Approximating the Expressive Power of Logics in Finite Models

Journal of Computer and System Sciences, 1996

The relationship between counting functions and logical expressibility is explored. The most well studied class of counting functions is *P, which consists of the functions counting the accepting computation paths of a nondeterministic polynomial-time Turing machine. For a logic L, *L is the class of functions on finite structures counting the tuples (T , cÄ ) satisfying a given formula (T , cÄ ) in L. Saluja, Subrahmanyam, and Thakur showed that on classes of ordered structures *FO=*P (where FO denotes first-order logic) and that every function in * 1 has a fully polynomial randomized approximation scheme. We give a probabilistic criterion for membership in * 1 . A consequence is that functions counting the number of cliques, the number of Hamilton cycles, and the number of pairs with distance greater than two in a graph, are not contained in * 1 . It is shown that on ordered structures * 1 1 captures the previously studied class spanP. On unordered structures *FO is a proper subclass of *P and * 1 1 is a proper subclass of spanP; in fact, no class *L contains all polynomial-time computable functions on unordered structures. However, it is shown that on unordered structures every function in *P is identical almost everywhere with some function *FO, and similarly for * 1 1 and spanP. Finally, we discuss the closure properties of *FO under arithmetical operations. ]

We present a formal syntax of approximate formulas suited for the logic with counting quantifiers SOLP. This logic was studied by us in [1] where, among other properties, we showed: (i) In the presence of a built–in (linear) order, SOLP can describe NP–complete problems and fragments of it capture classes like P and NL; (ii) weakening the ordering relation to an almost order we can separate meaningful fragments, using a combinatorial tool adapted to these languages. The purpose of the approximate formulas presented here, is to provide a syntactic approximation to logics contained in SOLP with built-in order, that should be complementary of the semantic approximation based on almost orders, by producing approximating logics where problems are described within a small counting error. We state and prove a Bridge Theorem that links expressibility in fragments of SOLP over almostordered structures to expressibility with respect to approximate formulas for the corresponding fragments over...

Archive for Mathematical Logic, 1999

Numerous results about capturing complexity classes of queries by means of logical languages work for ordered structures only, and deal with non-generic, or order-dependent, queries. Recent attempts to improve the situation by characterizing wide classes of finite models where linear order is definable by certain simple means have not been very promising, as certain commonly believed conjectures were recently refuted (Dawar's Conjecture). We take on another approach that has to do with normalization of a given order (rather than with defining a linear order from scratch). To this end, we show that normalizability of linear order is a strictly weaker condition than definability (say, in the least fixpoint logic), and still allows for extending Immerman-Vardi-style results to generic queries. It seems to be the weakest such condition. We then conjecture that linear order is normalizable in the least fixpoint logic for any finitely axiomatizable class of rigid structures. Truth of this conjecture, which is a strengthened version of Stolboushkin's conjecture, would have the same practical implications as Dawar's Conjecture. Finally, we suggest a series of reductions of the two conjectures to specialized classes of graphs, which we believe should simplify further work.

2011

Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorov-random strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of problems C defined in terms of polynomial-time truth-table reducibility to R K (the set of Kolmogorov-random strings) that lies between BPP and PSPACE . In this paper, we investigate improving this upper bound from PSPACE to PSPACE ∩ P/poly. More precisely, we present a collection of true statements in the language of arithmetic, (each provable in ZF) and show that if these statements can be proved in certain extensions of Peano arithmetic, then BPP ⊆ C ⊆ PSPACE ∩ P/poly.

arXiv (Cornell University), 2022

We prove that, on bounded expansion classes, every first-order formula with modulo counting is equivalent, in a linear-time computable monadic expansion, to an existential first-order formula. As a consequence, we derive, on bounded expansion classes, that first-order transductions with modulo counting have the same encoding power as existential first-order transductions. Also, modulo-counting first-order model checking and computation of the size of sets definable in modulo-counting first-order logic can be achieved in linear time on bounded expansion classes. As an application, we prove that a class has structurally bounded expansion if and only if it is a class of bounded depth vertex-minors of graphs in a bounded expansion class. We also show how our results can be used to implement fast matrix calculus on bounded expansion matrices over a finite field.

Logic Journal of the IGPL, 2020

In this article, we formally define and investigate the computational complexity of the definability problem for open first-order formulas (i.e. quantifier free first-order formulas) with equality. Given a logic $\boldsymbol{\mathcal{L}}$, the $\boldsymbol{\mathcal{L}}$-definability problem for finite structures takes as an input a finite structure $\boldsymbol{A}$ and a target relation $T$ over the domain of $\boldsymbol{A}$ and determines whether there is a formula of $\boldsymbol{\mathcal{L}}$ whose interpretation in $\boldsymbol{A}$ coincides with $T$. We show that the complexity of this problem for open first-order formulas (open definability, for short) is coNP-complete. We also investigate the parametric complexity of the problem and prove that if the size and the arity of the target relation $T$ are taken as parameters, then open definability is $\textrm{coW}[1]$-complete for every vocabulary $\tau $ with at least one, at least binary, relation.

Monographs in Theoretical Computer Science An EATCS Series, 2002

Lecture Notes in Mathematics, 1979

Lecture Notes in Computer Science, 2002

We systematically investigate the connections between constraint satisfaction problems, structures of bounded treewidth, and definability in logics with a finite number of variables. We first show that constraint satisfaction problems on inputs of treewidth less than k are definable using Datalog programs with at most k variables; this provides a new explanation for the tractability of these classes of problems. After this, we investigate constraint satisfaction on inputs that are homomorphically equivalent to structures of bounded treewidth. We show that these problems are solvable in polynomial time by establishing that they are actually definable in Datalog; moreover, we obtain a logical characterization of the property "being homomorphically equivalent to a structure of bounded treewidth" in terms of definability in finite-variable logics. Unfortunately, this expansion of the tractability landscape comes at a price, because we also show that, for each k ≥ 2, determining whether a structure is homomorphically equivalent to a structure of treewidth less than k is an NP-complete problem. In contrast, it is well known that, for each k ≥ 2, there is a polynomial-time algorithm for testing whether a given structure is of treewidth less than k. Finally, we obtain a logical characterization of the property "having bounded treewidth" that sheds light on the complexity-theoretic difference between this property and the property 'being homomorphically equivalent to a structure of bounded treewidth".

2012

Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorov-random strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of problems C defined in terms of polynomial-time truth-table reducibility to RK (the set of Kolmogorov-random strings) that lies between BPP and PSPACE [4, 3]. In this paper, we investigate improving this upper bound from PSPACE to PSPACE ∩ P/poly. More precisely, we present a collection of true statements in the language of arithmetic, (each provable in ZF) and show that if these statements can be proved in certain extensions of Peano arithmetic, then BPP ⊆C⊆PSPACE ∩ P/poly. We conjecture that C is equal to P, and discuss the possibility this might be an avenue for trying to prove the equality of BPP and P.