universal sentence μ=x1xkμ0\mu=\forall x_{1}\ldots\forall x_{k}\,\mu_{0} expresses that certain patterns of induced subgraphs with at most kk vertices are forbidden. In fact, let \mathscr{F} be a finite set of finite graphs and denote by Forb()\textsc{Forb}(\mathscr{F}) the class of (finite and infinite) graphs that do not contain an induced subgraph isomorphic to a graph in \mathscr{F}. Then for a universal sentence μ\mu as above we have

Graph(μ)=Forb(k(μ)).\textsc{Graph}(\mu)=\textsc{Forb}\big{(}\mathscr{F}_{k}(\mu)\big{)}. (1)

Here for any FO-sentence φ\varphi and k1k\geq 1 by k(φ)\mathscr{F}_{k}(\varphi) we denote the class of graphs that are models of ¬φ\neg\varphi and whose universe is {1,,}\{1,\ldots,\ell\} for some \ell with 1k1\leq\ell\leq k. Clearly, k(φ)\mathscr{F}_{k}(\varphi) is finite.

We say that a class 𝒞\mathscr{C} of finite and infinite graphs is definable by a finite set of forbidden induced subgraphs if there is a finite set \mathscr{F} of finite graphs such that 𝒞=Forb()\mathscr{C}=\textsc{Forb}(\mathscr{F}). Hence the graph-theoretic version of the Łoś-Tarski Theorem can be restated in the form:

(I) Let 𝒞\mathscr{C} be a class of finite and infinite graphs. The following are equivalent:
    (i)  𝒞\mathscr{C} is closed under induced subgraphs and FO-axiomatizable.
(ii)  𝒞\mathscr{C} is axiomatizable by a universal sentence.
(iii)  𝒞\mathscr{C} is definable by a finite set of forbidden induced subgraphs.

This version of the Łoś-Tarski Theorem is already contained, at least implicitly, in the article [20] of Vaught published in 1954. In addition, it is easy to see that the equivalence between (ii) and (iii) holds for any class of finite graphs too.

Note that we have repeatedly mentioned that in the Łoś-Tarski Theorem graphs are allowed to be infinite. This is not merely a technicality. In [2], to obtain the forbidden induced subgraph characterization of graphs of bounded shrub-depth using the Łoś-Tarski Theorem, one simple but vital step is to extend the notion of shrub-depth to infinite graphs. Indeed, Tait [18] exhibited a class 𝒞\mathscr{C} of finite structures (which might be understood as colored directed graphs) which is closed under induced substructures and FO-axiomatizable. Yet, 𝒞\mathscr{C} is not definable by any universal sentence, thus cannot be characterized by a finite set of forbidden induced substructures. As the first result of this paper, we strengthen Tait’s result to graphs.

Theorem 1.1.

There is a class 𝒞\mathscr{C} of finite graphs with the following properties.

  1. (i)

    𝒞\mathscr{C} is closed under induced subgraphs and FO-axiomatizable,

  2. (ii)

    𝒞\mathscr{C} is not definable by a finite set of forbidden induced subgraphs.

Even though we are interested in structural and algorithmic results for classes of finite graphs, we see that in order to apply the Łoś-Tarski Theorem for such purposes we have to consider classes of finite and infinite graphs. So in this paper “graph” means finite or infinite graph. As in the preceding result we mention it explicitly if we only consider finite graphs.

Complementing Theorem 1.1 we show that it is even undecidable whether a given FO-definable class of finite graphs which is closed under induced subgraphs can be characterized by a finite set of forbidden induced subgraphs. More precisely:

Theorem 1.2.

There is no algorithm that for any FO-sentence φ\varphi such that

Graphfin(φ):={G|G is a finite graph and a model of φ}\textsc{Graph}_{\textup{fin}}(\varphi):=\big{\{}G\;\big{|}\;\text{$G$ is a finite graph and a model of $\varphi$}\big{\}}

is closed under induced subgraphs decides whether φ\varphi is equivalent to a universal sentence on finite graphs.

As mentioned at the beginning, for a class of finite graphs definable by a finite set of forbidden induced subgraphs, it is preferable to have an explicit construction of those graphs. This however turns out to be difficult for many natural classes of graphs. For example, the forbidden induced subgraphs are only known for tree-depth at most 33 [7]. Let us consider the kk-vertex cover problem for a constant k1k\geq 1. It asks whether a given graph has a vertex cover (i.e., a set of vertices that contains at least one endpoint of every edge) of size at most kk. The class of all yes-instances of this problem, finite and infinite, is closed under induced subgraphs and FO-axiomatizable by the FO-sentence

φVCk:=φGraphx1xkyz(Eyz1k(x=yx=z)),\varphi^{k}_{\textup{VC}}:=\varphi_{\textsc{Graph}}\wedge\exists x_{1}\ldots\exists x_{k}\forall y\forall z\Big{(}Eyz\to\bigvee_{1\leq\ell\leq k}(x_{\ell}=y\vee x_{\ell}=z)\Big{)},

where φGraph\varphi_{\textsc{Graph}} axiomatizes the class of graphs. Hence, by (I) the class of yes-instances can be defined by a finite set of forbidden induced subgraphs. As the reader will notice it is by no means trivial to find a universal sentence equivalent to φVCk\varphi^{k}_{\textup{VC}}. But on the other hand, by the Completeness Theorem, we can search for such a universal sentence by enumerating all possible universal sentences μ\mu and all possible proofs for φVCkμ\vdash\varphi^{k}_{\textup{VC}}\leftrightarrow\mu, and then extract the corresponding forbidden induced subgraphs from μ\mu as in (1).

To explain the hardness of constructing forbidden induced subgraphs, we prove two negative results.

Theorem 1.3.

There is no algorithm that for any FO-sentence φ\varphi which is equivalent to a universal sentence μ\mu on finite graphs computes such a μ\mu.

Or equivalently, there is no algorithm that for any FO-sentence φ\varphi such that

Graphfin(φ)=Forbfin()\textsc{Graph}_{\textup{fin}}(\varphi)=\textsc{Forb}_{\textup{fin}}(\mathscr{F})

for a finite set \mathscr{F} of graphs computes such an \mathscr{F}. Here,

Forbfin():={G|G is a finite graph without induced subgraph isomorphic to a graph in }.\textsc{Forb}_{\textup{fin}}(\mathscr{F}):=\big{\{}G\;\big{|}\;\text{$G$ is a finite graph without induced subgraph isomorphic to a graph in $\mathscr{F}$}\big{\}}.
Theorem 1.4.

Let f:f:\mathbb{N}\to\mathbb{N} be a computable function. Then there is a class 𝒞\mathscr{C} of finite graphs and an FO-sentence φ\varphi such that

  1. (i)

    𝒞=Graphfin(φ)\mathscr{C}=\textsc{Graph}_{\textup{fin}}(\varphi).

  2. (ii)

    𝒞=Graphfin(μ)\mathscr{C}=\textsc{Graph}_{\textup{fin}}(\mu) for some universal sentence μ\mu, in particular 𝒞\mathscr{C} is closed under induced subgraphs.

  3. (iii)

    For every universal sentence μ\mu with 𝒞=Graphfin(μ)\mathscr{C}=\textsc{Graph}_{\textup{fin}}(\mu) we have |μ|f(|φ|)|\mu|\geq f(|\varphi|).

Theorem 1.3 significantly strengthens the aforementioned result of [10]. Even if a class 𝒞\mathscr{C} of finite graphs definable by a finite set of forbidden induced subgraphs is given by an FO-sentence φ\varphi with 𝒞=Graphfin(φ)\mathscr{C}=\textsc{Graph}_{\textup{fin}}(\varphi), instead of a much more powerful Turing machine, we still cannot compute an appropriate finite set of forbidden induced subgraphs for 𝒞\mathscr{C} from φ\varphi. On top of it, Theorem 1.4 implies that the size of forbidden subgraphs for 𝒞\mathscr{C} cannot be bounded by any computable function in terms of the size of φ\varphi.

There is an important precursor for Theorem 1.4,

Theorem 1.5 (Gurevich’s Theorem [14]).

Let f:f:\mathbb{N}\to\mathbb{N} be computable. Then there is an FO-sentence φ\varphi such that the class Mod(φ)\textsc{Mod}(\varphi) of models of φ\varphi is closed under induced substructures but for every universal sentence μ\mu with Modfin(μ)=Modfin(φ)\textsc{Mod}_{\textup{fin}}(\mu)=\textsc{Mod}_{\textup{fin}}(\varphi) we have |μ|f(|φ|)|\mu|\geq f(|\varphi|).

Hence, Theorem 1.4 can be viewed as the graph-theoretic version of Theorem 1.5.

Besides its importance in graph theory, Theorem 1.4 is also relevant in the context of algorithmic model theory. For algorithmic applications, the Łoś-Tarski theorem provides a normal form (i.e., a universal sentence) for any FO-sentence preserved under induced substructures. In [3], it is shown that on labelled trees there is no elementary bound on the length of the equivalent universal sentence in terms of the original one. We should point out that Theorem 1.4 is not comparable to Theorem 6.1 in [3], since our lower bound is uncomputable (and thus, much higher than non-elementary) while the classes of graphs we construct in the proof are dense (thus very far from trees).

Our technical contributions.

For every vocabulary it is well known that the class of structures of this vocabulary is FO-interpretable in graphs (see for example [8]). So one might expect that Theorem 1.1 and Theorem 1.4 can be derived easily from Tait’s Theorem and Gurevich’s Theorem using the standard FO-interpretations. However, an easy analysis shows that those interpretations result in classes of graphs that are not closed under induced subgraphs. So we introduce the notion of strongly existential interpretation which translates any class of structures preserved under induced substructures to a class of graphs closed under induced subgraphs. A lot of care is needed to construct strongly existential interpretations.

Related research.

Let us briefly mention some further results related to the Łoś-Tarski Theorem. Essentially one could divide them into three categories: (a) The positive results showing that for certain classes 𝒞\mathscr{C} of finite structures the analogue of the Łoś-Tarski Theorem holds if we restrict to structures in 𝒞\mathscr{C}. For example, this is the case if 𝒞\mathscr{C} is the class of all finite structures of tree-width kk or less for some kk\in\mathbb{N} [1] or if 𝒞\mathscr{C} is the class of all finite structures whose hypergraph satisfies certain properties [6]. (b) Both just mentioned papers contain also negative results, i.e, classes for which the analogue of the Łoś-Tarski Theorem fails: For example, in [1] this is shown for the class of finite planar graphs. (c) The third category contains generalizations of the Łoś-Tarski Theorem. For example, in [17] the authors, for k1k\geq 1 consider sentences of the form x1xkμ\exists x_{1}\ldots\exists x_{k}\mu, where μ\mu is universal. Then the role of the closure under induced substructures is taken over by a semantic “core property PS(kk)” which for k=0k=0 coincides with closure under induced substructures. Finally, we mention that in [4] the authors strengthen Tait’s result by showing that for every n1n\geq 1 there are first-order definable classes of finite structures closed under substructures which are not definable with nn quantifier alternations.

Organization of this paper.

In Section 2 we fix some notations and recall or derive some results about universal sentences we need in this paper. For the reader’s convenience, in Section 3 we include a proof of Tait’s result. Moreover, we prove a technical result, i.e., Proposition 3.11, which is an important tool in Gurevich’s Theorem. We introduce the concept of strongly existential interpretation in Section 4 and show that the results of the preceding section remain true under such interpretations. We present an appropriate strongly existential interpretation for graphs (in Section 5). Hence, we get the results of Section 3 for graphs. In Section 6 we first derive Gurevich’s Theorem and apply our interpretations to get the results for graphs. Finally, in Section 7, we prove that various problems related to our results are undecidable.

2 Preliminaries

We denote by \mathbb{N} the set of natural numbers greater or equal to 0. For nn\in\mathbb{N} let [n]:={1,2,,n}[n]:=\{1,2,\ldots,n\}.

First-order logic FO.

A vocabulary τ\tau is a finite set of relation symbols. Each relation symbol has an arity. A structure 𝒜\mathcal{A} of vocabulary τ\tau, or τ\tau-structure, consists of a (finite or infinite) nonempty set AA, called the universe of 𝒜\mathcal{A} and of an interpretation R𝒜ArR^{\mathcal{A}}\subseteq A^{r} of each rr-ary relation symbol RτR\in\tau. If 𝒜\mathcal{A} and \mathcal{B} are τ\tau-structures, then 𝒜\mathcal{A} is a substructure of \mathcal{B}, denoted by 𝒜\mathcal{A}\subseteq\mathcal{B}, if ABA\subseteq B and R𝒜RR^{\mathcal{A}}\subseteq R^{\mathcal{B}}, and 𝒜\mathcal{A} is an induced substructure of \mathcal{B}, denoted by 𝒜ind\mathcal{A}\subseteq_{\textup{ind}}\mathcal{B}, if ABA\subseteq B and R𝒜=RArR^{\mathcal{A}}=R^{\mathcal{B}}\cap A^{r}, where rr is the arity of RR. If, in addition, ABA\subsetneq B, then 𝒜\mathcal{A} is an proper induced substructure of \mathcal{B}. By Str[τ]\textsc{Str}[\tau] (Strfin[τ]\textsc{Str}_{\textup{fin}}[\tau] ) we denote the class of all (of all finite) τ\tau-structures.

Formulas φ\varphi of first-order logic FO of vocabulary τ\tau are built up from atomic formulas x1=x2x_{1}=x_{2} and Rx1xrRx_{1}\ldots x_{r} (where RτR\in\tau is of arity rr and x1,x2,,xrx_{1},x_{2},\ldots,x_{r} are variables) using the boolean connectives ¬\neg, \wedge, and \vee and the universal \forall and existential \exists quantifiers. A relation symbol RR is positive (negative) in φ\varphi if all atomic subformulas RR\ldots in φ\varphi appear in the scope of an even (odd) number of negation symbols. By the notation φ(x¯)\varphi(\bar{x}) with x¯=x1,,xe\bar{x}=x_{1},\ldots,x_{e} we indicate that the variables free in φ\varphi are among x1,,xex_{1},\ldots,x_{e}. If then 𝒜\mathcal{A} is a τ\tau-structure and a1,,aeAa_{1},\ldots,a_{e}\in A, then 𝒜φ(a1,,ae)\mathcal{A}\models\varphi(a_{1},\ldots,a_{e}) means that φ(x¯)\varphi(\bar{x}) holds in 𝒜\mathcal{A} if xix_{i} is interpreted by aia_{i} for i[k]i\in[k].

A sentence is a formula without free variables. For a sentence φ\varphi we denote by Mod(φ)\textsc{Mod}(\varphi) the class of models of φ\varphi and Modfin(φ)\textsc{Mod}_{\textup{fin}}(\varphi) is its subclass consisting of the finite models of φ\varphi. Sentences φ\varphi and ψ\psi are equivalent if Mod(φ)=Mod(ψ)\textsc{Mod}(\varphi)=\textsc{Mod}(\psi) and finitely equivalent if Modfin(φ)=Modfin(ψ)\textsc{Mod}_{\textup{fin}}(\varphi)=\textsc{Mod}_{\textup{fin}}(\psi).

Graphs.

Let τE:={E}\tau_{E}:=\{E\} with binary EE. For all τE\tau_{E}-structures we use the notation G=(V(G),E(G))G=(V(G),E(G)) common in graph theory. Here V(G)V(G), the universe of GG, is the set of vertices, and E(G)E(G), the interpretation of the relation symbol EE, is the set of edges. The τE\tau_{E}-structure G=(V(G),E(G))G=(V(G),E(G)) is a directed graph if E(G)E(G) does not contain self-loops, i.e., (v,v)E(G)(v,v)\notin E(G) for any vV(G)v\in V(G). If moreover (u,v)E(G)(u,v)\in E(G) implies (v,u)E(G)(v,u)\in E(G) for any pair (u,v)(u,v), then GG is an (undirected) graph. The graph H=(V(H),E(H))H=(V(H),E(H)) is an induced subgraph of GG if

V(H)V(G)\displaystyle V(H)\subseteq V(G) and E(H)=E(G)(V(H)×V(H)).\displaystyle E(H)=E(G)\cap\big{(}V(H)\times V(H)\big{)}.

We denote by Graph and Graphfin\textsc{Graph}_{\textup{fin}} the class of all graphs and the class of finite graphs, respectively. Furthermore, for an FO[τE]\textup{FO}[\tau_{E}]-sentence φ\varphi by Graph(φ)\textsc{Graph}(\varphi)(and Graphfin(φ)\textsc{Graph}_{\textup{fin}}(\varphi)) we denote the class of graphs (and the class of finite graphs) that are models of φ\varphi.

Universal sentences and forbidden induced substructures.

An FO-formula is universal if it is built up from atomic and negated atomic formulas by means of the connectives \wedge and \vee and the universal quantifier \forall. Often we say that a formula, say, containing the connective \to is universal if by replacing φψ\varphi\to\psi by ¬φψ\neg\varphi\vee\psi (and “simple manipulations”) we get an equivalent universal sentence. Every universal sentence μ\mu is equivalent to a sentence of the form x1xkμ0\forall x_{1}\ldots\forall x_{k}\,\mu_{0} for some kk\in\mathbb{N} and some quantifier-free μ0\mu_{0} and moreover the length |μ||\mu| of μ\mu is at most |φ||\varphi|. If in the definition of universal formula we replace the universal quantifier by the existential one we get the definition of an existential formula.

One easily verifies that the class of models of a universal sentence is closed under induced substructures. As already mentioned in the Introduction for classes of graphs, Łoś [15] and Tarski [19] proved:

Theorem 2.1 (Łoś-Tarski Theorem).

Let τ\tau be a vocabulary and φ\varphi an FO[τ]\textup{FO}[\tau]-sentence. Then Mod(φ)\textsc{Mod}(\varphi) is closed under induced substructures if and only if φ\varphi is equivalent to a universal sentence.

We fix a vocabulary τ\tau. Let \mathscr{F} be a finite set of finite τ\tau-structures and denote by Forb()\textsc{Forb}(\mathscr{F}) (and Forbfin()\textsc{Forb}_{\textup{fin}}(\mathscr{F})) the class of structures (of finite structures) that do not contain an induced substructure isomorphic to a structure in \mathscr{F}. Clearly for finite sets FF and FF^{\prime} of finite τ\tau-structures we have

if \mathscr{F}\subseteq\mathscr{F}^{\prime}, then Forb()Forb()\textsc{Forb}(\mathscr{F}^{\prime})\subseteq\textsc{Forb}(\mathscr{F}). (2)

We say that a class 𝒞\mathscr{C} of τ\tau-structures (of finite τ\tau-structures) is definable by a finite set of forbidden induced substructures if there is a finite set \mathscr{F} of finite structures such that 𝒞=Forb()\mathscr{C}=\textsc{Forb}(\mathscr{F}) (𝒞=Forbfin()\mathscr{C}=\textsc{Forb}_{\textup{fin}}(\mathscr{F})).

Recall that τE={E}\tau_{E}=\{E\} with binary EE.

φDG:=x¬Exx\displaystyle\varphi_{\textup{DG}}:=\forall x\neg Exx and φGraph:=x¬Exxxy(ExyEyx)\displaystyle\varphi_{\textsc{Graph}}:=\forall x\neg Exx\wedge\forall x\forall y(Exy\to Eyx) (3)

axiomatize the classes of directed graphs and of graphs, respectively. Let the τE\tau_{E}-structures H0=(V(H0),E(H0))H_{0}=(V(H_{0}),E(H_{0})) and H1=(V(H1),E(H1))H_{1}=(V(H_{1}),E({H_{1}})) be given by

V(H0):={1},E(H0):={(1,1)}\displaystyle V(H_{0}):=\{1\},\ E(H_{0}):=\big{\{}(1,1)\big{\}} and V(H1):={1,2},E(H0):={(1,2)}.\displaystyle V(H_{1}):=\{1,2\},\ E(H_{0}):=\big{\{}(1,2)\big{\}}.

Then Forb({H0})\textsc{Forb}\big{(}\{H_{0}\}\big{)} and Forb({H0,H1})\textsc{Forb}\big{(}\{H_{0},H_{1}\}\big{)} are the class of directed graphs and the class of graphs, respectively, i.e., Mod(φDG)=Forb({H0})\textsc{Mod}(\varphi_{\textup{DG}})=\textsc{Forb}\big{(}\{H_{0}\}\big{)} and Mod(φGraph)=Forb({H0,H1})\textsc{Mod}(\varphi_{\textsc{Graph}})=\textsc{Forb}\big{(}\{H_{0},H_{1}\}\big{)}.

The following result generalizes this simple fact and establishes the equivalence between axiomatizability by a universal sentence and definability by a finite set of forbidden induced substructures. For an arbitrary vocabulary τ\tau, an FO[τ]\textup{FO}[\tau]-sentence φ\varphi, and k1k\geq 1 let

k(φ):={𝒜Str[τ]|𝒜¬φ and A=[] for some [k]}.\mathscr{F}_{k}(\varphi):=\big{\{}\mathcal{A}\in\textsc{Str}[\tau]\;\big{|}\;\mathcal{A}\models\neg\varphi\text{\ and $A=[\ell]$ for some $\ell\in[k]$}\big{\}}. (4)

Thus, k(φ)\mathscr{F}_{k}(\varphi) is, up to isomorphism, the class of structures with at most kk elements which fail to be a model of φ\varphi. Note that 1(φDG)={H0}\mathscr{F}_{1}(\varphi_{\textup{DG}})=\{H_{0}\} and 1(φGraph)={H0,H1}\mathscr{F}_{1}(\varphi_{\textsc{Graph}})=\{H_{0},H_{1}\}. Clearly, for a τ\tau-sentence we have:

if Mod(φ)\textsc{Mod}(\varphi) is closed under indu ced substructures,\displaystyle\text{ced substructures},
then Mod(φ)Forb(k(φ)) for all k1.\displaystyle\text{then $\textsc{Mod}(\varphi)\subseteq\textsc{Forb}(\mathscr{F}_{k}(\varphi))$ for all $k\geq 1$}. (5)
Proposition 2.2.

For a class 𝒞\mathscr{C} of τ\tau-structures and k1k\geq 1 the statements (i) and (ii) are equivalent.

  1. (i)

    𝒞=Mod(μ)\mathscr{C}=\textsc{Mod}(\mu) for some universal sentence μ:=x1xkμ0\mu:=\forall x_{1}\ldots\forall x_{k}\,\mu_{0} with quantifier-free μ0\mu_{0}.

  2. (ii)

    𝒞=Forb()\mathscr{C}=\textsc{Forb}(\mathscr{F}) for some finite set \mathscr{F} of structures, all of at most kk elements.

If (i) holds for μ\mu, then 𝒞=Forb(k(μ))\mathscr{C}=\textsc{Forb}(\mathscr{F}_{k}(\mu)).

Proof : (i) \Rightarrow (ii) Let 𝒞=Mod(μ)\mathscr{C}=\textsc{Mod}(\mu) for μ\mu as in (i). Then Mod(μ)\textsc{Mod}(\mu) is closed under induced substructures and hence, 𝒞Forb(k(μ))\mathscr{C}\subseteq\textsc{Forb}\big{(}\mathscr{F}_{k}(\mu)\big{)} by (5).

Now assume that 𝒜𝒞\mathcal{A}\notin\mathscr{C}. Then 𝒜¬μ\mathcal{A}\models\neg\mu and hence there are a1,,akAa_{1},\ldots,a_{k}\in A with 𝒜¬μ0(a1,,ak)\mathcal{A}\models\neg\mu_{0}(a_{1},\ldots,a_{k}). For :=[a1,,ak]𝒜\mathcal{B}:=[a_{1},\ldots,a_{k}]^{\mathcal{A}}, the substructure of 𝒜\mathcal{A} induced by a1,,aka_{1},\ldots,a_{k}, we have ¬μ0(a1,,ak)\mathcal{B}\models\neg\mu_{0}(a_{1},\ldots,a_{k}) (as μ0\mu_{0} is quantifier-free) and thus, ¬μ\mathcal{B}\models\neg\mu. Therefore, \mathcal{B} is isomorphic to a structure in k(μ)\mathscr{F}_{k}(\mu) and therefore, 𝒜Forb(k(μ))\mathcal{A}\notin\textsc{Forb}\big{(}\mathscr{F}_{k}(\mu)\big{)}.

(ii) \Rightarrow (i) Let the τ\tau-structure 𝒜\mathcal{A} have at most kk elements and let a1,,aka_{1},\ldots,a_{k} be an enumeration of the elements of AA (possibly with repetitions). Let δ(𝒜;a1,,ak)\delta(\mathcal{A};a_{1},\ldots,a_{k}) be the conjunction of all literals (i.e., atomic or negated atomic formulas) λ(x1,,xk)\lambda(x_{1},\ldots,x_{k}) such that 𝒜λ(a1,,ak)\mathcal{A}\models\lambda(a_{1},\ldots,a_{k}). Then for every τ\tau-structure \mathcal{B} and b1,,bkBb_{1},\ldots,b_{k}\in B we have

δ(𝒜;a1,,ak)(b1,,bk)\displaystyle\mathcal{B}\models\delta(\mathcal{A};a_{1},\ldots,a_{k})(b_{1},\ldots,b_{k})\iff the clauses π(ai)=bi\pi(a_{i})=b_{i} for i[k]i\in[k]
define an isomorphism from 𝒜\mathcal{A} onto [b1,,bk][b_{1},\ldots,b_{k}]^{\mathcal{B}}. (6)

Now assume (ii), i.e., 𝒞=Forb()\mathscr{C}=\textsc{Forb}(\mathscr{F}) for some finite set \mathscr{F} of structures, all of at most kk elements. If \mathscr{F} is empty, then 𝒞=Mod(xx=x)\mathscr{C}=\textsc{Mod}(\forall x\,x=x). Otherwise for every 𝒜\mathcal{A}\in\mathscr{F} we fix an enumeration a1𝒜,,ak𝒜a^{\mathcal{A}}_{1},\ldots,a^{\mathcal{A}}_{k} of the elements of AA. We set

μ:=x1xk𝒜¬δ(𝒜;a1𝒜,,ak𝒜).\mu:=\forall x_{1}\ldots\forall x_{k}\bigwedge_{\mathcal{A}\in\mathscr{F}}\neg\delta(\mathcal{A};a^{\mathcal{A}}_{1},\ldots,a^{\mathcal{A}}_{k}).

Then Forb()=Mod(μ)\textsc{Forb}(\mathscr{F})=\textsc{Mod}(\mu). In fact, assume first that Mod(μ)\mathcal{B}\notin\textsc{Mod}(\mu). Then there are b1,,bkBb_{1},\ldots,b_{k}\in B and an 𝒜\mathcal{A}\in\mathscr{F} such that δ(𝒜;a1𝒜,,ak𝒜)(b1,,bk)\mathcal{B}\models\delta(\mathcal{A};a^{\mathcal{A}}_{1},\ldots,a^{\mathcal{A}}_{k})(b_{1},\ldots,b_{k}). By (6), then 𝒜\mathcal{A} is isomorphic to the induced substructure [b1,,bk][b_{1},\ldots,b_{k}]^{\mathcal{B}} of \mathcal{B}; hence, Forb()\mathcal{B}\notin\textsc{Forb}(\mathscr{F}).

Now assume Forb()\mathcal{B}\notin\textsc{Forb}(\mathscr{F}). Then there is an 𝒜\mathcal{A}\in\mathscr{F} and elements b1,,bkBb_{1},\ldots,b_{k}\in B such that the clauses π(ai𝒜)=bi\pi(a^{\mathcal{A}}_{i})=b_{i} for i[k]i\in[k] define an isomorphism from 𝒜\mathcal{A} onto [b1,,bk][b_{1},\ldots,b_{k}]^{\mathcal{B}}. By (6), then δ(𝒜;a1𝒜,,ak𝒜)(b1,,bk)\mathcal{B}\models\delta(\mathcal{A};a^{\mathcal{A}}_{1},\ldots,a^{\mathcal{A}}_{k})(b_{1},\ldots,b_{k}). Therefore, ¬μ\mathcal{B}\models\neg\mu, i.e., Mod(μ)\mathcal{B}\notin\textsc{Mod}(\mu). \Box

Corollary 2.3.

Let φ\varphi be a τ\tau-sentence and k1k\geq 1. Then

Mod(φ)=Forb(k(φ))\displaystyle\textsc{Mod}(\varphi)=\textsc{Forb}\big{(}\mathscr{F}_{k}(\varphi)\big{)} \displaystyle\iff φ\varphi is equivalent to a universal sentence
of the form x1xkμ0\forall x_{1}\ldots\forall x_{k}\,\mu_{0} with quantifier-free μ0\mu_{0}.

By (2) and (5) we get:

Corollary 2.4.

If Mod(μ)=Forb(k(μ))\textsc{Mod}(\mu)=\textsc{Forb}\big{(}\mathscr{F}_{k}(\mu)\big{)} for some universal μ\mu and some kk\in\mathbb{N}, then Mod(μ)=Forb((μ))\textsc{Mod}(\mu)=\textsc{Forb}\big{(}\mathscr{F}_{\ell}(\mu)\big{)} for all k\ell\geq k.

Corollary 2.5.

It is decidable whether two universal sentences are equivalent.

Proof : Let μ\mu and μ\mu^{\prime} be universal sentences. W.l.o.g. we may assume that μ=x1xkμ0\mu=\forall x_{1}\ldots\forall x_{k}\,\mu_{0} and μ=x1xμ0\mu^{\prime}=\forall x_{1}\ldots\forall x_{\ell}\,\mu^{\prime}_{0} with kk\leq\ell. By Corollary 2.3 and Corollary 2.4, we have

Mod(μ)=Forb((μ))\displaystyle\textsc{Mod}(\mu)=\textsc{Forb}\big{(}\mathscr{F}_{\ell}(\mu)\big{)} and Mod(μ)=Forb((μ)).\displaystyle\textsc{Mod}(\mu^{\prime})=\textsc{Forb}\big{(}\mathscr{F}_{\ell}(\mu^{\prime})\big{)}.

Thus μ\mu and μ\mu^{\prime} are equivalent if and only if (μ)=(μ)\mathscr{F}_{\ell}(\mu)=\mathscr{F}_{\ell}(\mu^{\prime}). The right hand side of this equivalence is clearly decidable. \Box

The last equivalence of this corollary shows:

Corollary 2.6.

For universal sentences μ\mu and μ\mu^{\prime} we have

μ and μ are equivalentμ and μ are finitely equivalent.\text{$\mu$ and $\mu^{\prime}$ are equivalent}\iff\text{$\mu$ and $\mu^{\prime}$ are finitely equivalent.}

The following consequence of Corollary 2.2 will be used in the next section.

Corollary 2.7.

Let m,km,k\in\mathbb{N} with m>km>k and let ψ0\psi_{0} and ψ1\psi_{1} be FO[τ]\textup{FO}[\tau]-sentences. Assume that 𝒜\mathcal{A} is a finite model of ψ0ψ1\psi_{0}\wedge\psi_{1} with at least mm elements and all its proper induced substructures with at most kk elements are models of ψ0¬ψ1\psi_{0}\wedge\neg\psi_{1}. Then ψ0¬ψ1\psi_{0}\wedge\neg\psi_{1} is not finitely equivalent to a universal sentence of the form μ:=x1xkμ0\mu:=\forall x_{1}\ldots\forall x_{k}\,\mu_{0} with quantifier-free μ0\mu_{0}.

Proof : For a contradiction assume Modfin(ψ0¬ψ1)=Modfin(μ)\textsc{Mod}_{\textup{fin}}(\psi_{0}\wedge\neg\psi_{1})=\textsc{Mod}_{\textup{fin}}(\mu) for μ\mu as above. As Mod(μ)=Forb(k(μ))\textsc{Mod}(\mu)=\textsc{Forb}\big{(}\mathscr{F}_{k}(\mu)\big{)} by Proposition 2.2, we get (applying the finitely equivalence of ψ0¬ψ1\psi_{0}\wedge\neg\psi_{1} and μ\mu to obtain the last equality)

Modfin(ψ0¬ψ1)=Modfin(μ)=Forbfin(k(μ))=Forbfin(k(ψ0¬ψ1)).\textsc{Mod}_{\textup{fin}}(\psi_{0}\wedge\neg\psi_{1})=\textsc{Mod}_{\textup{fin}}(\mu)=\textsc{Forb}_{\textup{fin}}\big{(}\mathscr{F}_{k}(\mu)\big{)}=\textsc{Forb}_{\textup{fin}}\big{(}\mathscr{F}_{k}(\psi_{0}\wedge\neg\psi_{1})\big{)}.

However, by the assumptions the structure 𝒜\mathcal{A} is contained in Modfin(ψ0¬ψ1)\textsc{Mod}_{\textup{fin}}(\psi_{0}\wedge\neg\psi_{1}) but not in the class Forbfin(k(ψ0¬ψ1))\textsc{Forb}_{\textup{fin}}(\mathscr{F}_{k}(\psi_{0}\wedge\neg\psi_{1})). \Box

Remark 2.8.

Let 𝒞\mathscr{C} be a class of τ\tau-structures closed under induced substructures. For an FO[τ]\textup{FO}[\tau]-sentence φ\varphi we set Mod𝒞(φ):={𝒜𝒞𝒜φ}\textsc{Mod}_{\mathscr{C}}(\varphi):=\{\mathcal{A}\in\mathscr{C}\mid\mathcal{A}\models\varphi\}. We say that the Łoś-Tarski Theorem holds for 𝒞\mathscr{C} if for every FO[τ]\textup{FO}[\tau]-sentence φ\varphi such that the class Mod𝒞(φ)\textsc{Mod}_{\mathscr{C}}(\varphi) is closed under induced substructures there is a universal sentence μ\mu such that

Mod𝒞(φ)=Mod𝒞(μ).\textsc{Mod}_{\mathscr{C}}(\varphi)=\textsc{Mod}_{\mathscr{C}}(\mu).

The following holds:

Let 𝒞\mathscr{C} and 𝒞\mathscr{C}^{\prime} be classes of τ\tau-structures closed under induced substructures with 𝒞𝒞\mathscr{C}^{\prime}\subseteq\mathscr{C}. Furthermore assume that there is a universal sentence μ0\mu_{0} such that 𝒞=Mod𝒞(μ0)\mathscr{C}^{\prime}=\textsc{Mod}_{\mathscr{C}}(\mu_{0}). If the analogue of the Łoś-Tarski Theorem holds for 𝒞\mathscr{C}, then it holds for 𝒞\mathscr{C}^{\prime}, too

In fact, for every FO[τ]\textup{FO}[\tau]-sentence φ\varphi we have Mod𝒞(φ)=Mod𝒞(μ0φ)\textsc{Mod}_{\mathscr{C}^{\prime}}(\varphi)=\textsc{Mod}_{\mathscr{C}}(\mu_{0}\wedge\varphi). Hence, if Mod𝒞(φ)\textsc{Mod}_{\mathscr{C}^{\prime}}(\varphi) is closed under induced substructures, then by assumption there is a universal μ\mu such that Mod𝒞(μ0φ)=Mod𝒞(μ)\textsc{Mod}_{\mathscr{C}}(\mu_{0}\wedge\varphi)=\textsc{Mod}_{\mathscr{C}}(\mu). Therefore, Mod𝒞(φ)=Mod𝒞(μ)=Mod𝒞(μ)\textsc{Mod}_{\mathscr{C}^{\prime}}(\varphi)=\textsc{Mod}_{\mathscr{C}}(\mu)=\textsc{Mod}_{\mathscr{C}^{\prime}}(\mu).

3 Basic ideas underlying the classical results

This section contains a proof of Tait’s Theorem telling us that the analogue of the Łoś-Tarski-Theorem fails if we only consider finite structures. Afterwards we refine the argument to derive a generalization, namely Proposition 3.11, which is a key result to get Gurevich’s Theorem.

We consider the vocabulary τ0:={<,Umin,Umax,S}\tau_{0}:=\{<,U_{\textup{min}},U_{\textup{max}},S\}, where << and SS (the successor relation) are binary relation symbols and UminU_{\textup{min}} and UmaxU_{\textup{max}} are unary.

Let φ0\varphi_{0}  be the conjunction of the universal sentences

  • x¬x<x\forall x\neg x<x,   xy(x<yx=yy<x)\forall x\forall y(x<y\vee x=y\vee y<x),  xyz((x<yy<z)x<z)\forall x\forall y\forall z((x<y\wedge y<z)\to x<z),  i.e., “<< is an ordering”

  • xy((Uminx(x=yx<y))\forall x\forall y\big{(}(U_{\textup{min}}\,x\to(x=y\vee x<y)\big{)}  i.e., “every element in UminU_{\textup{min}} is a minimum w.r.t. <<

  • xy((Umaxx(x=yy<x))\forall x\forall y\big{(}(U_{\textup{max}}\,x\to(x=y\vee y<x)\big{)}  i.e., “every element in UmaxU_{\textup{max}} is a maximum w.r.t. <<

  • xy(Sxyx<y)\forall xy(Sxy\to x<y)

  • xyz(x<y<z¬Sxz)\forall x\forall y\forall z(x<y<z\to\neg Sxz).

Note that from the axioms it follows that there is at most one element in UminU_{\textup{min}}, at most one in UmaxU_{\textup{max}}, and that SS is a subset of the successor relation w.r.t. <<. We call τ0\tau_{0}-orderings the models of φ0\varphi_{0}.

For τ0\tau_{0}-structures 𝒜\mathcal{A} and \mathcal{B} we write <𝒜\mathcal{B}\subseteq_{<}\mathcal{A} and say that \mathcal{B} is a <<-substructure of 𝒜\mathcal{A} if 𝒜\mathcal{A} is a substructure of \mathcal{B} with <=<𝒜(B×B)<^{\mathcal{B}}=<^{\mathcal{A}}\cap\,(B\times B).

We remark that the relation symbols Umin,UmaxU_{\textup{min}},\ U_{\textup{max}}, and SS are negative in φ0\varphi_{0}. Therefore we have:

Lemma 3.1.

Let <𝒜\mathcal{B}\subseteq_{<}\mathcal{A}. If 𝒜φ0\mathcal{A}\models\varphi_{0}, then φ0\mathcal{B}\models\varphi_{0}.

Let

φ1:=xUminxxUmaxxxy(x<yzSxz).\varphi_{1}:=\exists x\,U_{\textup{min}}\,x\wedge\exists xU_{\textup{max}}\,x\wedge\forall x\forall y(x<y\to\exists zSxz). (7)

We call models of φ0φ1\varphi_{0}\wedge\varphi_{1} complete τ0\tau_{0}-orderings. Clearly, for every k1k\geq 1 there is a unique, up to isomorphism, complete τ0\tau_{0}-ordering with exactly kk elements. The next lemma shows that all its proper <<-substructures are models of φ0¬φ1\varphi_{0}\wedge\neg\varphi_{1}.

Lemma 3.2.

Let 𝒜\mathcal{A} and \mathcal{B} be τ0\tau_{0}-structures. Assume that 𝒜φ0\mathcal{A}\models\varphi_{0} and \mathcal{B} is a finite <<-substructure of 𝒜\mathcal{A} that is a model of φ1\varphi_{1}. Then =𝒜\mathcal{B}=\mathcal{A} (in particular, 𝒜φ1\mathcal{A}\models\varphi_{1}).

Proof : By the previous lemma we know that φ0\mathcal{B}\models\varphi_{0}. Let B:={b1,,bn}B:=\{b_{1},\ldots,b_{n}\}. As <<^{\mathcal{B}} is an ordering, we may assume that

b1<b2<<bn1<bn.b_{1}<^{\mathcal{B}}b_{2}<^{\mathcal{B}}\ldots<^{\mathcal{B}}b_{n-1}<^{\mathcal{B}}b_{n}.

As (φ0φ1)\mathcal{B}\models(\varphi_{0}\wedge\varphi_{1}), we have Uminb1U_{\textup{min}}^{\mathcal{B}}b_{1}, UmaxbnU_{\textup{max}}^{\mathcal{B}}b_{n}, and Sbibi+1S^{\mathcal{B}}b_{i}b_{i+1} for i[n1]i\in[n-1]. As 𝒜\mathcal{B}\subseteq\mathcal{A}, everywhere we can replace the upper index B by A.

We show A=BA=B: Let aAa\in A. By 𝒜φ0\mathcal{A}\models\varphi_{0}, we have b1𝒜a𝒜bnb_{1}\leq^{\mathcal{A}}a\leq^{\mathcal{A}}b_{n}. Let i[n]i\in[n] be maximal with bi𝒜ab_{i}\leq^{\mathcal{A}}a. If i=ni=n, then bn=ab_{n}=a. Otherwise bi𝒜a<𝒜bi+1b_{i}\leq^{\mathcal{A}}a<^{\mathcal{A}}b_{i+1}. As S𝒜bibi+1S^{\mathcal{A}}b_{i}b_{i+1}, we see that bi=ab_{i}=a (by the last conjunct of φ0\varphi_{0}). Now 𝒜=\mathcal{A}=\mathcal{B} follows from 𝒜φ0\mathcal{A}\models\varphi_{0}. \Box

Corollary 3.3.

Every proper <<-substructure of a finite model of φ0φ1\varphi_{0}\wedge\varphi_{1} is a model of φ0¬φ1\varphi_{0}\wedge\neg\varphi_{1}.

The class of finite τ0\tau_{0}-orderings that are not complete is closed under <<-substructures but not axiomatizable by a universal sentence:

Theorem 3.4 (Tait’s Theorem).

The class Modfin(φ0¬φ1)\textsc{Mod}_{\textup{fin}}(\varphi_{0}\wedge\neg\varphi_{1}) is closed under <<-substructures (and hence, closed under induced substructures) but φ0¬φ1\varphi_{0}\wedge\neg\varphi_{1} is not finitely equivalent to a universal sentence.

Proof : Modfin(φ0¬φ1)\textsc{Mod}_{\textup{fin}}(\varphi_{0}\wedge\neg\varphi_{1}) is closed under <<-substructures: If 𝒜φ0¬φ1\mathcal{A}\models\varphi_{0}\wedge\neg\varphi_{1} and \mathcal{B} is a finite <<-substructure of 𝒜\mathcal{A}, then φ0\mathcal{B}\models\varphi_{0} (by Lemma 3.1). If ¬φ1\mathcal{B}\models\neg\varphi_{1}, we are done. If φ1\mathcal{B}\models\varphi_{1}, then 𝒜φ1\mathcal{A}\models\varphi_{1} by Lemma 3.2, which contradicts our assumption 𝒜¬φ1\mathcal{A}\models\neg\varphi_{1}.

Let kk\in\mathbb{N}. It is clear that there is a finite model 𝒜\mathcal{A} of φ0φ1\varphi_{0}\wedge\varphi_{1} with at least k+1k+1 elements. By Corollary 3.3 every proper induced substructure of 𝒜\mathcal{A} is a model of φ0¬φ1\varphi_{0}\wedge\neg\varphi_{1}. Therefore, by Corollary 2.7, the sentence φ0¬φ1\varphi_{0}\wedge\neg\varphi_{1} is not finitely equivalent to a universal sentence of the form μ:=x1xkμ0\mu:=\forall x_{1}\ldots\forall x_{k}\,\mu_{0} with quantifier-free μ0\mu_{0}. As kk was arbitrary, we get our claim. \Box

Remark 3.5.

A slight generalization of the previous proof shows that Modfin(φ0¬φ1)\textsc{Mod}_{\textup{fin}}(\varphi_{0}\wedge\neg\varphi_{1}) is not even axiomatizable by a Π2\Pi_{2}-sentence, i.e., by a sentence χ\chi of the form x1xky1yχ0\forall x_{1}\ldots\forall x_{k}\exists y_{1}\ldots\exists y_{\ell}\,\chi_{0} for some k,1k,\ell\geq 1 and quantifier-free χ0\chi_{0}. In fact, assume that Modfin(φ0¬φ1)=Modfin(χ)\textsc{Mod}_{\textup{fin}}(\varphi_{0}\wedge\neg\varphi_{1})=\textsc{Mod}_{\textup{fin}}(\chi). Again we choose a finite model 𝒜\mathcal{A} of φ0φ1\varphi_{0}\wedge\varphi_{1} with at least k+1k+1 elements. Then 𝒜⊧̸χ\mathcal{A}\not\models\chi. Hence there are a1,,akAa_{1},\ldots,a_{k}\in A with 𝒜¬y1yχ0(a1,,ak)\mathcal{A}\models\neg\exists y_{1}\ldots\exists y_{\ell}\,\chi_{0}(a_{1},\ldots,a_{k}). Then ¬y1yχ0(a1,,ak)\mathcal{B}\models\neg\exists y_{1}\ldots\exists y_{\ell}\,\chi_{0}(a_{1},\ldots,a_{k}), where :=[a1,,ak]𝒜\mathcal{B}:=[a_{1},\ldots,a_{k}]^{\mathcal{A}} is the substructure of 𝒜\mathcal{A} induced by a1,,aka_{1},\ldots,a_{k}. Hence, ⊧̸χ\mathcal{B}\not\models\chi and therefore, ⊧̸φ0¬φ1\mathcal{B}\not\models\varphi_{0}\wedge\neg\varphi_{1}. But this contradicts Corollary 3.3 as \mathcal{B} is a proper induced substructure of 𝒜\mathcal{A}.

Note that φ0¬φ1\varphi_{0}\wedge\neg\varphi_{1} is (equivalent to) a Σ2\Sigma_{2}-sentence, i.e., equivalent to the negation of a Π2\Pi_{2}-sentence.

We turn to a refinement of the previous statement that will be helpful to get Gurevich’s Theorem.

Definition 3.6.
  1. (a)

    Let τ\tau be obtained from the vocabulary τ0\tau_{0} by adding finitely many relation symbols “in pairs,” the standard RR together with its complement RcompR^{\textup{comp}} (intended as the complement of RR). The symbols RR and RcompR^{\textup{comp}} have the same arity and for our purposes we can restrict ourselves to unary or binary relation symbols (even though all results can be generalized to arbitrary arities). We briefly say that τ\tau is obtained from τ0\tau_{0} by adding pairs.

  2. (b)

    Let τ\tau be obtained from τ0\tau_{0} by adding pairs. We say that φ0τ\varphi_{0\tau} is a τ\tau-extension of φ0\varphi_{0} (where φ0\varphi_{0} is as above) if it is a universal sentence such that

    • (i)

      the sentence φ0\varphi_{0} is a conjunct of φ0τ\varphi_{0\tau},

    • (ii)

      the sentence R standardx¯(¬Rx¯¬Rcompx¯)\bigwedge_{R\textup{ standard}}\forall\bar{x}(\neg R\bar{x}\vee\neg R^{\textup{comp}}\bar{x}) is a conjunct of φ0τ\varphi_{0\tau},

    • (iii)

      besides << all relation symbols are negative in φ0τ\varphi_{0\tau} (if this is not the case for some new RR or RcompR^{\textup{comp}}, the idea is to replace any positive occurrence of RR or RcompR^{\textup{comp}} by ¬Rcomp\neg R^{\textup{comp}} and ¬R\neg R, respectively). For instance, we replace a subformula

      x<yRxy\displaystyle x<y\wedge Rxy\ \ by x<y¬Rcompxy.\displaystyle\ \ x<y\wedge\neg R^{\textup{comp}}xy.
  3. (c)

    Let τ\tau be obtained from τ0\tau_{0} by adding pairs. Then we set

    φ1τ:=φ1R standardx¯(Rx¯Rcompx¯),\varphi_{1\tau}:=\varphi_{1}\wedge\bigwedge_{R\textup{ standard}}\forall\bar{x}(R\bar{x}\vee R^{\textup{comp}}\bar{x}), (8)

    where φ1\varphi_{1} is as above (see (7)).

For a τ\tau-structure \mathcal{B} with φ0τφ1τ\mathcal{B}\models\varphi_{0\tau}\wedge\varphi_{1\tau} we have

R standard(x¯(¬Rx¯¬Rcompx¯)x¯(Rx¯Rcompx¯)).\mathcal{B}\models\bigwedge_{R\textup{ standard}}\Big{(}\forall\bar{x}(\neg R\bar{x}\vee\neg R^{\textup{comp}}\bar{x})\wedge\forall\bar{x}(R\bar{x}\vee R^{\textup{comp}}\bar{x})\Big{)}.

Hence,

if φ0τφ1τ, then (Rcomp) is the complement of R for standard Rτ.\text{if $\mathcal{B}\models\varphi_{0\tau}\wedge\varphi_{1\tau}$, then $(R^{\textup{comp}})^{\mathcal{B}}$ is the complement of $R^{\mathcal{B}}$ for standard $R\in\tau$}. (9)

Now we derive the analogues of Lemma 3.1–Theorem 3.4 essentially by the same proofs.

Lemma 3.7.

Let τ\tau be obtained from τ0\tau_{0} by adding pairs and let φ0τ\varphi_{0\tau} be an extension of φ0\varphi_{0}. If <𝒜\mathcal{B}\subseteq_{<}\mathcal{A} and 𝒜φ0τ\mathcal{A}\models\varphi_{0\tau}, then φ0τ\mathcal{B}\models\varphi_{0\tau}.

Proof : By Definition 3.6 (b) (iii) all relation symbols distinct from << are negative in φ0τ\varphi_{0\tau}. \Box

Lemma 3.8.

Let τ\tau be obtained from τ0\tau_{0} by adding pairs and let φ0τ\varphi_{0\tau} be an extension of φ0\varphi_{0}. Assume that 𝒜φ0τ\mathcal{A}\models\varphi_{0\tau} and that the finite <<-substructure \mathcal{B} of 𝒜\mathcal{A} is a model of φ1τ\varphi_{1\tau}. Then =𝒜\mathcal{B}=\mathcal{A} (in particular, 𝒜φ1τ\mathcal{A}\models\varphi_{1\tau}).

Proof : Let 𝒜τ0\mathcal{A}\upharpoonright\tau_{0} (and τ0\mathcal{B}\upharpoonright\tau_{0}) be the τ0\tau_{0}-structure obtained from 𝒜\mathcal{A} (from \mathcal{B}) by removing all relations in ττ0\tau\setminus\tau_{0}.

By Lemma 3.2 we know that τ0=𝒜τ0\mathcal{B}\upharpoonright\tau_{0}=\mathcal{A}\upharpoonright\tau_{0}. Furthermore, φ0τ\mathcal{B}\models\varphi_{0\tau} by the previous lemma; thus, φ0τφ1τ\mathcal{B}\models\varphi_{0\tau}\wedge\varphi_{1\tau}. Hence, by (9), (Rcomp)(R^{\textup{comp}})^{\mathcal{B}} is the complement of RR^{\mathcal{B}} for standard RR. Clearly, RR𝒜R^{\mathcal{B}}\subseteq R^{\mathcal{A}} and (Rcomp)(Rcomp)𝒜(R^{\textup{comp}})^{\mathcal{B}}\subseteq(R^{\textup{comp}})^{\mathcal{A}}. As A=BA=B and 𝒜\mathcal{A} is a model of the sentence R standardx¯(¬Rx¯¬Rcompx¯)\bigwedge_{R\textup{ standard}}\forall\bar{x}(\neg R\bar{x}\vee\neg R^{\textup{comp}}\bar{x}), we get R=R𝒜R^{\mathcal{B}}=R^{\mathcal{A}} and (Rcomp)=(Rcomp)𝒜(R^{\textup{comp}})^{\mathcal{B}}=(R^{\textup{comp}})^{\mathcal{A}}. \Box

Corollary 3.9.

Every proper <<-substructure of a finite model of φ0τφ1τ\varphi_{0\tau}\wedge\varphi_{1\tau} is a model of φ0τ¬φ1τ\varphi_{0\tau}\wedge\neg\varphi_{1\tau}.

By replacing in the proof of Tait’s Theorem the use of Lemma 3.1, Lemma 3.2, and Corollary 3.3 by Lemma 3.7, Lemma 3.8, and Corollary 3.9 respectively, we get:

Lemma 3.10.

Let τ\tau be obtained from τ0\tau_{0} by adding pairs and let φ0τ\varphi_{0\tau} be an extension of φ0\varphi_{0}. The class Modfin(φ0τ¬φ1τ)\textsc{Mod}_{\textup{fin}}(\varphi_{0\tau}\wedge\neg\varphi_{1\tau}) is closed under <<-substructures (and hence, closed under induced substructures) but φ0τ¬φ1τ\varphi_{0\tau}\wedge\neg\varphi_{1\tau} is not finitely equivalent to a universal sentence.

Perhaps the reader will ask why we do not introduce for << the “complement relation symbol” <comp<^{\textup{comp}} and add the corresponding conjuncts to φ0τ\varphi_{0\tau} and φ1τ\varphi_{1\tau} (or, to φ0\varphi_{0} and φ1\varphi_{1}) in order to get a result of the type of Lemma 3.8 (or already of the type of Lemma 3.2) where we can replace “<<-substructure” by “substructure.” The reader will realize that corresponding proofs of B=AB=A break down.

The next proposition provides a uniform way to construct FO-sentences that are only equivalent to universal sentences of large size, which is the core of the proof of Gurevich’s Theorem.

Proposition 3.11.

Again let τ\tau be obtained from τ0\tau_{0} by adding pairs and φ0τ\varphi_{0\tau} be an extension of φ0\varphi_{0}. Let m1m\geq 1 and γ\gamma be an FO[τ]\textup{FO}[\tau]-sentence such that

φ0τφ1τγ has no infinite model but a finite model with at least m elements.\text{$\varphi_{0\tau}\wedge\varphi_{1\tau}\wedge\gamma$ has no infinite model but a finite model with at least $m$ elements}. (10)

For

χ:=φ0τ(φ1τ¬γ)\chi:=\varphi_{0\tau}\wedge(\varphi_{1\tau}\to\neg\gamma)

the statements (a) and (b) hold.

  • (a)

    The class Mod(χ)\textsc{Mod}(\chi) is closed under <<-substructures.

  • (b)

    If μ:=x1xkμ0\mu:=\forall x_{1}\ldots\forall x_{k}\,\mu_{0} with quantifier-free μ0\mu_{0} is finitely equivalent to χ\chi, then kmk\geq m.

Proof : (a) Let 𝒜χ\mathcal{A}\models\chi and <𝒜\mathcal{B}\subseteq_{<}\mathcal{A}. Thus, φ0τ\mathcal{B}\models\varphi_{0\tau}. If ⊧̸φ1τ\mathcal{B}\not\models\varphi_{1\tau}, we are done. Assume φ1τ\mathcal{B}\models\varphi_{1\tau}. In case BB is infinite, we conclude by (10) that \mathcal{B} is a model of ¬γ\neg\gamma and hence of χ\chi. Otherwise BB is finite; then =𝒜\mathcal{B}=\mathcal{A} (by Lemma 3.8) and thus, χ\mathcal{B}\models\chi.

(b) According to (10) there is a finite model 𝒜\mathcal{A} of φ0τφ1τγ\varphi_{0\tau}\wedge\varphi_{1\tau}\wedge\gamma, i.e., of φ0τ¬(φ1τ¬γ)\varphi_{0\tau}\wedge\neg(\varphi_{1\tau}\to\neg\gamma), with at least mm elements. By Corollary 3.9 every proper induced substructure of 𝒜\mathcal{A} is not a model of φ1τ\varphi_{1\tau} and therefore, it is a model of φ0τ(φ1τ¬γ)\varphi_{0\tau}\wedge(\varphi_{1\tau}\to\neg\gamma). Hence by Corollary 2.7, φ0τ(φ1τ¬γ)\varphi_{0\tau}\wedge(\varphi_{1\tau}\to\neg\gamma) is not finitely equivalent to a universal sentence of the form μ:=x1xkμ0\mu:=\forall x_{1}\ldots\forall x_{k}\,\mu_{0} with k<mk<m and quantifier-free μ0\mu_{0}. \Box

Remark 3.12.

We can strengthen the statement (b) of the preceding proposition to:

If the Π2\Pi_{2}-sentence x1xky1yχ0\forall x_{1}\ldots\forall x_{k}\exists y_{1}\ldots\exists y_{\ell}\;\chi_{0} with quantifier-free χ0\chi_{0} is finitely equivalent to χ\chi, then kmk\geq m.

The proof is similar to that of the result in Remark 3.5 and is left to the reader.

4 The general machinery: strongly existential interpretations

We show that appropriate interpretations preserve the validity of Tait’s theorem and of the statement of Proposition 3.11. Later on these interpretations will allow us to get versions of the results for graphs.

Let τE:={E}\tau_{E}:=\{E\} with binary EE. As already remarked in the Preliminaries for all τE\tau_{E}-structures we use the notation G=(V(G),E(G))G=(V(G),E(G)) common in graph theory.

Let τ\tau be obtained from τ0\tau_{0} by adding pairs. Furthermore, let II be an interpretation of width 22 (we only need this case) of τ\tau-structures in τE\tau_{E}-structures. This means that II assigns to every unary relation symbol TτT\in\tau an FO[τE]\textup{FO}[\tau_{E}]-formula φT(x1,x2)\varphi_{T}(x_{1},x_{2}) and to every binary relation symbol TτT\in\tau an FO[τE]\textup{FO}[\tau_{E}]-formula φT(x1,x2,y1,y2)\varphi_{T}(x_{1},x_{2},y_{1},y_{2}); moreover, II selects an FO[τE]\textup{FO}[\tau_{E}]-formula φuni(x1,x2)\varphi_{\textup{uni}}(x_{1},x_{2}).

Then II assigns to every τE\tau_{E}-structure GG with Gx¯φuni(x¯)G\models\exists\bar{x}\varphi_{\textup{uni}}(\bar{x}) a τ\tau-structure GIG_{I}, which we often denote by 𝒪I(G)\mathcal{O}_{I}(G), defined by

  • OI(G):={a¯V(G)×V(G)|Gφuni(a¯)}O_{I}(G):=\big{\{}\bar{a}\in V(G)\times V(G)\;\big{|}\;G\models\varphi_{\textup{uni}}(\bar{a})\big{\}}

  • TOI(G):={a¯OI(G)|GφT(a¯)}T^{O_{I}(G)}:=\big{\{}\bar{a}\in O_{I}(G)\;\big{|}\;G\models\varphi_{T}(\bar{a})\big{\}}  for unary TτT\in\tau

  • TOI(G):={(a¯,b¯)OI(G)×OI(G)|GφT(a¯,b¯)}T^{O_{I}(G)}:=\big{\{}(\bar{a},\bar{b})\in O_{I}(G)\times O_{I}(G)\;\big{|}\;G\models\varphi_{T}(\bar{a},\bar{b})\big{\}}  for binary TτT\in\tau.

As the interpretation II is of width 22, we have

|OI(G)||V(G)|2.|O_{I}(G)|\leq|V(G)|^{2}. (11)

Recall that for every sentence φFO[τ]\varphi\in\textup{FO}[\tau] there is a sentence φIFO[τE]\varphi^{I}\in\textup{FO}[\tau_{E}] such that for all τE\tau_{E}-structures GG with Gx¯φuni(x¯)G\models\exists\bar{x}\varphi_{\textup{uni}}(\bar{x}) we have

(GI=)𝒪I(G)φGφI.\left(G_{I}=\right)\;\mathcal{O}_{I}(G)\models\varphi\iff G\models\varphi^{I}. (12)

For example, for the sentence φ=xyTxy\varphi=\forall x\forall y\,Txy we have

φI=x¯(φuni(x¯)y¯(φuni(y¯)φT(x¯,y¯))).\varphi^{I}=\forall\bar{x}\Big{(}\varphi_{\textup{uni}}(\bar{x})\to\forall\bar{y}\big{(}\varphi_{\textup{uni}}(\bar{y})\to\varphi_{T}(\bar{x},\bar{y})\big{)}\Big{)}.

Furthermore there is a constant cIc_{I}\in\mathbb{N} such that for all φFO[τ]\varphi\in\textup{FO}[\tau],

|φI|cI|φ|.|\varphi^{I}|\leq c_{I}\cdot|\varphi|. (13)
Definition 4.1.

Let τ\tau be obtained from τ0\tau_{0} by adding pairs and let II be an interpretation of τ0\tau_{0}-structures in τE\tau_{E} as just described. We say that II is strongly existential if all formulas of II are existential and φ<\varphi_{<} is even quantifier-free.

Lemma 4.2.

Let τ\tau be obtained from τ0\tau_{0} by adding pairs and let φ0τ\varphi_{0\tau} be an extension of φ0\varphi_{0}. Then for every strongly existential interpretation II the sentence φ0τI\varphi^{I}_{0\tau} is (equivalent to) a universal sentence.

Proof : The claim holds as all relation symbols distinct from << are negative in φ0τ\varphi_{0\tau}. For example, for φ:=xy(Uminx(x=yx<y))\varphi:=\forall x\forall y\big{(}U_{\textup{min}}\,x\to(x=y\vee x<y)\big{)}, we have

φI=x¯(φuni(x¯)y¯(φuni(y¯)(φUmin(x¯)((x1=y1x2=y2)φ<(x¯,y¯))))).\varphi^{I}=\forall\bar{x}\Big{(}\varphi_{\textup{uni}}(\bar{x})\to\forall\bar{y}\big{(}\varphi_{\textup{uni}}(\bar{y})\to(\varphi_{U_{\textup{min}}}(\bar{x})\to((x_{1}=y_{1}\wedge x_{2}=y_{2})\vee\varphi_{<}(\bar{x},\bar{y})))\big{)}\Big{)}.

The following result shows that strongly existential interpretations preserve induced substructures in such a way that we can translate the results of the preceding section to the actual context.

Lemma 4.3.

Assume that II is strongly existential. Then for all τE\tau_{E}-structures GG and HH with HindGH\subseteq_{\textup{ind}}G and OI(H)O_{I}(H)\neq\emptyset, we have 𝒪I(H)<𝒪I(G)\mathcal{O}_{I}(H)\subseteq_{<}\mathcal{O}_{I}(G).

Proof : As φuni\varphi_{\textup{uni}} is existential, we have OI(H)OI(G)O_{I}(H)\subseteq O_{I}(G). Let TτT\in\tau be distinct from << and b¯T𝒪I(H)\bar{b}\in T^{\mathcal{O}_{I}(H)}. Then HφT(b¯)H\models\varphi_{T}(\bar{b}). As φT\varphi_{T} is existential, GφT(b¯)G\models\varphi_{T}(\bar{b}) and thus, b¯T𝒪I(G)\bar{b}\in T^{\mathcal{O}_{I}(G)}. Moreover, for b¯,b¯OI(H)\bar{b},\bar{b}^{\prime}\in O_{I}(H) we have

b¯<𝒪I(H)b¯\displaystyle\bar{b}<^{\mathcal{O}_{I}(H)}\bar{b}^{\prime} \displaystyle\iff Hφ<(b¯,b¯)\displaystyle H\models\varphi_{<}(\bar{b},\bar{b}^{\prime})
\displaystyle\iff Gφ<(b¯,b¯)(as HindG and φ< is quantifier-free)\displaystyle G\models\varphi_{<}(\bar{b},\bar{b}^{\prime})\qquad\text{(as $H\subseteq_{\textup{ind}}G$ and $\varphi_{<}$ is quantifier-free)}
\displaystyle\iff b¯<𝒪I(G)b¯.\displaystyle\bar{b}<^{\mathcal{O}_{I}(G)}\bar{b}^{\prime}.

Putting all together we see that 𝒪I(H)<𝒪I(G)\mathcal{O}_{I}(H)\subseteq_{<}\mathcal{O}_{I}(G). \Box

We obtain from Lemma 3.8 the corresponding result in our framework.

Lemma 4.4.

Assume that II is strongly existential. Let φ0τ\varphi_{0\tau} be an extension of φ0\varphi_{0}. Let GG be a τE\tau_{E}-structure and Gφ0τIG\models\varphi_{0\tau}^{I}. Let HindGH\subseteq_{\textup{ind}}G with finite OI(H)O_{I}(H). If Hφ1τIH\models\varphi_{1\tau}^{I}, then 𝒪I(H)=𝒪I(G)\mathcal{O}_{I}(H)=\mathcal{O}_{I}(G) and Gφ1τIG\models\varphi^{I}_{1\tau}.

Proof : As Hφ1τIH\models\varphi_{1\tau}^{I}, in particular H(xUminx)IH\models(\exists x\,U_{\textup{min}}\,x)^{I}; thus, OI(H)O_{I}(H)\neq\emptyset. Therefore, 𝒪I(H)<𝒪I(G)\mathcal{O}_{I}(H)\subseteq_{<}\mathcal{O}_{I}(G) by Lemma 4.3. By assumption and (12), 𝒪I(G)φ0τ\mathcal{O}_{I}(G)\models\varphi_{0\tau} and 𝒪I(H)φ1τ\mathcal{O}_{I}(H)\models\varphi_{1\tau}. As OI(H)O_{I}(H) is finite, Lemma 3.8 implies 𝒪I(H)=𝒪I(G)\mathcal{O}_{I}(H)=\mathcal{O}_{I}(G), and in particular 𝒪I(G)φ1τ\mathcal{O}_{I}(G)\models\varphi_{1\tau}. Hence, Gφ1τIG\models\varphi^{I}_{1\tau} by (12). \Box

We now prove for strongly existential interpretations two results, Proposition 4.5 corresponds to Tait’s Theorem (Theorem 3.4), and Proposition 4.6 corresponds to Proposition 3.11 (relevant to Gurevich’s Theorem). In our application of these results to graphs in the next section the sentence ψ\psi will be x¬Exxxy(ExyEyx)\forall x\neg Exx\wedge\forall x\forall y(Exy\to Eyx), i.e., the sentence φGraph\varphi_{\textsc{Graph}} (cf. (3)) axiomatizing the class of graphs.

Proposition 4.5.

Let ψ\psi be a universal τE\tau_{E}-sentence. Assume that the interpretation II of τ0\tau_{0}-structures in τE\tau_{E}-structures is strongly existential. Furthermore, assume that for every sufficiently large finite complete τ0\tau_{0}-ordering 𝒜\mathcal{A} there is a finite τE\tau_{E}-structure GG with 𝒪I(G)𝒜\mathcal{O}_{I}(G)\cong\mathcal{A} and GψG\models\psi. Then there is an FO[τE]\textup{FO}[\tau_{E}]-sentence φ\varphi such that Modfin(ψφ)\textsc{Mod}_{\textup{fin}}(\psi\wedge\varphi) is closed under induced substructures, but ψφ\psi\wedge\varphi is not finitely equivalent to a universal sentence.

As φ\varphi we an take the sentence

φ:=x¯¬φuni(x¯)(φ0I¬φ1I)\varphi:=\forall\bar{x}\neg\varphi_{\textup{uni}}(\bar{x})\vee\big{(}\varphi^{I}_{0}\wedge\neg\varphi^{I}_{1}\big{)}

(for the definition of φ0\varphi_{0} and φ1\varphi_{1} see page 3 and (7), respectively).

Proof : First we verify that the class Modfin(ψφ)\textsc{Mod}_{\textup{fin}}(\psi\wedge\varphi) is closed under induced substructures. Assume GψφG\models\psi\wedge\varphi and HindGH\subseteq_{\textup{ind}}G. Since ψ\psi is universal, we have HψH\models\psi. If Gx¯¬φuni(x¯)G\models\forall\bar{x}\neg\varphi_{\textup{uni}}(\bar{x}), then Hx¯¬φuni(x¯)H\models\forall\bar{x}\neg\varphi_{\textup{uni}}(\bar{x}). Now assume that Gφ0I¬φ1IG\models\varphi^{I}_{0}\wedge\neg\varphi^{I}_{1}. Then Hφ0IH\models\varphi^{I}_{0}, as φ0I\varphi_{0}^{I} is universal by Lemma 4.2. If Hx¯¬φuni(x¯)H\models\forall\bar{x}\neg\varphi_{\textup{uni}}(\bar{x}) or H¬φ1IH\models\neg\varphi^{I}_{1}, we are done. Otherwise OI(H)O_{I}(H)\neq\emptyset and Hφ1IH\models\varphi^{I}_{1}. Then Gφ1IG\models\varphi^{I}_{1} (see Lemma 4.4), a contradiction.

Finally we show that for every kk\in\mathbb{N} the sentence ψφ\psi\wedge\varphi is not finitely equivalent to a sentence of the form μ=z1zkμ0\mu=\forall z_{1}\ldots\forall z_{k}\,\mu_{0} with quantifier-free μ0\mu_{0}. Let

𝒜:=(A,<𝒜,Umin𝒜,Umax𝒜,S𝒜)\mathcal{A}:=\big{(}A,<^{\mathcal{A}},U_{\textup{min}}^{\mathcal{A}},U_{\textup{max}}^{\mathcal{A}},S^{\mathcal{A}}\big{)}

be a complete τ0\tau_{0}-ordering with at least k2+1k^{2}+1 elements. In particular, 𝒜φ0φ1\mathcal{A}\models\varphi_{0}\wedge\varphi_{1}. By assumption we can choose 𝒜\mathcal{A} in such a way that there is a finite τE\tau_{E}-structure GG such that 𝒪I(G)𝒜\mathcal{O}_{I}(G)\cong\mathcal{A} and GψG\models\psi. Then 𝒪I(G)φ0φ1\mathcal{O}_{I}(G)\models\varphi_{0}\wedge\varphi_{1}, hence, Gφ0Iφ1IG\models\varphi_{0}^{I}\wedge\varphi^{I}_{1}. Thus Gψ¬φG\models\psi\wedge\neg\varphi. As |OI(G)|=|A|k2+1|O_{I}(G)|=|A|\geq k^{2}+1, the graph GG must contain more than kk elements by (11).

We want to show that every induced substructure of GG with at most kk elements is a model of ψφ\psi\wedge\varphi. Then the result follows from Corollary 2.7. So let HH be an induced substructure of GG with at most kk elements. Clearly, H(ψφ0I)H\models(\psi\wedge\varphi^{I}_{0}). If Hx¯¬φuni(x¯)H\models\forall\bar{x}\neg\varphi_{\textup{uni}}(\bar{x}) or H¬φ1IH\models\neg\varphi^{I}_{1}, we are done. Otherwise OI(H)O_{I}(H)\neq\emptyset and Hφ1IH\models\varphi^{I}_{1}. Then, Lemma 4.4 implies OI(H)=OI(G)O_{I}(H)=O_{I}(G). Recall |V(H)|k|V(H)|\leq k, so OI(H)O_{I}(H) has at most k2k^{2} elements by (11), a contradiction as |OI(G)|k2+1|O_{I}(G)|\geq k^{2}+1. \Box

Proposition 4.6.

Assume that ψ\psi is a universal τE\tau_{E}-sentence. Let τ\tau be obtained from τ0\tau_{0} by adding pairs and let φ0τ\varphi_{0\tau} be an extension of φ0\varphi_{0}. Let II be a strongly existential interpretation of τ\tau-structures in τE\tau_{E}-structures with the property that for every finite τ\tau-structure 𝒜\mathcal{A}, which is a model of φ0τφ1τ\varphi_{0\tau}\wedge\varphi_{1\tau}, there is a finite τE\tau_{E}-structure GG with 𝒪I(G)𝒜\mathcal{O}_{I}(G)\cong\mathcal{A} and GψG\models\psi.

Let m1m\geq 1 and γ\gamma be an FO[τ]\textup{FO}[\tau]-sentence such that

φ0τφ1τγ has no infinite model but a finite model with at least m elements.\text{$\varphi_{0\tau}\wedge\varphi_{1\tau}\wedge\gamma$ has no infinite model but a finite model with at least $m$ elements}. (13)

For

ρ:=x¯¬φuni(x¯)(φ0τ(φ1τ¬γ))I\rho:=\forall\bar{x}\neg\varphi_{\textup{uni}}(\bar{x})\vee\big{(}\varphi_{0\tau}\wedge(\varphi_{1\tau}\to\neg\gamma)\big{)}^{I} (14)

the statements (a) and (b) hold.

  • (a)

    The class Mod(ψρ)\textsc{Mod}(\psi\wedge\rho) is closed under induced substructures.

  • (b)

    If μ:=x1xkμ0\mu:=\forall x_{1}\ldots\forall x_{k}\,\mu_{0} with quantifier-free μ0\mu_{0} is finitely equivalent to ψρ\psi\wedge\rho, then k2mk^{2}\geq m.

Proof : (a) Assume that GψρG\models\psi\wedge\rho and HindGH\subseteq_{\textup{ind}}G. Clearly HψH\models\psi. If Hx¯¬φuni(x¯)H\models\forall\bar{x}\neg\varphi_{\textup{uni}}(\bar{x}), then we are done. Otherwise, the universe of 𝒪I(H)\mathcal{O}_{I}(H) and hence, that of 𝒪I(G)\mathcal{O}_{I}(G), are not empty. Then Gφ0τIG\models\varphi^{I}_{0\tau} and as HindGH\subseteq_{\textup{ind}}G, we have Hφ0τIH\models\varphi^{I}_{0\tau} by Lemma 4.2.

If H⊧̸φ1τIH\not\models\varphi^{I}_{1\tau}, we are done. Otherwise, Hφ1τIH\models\varphi^{I}_{1\tau}. If HIH_{I} is infinite, then HI¬γH_{I}\models\neg\gamma by (13) and we are again done. If HIH_{I} is finite, then 𝒪I(H)=𝒪I(G)\mathcal{O}_{I}(H)=\mathcal{O}_{I}(G) by Lemma 4.4. Thus 𝒪I(G)φ1τ\mathcal{O}_{I}(G)\models\varphi_{1\tau} and hence, 𝒪I(G)¬γ\mathcal{O}_{I}(G)\models\neg\gamma as GρG\models\rho. Therefore, 𝒪I(H)¬γ\mathcal{O}_{I}(H)\models\neg\gamma and thus, HρH\models\rho.

(b) By (13) there is a finite model 𝒜\mathcal{A} of φ0τφ1τγ\varphi_{0\tau}\wedge\varphi_{1\tau}\wedge\gamma with at least mm elements. By assumption there is a finite τE\tau_{E}-structure GG with 𝒪I(G)𝒜\mathcal{O}_{I}(G)\cong\mathcal{A} and GψG\models\psi. Clearly, G¬x¯¬φuni(x¯)G\models\neg\forall\bar{x}\neg\varphi_{\textup{uni}}(\bar{x}) and G(φ0τφ1τγ)IG\models(\varphi_{0\tau}\wedge\varphi_{1\tau}\wedge\gamma)^{I}. Hence, Gψ¬ρG\models\psi\wedge\neg\rho. Assume that k2<mk^{2}<m. We want to show that every induced substructure of GG with at most kk elements is a model of ψρ\psi\wedge\rho. Then the claim (b) follows from Corollary 2.7.

So let HH be an induced substructure of GG with at most kk elements. Clearly, H(ψφ0τI)H\models(\psi\wedge\varphi^{I}_{0\tau}). If Hx¯¬φuni(x¯)H\models\forall\bar{x}\neg\varphi_{\textup{uni}}(\bar{x}) or H¬φ1τIH\models\neg\varphi^{I}_{1\tau}, we are done. Otherwise OI(H)O_{I}(H)\neq\emptyset and Hφ1τIH\models\varphi^{I}_{1\tau}. Then, OI(H)=OI(G)O_{I}(H)=O_{I}(G) by Lemma 4.4. This leads to a contradiction, as OI(H)O_{I}(H) has at most k2k^{2} elements by (12), while OI(G)O_{I}(G) has mm elements and we assumed k2<mk^{2}<m. \Box

Remark 4.7.

The results corresponding to Remark 3.5 and Remark 3.12 are valid for Proposition 4.5 and Proposition 4.6 too. In particular, the sentence ψφ(=ψx¯¬φuni(x¯)(φ0I¬φ1I))\psi\wedge\varphi\ \big{(}=\psi\wedge\forall\bar{x}\neg\varphi_{\textup{uni}}(\bar{x})\vee\big{(}\varphi^{I}_{0}\wedge\neg\varphi^{I}_{1}\big{)}\big{)} is not equivalent to a Π2\Pi_{2}-sentence. Furthermore ψφ\psi\wedge\varphi itself is equivalent to a Σ2\Sigma_{2}-sentence. In fact, as all relation symbols besides << are negative in φ0\varphi_{0}, the sentence φ0I\varphi_{0}^{I} is universal. Moreover, as UminU_{\textup{min}}, UmaxU_{\textup{max}}, and SS are positive in φ1\varphi_{1}, the sentence φ1I\varphi_{1}^{I} (as φ1\varphi_{1}) is equivalent to a Π2\Pi_{2}-sentence. Hence ψφ\psi\wedge\varphi is equivalent to a Σ2\Sigma_{2}-sentence.

5 Tait’s Theorem for finite graphs

In this section we introduce a strongly existential interpretation, which allows us to get Tait’s Theorem for graphs. The corresponding result for Gurevich’s Theorem will be derived in Section 6.

We first introduce a further concept. Let GG be a graph and a,bV(G)a,b\in V(G). For r,s3r,s\geq 3 a path from vertex aa to vertex bb of length rr with an ss-ear is a path between aa and bb with a cycle of length ss; one vertex of this cycle is adjacent to the vertex adjacent to bb on the path. Figure 1 is a path from aa to bb of length 66 with a 44-ear.

aabb
Figure 1: A path of length 6 with a 44-ear.
Lemma 5.1.

For r,s3r,s\geq 3 there are quantifier-free formulas φcr(x,z¯)\varphi_{cr}(x,\bar{z}) and φpe,r,s(x,y,z¯,w¯)\varphi_{pe,r,s}(x,y,\bar{z},\bar{w}) such that for all graphs GG we have

  • (a)

    Gφcr(a,u¯)u¯G\models\varphi_{cr}(a,\bar{u})\iff\bar{u} is a cycle of length rr containing aa.

  • (b)

    Gφpe,r,s(a,b,u¯,v¯)u¯G\models\varphi_{pe,r,s}(a,b,\bar{u},\bar{v})\iff\bar{u} is path from aa to bb of length rr with the ss-ear v¯\bar{v}.

Proof : (a) We can take as φcr(x,z1,,zr)\varphi_{cr}(x,z_{1},\ldots,z_{r}) the formula

x=z1Ezrz11i<rEzizi+11i<jr¬zi=zj.x=z_{1}\wedge Ez_{r}z_{1}\wedge\bigwedge_{1\leq i<r}Ez_{i}z_{i+1}\wedge\bigwedge_{1\leq i<j\leq r}\neg z_{i}=z_{j}.

(b) We can take as φpe,r,s(x,y,z0,,zr,w1,,ws)\varphi_{pe,r,s}(x,y,z_{0},\ldots,z_{r},w_{1},\ldots,w_{s}) the formula

x=z0y=zr0i<r1Ezizi+10i<jr¬zi\displaystyle x=z_{0}\wedge y=z_{r}\wedge\bigwedge_{0\leq i<r-1}Ez_{i}z_{i+1}\wedge\bigwedge_{0\leq i<j\leq r}\neg z_{i} =zj0ir,j[s]¬zi=wj\displaystyle=z_{j}\wedge\bigwedge_{0\leq i\leq r,\ j\in[s]}\neg z_{i}=w_{j}
φcs(w1,w1,,wr)Ezr1w1.\displaystyle\wedge\varphi_{cs}(w_{1},w_{1},\ldots,w_{r})\wedge Ez_{r-1}w_{1}. \Box

To understand better how we obtain the desired interpretation we first assign to every complete τ0\tau_{0}-ordering 𝒜\mathcal{A}, i.e., to every model of φ0φ1\varphi_{0}\wedge\varphi_{1}, a τE\tau_{E}-structure G:=G(𝒜)G:=G(\mathcal{A}) which is a graph.

In a first step we extend 𝒜\mathcal{A} to a τ0\tau^{*}_{0}-structure 𝒜\mathcal{A}^{*}, where τ0:=τ0{B,C,L,F}\tau^{*}_{0}:=\tau_{0}\cup\{B,C,L,F\} in the following way. Here B,CB,C are unary and L,FL,F are binary relation symbols.

For every original (or, basic) element aa, i.e., for every aAa\in A, we introduce a new element aa^{\prime}, the companion of aa. We set

  • A:=A{aaA}A^{*}:=A\cup\{a^{\prime}\mid a\in A\},

  • B𝒜:=AB^{\mathcal{A}^{*}}:=A,   C𝒜:={aaA}C^{\mathcal{A}^{*}}:=\{a^{\prime}\mid a\in A\},

  • L𝒜:={(a,a)|aA},F𝒜:={(a,b),(b,a)|a,bA,a<𝒜b}L^{\mathcal{A}^{*}}:=\big{\{}(a,a^{\prime})\;\big{|}\;a\in A\big{\}},\qquad F^{\mathcal{A}^{*}}:=\big{\{}(a^{\prime},b),(b,a^{\prime})\;\big{|}\;a,b\in A,\ a<^{\mathcal{A}}b\big{\}}.

Note that the relation FF is irreflexive and symmetric, i.e., (A,F𝒜)\big{(}A^{*},F^{\mathcal{A}^{*}}\big{)} is already a graph, which is illustrated by Figure 2. Observe that FF contains the whole information of the ordering <𝒜<^{\mathcal{A}} up to isomorphism.

a0a_{0}a1a_{1}a2a_{2}a3a_{3}a4a_{4}a5a_{5}a0a_{0}a1a_{1}a2a_{2}a3a_{3}a4a_{4}a5a_{5}a0a^{\prime}_{0}a1a^{\prime}_{1}a2a^{\prime}_{2}a3a^{\prime}_{3}a4a^{\prime}_{4}a5a^{\prime}_{5}
Figure 2: Turning an ordering to the relation FF.

We use 𝒜\mathcal{A}^{*} to define the desired graph G=G(𝒜)G=G(\mathcal{A}). The vertex set V(G)V(G) contains the elements of AA^{*}, and the edge relation E(G)E(G) contains F𝒜F^{\mathcal{A}^{*}}. Furthermore GG contains just all the vertices and edges required by the following items:

  • To aUmin𝒜a\in U_{\textup{min}}^{\mathcal{A}} we add a cycle of length 55 consisting of new vertices, i.e., not in AA^{*} (besides aa).

  • To aUmax𝒜a\in U_{\textup{max}}^{\mathcal{A}} we add a cycle of length 77 consisting of new vertices (besides aa).

  • To aB𝒜a\in B^{\mathcal{A}^{*}} we add a cycle of length 99 consisting of new vertices (besides aa).

  • To aC𝒜a\in C^{\mathcal{A}^{*}} we add a cycle of length 1111 consisting of new vertices (besides aa).

  • To (a,b)S𝒜(a,b)\in S^{\mathcal{A}} we add a path from aa to bb of length 1717 with a 1313-ear consisting of new vertices (besides aa and bb).

  • To (a,a)L𝒜(a,a^{\prime})\in L^{\mathcal{A}^{*}} we add a path from aa to aa^{\prime} of length 1717 with a 1515-ear consisting of new vertices (besides aa and aa^{\prime}).

Hereby we meant by “add a cycle” or “add a path with an ear” that we only add the edges required by the corresponding formulas in Lemma 5.1.

To ease the discussion, we divide cycles in G(=G(𝒜))G\ (=G(\mathcal{A})) into four categories.

[FF-cycle]  These are cycles in (A,F𝒜)\big{(}A^{*},F^{\mathcal{A}^{*}}\big{)}, i.e., cycles using only edges of F𝒜F^{\mathcal{A}^{*}}.

[TT-cycle]  For every unary T{Umin,Umax,B,C}T\in\big{\{}U_{\textup{min}},U_{\textup{max}},B,C\big{\}}, a TT-cycle is the cycle introduced for an aT𝒜a\in T^{\mathcal{A}}.

[ear-cycle]  These are the cycles constructed as ears on the gadgets for the relations S𝒜S^{\mathcal{A}^{*}} and L𝒜L^{\mathcal{A}^{*}}.

[mixed-cycle]  All the other cycles are mixed.

For example, we get a mixed cycle if we start with a2a_{2}, a0a^{\prime}_{0}, a1a_{1} in Figure 2 and then add the path introduced for (a1,a2)S𝒜(a_{1},a_{2})\in S^{\mathcal{A}} (ignoring the ear).

A number of observations for these types of cycles are in order.

Lemma 5.2.
  1. (i)

    All the FF-cycles are of even length.111Moreover one can show that every chordless FF-cycle has length 44.

  2. (ii)

    Every UminU_{\textup{min}}-, UmaxU_{\textup{max}}-, BB-, and CC-cycle is of length 55, 77, 99, and 1111, respectively.

  3. (iii)

    Every ear-cycle is of length 1313 or 1515.

  4. (iv)

    Every mixed-cycle neither uses new vertices of any TT-cycle for T{Umin,Umax,B,C}T\in\big{\{}U_{\textup{min}},U_{\textup{max}},B,C\big{\}} nor any vertex of any ear-cycle.

  5. (v)

    Every mixed-cycle has length at least 1717.

Proof : (i) follows easily from the fact that (A,F𝒜)\big{(}A^{*},F^{\mathcal{A}^{*}}\big{)} is a bipartite graph; (ii) and (iii) are trivial.

For (iv) assume that a mixed-cycle uses a new vertex bb of a TT-cycle 𝒞\mathcal{C} introduced for some aT𝒜a\in T^{\mathcal{A}^{*}}, where T{Umin,Umax,B,C}T\in\big{\{}U_{\textup{min}},U_{\textup{max}},B,C\big{\}}. As 𝒞\mathcal{C} is mixed, it must contain a vertex cT𝒜c\notin T^{\mathcal{A}^{*}}. To reach bb from cc the mixed cycle must pass through aa and hence must contain one of the two segments of 𝒞\mathcal{C} between bb and aa. As a consequence, in order for the mixed-cycle to go back from bb to cc, it must also use the other segment of 𝒞\mathcal{C} between aa and bb. This means that it must be the TT-cycle 𝒞\mathcal{C} itself, instead of a mixed one. A similar argument shows that mixed cycles do not contain vertices of any ear-cycle.

To prove (v), let 𝒞\mathcal{C} be a mixed-cycle. By (iv), 𝒞\mathcal{C} must contain all vertices of a (at least one) path introduced for a pair (a,a)L𝒜(a,a^{\prime})\in L^{\mathcal{A}*} or (a,b)S𝒜(a,b)\in S^{\mathcal{A}^{*}} (ignoring the ear). As this path has length 1717, we get our claim. \Box

Conversely, given a τE\tau_{E}-structure GG, which is a graph, we construct a τ0\tau_{0}-structure which we denote by 𝒪(G)\mathcal{O}(G), possibly the empty structure. Recall the definitions of “cycle” and of “path with ear” given by Lemma 5.1.

  • O(G):={(a1,a2)V(G)×V(G)|a1 is a member of a cycle of length 9a2 is a member
    of a cycle of length 11, and there is a path from a1 to a2 of length 17 with a 15-ear
    }
    O(G):=\big{\{}(a_{1},a_{2})\in V(G)\times V(G)\;\big{|}\;\text{$a_{1}$ is a member of a cycle of length $9$, \ $a_{2}$ is a member}\\ \text{of a cycle of length $11$, and there is a path from $a_{1}$ to $a_{2}$ of length $17$ with a $15$-ear}\big{\}}

  • <𝒪(G):={((a1,a2),(b1,b2))O(G)×O(G)|{a2,b1}E(G)}<^{\mathcal{O}(G)}:=\big{\{}((a_{1},a_{2}),(b_{1},b_{2}))\in O(G)\times O(G)\;\big{|}\;\{a_{2},b_{1}\}\in E(G)\big{\}}

  • Umin𝒪(G):={(a1,a2)O(G)|a1 is a member of a cycle of 5 elements}U_{\textup{min}}^{\mathcal{O}(G)}:=\big{\{}(a_{1},a_{2})\in O(G)\;\big{|}\;\text{$a_{1}$ is a member of a cycle of $5$ elements}\big{\}}

  • Umax𝒪(G):={(a1,a2)O(G)|a1 is a member of a cycle of 7 elements}U_{\textup{max}}^{\mathcal{O}(G)}:=\big{\{}(a_{1},a_{2})\in O(G)\;\big{|}\;\text{$a_{1}$ is a member of a cycle of $7$ elements}\big{\}}

  • S𝒪(G):={((a1,a2),(b1,b2))O(G)×O(G)there is a path from a1 to b1 of length 17
    with a 13-ear
    }
    S^{\mathcal{O}(G)}:=\big{\{}((a_{1},a_{2}),(b_{1},b_{2}))\in O(G)\times O(G)\mid\text{there is a path from $a_{1}$ to $b_{1}$ of length $17$}\\ \text{with a $13$-ear}\big{\}}
    .

Lemma 5.3.

For every complete τ0\tau_{0}-ordering 𝒜\mathcal{A} we have 𝒪(G(𝒜))𝒜\mathcal{O}(G(\mathcal{A}))\cong\mathcal{A}.

Proof : Let G:=G(𝒜)G:=G(\mathcal{A}) and 𝒜+:=𝒪(G)\mathcal{A}^{+}:=\mathcal{O}(G). We claim that the mapping h:AA+h:A\to A^{+} defined by

h(a):=(a,a)for aAh(a):=(a,a^{\prime})\quad\text{for $a\in A$}

is an isomorphism from 𝒜\mathcal{A} to 𝒜+\mathcal{A}^{+}. To that end, we first prove that

A+={(a,a)|aA},A^{+}=\big{\{}(a,a^{\prime})\;\big{|}\;a\in A\big{\}},

which implies that hh is well defined and a bijection. For every aAa\in A it is easy to see that (a,a)O(G)(=A+)(a,a^{\prime})\in O(G)\ (=A^{+}). For the converse, let (a1,a2)O(G)(a_{1},a_{2})\in O(G). In particular, a1a_{1} is a member of a cycle of length 99. By Lemma 5.2, this must be a BB-cycle which contains some aAa\in A. Using the same argument, a2a_{2} is a member of a CC-cycle which contains a vertex bb^{\prime} being the companion of some bAb\in A. Furthermore, there is a path from a1a_{1} to a2a_{2} of length 1717 with a 1515-ear. The 1515-ear is a cycle of length 1515. Again by Lemma 5.2 this cycle is an ear-cycle which belongs to the gadget we introduced for some (c,c)L𝒜(c,c^{\prime})\in L^{\mathcal{A}^{*}} with cAc\in A. Then it is easy to see that a=c=ba=c=b. This finishes the proof that hh is a bijection from AA to A+A^{+}.

Similarly, we can prove that hh preserves all the relations. \Box

We want to show that we can obtain 𝒪(G)\mathcal{O}(G) from GG by a strongly existential FO-interpretation. We set

η(x,x,x¯,x¯,z¯,w¯):=\displaystyle\eta(x,x^{\prime},\bar{x},\bar{x}^{\prime},\bar{z},\bar{w}):= x¯ is a cycle of length 9 containing xx¯ is a cycle of length 11 containing x,\displaystyle\ \text{``$\bar{x}$ is a cycle of length $9$ containing $x$, \ $\bar{x}^{\prime}$ is a cycle of length $11$ containing $x^{\prime}$},
and z¯\bar{z} is a path from xx to xx^{\prime} of length 1717 with the 1515-ear w¯\bar{w}
=\displaystyle= φc9(x,x¯)φc11(x,x¯)φpe,17,15(x,xz¯,w¯).\displaystyle\ \varphi_{c9}(x,\bar{x})\wedge\varphi_{c11}(x^{\prime},\bar{x}^{\prime})\wedge\varphi_{pe,17,15}(x,x^{\prime}\bar{z},\bar{w}).

We define the desired interpretation II of width 22 of τ0\tau_{0}-structures in graphs. We set

φuni(x,x):=x¯x¯z¯w¯η(x,x,x¯,x¯,z¯,w¯).\varphi_{\textup{uni}}(x,x^{\prime}):=\exists\bar{x}\exists\bar{x}^{\prime}\exists\bar{z}\exists\bar{w}\,\eta(x,x^{\prime},\bar{x},\bar{x}^{\prime},\bar{z},\bar{w}).

Hence for every graph GG,

OI(G)={(a1,a2)V(G)×V(G)|Gx¯x¯z¯w¯η(a1,a2,x¯,x¯,z¯,w¯)}.O_{I}(G)=\big{\{}(a_{1},a_{2})\in V(G)\times V(G)\;\big{|}\;\text{$G\models\exists\bar{x}\exists\bar{x}^{\prime}\exists\bar{z}\exists\bar{w}\,\eta(a_{1},a_{2},\bar{x},\bar{x}^{\prime},\bar{z},\bar{w})$}\big{\}}.

Furthermore we define

  • φUmin(x,x):=z¯φc5(x,z¯)\varphi_{U_{\textup{min}}}(x,x^{\prime}):=\exists\bar{z}\,\varphi_{c5}(x,\bar{z}),

  • φUmax(x,x):=z¯φc7(x,z¯)\varphi_{U_{\textup{max}}}(x,x^{\prime}):=\exists\bar{z}\,\varphi_{c7}(x,\bar{z}),

  • φS(x,x,y,y):=z¯w¯z¯ is a path of length 17 from x to y with a 13-ear w¯\varphi_{S}(x,x^{\prime},y,y^{\prime}):=\exists\bar{z}\exists\bar{w}\;\text{``$\bar{z}$ is a path of length $17$ from $x$ to $y$ with a $13$-ear $\bar{w}$''}
                          =z¯w¯φpe,17,13(x,z¯,w¯)=\exists\bar{z}\exists\bar{w}\varphi_{pe,17,13}(x,\bar{z},\bar{w}).

Then we have:

Lemma 5.4.

The interpretation II given by (φuni,φ<,φUmin,φUmax,φS)\big{(}\varphi_{\textup{uni}},\varphi_{<},\varphi_{U_{\textup{min}}},\varphi_{U_{\textup{max}}},\varphi_{S}\big{)} is strongly existential. For every complete τ0\tau_{0}-ordering 𝒜\mathcal{A} we have 𝒪I(G(𝒜))=𝒪(G(𝒜))\mathcal{O}_{I}(G(\mathcal{A}))=\mathcal{O}(G(\mathcal{A})) and hence, by Lemma 5.3,

𝒪I(G(𝒜))𝒜.\mathcal{O}_{I}(G(\mathcal{A}))\cong\mathcal{A}.

Setting ψ:=φGraph\psi:=\varphi_{\textsc{Graph}}, the sentence axiomatizing the class of graphs, we get from Proposition 4.5:

Theorem 5.5 (Tait’s Theorem for graphs).

There is a τE\tau_{E}-sentence φ\varphi such that Graphfin(φ)\textsc{Graph}_{\textup{fin}}(\varphi), the class of finite graphs that are models of φ\varphi, is closed under induced subgraphs but φ\varphi is not equivalent to a universal sentence in finite graphs.

In this section we presented a strongly existential interpretation of τ0\tau_{0}-structures and applied it to finite complete τ0\tau_{0}-orderings, i.e, to models of φ0φ1\varphi_{0}\wedge\varphi_{1}. A straightforward generalization of the preceding proofs allows to show the following result for vocabularies obtained from τ0\tau_{0} by adding pairs. We shall use it in Section 6.

Lemma 5.6.

Let τ\tau be obtained from τ0\tau_{0} by adding pairs. There is a strongly existential interpretation I(=Iτ)I\ (=I_{\tau}) that for every extension φ0τ\varphi_{0\tau} of φ0\varphi_{0} assigns to every τ\tau-structure 𝒜\mathcal{A} that is a model of φ0τφ1τ\varphi_{0\tau}\wedge\varphi_{1\tau} a graph G(𝒜)G(\mathcal{A}) with 𝒪I(G(𝒜))𝒜\mathcal{O}_{I}(G(\mathcal{A}))\cong\mathcal{A}. For finite 𝒜\mathcal{A} the graph G(𝒜)G(\mathcal{A}) is finite.

Proof : We get the graph G(𝒜)G(\mathcal{A}) as in the case τ:=τ0\tau:=\tau_{0}: For the elements of new unary relations we add cycles such that the lengths of the cycles are odd and distinct for distinct unary relations in τ\tau. Let cc be the maximal length of these cycles. Then we add paths with ears to the tuples of binary relations as above. For distinct binary relations the ears should have distinct length and again this length should be odd and greater than cc. On the other hand, the length of added new paths can be the same for all binary relations but should be greater than the length of all the cycles. \Box

Remark 5.7.

(a) Let 𝒞:=Modfin(x¬Exx)\mathscr{C}:=\textsc{Mod}_{\textup{fin}}(\forall x\neg Exx) be the class of directed graphs. Then 𝒞:=Graphfin\mathscr{C}^{\prime}:=\textsc{Graph}_{\textup{fin}}, the class of finite graphs, is a subclass of 𝒞\mathscr{C} closed under induced substructures and definable in 𝒞\mathscr{C} by the universal sentence xy(ExyEyx)\forall x\forall y(Exy\to Eyx). As the Łoś-Tarski Theorem fails for the class of finite graphs, it fails for the class of directed graphs by Remark 2.8.

(b) Now let 𝒞:=Graphfin\mathscr{C}:=\textsc{Graph}_{\textup{fin}} and 𝒞:=Planarfin\mathscr{C}^{\prime}:=\textsc{Planar}_{\textup{fin}} be the class of finite planar graphs, a subclass of Graphfin\textsc{Graph}_{\textup{fin}} closed under induced subgraphs. As mentioned in the Introduction, in [1] it is shown that the Łoś-Tarski Theorem fails for Planarfin\textsc{Planar}_{\textup{fin}}. As Planarfin\textsc{Planar}_{\textup{fin}} is not axiomatizable in Graphfin\textsc{Graph}_{\textup{fin}} by a universal sentence, not even by a first-order sentence, we do not get the failure of the Łoś-Tarski Theorem for the class of finite graphs, i.e., Tait’s Theorem for graphs, by applying the result of Remark 2.8. We show that Planarfin=Forbfin()\textsc{Planar}_{\textup{fin}}=\textsc{Forb}_{\textup{fin}}(\mathscr{F}) for a finite set \mathscr{F} of finite graphs (or, equivalently, Planarfin=Modfin(μ)\textsc{Planar}_{\textup{fin}}=\textsc{Mod}_{\textup{fin}}(\mu) for a universal μ\mu) leads to a contradiction. Let kk be the maximum size of the set of vertices of graphs in \mathscr{F}. Let GG be the graph obtained from the clique K5K_{5} of 5 vertices by subdividing each edge by k+1k+1. Clearly, GPlanarfinG\notin\textsc{Planar}_{\textup{fin}}. However, every subgraph of GG induced on at most kk elements is planar. Hence, GForbfin()G\in\textsc{Forb}_{\textup{fin}}(\mathscr{F}).

(c) Let τ\tau be any vocabulary with at least one at least binary relation TT. Then the Łoś-Tarski Theorem fails for the class 𝒞:=Strfin[τ]\mathscr{C}:=\textsc{Str}_{\textup{fin}}[\tau], the class of all finite τ\tau-structures. By Remark 2.8 it suffices to show the existence of a universally definable subclass 𝒞\mathscr{C}^{\prime} of 𝒞\mathscr{C} which “essentially is the class of graphs.” We set

μ:=xu¯¬Txxu¯xyu¯v¯(Txyu¯Tyxv¯)Rτ,RTu¯¬Ru¯\mu:=\forall x\forall\bar{u}\neg Txx\bar{u}\wedge\forall x\forall y\forall\bar{u}\forall\bar{v}(Txy\bar{u}\to Tyx\bar{v})\wedge\bigwedge_{R\in\tau,\ R\neq T}\forall\bar{u}\neg R\bar{u}

and let 𝒞\mathscr{C}^{\prime} be Modfin(μ)\textsc{Mod}_{\textup{fin}}(\mu).

If τ\tau only contains unary relation symbols, the Łoś-Tarski Theorem holds for Strfin[τ]\textsc{Str}_{\textup{fin}}[\tau]. It is easy to see for an FO(τ)\textup{FO}(\tau)-sentence φ\varphi that the closure under induced substructures of Modfin(φ)\textsc{Mod}_{\textup{fin}}(\varphi) implies that of Mod(φ)\textsc{Mod}(\varphi).

6 Gurevich’s Theorem

The following discussion will eventually lead to a proof of Gurevich’s Theorem, i.e., Theorem 1.5. Our proof essentially follows Gurevich’s proof in [14], but it contains some elements of Rossman’s proof of the same result in [16]222The reader of [14] will realize that the definition of φn\varphi^{n} on page 190 of [14] must be modified in order to ensure that the class of models of φn\varphi^{n} is closed under induced substructures. Afterwards we show that it remains true if we restrict ourselves to graphs.

Our main tool is Proposition 3.11, and the goal is to construct a formula γ\gamma in (10) whose size is much smaller than the number mm. Basically γ\gamma will describe a very long computation of a Turing machine on a short input. We fix a universal Turing machine MM operating on an one-way infinite tape, the tape alphabet is {0,1}\{0,1\}, where 0 is also considered as blank, and QQ is the set of states of MM. The initial state is q0q_{0}, and qhq_{h} is the halting state; thus q0,qhQq_{0},q_{h}\in Q and we assume that q0qhq_{0}\neq q_{h}. An instruction of MM has the form

qapbd,qapbd,

where q,pQq,p\in Q, a,b{0,1}a,b\in\{0,1\} and d{1,0,1}d\in\{-1,0,1\}. It indicates that if MM is in state qq and the head of MM reads an aa, then the head replaces aa by bb and moves to the left (if d=1d=-1), stays still (if d=0d=0), or moves to the right (if d=1d=1). In order to describe computations of MM by FO-formulas we introduce binary predicates Hq(x,t)H_{q}(x,t) for qQq\in Q to indicate that at time tt the machine is in state qq and the head scans cell xx, and a binary predicate C0(x,t)C_{0}(x,t) to indicate that the content of cell xx at time tt is 0.

The vocabulary τM\tau_{M} is obtained from τ0\tau_{0} by adding pairs (see Definition 3.6 (a)),

τM:=τ0{Hq,Hqcomp|qQ}{C0,C0comp}.\tau_{M}:=\tau_{0}\cup\big{\{}H_{q},H_{q}^{\textup{comp}}\;\big{|}\;q\in Q\big{\}}\cup\big{\{}C_{0},C_{0}^{\textup{comp}}\big{\}}.

Intuitively, Hqcomp(x,t)H_{q}^{\textup{comp}}(x,t) says that “at time tt the machine is not in state qq or the head is not in cell xx;” and C0comp(x,t)C_{0}^{\textup{comp}}(x,t) says that “at time tt the content of cell xx is (not 0 and thus is) 1.” Sometimes we write C1C_{1} instead of C0compC_{0}^{\textup{comp}} (e.g., below in φ2\varphi_{2} if a=1a=1 or b=0b=0).

Let φ0\varphi_{0} and φ1\varphi_{1} be the sentences already introduced in Section 3. For w{0,1}w\in\{0,1\}^{*} the sentence φ0w\varphi_{0w} will be an extension of φ0\varphi_{0} (compare Definition 3.6 (b)); hence, φ0w\varphi_{0w} will be a universal sentence and all relations symbols besides << are negative in φ0w\varphi_{0w}; in particular, it contains as conjuncts φ0\varphi_{0} and

xt(¬C0(x,t)¬C0comp(x,t))qQxt(¬Hq(x,t)¬Hqcomp(x,t)).\forall x\forall t\big{(}\neg C_{0}(x,t)\vee\neg C^{\textup{comp}}_{0}(x,t)\big{)}\wedge\bigwedge_{q\in Q}\forall x\forall t\big{(}\neg H_{q}(x,t)\vee\neg H_{q}^{\textup{comp}}(x,t)\big{)}.

Finally, φ0w\varphi_{0w} will contain the following sentences φ2\varphi_{2} and φw\varphi_{w} as conjuncts. The sentence φ2\varphi_{2} describes one computation step. It contains for each instruction of MM one conjunct. For example, the instruction qapb1qapb1 contributes the conjunct

xxtty((Hq\displaystyle\forall xx^{\prime}\forall tt^{\prime}\forall y\Big{(}\big{(}H_{q} (x,t)Ca(x,t)S(x,x)S(t,t))\displaystyle(x,t)\wedge C_{a}(x,t)\wedge S(x,x^{\prime})\wedge S(t,t^{\prime})\big{)}
(\displaystyle\to\big{(} (¬C1b(x,t)¬Hpcomp(x,t))\displaystyle(\neg C_{1-b}(x,t^{\prime})\wedge\neg H^{\textup{comp}}_{p}(x^{\prime},t^{\prime}))
(yxrQ¬Hr(y,t))\displaystyle\wedge(y\neq x^{\prime}\to\bigwedge_{r\in Q}\neg H_{r}(y,t^{\prime}))
(yx((C0(y,t)¬C0comp(y,t))(C0comp(y,t)¬C0(y,t)))))).\displaystyle\wedge(y\neq x\to((C_{0}(y,t)\to\neg C^{\textup{comp}}_{0}(y,t))\wedge(C^{\textup{comp}}_{0}(y,t^{\prime})\to\neg C_{0}(y,t^{\prime}))))\big{)}\Big{)}.

For w{0,1}w\in\{0,1\}^{*} the sentence φw\varphi_{w} describes the initial configuration of MM with input ww (if w=w1w|w|w=w_{1}\ldots w_{|w|}, the first |w||w| cells contain w1,,w|w|w_{1},\ldots,w_{|w|}, the remaining cells contain 0, and the head scans the first cell in the starting state q0q_{0}). Hence, as φw\varphi_{w} we can take the conjunction of

  • x1x|w|((Uminx1i[|w|1]Sxixi+1)(i[|w|],wi=0¬C0comp(xi,x1)i[|w|],wi=1¬C0(xi,x1)))\forall x_{1}\ldots\forall x_{|w|}\big{(}(U_{\textup{min}}\,x_{1}\wedge\bigwedge_{i\in[|w|-1]}S{x_{i}}x_{i+1})\\ {\hskip 113.81102pt}\to(\bigwedge_{\begin{subarray}{c}i\in[|w|],\\ w_{i}=0\end{subarray}}\neg C^{\textup{comp}}_{0}(x_{i},x_{1})\wedge\bigwedge_{\begin{subarray}{c}i\in[|w|],\\ w_{i}=1\end{subarray}}\neg C_{0}(x_{i},x_{1}))\big{)}

  • x1x|w|x((Uminx1i[|w|1]Sxixi+1x|w|<x)¬C0comp(x,x1))\forall x_{1}\ldots\forall x_{|w|}\forall x\big{(}(U_{\textup{min}}\,x_{1}\wedge\bigwedge_{i\in[|w|-1]}S{x_{i}}x_{i+1}\wedge x_{|w|}<x)\to\neg C^{\textup{comp}}_{0}(x,x_{1})\big{)}

  • xy(Uminx(¬Hq0comp(x,x)(yxqQ¬Hq(y,x))))\forall x\forall y\big{(}U_{\textup{min}}\,x\to(\neg H_{q_{0}}^{\textup{comp}}(x,x)\wedge(y\neq x\to\bigwedge_{q\in Q}\neg H_{q}(y,x)))\big{)}.

Note that UminU_{\textup{min}}, UmaxU_{\textup{max}}, and SS are negative in φ0w\varphi_{0w}. We set φ1M:=φ1τM\varphi_{1M}:=\varphi_{1\tau_{M}}; recall that by Definition 3.6 (c),

φ1M=φ1xt(C0(x,t)C0comp(x,t))qQxt(Hq(x,t)Hqcomp(x,t)).\varphi_{1M}=\varphi_{1}\wedge\forall x\forall t\big{(}C_{0}(x,t)\vee C^{\textup{comp}}_{0}(x,t)\big{)}\wedge\bigwedge_{q\in Q}\forall x\forall t\big{(}H_{q}(x,t)\vee H_{q}^{\textup{comp}}(x,t)\big{)}.

Let w{0,1}w\in\{0,1\}^{*} and rr\in\mathbb{N}. Furthermore, let 𝒜\mathcal{A} be a τM\tau_{M}-structure where <𝒜<^{\mathcal{A}} is an ordering and |A|r+1|A|\geq r+1. Let a0,,ara_{0},\ldots,a_{r} be the first r+1r+1 elements of <𝒜<^{\mathcal{A}}. Assume that MM on the input w{0,1}w\in\{0,1\}^{*} runs at least rr steps. We say that 𝒜\mathcal{A} correctly encodes rr steps of the computation of MM on ww if for i,ji,j with 0i,jr0\leq i,j\leq r,

(ai,aj)C0𝒜\displaystyle(a_{i},a_{j})\in C_{0}^{\mathcal{A}} \displaystyle\iff the content of cell ii after jj steps is 0 (15)

and for qQq\in Q,

(ai,aj)Hq𝒜\displaystyle(a_{i},a_{j})\in H_{q}^{\mathcal{A}} \displaystyle\iff after jj steps MM is in state qq and the head scans cell jj. (16)

From the definitions of the sentences φ0w\varphi_{0w} and φ1M\varphi_{1M}, we see:

Lemma 6.1.

Let w{0,1}w\in\{0,1\}^{*} and 𝒜\mathcal{A} be a model φ0wφ1M\varphi_{0w}\wedge\varphi_{1M}. If for rr\in\mathbb{N} we have r+1|A|r+1\leq|A| (in particular, if AA is infinite) and MM on ww runs at least rr steps, then 𝒜\mathcal{A} correctly encodes rr steps of the computation of MM on ww.

Finally, let γM\gamma_{M} be a sentence expressing that “the machine MM reaches the halting state qhq_{h} in exactly ‘max’ steps,” more precisely,

γM:=tx(UmaxtHqh(x,t)ty(t<t¬Hqh(y,t))).\gamma_{M}:=\exists t\exists x\big{(}U_{\textup{max}}t\wedge H_{q_{h}}(x,t)\wedge\forall t^{\prime}\forall y(t^{\prime}<t\to\neg H_{q_{h}}(y,t^{\prime}))\big{)}. (17)

As a consequence of the preceding lemma, we obtain:

Corollary 6.2.

Let w{0,1}w\in\{0,1\}^{*} and assume that MM on input ww eventually halts, say in h(w)h(w) steps, then

φ0wφ1MγM\varphi_{0w}\wedge\varphi_{1M}\wedge\gamma_{M}

has no infinite model but a model with exactly h(w)+1h(w)+1 elements (this model is unique up to isomorphism).

Proof : Let 𝒜φ0wφ1MγM\mathcal{A}\models\varphi_{0w}\wedge\varphi_{1M}\wedge\gamma_{M}. Then 𝒜τ0\mathcal{A}\upharpoonright\tau_{0} is a complete τ0\tau_{0}-ordering and 𝒜\mathcal{A} contains the description of the complete halting computation of MM on the input ww. As the machine MM reaches the halting state in exactly h(w)h(w) steps, we see that |A|=h(w)+1|A|=h(w)+1; in particular, AA is finite.

On the other hand, we can interpret (15) and (16) as defining relations C0𝒜C_{0}^{\mathcal{A}} and Hq𝒜H_{q}^{\mathcal{A}} on the set A:={a0,,ah(w)}A:=\big{\{}a_{0},\ldots,a_{h(w)}\big{\}} equipped with the “natural” ordering and its corresponding relations UminU_{\textup{min}}, UmaxU_{\textup{max}}, and SS. If furthermore we let (C0comp)𝒜(C^{\textup{comp}}_{0})^{\mathcal{A}} and (Hqcomp)𝒜(H_{q}^{\textup{comp}})^{\mathcal{A}} be the complements in A×AA\times A of C0𝒜C_{0}^{\mathcal{A}} and Hq𝒜H_{q}^{\mathcal{A}}, respectively, we get a model of φ0wφ1MγM\varphi_{0w}\wedge\varphi_{1M}\wedge\gamma_{M} with exactly h(w)+1h(w)+1 elements. \Box

We set

χw:=φ0w(φ1M¬γM).\chi_{w}:=\varphi_{0w}\wedge(\varphi_{1M}\to\neg\gamma_{M}). (18)

By Proposition 3.11 and Corollary 6.2, we get:

Lemma 6.3.

Let MM on input ww eventually halt, say in h(w)h(w) steps. Then:

  1. (a)

    Mod(χw)\textsc{Mod}(\chi_{w}) is closed under <<-substructures.

  2. (b)

    If χw\chi_{w} is finitely equivalent to a universal sentence μ\mu, then |μ|h(w)+1|\mu|\geq h(w)+1.

Now we show the following version of Gurevich’s Theorem.

Theorem 6.4.

Let f:f:\mathbb{N}\to\mathbb{N} be a computable function. Then there is a w{0,1}w\in\{0,1\}^{*} such that Mod(χw)\textsc{Mod}(\chi_{w}) is closed under <<-substructures but χw\chi_{w} is not finitely equivalent to a universal sentence of length less than f(|χw|)f(|\chi_{w}|).

Proof : By the previous lemma it suffices to find a w{0,1}w\in\{0,1\}^{*} such that MM on input ww halts in h(w)h(w) steps with

h(w)f(|χw|).h(w)\geq f(|\chi_{w}|).

W.l.o.g. we assume that ff is increasing. An analysis of the formula χw\chi_{w} shows that for some cMc_{M}\in\mathbb{N} we have for all w{0,1}w\in\{0,1\}^{*},

|χw|cM|w|.|\chi_{w}|\leq c_{M}\cdot|w|. (19)

We define g:g:\mathbb{N}\to\mathbb{N} by

g(k):=f(5cMk).g(k):=f(5\cdot c_{M}\cdot k). (20)

Let M0M_{0} be a Turing machine computing gg, more precisely, the function 1k1g(k)1^{k}\mapsto 1^{g(k)}. We code M0M_{0} and 1k1^{k} by a {0,1}\{0,1\}-string code(M0,1k)\textit{code}(M_{0},1^{k}) such that MM on code(M0,1k)\textit{code}(M_{0},1^{k}) simulates the computation of M0M_{0} on 1k1^{k}.

Choose the least kk such that for w:=code(M0,1k)w:=\textit{code}(M_{0},1^{k}) we have

|w|5k.|w|\leq 5k. (21)

The universal Turing machine MM on input ww computes 1g(k)1^{g(k)} and thus runs at least g(k)g(k) steps, say, exactly h(w)h(w) steps. By (19) – (21)

h(w)g(k)=f(5cMk)f(cM|w|)f(|χw|).h(w)\geq g(k)=f(5\cdot c_{M}\cdot k)\geq f(c_{M}\cdot|w|)\geq f(|\chi_{w}|).

Finally we prove Gurevich’s Theorem for graphs. For τ:=τM\tau:=\tau_{M} let II be an interpretation according to Lemma 5.6. For w{0,1}w\in\{0,1\}^{*} we consider the sentence

ρw:=x¯¬φuni(x¯)(φ0w(φ1M¬γM))I=x¯¬φuni(x¯)χwI.\rho_{w}:=\forall\bar{x}\neg\varphi_{\textup{uni}}(\bar{x})\vee(\varphi_{0w}\wedge(\varphi_{1M}\to\neg\gamma_{M}))^{I}=\forall\bar{x}\neg\varphi_{\textup{uni}}(\bar{x})\vee\chi_{w}^{I}. (21)

That is, for GρwG\models\rho_{w}, either the graph GG interprets an empty τM\tau_{M}-structure, or a τM\tau_{M}-structure which is a model of χw\chi_{w}. If MM halts in h(w)h(w) steps on input ww, then φ0wφ1MγM\varphi_{0w}\wedge\varphi_{1M}\wedge\gamma_{M} has no infinite model but a finite model with h(w)+1h(w)+1 elements by Corollary 6.2. Hence taking in Proposition 4.6 as ψ\psi the sentence ψGraph\psi_{\textsc{Graph}} axiomatizing the class of graphs we get the following analogue of Lemma 6.3.

Lemma 6.5.

Let MM on input ww halt in h(w)h(w) steps. Then:

  1. (a)

    Graph(ρw)\textsc{Graph}(\rho_{w}), the class of graphs that are model of ρw\rho_{w}, is closed under induced subgraphs (and hence equivalent in the class of graphs to a universal sentence).

  2. (b)

    If ρw\rho_{w} is equivalent in the class of finite graphs to a universal sentence μ\mu, then |μ|2h(w)|\mu|^{2}\geq h(w).

Theorem 6.6 (Gurevich’s Theorem for graphs).

Let f:f:\mathbb{N}\to\mathbb{N} be a computable function. Furthermore, let ρw\rho_{w} be defined by (21), where II is an interpretation for τ:=τM\tau:=\tau_{M} according to Lemma 5.6. Then there is a w{0,1}w\in\{0,1\}^{*} such that Graph(ρw)\textsc{Graph}(\rho_{w}) is closed under induced subgraphs but ρw\rho_{w} is not equivalent in the class of finite graphs to a universal sentence of length less than f(|ρw|)f(|\rho_{w}|).

Proof : Again we assume that ff is increasing. By the previous lemma it suffices to find a w{0,1}w\in\{0,1\}^{*} such that MM on input ww halts in h(w)h(w) steps with

h(w)f(|ρw|)2.h(w)\geq f(|\rho_{w}|)^{2}.

There is a cc\in\mathbb{N}, which depends on II but not on ww, such that for cIc_{I} as in (13) and cMc_{M} as in (19) we have for dM:=c+cIcMd_{M}:=c+c_{I}\cdot c_{M},

|ρw|c+cI|χw|c+cIcM|w|dM|w|.|\rho_{w}|\leq c+c_{I}\cdot|\chi_{w}|\leq c+c_{I}\cdot c_{M}\cdot|w|\leq d_{M}\cdot|w|. (22)

We define g:g:\mathbb{N}\to\mathbb{N} by

g(k):=f(5dMk)2g(k):=f(5\cdot d_{M}\cdot k)^{2} (23)

and then proceed as in the proof of Theorem 6.4. Let M0M_{0} be a Turing machine computing the function 1k1g(k)1^{k}\mapsto 1^{g(k)}. We code M0M_{0} and 1k1^{k} by a {0,1}\{0,1\}-string code(M0,1k)\textit{code}(M_{0},1^{k}) such that MM on code(M0,1k)\textit{code}(M_{0},1^{k}) simulates the computation of M0M_{0} on 1k1^{k}.

Choose the least kk such that for w:=code(M0,1k)w:=\textit{code}(M_{0},1^{k}) we have

|w|5k.|w|\leq 5k. (24)

The universal Turing machine MM on input ww computes 1g(k)1^{g(k)} and thus runs at least g(k)g(k) steps, say, exactly h(w)h(w) steps. We have

h(w)g(k)=f(5dMk)2f(dM|w|)2f(|ρw|)2h(w)\geq g(k)=f(5\cdot d_{M}\cdot k)^{2}\geq f(d_{M}\cdot|w|)^{2}\geq f(|\rho_{w}|)^{2}

by (22) – (24). \Box

Remark 6.7.

Using previous remarks (Remark 3.12 and Remark 4.7) one can even show that for every computable function f:f:\mathbb{N}\to\mathbb{N} the sentence χw\chi_{w} is not finitely equivalent to a Π2\Pi_{2}-sentence of length less than f(|χw|)f(|\chi_{w}|) and the sentence ρw\rho_{w} is not finitely equivalent in graphs to a Π2\Pi_{2}-sentence of length less than f(|χw|)f(|\chi_{w}|). Moreover, χw\chi_{w} and ρw\rho_{w} are equivalent to Σ2\Sigma_{2}.

For this purpose note that in models of φ0w\varphi_{0w} the sentence γM\gamma_{M} is equivalent to

tx(UmaxtHqh(x,t))t1t2y(t1<t2¬Hqh(y,t2)).\exists t\exists x\big{(}U_{\textup{max}}t\wedge H_{q_{h}}(x,t)\big{)}\wedge\forall t_{1}\forall t_{2}\forall y\big{(}t_{1}<t_{2}\to\neg H_{q_{h}}(y,t_{2})\big{)}.

and hence equivalent to a Σ2\Sigma_{2} and to a Π2\Pi_{2}-sentence. One easily verifies that the same holds for γMI\gamma_{M}^{I}.

7 Some undecidable problems

In this section we show that various problems related to the results of the preceding sections are undecidable. Among others, these results explain why it might be hard, in fact impossible in general, to obtain forbidden induced subgraphs for various classes of graphs.

Proposition 7.1.

There is no algorithm that applied to any FO[τE]\textup{FO}[\tau_{E}]-sentence φ\varphi decides whether the class Graph(φ)\textsc{Graph}(\varphi) is closed under induced subgraphs.

Proof : Assume 𝔸\mathbb{A} is such an algorithm. By the Completeness Theorem there is an algorithm 𝔹\mathbb{B} that assigns to every sentence φ\varphi with Graph(φ)\textsc{Graph}(\varphi) closed under induced subgraphs a universal sentence equivalent to φ\varphi in graphs. Define the function gg by

g(φ):={0,if 𝔸 rejects φm,𝔹 needs m steps to produce a universal sentence equivalent to φg(\varphi):=\begin{cases}0,&\text{if $\mathbb{A}$ rejects $\varphi$}\\ m,&\text{$\mathbb{B}$ needs $m$ steps to produce a universal sentence equivalent to $\varphi$}\end{cases}

and set f(k):=max{g(φ)|φ|k}f(k):=\textup{max}\{g(\varphi)\mid|\varphi|\leq k\}. Then ff would contradict Gurevich’s Theorem for graphs, i.e., Theorem 6.6. \Box

Corollary 7.2.

There is no algorithm that applied to any FO[τE]\textup{FO}[\tau_{E}]-sentence φ\varphi either reports that Graph(φ)\textsc{Graph}(\varphi) is not closed under induced subgraphs or it computes for Graph(φ)\textsc{Graph}(\varphi) a class of forbidden induced subgraphs.

Proof : Otherwise we could use this algorithm as a decision algorithm for the previous result. \Box

The following proposition is the analog of Proposition 7.1 for classes of finite graphs. We state it for FO[τE]\textup{FO}[\tau_{E}]-sentences and graphs even though we prove it for FO[τM]\textup{FO}[\tau_{M}]-sentences. One gets the version for graphs using the machinery we developed in previous sections similarly as we do it to get Corollary 7.5 from Proposition 7.4 below.

We write M:wM:w\mapsto\infty for the universal Turing machine MM and a word w{0,1}w\in\{0,1\}^{*} if MM on input ww does not halt. We make use of the sentences φ0w\varphi_{0w}, φ1M\varphi_{1M}, and γM\gamma_{M} defined in the previous section.

Proposition 7.3.

There is no algorithm that applied to any FO[τE]\textup{FO}[\tau_{E}]-sentence φ\varphi decides whether the class Graphfin(φ)\textsc{Graph}\,_{\textup{fin}}(\varphi) is closed under induced subgraphs.

Proof : For the universal Turing machine MM and a word w{0,1}w\in\{0,1\}^{*} consider the sentence

πw:=φ0wφ1MγM.\pi_{w}:=\varphi_{0w}\wedge\varphi_{1M}\wedge\gamma_{M}.

Then

Modfin(πw)\textsc{Mod}\,_{\textup{fin}}(\pi_{w}) is closed under induced subgraphs \displaystyle\iff M:w.\displaystyle M:w\mapsto\infty. (25)

In fact, if M:wM:w\mapsto\infty, then Modfin(πw)=\textsc{Mod}\,_{\textup{fin}}(\pi_{w})=\emptyset, hence Modfin(πw)\textsc{Mod}\,_{\textup{fin}}(\pi_{w}) is trivially closed under induced subgraphs. If MM on input ww halts after h(w)h(w) steps, then, up to isomorphism, there is a unique model 𝒜w\mathcal{A}_{w} of πw\pi_{w} and it has h(w)+1h(w)+1 elements. By Lemma 3.8 every proper induced substructure of 𝒜w\mathcal{A}_{w} is not a model of πw\pi_{w}. Hence Modfin(πw)\textsc{Mod}\,_{\textup{fin}}(\pi_{w}) is not closed under induced subgraphs. As the halting problem for every universal Turing machine is not decidable, by (25) we get our claim. \Box

Proposition 7.4.

There is no algorithm that applied to any FO[τM]\textup{FO}[\tau_{M}]-sentence, which is finitely equivalent to a universal sentence, computes such a universal sentence.

Proof : Again we show that such an algorithm would allow us to decide for every w{0,1}w\in\{0,1\}^{*} whether the universal Turing machine MM halts on input ww. In (18) we defined χw\chi_{w} by

χw=φ0w(φ1M¬γM).\chi_{w}=\varphi_{0w}\wedge(\varphi_{1M}\to\neg\gamma_{M}).

If MM halts on ww, by Lemma 6.3 we know that Mod(χw)\textsc{Mod}(\chi_{w}) is closed under <<-substructures and thus equivalent to a universal sentence. The claimed algorithm (or, even the Completeness Theorem) will produce such a universal μ\mu. Furthermore, by Corollary 6.2 we know that there is a finite model with h(w)+1h(w)+1 elements, which is a model of φ0w¬χw\varphi_{0w}\wedge\neg\chi_{w}, hence it is a model of φ0w¬μ\varphi_{0w}\wedge\neg\mu.

If MM does not halt on ww, then we show that Modfin(χw)=Modfin(φ0w)\textsc{Mod}\,_{\textup{fin}}(\chi_{w})=\textsc{Mod}\,_{\textup{fin}}(\varphi_{0w}). Clearly Modfin(χw)Modfin(φ0w)\textsc{Mod}_{\textup{fin}}(\chi_{w})\subseteq\textsc{Mod}_{\textup{fin}}(\varphi_{0w}). Now let 𝒜\mathcal{A} be a finite model of φ0w\varphi_{0w}. If 𝒜⊧̸φ1M\mathcal{A}\not\models\varphi_{1M}, then 𝒜χw\mathcal{A}\models\chi_{w}. Otherwise 𝒜φ1M\mathcal{A}\models\varphi_{1M}, then 𝒜\mathcal{A} correctly represents the first |A|1|A|-1 steps of the computation of MM on ww by Lemma 6.1. Thus 𝒜\mathcal{A} is a model of ¬γM\neg\gamma_{M} as MM does not halt on ww. Therefore, 𝒜\mathcal{A} is a model of χw\chi_{w}.

Now we can see whether MM does not halt on ww by checking whether the universal sentence produced by the claimed algorithm is finitely equivalent to the universal sentence φ0w\varphi_{0w}. This can be checked effectively by Corollary 2.5 and Corollary 2.6. \Box

Corollary 7.5.

There is no algorithm that applied to any FO[τE]\textup{FO}[\tau_{E}]-sentence φ\varphi such that Graphfin(φ)\textsc{Graph}\,_{\textup{fin}}(\varphi) has a finite set of forbidden induced subgraphs computes such a set.

Proof : Equivalently we show that there is no algorithm that applied to any FO[τE]\textup{FO}[\tau_{E}]-sentence φ\varphi such that Graphfin(φ)=Graphfin(μ)\textsc{Graph}\,_{\textup{fin}}(\varphi)=\textsc{Graph}\,_{\textup{fin}}(\mu) for some universal sentence μ\mu computes such a μ\mu.

For graphs let I(=IτM)I\ (=I_{\tau_{M}}) be a strongly existential interpretation of τM\tau_{M}-structures in graphs according to Lemma 5.6. We know that for every finite τM\tau_{M}-structure 𝒜\mathcal{A} there is a finite graph GG such that GI𝒜G_{I}\cong\mathcal{A}.

For w{0,1}w\in\{0,1\}^{*} we consider the sentence ρw\rho_{w} defined in (21) in the proof of Theorem 6.4,

ρw=x¯¬φuni(x¯)(φ0w(φ1M¬γM))I=x¯¬φuni(x¯)χwI.\rho_{w}=\forall\bar{x}\neg\varphi_{\textup{uni}}(\bar{x})\vee(\varphi_{0w}\wedge(\varphi_{1M}\to\neg\gamma_{M}))^{I}=\forall\bar{x}\neg\varphi_{\textup{uni}}(\bar{x})\vee\chi_{w}^{I}.

We show that ρw\rho_{w} is equivalent to a universal sentence μ\mu on finite graphs. Moreover, MM does not halt on input ww if and only if μ\mu is finitely equivalent to the universal sentence x¬φuni(x¯)φ0wI\forall x\neg\varphi_{\textup{uni}}(\bar{x})\vee\varphi_{0w}^{I}

If MM halts on ww, then φ0wφ1MγM\varphi_{0w}\wedge\varphi_{1M}\wedge\gamma_{M} has no infinite model but a finite model 𝒜\mathcal{A}. Hence, by Proposition 4.6 we know that Graph(ρw)\textsc{Graph}(\rho_{w}) is closed under induced subgraphs. Therefore, ρw\rho_{w} is equivalent to a universal sentence μ\mu in Graph. Let GG be a finite graph with GI𝒜G_{I}\cong\mathcal{A}. Then G(φ0wφ1MγM)IG\models(\varphi_{0w}\wedge\varphi_{1M}\wedge\gamma_{M})^{I} and thus, G¬ρwG\models\neg\rho_{w}. Hence GG is a finite graph which is a model of φ0wI¬μ\varphi^{I}_{0w}\wedge\neg\mu. This means that μ\mu is not equivalent to x¬φuni(x¯)φ0wI\forall x\neg\varphi_{\textup{uni}}(\bar{x})\vee\varphi_{0w}^{I} on all finite graphs, as GG is also a model of x¬φuni(x¯)φ0wI\forall x\neg\varphi_{\textup{uni}}(\bar{x})\vee\varphi_{0w}^{I}.

If M:wM:w\to\infty, then we show that Graphfin(ρw)=Graphfin(x¬φuni(x¯)φ0wI)\textsc{Graph}\,_{\textup{fin}}(\rho_{w})=\textsc{Graph}\,_{\textup{fin}}(\forall x\neg\varphi_{\textup{uni}}(\bar{x})\vee\varphi_{0w}^{I}). Clearly Graphfin(ρw)Graphfin(x¬φuni(x¯)φ0wI)\textsc{Graph}_{\textup{fin}}(\rho_{w})\subseteq\textsc{Graph}_{\textup{fin}}(\forall x\neg\varphi_{\textup{uni}}(\bar{x})\vee\varphi_{0w}^{I}). Now let the graph GG be a model of x¬φuni(x¯)φ0wI\forall x\neg\varphi_{\textup{uni}}(\bar{x})\vee\varphi_{0w}^{I}. Further we can assume that Gxφuni(x¯)G\models\exists x\varphi_{\textup{uni}}(\bar{x}). In particular, 𝒜:=GI\mathcal{A}:=G_{I} is well defined. If G⊧̸φ1MIG\not\models\varphi^{I}_{1M}, then GχwIG\models\chi^{I}_{w} and therefore, GρwG\models\rho_{w}. If Gφ1MIG\models\varphi^{I}_{1M}, then 𝒜φ0wφ1MI\mathcal{A}\models\varphi_{0w}\wedge\varphi^{I}_{1M}. As M:wM:w\to\infty, by Lemma 6.1 the structure 𝒜\mathcal{A} correctly represents the first |A|1|A|-1 steps of the computation of MM on ww. Thus, 𝒜\mathcal{A} is a model of ¬γM\neg\gamma_{M}, again as MM does not halt on input ww. It follows that GG is a model of ¬γMI\neg\gamma_{M}^{I}, and then GρwG\models\rho_{w}.

Now we can decide the halting problem for MM. Given a word ww, we use the claimed algorithm to get a universal sentence μ\mu equivalent to ρw\rho_{w} in the class of graphs. Finally we check whether μ\mu is finitely equivalent to x¬φuni(x¯)φ0wI\forall x\neg\varphi_{\textup{uni}}(\bar{x})\vee\varphi_{0w}^{I}. This can be checked effectively again by Corollary 2.5 and Corollary 2.6. \Box

Observe that Corollary 7.5 is precisely Theorem 1.3 as stated in the Introduction. Finally we prove Theorem 1.2, which is equivalent to the following result.

Theorem 7.6.

There is no algorithm that applied to an FO[τE]\textup{FO}[\tau_{E}]-sentence φ\varphi such that Graphfin(φ)\textsc{Graph}_{\textup{fin}}(\varphi) is closed under induced subgraphs decides whether there is a finite set \mathscr{F} of graphs such that

Graphfin(φ)=Forbfin().\textsc{Graph}_{\textup{fin}}(\varphi)=\textsc{Forb}_{\textup{fin}}(\mathscr{F}).

Proof : Again we prove the corresponding result for τM\tau_{M}-sentences and τM\tau_{M}-structures and leave it to the reader to translate it to graphs as in the previous proof. That is, we show:

There is no algorithm that applied to an FO[τM]\textup{FO}[\tau_{M}]-sentence φ\varphi such that Modfin(φ)\textsc{Mod}_{\textup{fin}}(\varphi) is closed under induced substructures decides whether there is a finite set FF of finite τM\tau_{M}-structures such that

Modfin(φ)=Forbfin().\textsc{Mod}_{\textup{fin}}(\varphi)=\textsc{Forb}_{\textup{fin}}(\mathscr{F}).

For w{0,1}w\in\{0,1\}^{*} let

αw:=φ0w(φ1MγM).\alpha_{w}:=\varphi_{0w}\wedge(\varphi_{1M}\to\gamma_{M}).

We show that Modfin(αw)\textsc{Mod}_{\textup{fin}}(\alpha_{w}) is closed under induced subgraphs and that

M:w\displaystyle M:w\to\infty \displaystyle\iff αw is not finitely equivalent to a universal sentence.\displaystyle\text{$\alpha_{w}$ is not finitely equivalent to a universal sentence}.

Assume first that M:wM:w\to\infty. Then φ0wφ1MγM\varphi_{0w}\wedge\varphi_{1M}\wedge\gamma_{M} has no finite model by Lemma 6.1 and the definition (17) of γM\gamma_{M}. Therefore, Modfin(αw)=Modfin(φ0w¬φ1M)\textsc{Mod}_{\textup{fin}}(\alpha_{w})=\textsc{Mod}_{\textup{fin}}(\varphi_{0w}\wedge\neg\varphi_{1M}). By Lemma 3.10 we know that Modfin(φ0w¬φ1M)\textsc{Mod}_{\textup{fin}}(\varphi_{0w}\wedge\neg\varphi_{1M}) is closed under induced substructures but not finitely equivalent to a universal sentence.

Now assume that MM on input ww halts in h(w)h(w) steps. Then Corollary 6.2 guarantees that there is a unique model 𝒜w\mathcal{A}_{w} of φ0wφ1MγM\varphi_{0w}\wedge\varphi_{1M}\wedge\gamma_{M} with |Aw|=h(w)+1|A_{w}|=h(w)+1. We present a finite set \mathscr{F} of finite τM\tau_{M}-structures such that

Modfin(αw)=Forbfin().\textsc{Mod}_{\textup{fin}}(\alpha_{w})=\textsc{Forb}_{\textup{fin}}(\mathscr{F}). (26)

As φ0w\varphi_{0w} is universal, there is a finite set 0\mathscr{F}_{0} of finite τM\tau_{M}-structures such that

Modfin(φ0w)=Forbfin(0).\textsc{Mod}_{\textup{fin}}(\varphi_{0w})=\textsc{Forb}_{\textup{fin}}(\mathscr{F}_{0}).

Moveover, we set

1:={Str[τM]|φ0wφ1M and B=[] for some h(w)}\mathscr{F}_{1}:=\big{\{}\mathcal{B}\in\textsc{Str}[\tau_{M}]\;\big{|}\;\mathcal{B}\models\varphi_{0w}\wedge\varphi_{1M}\text{\ and $B=[\ell]$ for some $\ell\leq h(w)$}\big{\}}

and

2:={Str[τM]|φ0wφ1Mtt(t<ty\displaystyle\mathscr{F}_{2}:=\big{\{}\mathcal{B}\in\textsc{Str}[\tau_{M}]\;\big{|}\;\mathcal{B}\models\varphi_{0w}\wedge\varphi^{*}_{1M}\wedge\forall t\forall t^{\prime}(t<t^{\prime}\to\forall y ¬Hqh(y,t))\displaystyle\neg H_{q_{h}}(y,t))
and B=[h(w)+2]}.\displaystyle\text{and $B=[h(w)+2]$}\big{\}}.

Here φ1M\varphi^{*}_{1M}is obtained from φ1M\varphi_{1M} by replacing the conjunct φ1\varphi_{1} (see (7)) by

φ1:=xUminxxy(x<yzSxz).\varphi^{*}_{1}:=\exists xU_{\textup{min}}x\wedge\forall x\forall y(x<y\to\exists zSxz).

The difference is that φ1\varphi^{*}_{1} does not require the set UmaxU_{\textup{max}} to be nonempty. Hence,

φ1M=φ1xt(C0(x,t)C0comp(x,t))qQxt(Hq(x,t)Hqcomp(x,t)).\varphi^{*}_{1M}=\varphi^{*}_{1}\wedge\forall x\forall t\big{(}C_{0}(x,t)\vee C^{\textup{comp}}_{0}(x,t)\big{)}\wedge\bigwedge_{q\in Q}\forall x\forall t\big{(}H_{q}(x,t)\vee H_{q}^{\textup{comp}}(x,t)\big{)}.

Note that Lemma 6.1 remains true if in its statement we replace φ1M\varphi_{1M} by φ1M\varphi^{*}_{1M}.

For :=012\mathscr{F}:=\mathscr{F}_{0}\cup\mathscr{F}_{1}\cup\mathscr{F}_{2} we show (26). Assume first that a finite structure 𝒞\mathcal{C} is a model of αw\alpha_{w}. In particular, 𝒞φ0w\mathcal{C}\models\varphi_{0w} and therefore, 𝒞\mathcal{C} has no induced substructure isomorphic to a structure in 0\mathscr{F}_{0}.

Now, for a contradiction suppose that \mathcal{B} is an induced substructure of 𝒞\mathcal{C} isomorphic to a structure in 1\mathscr{F}_{1}. Then φ1M\mathcal{B}\models\varphi_{1M} and thus, by Lemma 3.8, 𝒞=\mathcal{C}=\mathcal{B}. As 𝒞αw\mathcal{C}\models\alpha_{w}, we get 𝒞φ0wφ1MγM\mathcal{C}\models\varphi_{0w}\wedge\varphi_{1M}\wedge\gamma_{M}. Hence, 𝒞𝒜w\mathcal{C}\cong\mathcal{A}_{w}, a contradiction, as on the one hand |C|=|B|h(w)|C|=|B|\leq h(w) and on the other hand |C|=|Aw|=h(w)+1|C|=|A_{w}|=h(w)+1.

Next we show that 𝒞\mathcal{C} has no induced substructure \mathcal{B} isomorphic to a structure in 2\mathscr{F}_{2}. As φ0wφ1M\mathcal{B}\models\varphi_{0w}\wedge\varphi^{*}_{1M} and has h(w)+2h(w)+2 elements, the first h(w)+1h(w)+1 elements of \mathcal{B} correctly encode the first h(w)h(w) steps of the computation of MM on ww, hence the full computation. As |B|=h(w)+2|B|=h(w)+2, this contradicts tt(t<ty¬Hqh(y,t))\mathcal{B}\models\forall t\forall t^{\prime}\big{(}t<t^{\prime}\to\forall y\neg H_{q_{h}}(y,t)\big{)}.

As the final step let 𝒞Forbfin()\mathcal{C}\in\textsc{Forb}_{\textup{fin}}(\mathscr{F}). We show that 𝒞αw\mathcal{C}\models\alpha_{w}. As 𝒞\mathcal{C} omits the structures in 0\mathscr{F}_{0} as induced substructures, we see that 𝒞φ0w\mathcal{C}\models\varphi_{0w}. If 𝒞⊧̸φ1M\mathcal{C}\not\models\varphi_{1M}, we are done.

Recall that by Lemma 6.1 (more precisely, by the extension of Lemma 6.1 mentioned above) for finite structures \mathcal{B} of φ0wφ1M\varphi_{0w}\wedge\varphi^{*}_{1M} we know:

  • (a)

    if |B|h(w)+1|B|\leq h(w)+1, then \mathcal{B} encodes |B|1|B|-1 steps of the computation of MM on ww,

  • (b)

    if |B|>h(w)+1|B|>h(w)+1, then the first h(w)+1h(w)+1 elements in the ordering <<^{\mathcal{B}} correctly encode the (full) computation of MM on ww.

Now assume that 𝒞φ1M\mathcal{C}\models\varphi_{1M}, then (a) and (b) apply to 𝒞\mathcal{C}. As no structure in 1\mathscr{F}_{1} is isomorphic to an induced substructure of 𝒞\mathcal{C}, we see that |C|h(w)+1|C|\geq h(w)+1. But 𝒞\mathcal{C} cannot have more than h(w)+1h(w)+1 elements, as otherwise the substructure of 𝒞\mathcal{C} induced on the first h(w)+2h(w)+2 elements would be isomorphic to a structure \mathcal{B} in F2F_{2}, a contradiction. \Box

Remark 7.7.

Mainly using Remark 6.7 one easily verifies that in all results but Proposition 7.3 of this section we can replace

There is no algorithm that applied to an FO[τE]\textup{FO}[\tau_{E}]-sentence φ\varphi

by

There is no algorithm that applied to a Σ2\Sigma_{2}-sentence φ\varphi

In Proposition 7.3 we have to replace it by

There is no algorithm that applied to a Π2\Pi_{2}-sentence φ\varphi

as φ1M\varphi_{1M} (and φ1MI\varphi^{I}_{1M}) are Π2\Pi_{2}-sentences.

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