Forbidden Induced Subgraphs and the Łoś-Tarski Theorem
Abstract
Let 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 is definable in first-order logic (FO) by a sentence if and only if 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 the corresponding forbidden induced subgraphs. We show that this machinery fails on finite graphs.
-
–
There is a class of finite graphs which is definable in FO and closed under induced subgraphs but has no finite set of forbidden induced subgraphs.
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–
Even if we only consider classes 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 , which defines , and the size of the characterization cannot be bounded by for any computable function .
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 and consist of all graphs with vertex set and maximum degree exactly . Then a graph has degree at most if and only if no graph in is isomorphic to an induced subgraph of . 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 of finite and infinite graphs that are models of a sentence of first-order logic is closed under induced subgraphs (or, that is preserved under induced subgraphs) if and only if there is a universal FO-sentence with . Recall that a universal sentence is a sentence of the form , where is quantifier-free.
Such a universal sentence expresses that certain patterns of induced subgraphs with at most vertices are forbidden. In fact, let be a finite set of finite graphs and denote by the class of (finite and infinite) graphs that do not contain an induced subgraph isomorphic to a graph in . Then for a universal sentence as above we have
(1) |
Here for any FO-sentence and by we denote the class of graphs that are models of and whose universe is for some with . Clearly, is finite.
We say that a class of finite and infinite graphs is definable by a finite set of forbidden induced subgraphs if there is a finite set of finite graphs such that . Hence the graph-theoretic version of the Łoś-Tarski Theorem can be restated in the form:
(I) | Let be a class of finite and infinite graphs. The following are equivalent: |
(i) is closed under induced subgraphs and FO-axiomatizable. | |
(ii) is axiomatizable by a universal sentence. | |
(iii) 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 of finite structures (which might be understood as colored directed graphs) which is closed under induced substructures and FO-axiomatizable. Yet, 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 of finite graphs with the following properties.
-
(i)
is closed under induced subgraphs and FO-axiomatizable,
-
(ii)
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 such that
is closed under induced subgraphs decides whether 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 [7]. Let us consider the -vertex cover problem for a constant . 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 . The class of all yes-instances of this problem, finite and infinite, is closed under induced subgraphs and FO-axiomatizable by the FO-sentence
where 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 . But on the other hand, by the Completeness Theorem, we can search for such a universal sentence by enumerating all possible universal sentences and all possible proofs for , and then extract the corresponding forbidden induced subgraphs from 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 which is equivalent to a universal sentence on finite graphs computes such a .
Or equivalently, there is no algorithm that for any FO-sentence such that
for a finite set of graphs computes such an . Here,
Theorem 1.4.
Let be a computable function. Then there is a class of finite graphs and an FO-sentence such that
-
(i)
.
-
(ii)
for some universal sentence , in particular is closed under induced subgraphs.
-
(iii)
For every universal sentence with we have .
Theorem 1.3 significantly strengthens the aforementioned result of [10]. Even if a class of finite graphs definable by a finite set of forbidden induced subgraphs is given by an FO-sentence with , instead of a much more powerful Turing machine, we still cannot compute an appropriate finite set of forbidden induced subgraphs for from . On top of it, Theorem 1.4 implies that the size of forbidden subgraphs for cannot be bounded by any computable function in terms of the size of .
There is an important precursor for Theorem 1.4,
Theorem 1.5 (Gurevich’s Theorem [14]).
Let be computable. Then there is an FO-sentence such that the class of models of is closed under induced substructures but for every universal sentence with we have .
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 of finite structures the analogue of the Łoś-Tarski Theorem holds if we restrict to structures in . For example, this is the case if is the class of all finite structures of tree-width or less for some [1] or if 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 consider sentences of the form , where is universal. Then the role of the closure under induced substructures is taken over by a semantic “core property PS()” which for coincides with closure under induced substructures. Finally, we mention that in [4] the authors strengthen Tait’s result by showing that for every there are first-order definable classes of finite structures closed under substructures which are not definable with 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 the set of natural numbers greater or equal to 0. For let .
First-order logic FO.
A vocabulary is a finite set of relation symbols. Each relation symbol has an arity. A structure of vocabulary , or -structure, consists of a (finite or infinite) nonempty set , called the universe of and of an interpretation of each -ary relation symbol . If and are -structures, then is a substructure of , denoted by , if and , and is an induced substructure of , denoted by , if and , where is the arity of . If, in addition, , then is an proper induced substructure of . By ( ) we denote the class of all (of all finite) -structures.
Formulas of first-order logic FO of vocabulary are built up from atomic formulas and (where is of arity and are variables) using the boolean connectives , , and and the universal and existential quantifiers. A relation symbol is positive (negative) in if all atomic subformulas in appear in the scope of an even (odd) number of negation symbols. By the notation with we indicate that the variables free in are among . If then is a -structure and , then means that holds in if is interpreted by for .
A sentence is a formula without free variables. For a sentence we denote by the class of models of and is its subclass consisting of the finite models of . Sentences and are equivalent if and finitely equivalent if .
Graphs.
Let with binary . For all -structures we use the notation common in graph theory. Here , the universe of , is the set of vertices, and , the interpretation of the relation symbol , is the set of edges. The -structure is a directed graph if does not contain self-loops, i.e., for any . If moreover implies for any pair , then is an (undirected) graph. The graph is an induced subgraph of if
and |
We denote by Graph and the class of all graphs and the class of finite graphs, respectively. Furthermore, for an -sentence by (and ) we denote the class of graphs (and the class of finite graphs) that are models of .
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 and and the universal quantifier . Often we say that a formula, say, containing the connective is universal if by replacing by (and “simple manipulations”) we get an equivalent universal sentence. Every universal sentence is equivalent to a sentence of the form for some and some quantifier-free and moreover the length of is at most . 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 be a vocabulary and an -sentence. Then is closed under induced substructures if and only if is equivalent to a universal sentence.
We fix a vocabulary . Let be a finite set of finite -structures and denote by (and ) the class of structures (of finite structures) that do not contain an induced substructure isomorphic to a structure in . Clearly for finite sets and of finite -structures we have
if , then . | (2) |
We say that a class of -structures (of finite -structures) is definable by a finite set of forbidden induced substructures if there is a finite set of finite structures such that ().
Recall that with binary .
and | (3) |
axiomatize the classes of directed graphs and of graphs, respectively. Let the -structures and be given by
and |
Then and are the class of directed graphs and the class of graphs, respectively, i.e., and .
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 , an -sentence , and let
(4) |
Thus, is, up to isomorphism, the class of structures with at most elements which fail to be a model of . Note that and . Clearly, for a -sentence we have:
if is closed under indu | ||||
(5) |
Proposition 2.2.
For a class of -structures and the statements (i) and (ii) are equivalent.
-
(i)
for some universal sentence with quantifier-free .
-
(ii)
for some finite set of structures, all of at most elements.
If (i) holds for , then .
Proof : (i) (ii) Let for as in (i). Then is closed under induced substructures and hence, by (5).
Now assume that . Then and hence there are with . For , the substructure of induced by , we have (as is quantifier-free) and thus, . Therefore, is isomorphic to a structure in and therefore, .
(ii) (i) Let the -structure have at most elements and let be an enumeration of the elements of (possibly with repetitions). Let be the conjunction of all literals (i.e., atomic or negated atomic formulas) such that . Then for every -structure and we have
the clauses for | ||||
define an isomorphism from onto . | (6) |
Now assume (ii), i.e., for some finite set of structures, all of at most elements. If is empty, then . Otherwise for every we fix an enumeration of the elements of . We set
Then . In fact, assume first that . Then there are and an such that . By (6), then is isomorphic to the induced substructure of ; hence, .
Now assume . Then there is an and elements such that the clauses for define an isomorphism from onto . By (6), then . Therefore, , i.e., .
Corollary 2.3.
Let be a -sentence and . Then
is equivalent to a universal sentence | ||||
of the form with quantifier-free . |
Corollary 2.4.
If for some universal and some , then for all .
Corollary 2.5.
It is decidable whether two universal sentences are equivalent.
Proof : Let and be universal sentences. W.l.o.g. we may assume that and with . By Corollary 2.3 and Corollary 2.4, we have
and |
Thus and are equivalent if and only if . The right hand side of this equivalence is clearly decidable.
The last equivalence of this corollary shows:
Corollary 2.6.
For universal sentences and we have
The following consequence of Corollary 2.2 will be used in the next section.
Corollary 2.7.
Let with and let and be -sentences. Assume that is a finite model of with at least elements and all its proper induced substructures with at most elements are models of . Then is not finitely equivalent to a universal sentence of the form with quantifier-free .
Proof : For a contradiction assume for as above. As by Proposition 2.2, we get (applying the finitely equivalence of and to obtain the last equality)
However, by the assumptions the structure is contained in but not in the class .
Remark 2.8.
Let be a class of -structures closed under induced substructures. For an -sentence we set . We say that the Łoś-Tarski Theorem holds for if for every -sentence such that the class is closed under induced substructures there is a universal sentence such that
The following holds:
Let and be classes of -structures closed under induced substructures with . Furthermore assume that there is a universal sentence such that . If the analogue of the Łoś-Tarski Theorem holds for , then it holds for , too
In fact, for every -sentence we have . Hence, if is closed under induced substructures, then by assumption there is a universal such that . Therefore, .
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 , where and (the successor relation) are binary relation symbols and and are unary.
Let be the conjunction of the universal sentences
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–
, , , i.e., “ is an ordering”
-
–
i.e., “every element in is a minimum w.r.t. ”
-
–
i.e., “every element in is a maximum w.r.t. ”
-
–
-
–
.
Note that from the axioms it follows that there is at most one element in , at most one in , and that is a subset of the successor relation w.r.t. . We call -orderings the models of .
For -structures and we write and say that is a -substructure of if is a substructure of with .
We remark that the relation symbols , and are negative in . Therefore we have:
Lemma 3.1.
Let . If , then .
Let
(7) |
We call models of complete -orderings. Clearly, for every there is a unique, up to isomorphism, complete -ordering with exactly elements. The next lemma shows that all its proper -substructures are models of .
Lemma 3.2.
Let and be -structures. Assume that and is a finite -substructure of that is a model of . Then (in particular, ).
Proof : By the previous lemma we know that . Let . As is an ordering, we may assume that
As , we have , , and for . As , everywhere we can replace the upper index B by A.
We show : Let . By , we have . Let be maximal with . If , then . Otherwise . As , we see that (by the last conjunct of ). Now follows from .
Corollary 3.3.
Every proper -substructure of a finite model of is a model of .
The class of finite -orderings that are not complete is closed under -substructures but not axiomatizable by a universal sentence:
Theorem 3.4 (Tait’s Theorem).
The class is closed under -substructures (and hence, closed under induced substructures) but is not finitely equivalent to a universal sentence.
Proof : is closed under -substructures: If and is a finite -substructure of , then (by Lemma 3.1). If , we are done. If , then by Lemma 3.2, which contradicts our assumption .
Let . It is clear that there is a finite model of with at least elements. By Corollary 3.3 every proper induced substructure of is a model of . Therefore, by Corollary 2.7, the sentence is not finitely equivalent to a universal sentence of the form with quantifier-free . As was arbitrary, we get our claim.
Remark 3.5.
A slight generalization of the previous proof shows that is not even axiomatizable by a -sentence, i.e., by a sentence of the form for some and quantifier-free . In fact, assume that . Again we choose a finite model of with at least elements. Then . Hence there are with . Then , where is the substructure of induced by . Hence, and therefore, . But this contradicts Corollary 3.3 as is a proper induced substructure of .
Note that is (equivalent to) a -sentence, i.e., equivalent to the negation of a -sentence.
We turn to a refinement of the previous statement that will be helpful to get Gurevich’s Theorem.
Definition 3.6.
-
(a)
Let be obtained from the vocabulary by adding finitely many relation symbols “in pairs,” the standard together with its complement (intended as the complement of ). The symbols and 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 is obtained from by adding pairs.
-
(b)
Let be obtained from by adding pairs. We say that is a -extension of (where is as above) if it is a universal sentence such that
-
(i)
the sentence is a conjunct of ,
-
(ii)
the sentence is a conjunct of ,
-
(iii)
besides all relation symbols are negative in (if this is not the case for some new or , the idea is to replace any positive occurrence of or by and , respectively). For instance, we replace a subformula
by
-
(i)
- (c)
For a -structure with we have
Hence,
(9) |
Now we derive the analogues of Lemma 3.1–Theorem 3.4 essentially by the same proofs.
Lemma 3.7.
Let be obtained from by adding pairs and let be an extension of . If and , then .
Proof : By Definition 3.6 (b) (iii) all relation symbols distinct from are negative in .
Lemma 3.8.
Let be obtained from by adding pairs and let be an extension of . Assume that and that the finite -substructure of is a model of . Then (in particular, ).
Proof : Let (and ) be the -structure obtained from (from ) by removing all relations in .
By Lemma 3.2 we know that . Furthermore, by the previous lemma; thus, . Hence, by (9), is the complement of for standard . Clearly, and . As and is a model of the sentence , we get and .
Corollary 3.9.
Every proper -substructure of a finite model of is a model of .
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 be obtained from by adding pairs and let be an extension of . The class is closed under -substructures (and hence, closed under induced substructures) but is not finitely equivalent to a universal sentence.
Perhaps the reader will ask why we do not introduce for the “complement relation symbol” and add the corresponding conjuncts to and (or, to and ) 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 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 be obtained from by adding pairs and be an extension of . Let and be an -sentence such that
(10) |
For
the statements (a) and (b) hold.
-
(a)
The class is closed under -substructures.
-
(b)
If with quantifier-free is finitely equivalent to , then .
Proof : (a) Let and . Thus, . If , we are done. Assume . In case is infinite, we conclude by (10) that is a model of and hence of . Otherwise is finite; then (by Lemma 3.8) and thus, .
(b) According to (10) there is a finite model of , i.e., of , with at least elements. By Corollary 3.9 every proper induced substructure of is not a model of and therefore, it is a model of . Hence by Corollary 2.7, is not finitely equivalent to a universal sentence of the form with and quantifier-free .
Remark 3.12.
We can strengthen the statement (b) of the preceding proposition to:
If the -sentence with quantifier-free is finitely equivalent to , then .
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 with binary . As already remarked in the Preliminaries for all -structures we use the notation common in graph theory.
Let be obtained from by adding pairs. Furthermore, let be an interpretation of width (we only need this case) of -structures in -structures. This means that assigns to every unary relation symbol an -formula and to every binary relation symbol an -formula ; moreover, selects an -formula .
Then assigns to every -structure with a -structure , which we often denote by , defined by
-
–
-
–
for unary
-
–
for binary .
As the interpretation is of width , we have
(11) |
Recall that for every sentence there is a sentence such that for all -structures with we have
(12) |
For example, for the sentence we have
Furthermore there is a constant such that for all ,
(13) |
Definition 4.1.
Let be obtained from by adding pairs and let be an interpretation of -structures in as just described. We say that is strongly existential if all formulas of are existential and is even quantifier-free.
Lemma 4.2.
Let be obtained from by adding pairs and let be an extension of . Then for every strongly existential interpretation the sentence is (equivalent to) a universal sentence.
Proof : The claim holds as all relation symbols distinct from are negative in . For example, for , we have
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 is strongly existential. Then for all -structures and with and , we have .
Proof : As is existential, we have . Let be distinct from and . Then . As is existential, and thus, . Moreover, for we have
Putting all together we see that .
We obtain from Lemma 3.8 the corresponding result in our framework.
Lemma 4.4.
Assume that is strongly existential. Let be an extension of . Let be a -structure and . Let with finite . If , then and .
Proof : As , in particular ; thus, . Therefore, by Lemma 4.3. By assumption and (12), and . As is finite, Lemma 3.8 implies , and in particular . Hence, by (12).
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 will be , i.e., the sentence (cf. (3)) axiomatizing the class of graphs.
Proposition 4.5.
Let be a universal -sentence. Assume that the interpretation of -structures in -structures is strongly existential. Furthermore, assume that for every sufficiently large finite complete -ordering there is a finite -structure with and . Then there is an -sentence such that is closed under induced substructures, but is not finitely equivalent to a universal sentence.
Proof : First we verify that the class is closed under induced substructures. Assume and . Since is universal, we have . If , then . Now assume that . Then , as is universal by Lemma 4.2. If or , we are done. Otherwise and . Then (see Lemma 4.4), a contradiction.
Finally we show that for every the sentence is not finitely equivalent to a sentence of the form with quantifier-free . Let
be a complete -ordering with at least elements. In particular, . By assumption we can choose in such a way that there is a finite -structure such that and . Then , hence, . Thus . As , the graph must contain more than elements by (11).
We want to show that every induced substructure of with at most elements is a model of . Then the result follows from Corollary 2.7. So let be an induced substructure of with at most elements. Clearly, . If or , we are done. Otherwise and . Then, Lemma 4.4 implies . Recall , so has at most elements by (11), a contradiction as .
Proposition 4.6.
Assume that is a universal -sentence. Let be obtained from by adding pairs and let be an extension of . Let be a strongly existential interpretation of -structures in -structures with the property that for every finite -structure , which is a model of , there is a finite -structure with and .
Let and be an -sentence such that
(13) |
For
(14) |
the statements (a) and (b) hold.
-
(a)
The class is closed under induced substructures.
-
(b)
If with quantifier-free is finitely equivalent to , then .
Proof : (a) Assume that and . Clearly . If , then we are done. Otherwise, the universe of and hence, that of , are not empty. Then and as , we have by Lemma 4.2.
If , we are done. Otherwise, . If is infinite, then by (13) and we are again done. If is finite, then by Lemma 4.4. Thus and hence, as . Therefore, and thus, .
(b) By (13) there is a finite model of with at least elements. By assumption there is a finite -structure with and . Clearly, and . Hence, . Assume that . We want to show that every induced substructure of with at most elements is a model of . Then the claim (b) follows from Corollary 2.7.
So let be an induced substructure of with at most elements. Clearly, . If or , we are done. Otherwise and . Then, by Lemma 4.4. This leads to a contradiction, as has at most elements by (12), while has elements and we assumed .
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 is not equivalent to a -sentence. Furthermore itself is equivalent to a -sentence. In fact, as all relation symbols besides are negative in , the sentence is universal. Moreover, as , , and are positive in , the sentence (as ) is equivalent to a -sentence. Hence is equivalent to a -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 be a graph and . For a path from vertex to vertex of length with an -ear is a path between and with a cycle of length ; one vertex of this cycle is adjacent to the vertex adjacent to on the path. Figure 1 is a path from to of length with a -ear.
Lemma 5.1.
For there are quantifier-free formulas and such that for all graphs we have
-
(a)
is a cycle of length containing .
-
(b)
is path from to of length with the -ear .
Proof : (a) We can take as the formula
(b) We can take as the formula
To understand better how we obtain the desired interpretation we first assign to every complete -ordering , i.e., to every model of , a -structure which is a graph.
In a first step we extend to a -structure , where in the following way. Here are unary and are binary relation symbols.
For every original (or, basic) element , i.e., for every , we introduce a new element , the companion of . We set
-
–
,
-
–
, ,
-
–
.
Note that the relation is irreflexive and symmetric, i.e., is already a graph, which is illustrated by Figure 2. Observe that contains the whole information of the ordering up to isomorphism.
We use to define the desired graph . The vertex set contains the elements of , and the edge relation contains . Furthermore contains just all the vertices and edges required by the following items:
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–
To we add a cycle of length consisting of new vertices, i.e., not in (besides ).
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To we add a cycle of length consisting of new vertices (besides ).
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–
To we add a cycle of length consisting of new vertices (besides ).
-
–
To we add a cycle of length consisting of new vertices (besides ).
-
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To we add a path from to of length with a -ear consisting of new vertices (besides and ).
-
–
To we add a path from to of length with a -ear consisting of new vertices (besides and ).
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 into four categories.
[-cycle] These are cycles in , i.e., cycles using only edges of .
[-cycle] For every unary , a -cycle is the cycle introduced for an .
[ear-cycle] These are the cycles constructed as ears on the gadgets for the relations and .
[mixed-cycle] All the other cycles are mixed.
For example, we get a mixed cycle if we start with , , in Figure 2 and then add the path introduced for (ignoring the ear).
A number of observations for these types of cycles are in order.
Lemma 5.2.
-
(i)
All the -cycles are of even length.111Moreover one can show that every chordless -cycle has length .
-
(ii)
Every -, -, -, and -cycle is of length , , , and , respectively.
-
(iii)
Every ear-cycle is of length or .
-
(iv)
Every mixed-cycle neither uses new vertices of any -cycle for nor any vertex of any ear-cycle.
-
(v)
Every mixed-cycle has length at least .
Proof : (i) follows easily from the fact that is a bipartite graph; (ii) and (iii) are trivial.
For (iv) assume that a mixed-cycle uses a new vertex of a -cycle introduced for some , where . As is mixed, it must contain a vertex . To reach from the mixed cycle must pass through and hence must contain one of the two segments of between and . As a consequence, in order for the mixed-cycle to go back from to , it must also use the other segment of between and . This means that it must be the -cycle 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 be a mixed-cycle. By (iv), must contain all vertices of a (at least one) path introduced for a pair or (ignoring the ear). As this path has length , we get our claim.
Conversely, given a -structure , which is a graph, we construct a -structure which we denote by , possibly the empty structure. Recall the definitions of “cycle” and of “path with ear” given by Lemma 5.1.
-
–
-
–
-
–
-
–
-
–
.
Lemma 5.3.
For every complete -ordering we have .
Proof : Let and . We claim that the mapping defined by
is an isomorphism from to . To that end, we first prove that
which implies that is well defined and a bijection. For every it is easy to see that . For the converse, let . In particular, is a member of a cycle of length . By Lemma 5.2, this must be a -cycle which contains some . Using the same argument, is a member of a -cycle which contains a vertex being the companion of some . Furthermore, there is a path from to of length with a -ear. The -ear is a cycle of length . Again by Lemma 5.2 this cycle is an ear-cycle which belongs to the gadget we introduced for some with . Then it is easy to see that . This finishes the proof that is a bijection from to .
Similarly, we can prove that preserves all the relations.
We want to show that we can obtain from by a strongly existential FO-interpretation. We set
and is a path from to of length with the -ear ” | |||
We define the desired interpretation of width of -structures in graphs. We set
Hence for every graph ,
Furthermore we define
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–
,
-
–
,
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–
.
Then we have:
Lemma 5.4.
The interpretation given by is strongly existential. For every complete -ordering we have and hence, by Lemma 5.3,
Setting , the sentence axiomatizing the class of graphs, we get from Proposition 4.5:
Theorem 5.5 (Tait’s Theorem for graphs).
There is a -sentence such that , the class of finite graphs that are models of , is closed under induced subgraphs but is not equivalent to a universal sentence in finite graphs.
In this section we presented a strongly existential interpretation of -structures and applied it to finite complete -orderings, i.e, to models of . A straightforward generalization of the preceding proofs allows to show the following result for vocabularies obtained from by adding pairs. We shall use it in Section 6.
Lemma 5.6.
Let be obtained from by adding pairs. There is a strongly existential interpretation that for every extension of assigns to every -structure that is a model of a graph with . For finite the graph is finite.
Proof : We get the graph as in the case : 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 . Let 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 . 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.
Remark 5.7.
(a) Let be the class of directed graphs. Then , the class of finite graphs, is a subclass of closed under induced substructures and definable in by the universal sentence . 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 and be the class of finite planar graphs, a subclass of closed under induced subgraphs. As mentioned in the Introduction, in [1] it is shown that the Łoś-Tarski Theorem fails for . As is not axiomatizable in 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 for a finite set of finite graphs (or, equivalently, for a universal ) leads to a contradiction. Let be the maximum size of the set of vertices of graphs in . Let be the graph obtained from the clique of 5 vertices by subdividing each edge by . Clearly, . However, every subgraph of induced on at most elements is planar. Hence, .
(c) Let be any vocabulary with at least one at least binary relation . Then the Łoś-Tarski Theorem fails for the class , the class of all finite -structures. By Remark 2.8 it suffices to show the existence of a universally definable subclass of which “essentially is the class of graphs.” We set
and let be .
If only contains unary relation symbols, the Łoś-Tarski Theorem holds for . It is easy to see for an -sentence that the closure under induced substructures of implies that of .
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 on page 190 of [14] must be modified in order to ensure that the class of models of 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 in (10) whose size is much smaller than the number . Basically will describe a very long computation of a Turing machine on a short input. We fix a universal Turing machine operating on an one-way infinite tape, the tape alphabet is , where is also considered as blank, and is the set of states of . The initial state is , and is the halting state; thus and we assume that . An instruction of has the form
where , and . It indicates that if is in state and the head of reads an , then the head replaces by and moves to the left (if ), stays still (if ), or moves to the right (if ). In order to describe computations of by FO-formulas we introduce binary predicates for to indicate that at time the machine is in state and the head scans cell , and a binary predicate to indicate that the content of cell at time is 0.
The vocabulary is obtained from by adding pairs (see Definition 3.6 (a)),
Intuitively, says that “at time the machine is not in state or the head is not in cell ;” and says that “at time the content of cell is (not 0 and thus is) 1.” Sometimes we write instead of (e.g., below in if or ).
Let and be the sentences already introduced in Section 3. For the sentence will be an extension of (compare Definition 3.6 (b)); hence, will be a universal sentence and all relations symbols besides are negative in ; in particular, it contains as conjuncts and
Finally, will contain the following sentences and as conjuncts. The sentence describes one computation step. It contains for each instruction of one conjunct. For example, the instruction contributes the conjunct
For the sentence describes the initial configuration of with input (if , the first cells contain , the remaining cells contain , and the head scans the first cell in the starting state ). Hence, as we can take the conjunction of
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–
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–
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–
.
Note that , , and are negative in . We set ; recall that by Definition 3.6 (c),
Let and . Furthermore, let be a -structure where is an ordering and . Let be the first elements of . Assume that on the input runs at least steps. We say that correctly encodes steps of the computation of on if for with ,
the content of cell after steps is 0 | (15) |
and for ,
after steps is in state and the head scans cell . | (16) |
From the definitions of the sentences and , we see:
Lemma 6.1.
Let and be a model . If for we have (in particular, if is infinite) and on runs at least steps, then correctly encodes steps of the computation of on .
Finally, let be a sentence expressing that “the machine reaches the halting state in exactly ‘max’ steps,” more precisely,
(17) |
As a consequence of the preceding lemma, we obtain:
Corollary 6.2.
Let and assume that on input eventually halts, say in steps, then
has no infinite model but a model with exactly elements (this model is unique up to isomorphism).
Proof : Let . Then is a complete -ordering and contains the description of the complete halting computation of on the input . As the machine reaches the halting state in exactly steps, we see that ; in particular, is finite.
On the other hand, we can interpret (15) and (16) as defining relations and on the set equipped with the “natural” ordering and its corresponding relations , , and . If furthermore we let and be the complements in of and , respectively, we get a model of with exactly elements.
Lemma 6.3.
Let on input eventually halt, say in steps. Then:
-
(a)
is closed under -substructures.
-
(b)
If is finitely equivalent to a universal sentence , then .
Now we show the following version of Gurevich’s Theorem.
Theorem 6.4.
Let be a computable function. Then there is a such that is closed under -substructures but is not finitely equivalent to a universal sentence of length less than .
Proof : By the previous lemma it suffices to find a such that on input halts in steps with
W.l.o.g. we assume that is increasing. An analysis of the formula shows that for some we have for all ,
(19) |
We define by
(20) |
Let be a Turing machine computing , more precisely, the function . We code and by a -string such that on simulates the computation of on .
Choose the least such that for we have
(21) |
The universal Turing machine on input computes and thus runs at least steps, say, exactly steps. By (19) – (21)
Finally we prove Gurevich’s Theorem for graphs. For let be an interpretation according to Lemma 5.6. For we consider the sentence
(21) |
That is, for , either the graph interprets an empty -structure, or a -structure which is a model of . If halts in steps on input , then has no infinite model but a finite model with elements by Corollary 6.2. Hence taking in Proposition 4.6 as the sentence axiomatizing the class of graphs we get the following analogue of Lemma 6.3.
Lemma 6.5.
Let on input halt in steps. Then:
-
(a)
, the class of graphs that are model of , is closed under induced subgraphs (and hence equivalent in the class of graphs to a universal sentence).
-
(b)
If is equivalent in the class of finite graphs to a universal sentence , then .
Theorem 6.6 (Gurevich’s Theorem for graphs).
Proof : Again we assume that is increasing. By the previous lemma it suffices to find a such that on input halts in steps with
There is a , which depends on but not on , such that for as in (13) and as in (19) we have for ,
(22) |
We define by
(23) |
and then proceed as in the proof of Theorem 6.4. Let be a Turing machine computing the function . We code and by a -string such that on simulates the computation of on .
Choose the least such that for we have
(24) |
The universal Turing machine on input computes and thus runs at least steps, say, exactly steps. We have
Remark 6.7.
Using previous remarks (Remark 3.12 and Remark 4.7) one can even show that for every computable function the sentence is not finitely equivalent to a -sentence of length less than and the sentence is not finitely equivalent in graphs to a -sentence of length less than . Moreover, and are equivalent to .
For this purpose note that in models of the sentence is equivalent to
and hence equivalent to a and to a -sentence. One easily verifies that the same holds for .
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 -sentence decides whether the class is closed under induced subgraphs.
Proof : Assume is such an algorithm. By the Completeness Theorem there is an algorithm that assigns to every sentence with closed under induced subgraphs a universal sentence equivalent to in graphs. Define the function by
and set . Then would contradict Gurevich’s Theorem for graphs, i.e., Theorem 6.6.
Corollary 7.2.
There is no algorithm that applied to any -sentence either reports that is not closed under induced subgraphs or it computes for a class of forbidden induced subgraphs.
Proof : Otherwise we could use this algorithm as a decision algorithm for the previous result.
The following proposition is the analog of Proposition 7.1 for classes of finite graphs. We state it for -sentences and graphs even though we prove it for -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 for the universal Turing machine and a word if on input does not halt. We make use of the sentences , , and defined in the previous section.
Proposition 7.3.
There is no algorithm that applied to any -sentence decides whether the class is closed under induced subgraphs.
Proof : For the universal Turing machine and a word consider the sentence
Then
is closed under induced subgraphs | (25) |
In fact, if , then , hence is trivially closed under induced subgraphs. If on input halts after steps, then, up to isomorphism, there is a unique model of and it has elements. By Lemma 3.8 every proper induced substructure of is not a model of . Hence is not closed under induced subgraphs. As the halting problem for every universal Turing machine is not decidable, by (25) we get our claim.
Proposition 7.4.
There is no algorithm that applied to any -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 whether the universal Turing machine halts on input . In (18) we defined by
If halts on , by Lemma 6.3 we know that is closed under -substructures and thus equivalent to a universal sentence. The claimed algorithm (or, even the Completeness Theorem) will produce such a universal . Furthermore, by Corollary 6.2 we know that there is a finite model with elements, which is a model of , hence it is a model of .
If does not halt on , then we show that . Clearly . Now let be a finite model of . If , then . Otherwise , then correctly represents the first steps of the computation of on by Lemma 6.1. Thus is a model of as does not halt on . Therefore, is a model of .
Now we can see whether does not halt on by checking whether the universal sentence produced by the claimed algorithm is finitely equivalent to the universal sentence . This can be checked effectively by Corollary 2.5 and Corollary 2.6.
Corollary 7.5.
There is no algorithm that applied to any -sentence such that 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 -sentence such that for some universal sentence computes such a .
For graphs let be a strongly existential interpretation of -structures in graphs according to Lemma 5.6. We know that for every finite -structure there is a finite graph such that .
For we consider the sentence defined in (21) in the proof of Theorem 6.4,
We show that is equivalent to a universal sentence on finite graphs. Moreover, does not halt on input if and only if is finitely equivalent to the universal sentence
If halts on , then has no infinite model but a finite model . Hence, by Proposition 4.6 we know that is closed under induced subgraphs. Therefore, is equivalent to a universal sentence in Graph. Let be a finite graph with . Then and thus, . Hence is a finite graph which is a model of . This means that is not equivalent to on all finite graphs, as is also a model of .
If , then we show that . Clearly . Now let the graph be a model of . Further we can assume that . In particular, is well defined. If , then and therefore, . If , then . As , by Lemma 6.1 the structure correctly represents the first steps of the computation of on . Thus, is a model of , again as does not halt on input . It follows that is a model of , and then .
Now we can decide the halting problem for . Given a word , we use the claimed algorithm to get a universal sentence equivalent to in the class of graphs. Finally we check whether is finitely equivalent to . This can be checked effectively again by Corollary 2.5 and Corollary 2.6.
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 -sentence such that is closed under induced subgraphs decides whether there is a finite set of graphs such that
Proof : Again we prove the corresponding result for -sentences and -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 -sentence such that is closed under induced substructures decides whether there is a finite set of finite -structures such that
For let
We show that is closed under induced subgraphs and that
Assume first that . Then has no finite model by Lemma 6.1 and the definition (17) of . Therefore, . By Lemma 3.10 we know that is closed under induced substructures but not finitely equivalent to a universal sentence.
Now assume that on input halts in steps. Then Corollary 6.2 guarantees that there is a unique model of with . We present a finite set of finite -structures such that
(26) |
As is universal, there is a finite set of finite -structures such that
Moveover, we set
and
Here is obtained from by replacing the conjunct (see (7)) by
The difference is that does not require the set to be nonempty. Hence,
Note that Lemma 6.1 remains true if in its statement we replace by .
For we show (26). Assume first that a finite structure is a model of . In particular, and therefore, has no induced substructure isomorphic to a structure in .
Now, for a contradiction suppose that is an induced substructure of isomorphic to a structure in . Then and thus, by Lemma 3.8, . As , we get . Hence, , a contradiction, as on the one hand and on the other hand .
Next we show that has no induced substructure isomorphic to a structure in . As and has elements, the first elements of correctly encode the first steps of the computation of on , hence the full computation. As , this contradicts .
As the final step let . We show that . As omits the structures in as induced substructures, we see that . If , we are done.
Recall that by Lemma 6.1 (more precisely, by the extension of Lemma 6.1 mentioned above) for finite structures of we know:
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(a)
if , then encodes steps of the computation of on ,
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(b)
if , then the first elements in the ordering correctly encode the (full) computation of on .
Now assume that , then (a) and (b) apply to . As no structure in is isomorphic to an induced substructure of , we see that . But cannot have more than elements, as otherwise the substructure of induced on the first elements would be isomorphic to a structure in , a contradiction.
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 -sentence … |
by
There is no algorithm that applied to a -sentence … |
In Proposition 7.3 we have to replace it by
There is no algorithm that applied to a -sentence … |
as (and ) are -sentences.
References
- [1] A. Atserias, A. Dawar, and M. Grohe. Preservation under extensions on well-behaved finite structures. SIAM Journal on Computing, 38:1364–1381, 2008.
- [2] Y. Chen and J. Flum. FO-definability of shrub-depth. In 28th EACSL Annual Conference on Computer Science Logic, CSL 2020, January 13-16, 2020, Barcelona, Spain, pages 15:1–15:16, 2020.
- [3] A. Dawar, M. Grohe, S. Kreutzer, and N. Schweikardt. Model theory makes formulas large. In Automata, Languages and Programming, 34th International Colloquium, ICALP 2007, Wroclaw, Poland, July 9-13, 2007, Proceedings, pages 913–924, 2007.
- [4] A. Dawar and A. Sankaran. Extension preservation in the finite and prefix classes of first order logic. CoRR, abs/2007.05459, 2020.
- [5] G. Ding. Subgraphs and well-quasi-ordering. Journal of Graph Theory, 16(5):489–502, 1992.
- [6] D. Duris. Extension preservation theorems on classes of acyclic finite structures. SIAM Journal on Computing, 39(8):3670–3681, 2010.
- [7] Z. Dvorák, A. C. Giannopoulou, and D. M. Thilikos. Forbidden graphs for tree-depth. European Journal of Combinatorics, 33(5):969–979, 2012.
- [8] H.-D. Ebbinghaus and J. Flum. Finite Model Theory. Perspectives in Mathematical Logic. Springer, 1999.
- [9] M. R. Fellows. Private communication. 2019.
- [10] M. R. Fellows and M. A. Langston. On search, decision, and the efficiency of polynomial-time algorithms. Journal of Computer and System Sciences, 49(3):769–779, 1994.
- [11] J. Gajarský and S. Kreutzer. Computing shrub-depth decompositions. In 37th International Symposium on Theoretical Aspects of Computer Science, STACS 2020, March 10-13, 2020, Montpellier, France, pages 56:1–56:17, 2020.
- [12] R. Ganian, P. Hlinený, J. Nesetril, J. Obdrzálek, and P. Ossona de Mendez. Shrub-depth: Capturing height of dense graphs. Logical Methods in Computer Science, 15(1), 2019.
- [13] R. Ganian, P. Hlinený, J. Nesetril, J. Obdrzálek, P. Ossona de Mendez, and R. Ramadurai. When trees grow low: Shrubs and fast . In Mathematical Foundations of Computer Science 2012 - 37th International Symposium, MFCS 2012, Bratislava, Slovakia, August 27-31, 2012. Proceedings, pages 419–430, 2012.
- [14] Y. Gurevich. Toward logic tailored for computational complexity. Lecture Notes in Mathematics, 1104:175–216, 1984.
- [15] J. Łoś. On the extending of models I. Fundamenta Mathematicae, 42:38–54, 1955.
- [16] B. Rossman. Łoś-Tarski Theorem has non-recursive blow-up. Unpublished manuscript, pages 1–2, 2012.
- [17] A. Sankaran, B. Adsul, and S. Chakraborty. A generalization of the Łoś-Tarski preservation theorem. Annals of Pure and Applied Logic, 167(3):189–210, 2016.
- [18] W. W. Tait. A counterexample to a conjecture of Scott and Suppes. The Journal of Symbolic Logic, 24(1):15–16, 1959.
- [19] A. Tarski. Contributions to the theory of models I, II. Indagationes Mathematicae, 16:589–588, 1954.
- [20] R. Vaught. Remarks on universal classes of relational systems. Indagationes Mathematicae, 16:572–591, 1954.
- [21] T. Zaslavsky. Forbidden induced subgraphs. Electronic Notes in Discrete Mathematics, 63:3–10, 2017.