Some structural complexity results for $\exists{\mathbb{R}}$

The existential theory of the reals collects all true sentences in existential first-order logic over ${\mathbb{R}},$ i.e., sentences of the form $\exists x_{1}\ldots\exists x_{n}\Phi(x_{1},\ldots,x_{n}),$ where $\Phi$ is a quantifier-free formula expressing a Boolean combination of polynomial equalities and inequalities. The complexity of deciding the truth of a given sentence plays a prominent role for several computational models and complexity classes. In the Blum-Shub-Smale model - henceforth BSS for short - over ${\mathbb{R}}$ basic arithmetic operations can be executed with unit cost and the size of $\Phi$ is basically the algebraic size of a dense encoding of all polynomials occuring in it. The above decision problem then turns out to be ${\rm NP}_{{\mathbb{R}}}$-complete, where the latter class is the real number analogue of classical NP [BSS]. However, the decision problem also makes perfect sense in the Turing model of computation if we restrict all coefficients for the polynomials in $\Phi$ to be rational. Then, its size is the usual bit-size and the problem of deciding truth is easily seen to be NP-hard. In the Turing model, the exact complexity of the existential theory for the reals is (wide) open. Using deep results on quantifier elimination it has been shown that it can be decided within PSPACE, see [Canny] and also [Renegar]. The open placement of the problem somewhere between NP and PSPACE has led to the definition of an own complexity class denoted ${\exists{\mathbb{R}}},$ see a precise definition in the next subsection.

Beside many other things, in [Schaefer2024] the authors ask whether several structural results well known to hold for the classical class NP in the Turing model also can be settled for ${\exists{\mathbb{R}}}.$ The latter include an analogue of the Baker-Gill-Solovay result [BGS] on existence of oracles separating ${\exists{\mathbb{R}}}$ from NP and the question whether ${\exists{\mathbb{R}}}$ can be characterized by means of descriptive complexity as done in [Fagin] for NP. In this paper we give positive answers to these questions and also show that Ladner’s result [Ladner] can be transformed showing that if we assume ${\exists{\mathbb{R}}}\neq{\rm NP}$ there exist problems in the set difference not being ${\exists{\mathbb{R}}}$-complete.

A few words on the proofs might be appropriate. All above mentioned results for NP turn out to hold in the corresponding variants as well for ${\exists{\mathbb{R}}}$. In some cases, though not completely straightforward, proofs follow from not too complicated modifications of known proofs of related results in the BSS model. One basic ingredient for the proofs to work is the characterization of ${\exists{\mathbb{R}}}$ via the BSS class BP$({\rm NP}_{{\mathbb{R}}}^{0})$ in Theorem 1.1 below together with the most important observation that the class of ${\rm NP}_{{\mathbb{R}}}^{0}$-machines in the BSS model is countable. Therefore, we see the main value of the present paper in working out the necessary proof details of expected results in order to enlarge the body of structural results known to hold for ${\exists{\mathbb{R}}}.$

The first question from the compendium [Schaefer2024] we want to deal with is whether there exist oracles which do and do not separate ${\exists{\mathbb{R}}}$ from NP and PSPACE, respectively. Comparing P and NP in the Turing model, these questions were answered affirmatively long ago [BGS], in the BSS model an analogue result holds [Emerson], see also [Gassner]. Usually, the construction of an oracle separating the representative classes is done via a diagonalization argument. As consequence, the separating oracles in both settings are not very natural (with an exception in [Gassner], where a Knapsack problem is used to separate other classes). As we shall see, for NP and ${\exists{\mathbb{R}}}$ a quite natural oracle work. The definition of oracle classes like P^A and NP^A naturally relies on a basic model for algorithms defining a complexity class, like deterministic or non-deterministic polynomial time Turing machines. Then, one equips them with the additional feature to use intermediate results as queries to an oracle. Following the same nearby ongoing for ${\exists{\mathbb{R}}}$ we should work with the following definition.

Note that though this is the straightforward definition of oracle classes ${\exists{\mathbb{R}}}^{A}$, some subtleties are hidden. Since the basic machine is an ${\rm NP}_{{\mathbb{R}}}^{0}$-algorithm, it is allowed to produce real number oracle questions, something a Turing machine cannot. Vice versa, since a Turing machine only computes with discrete data, we know in advance that an NP-oracle machine will never produce arbitrary non-rational queries, an information which is undecidable for a BSS machine to know in general. Our results below establishing the integers as separating oracle implicitly make use of such effects. We further comment on this at the end of this section.

The next question from the compendium [Schaefer2024] is whether ${\exists{\mathbb{R}}}$ can be characterized by purely logical means in the sense of descriptive complexity theory. Corresponding results were given in the early years of finite model theory for classical NP [Fagin] and later on for the real number version ${\rm NP}_{{\mathbb{R}}}$ in the BSS model [GraedelMeer], based on the development of meta-finite model theory in [GraedelGurevich]; see [Graedeletal] for a more intensive introduction into such results. One special feature, given the above mentioned characterization of ${\exists{\mathbb{R}}}$ as BP$({\rm NP}_{{\mathbb{R}}}^{0})$, is the mixture of discrete inputs coded as in classical complexity theory over $\{0,1\}^{*}$, and an algebraic cost measure for the arithmetic computations performed by the underlying machine. It turns out that this split can be modelled by a restriction of so-called ${\mathbb{R}}$-structures used in [GraedelMeer] to characterize (full) ${\rm NP}_{{\mathbb{R}}}$ and respective logics on them. Thus, the focus below will be on defining the corresponding restriction which we call discrete ${\mathbb{R}}$-structures, and the suitable logics on them used to capture both ${\exists{\mathbb{R}}}$ and BP$({\rm P}_{{\mathbb{R}}}^{0})$. Corresponding proofs then are quite similar to those given in [GraedelMeer] and are only sketched in the Appendix. Though below we work with restricted ${\mathbb{R}}$-structures, we have to recall their definition in full generality, except for disallowing arbitrary real constants in the so-called secondary part due to the fact that the BSS algorithms studied here do not use such constants.

We now restrict ${\mathbb{R}}$-structures to so-called discrete ${\mathbb{R}}$-structures suitable for modelling computations in $\mbox{P}_{{\mathbb{R}}}^{0}$ and ${\rm NP}_{{\mathbb{R}}}^{0}$ on Boolean languages. In order to model bit-strings as structures the universe always will have the form $A=\{0,1,\ldots,n-1\}$ and $L_{s}$ contains a unary relation $X\subseteq A$ interpreted as a bit-vector in $\{0,1\}^{n}$ via $X(i)=1\leftrightarrow i\in X,X(i)=0\leftrightarrow i\not\in X.$ In addition, for well known reasons we include a linear order $<\in L_{s}$ in order to capture polynomial time computations below. Finally, as the only element in $L_{f}$ we need a function $real:A\mapsto{\mathbb{R}}$ which is able to change the type of an element in $A$ to become a real (like the cost operator in a programming language like $C_{++}):$ We define $real(i):=\left\{\begin{array}[]{ll}1\in{\mathbb{R}}&\mbox{for}\ i\neq 0,i\in A\\ 0\in{\mathbb{R}}&\mbox{for}\ i=0\in A\end{array}\right.$

Note that below when we use existential second-order (so) logic, both $<$ and $real$ can be avoided by claiming their existence as functions from $A$ to ${\mathbb{R}}$ using an existential quantifier together with a fo-formula.

Discrete ${\mathbb{R}}$-structures and logics on them provide the right framework to model BSS-computations on Boolean languages that do not use constants other than rational ones. Inputs being discrete and measured by their usual bit size, the used logics will transfer those inputs into the real number part of a discrete ${\mathbb{R}}$-structure so that computations are modelled therein, including the use of the algebraic unit cost measure. Let $V=\{v_{0},v_{1},\ldots\}$ denote a countable set of variables. The $v_{i}$ are supposed to be fo-variables ranging over the discrete universe of a discrete ${\mathbb{R}}$-structure.

In order to capture complexity class ${\exists{\mathbb{R}}}={\rm BP}({\rm NP}_{{\mathbb{R}}}^{0})$ as well as BP$(\mbox{P}_{{\mathbb{R}}}^{0})$ we have to extend fo-logic to existential second-order logic and first order logic, respectively. This is done similarly as in [GraedelMeer] and the proofs showing that the resulting logics capture the intended classes is a nearby variation of the corresponding ones in the full BSS model. The only technical aspect to respect is that for discrete ${\mathbb{R}}$-structures and the logics used we cannot introduce arbitrary real constants. This is rather a restriction for fixed point logic than for existential so-logic since for the latter we can existentially quantify such real objects and then express their desired properties via a fo-formula. Since fixed point logic also needs to work with so-objects, we first define existential so logic.

Dealing with a logical characterization of BP$({\rm P}_{{\mathbb{R}}}^{0})$ this is not possible, the reason why at some places a bit more care for technical details is necessary. Nevertheless, this is at most tedious. The extension of FO${}_{{\mathbb{R}}}^{0}$ to fixed point logic necessary in order to capture polynomial time in our setting once again was basically introduced already in [GraedelMeer], here we outline the technical differences due to the lack of arbitrary real constants. The only new technical demand in order to describe ${\rm P}_{{\mathbb{R}}}^{0}$-computations using a logic on discrete ${\mathbb{R}}$-structures is to change at certain places discrete data (i.e., objects formalized in the discrete part of an ${\mathbb{R}}$-structure) to real numbers. Suppose, for example, we use the given order on some $A^{k}$ and express by FO_R logic that $\underline{t}\in A^{k}$ is the successor of $\underline{s}$ in this order, i.e., we have a formula saying $\underline{t}=\underline{s}+1.$ We then often want to use the characteristic value $\chi[\underline{t}=\underline{s}+1]$ as real number. Formally, this can be expressed in our logics as follows: $\underline{t}=\underline{s}+1$ is an abbreviation of a formula on the discrete part of an ${\mathbb{R}}$-structure. We can introduce a new discrete variable $\sigma$ together with the fo-formula $\sigma\Leftrightarrow\underline{t}=\underline{s}+1.$ Now, $real(\sigma)$ is a number term expressing the truth value of $\underline{t}=\underline{s}+1$ as real (!) number $0$ or $1$. We express this by writing $real([\chi[\underline{t}=\underline{s}+1]).$ Similar constructions are used below and should be clear from the context.

We need the following definitions, compare [GraedelMeer]. In order to deal with partially defined functions from some $A^{k}\to{\mathbb{R}}$, enlarge ${\mathbb{R}}$ by an element $undef$ and extend the arithmetic operations via $r+undef:=r-undef:=undef,\\ r\cdot undef:=r/undef:=\left\{\begin{array}[]{cc}0&r=0\\ undef&r\neq 0\end{array}\right.$ and $sgn(undef):=undef.$ For a number term $F(\underline{s},\underline{t})$ with free first order variables $\underline{s},\underline{t}$, the $\max$ operator $\max\limits_{\underline{s}}F(\underline{s},\underline{t})$ is a number term with free variables $\underline{t}$ and obvious semantics. Below, we use $\max$ in order to cycle through some $A^{k}.$ Typically, we want $\underline{s}$ to be the direct predecessor of $\underline{t},$ then express the characteristic value $\chi[\underline{t}=\underline{s}+1]$ via FO${}^{0}_{{\mathbb{R}}}$-logic, take the corresponding number term $real(\chi[\underline{t}=\underline{s}+1])$ as explained above, and then identify $\underline{s}$ via $\max\limits_{\underline{s}}real(\chi[\underline{t}=\underline{s}+1]).$ That way, when a computation is in step $\underline{s}$ we can formalize the next step $\underline{t}.$ In order to define fixed point logic on discrete ${\mathbb{R}}$-structures ${\mathfrak{D}}=(L_{s},{\cal R},\{real\})$ let $Z$ be a function symbol of arity $r$ interpreted on ${\mathfrak{D}}$ as function from $A^{r}\to{\mathbb{R}}$ and let $F(Z,\underline{x})$ be a number term in FO${}^{0}_{{\mathbb{R}}}+\max$ logic on signature $(L_{s},\{real,Z\})$ (i.e., the closure of FO${}^{0}_{{\mathbb{R}}}$ logic under use of the $\max$ operator) with free first order variables $\underline{x}=(x_{1},\ldots,x_{r})\in V^{r}.$

We finally note that another approach to characterize a certain fragment of ${\exists{\mathbb{R}}}$ by means of a so-called probabilistic independence logic was given in [Hannula].

The next classical result which can be transformed relatively straightforwardly to ${\exists{\mathbb{R}}}$ is Ladner’s theorem [Ladner] guaranteeing the existence of intermediate problems between P and the NP-complete ones. The range of the classical proof technique has been analyzed in [Schoening]. In relation with the BSS model of computation over uncountable structures the theorem has been shown in [MalajovichMeer] to hold as well over the complex numbers. Over ${\mathbb{R}}$ it is open, but [BMM, ChapuisKoiran, Meer2012] give partial results. Especially, [BMM] addresses the importance of quantifier elimination QE algorithms when constructing intermediate problems. In the case of separating the class of ${\exists{\mathbb{R}}}$-complete problems from NP under the assumption ${\exists{\mathbb{R}}}\neq{\rm NP}$ a combination of the arguments from [BMM, MalajovichMeer] works as well.

For the rest of this section we outline the proof idea of how to construct such a problem. Technical details then easily can be filled following the presentations in [BMM, MalajovichMeer]. Our construction starts from the ${\exists{\mathbb{R}}}$-complete problem ${\rm QPS}^{0}$ of Definition 1. As mentioned before, the ${\rm NP}_{{\mathbb{R}}}$-completeness proof of the corresponding problem for the class ${\rm NP}_{{\mathbb{R}}}$ in [BSS] shows as well completeness of ${\rm QPS}^{0}$ in ${\exists{\mathbb{R}}}$ since given an ${\rm NP}_{{\mathbb{R}}}^{0}$-machine the reduction to the full $\rm QPS$ problem, i.e., where polynomials are allowed to have real coefficients as well, does not introduce real constants and runs in polynomial time also in the Turing model; intermediate results can be subsumed under real existential quantification.²²2Note that guessing a real solution might lead to real intermediate results; they can be existentially quantified in ${\exists{\mathbb{R}}}$ in the reduction proof.

The main reasons why a proof of Theorem 4.1 can be done similarly to showing related results are the decidability of ${\exists{\mathbb{R}}}$ within a computable time bound in the Turing model due to real quantifier elimination, and effective countability of NP-machines together with polynomial time bounds. Starting from ${\rm QPS}^{0}$ we build certain restricted languages $L(a)$ depending on strictly increasing sequences $(a):=(a_{i})_{i\in{\mathbb{N}}}$ of natural numbers. For input dimensions $n\in\{a_{2i-1},\ldots,a_{2i}-1\}$ we let $L(a)$ be the ${\rm QPS}^{0}$ problem, for the remaining input dimensions $L(a)$ equals the empty set. The strategy now is to define two such sequences $(a),(b)\in{\mathbb{N}}^{{\mathbb{N}}}$ so that both have the desired properties. We need two sequences in order to guarantee non-completeness.

Since ${\exists{\mathbb{R}}}$-completeness is defined downward from ${\rm QPS}^{0}$ by usual polynomial time reductions, it is easy to see that assuming ${\exists{\mathbb{R}}}\neq{\rm NP}$ neither $L(a)$ nor $L(b)$ can be ${\exists{\mathbb{R}}}$-complete if $(a)$ and $(b)$ have an exponential gap.