Existential Second-order Logic Research Articles

Tractability frontiers in probabilistic team semantics and existential second-order logic over the reals

Probabilistic team semantics is a framework for logical analysis of probabilistic dependencies. Our focus is on the axiomatizability, complexity, and expressivity of probabilistic inclusion logic and its extensions. We identify a natural fragment of existential second-order logic with additive real arithmetic that captures exactly the expressivity of probabilistic inclusion logic. We furthermore relate these formalisms to linear programming, and doing so obtain PTIME data complexity for the logics. Moreover, on finite structures, we show that the full existential second-order logic with additive real arithmetic can only express NP properties. Lastly, we present a sound and complete axiomatization for probabilistic inclusion logic at the atomic level.

Logics with Multiteam Semantics

Team semantics is the mathematical basis of modern logics of dependence and independence. In contrast to classical Tarski semantics, a formula is evaluated not for a single assignment of values to the free variables, but on a set of such assignments, called a team. Team semantics is appropriate for a purely logical understanding of dependency notions, where only the presence or absence of data matters, but being based on sets, it does not take into account multiple occurrences of data values. It is therefore insufficient in scenarios where such multiplicities matter, in particular for reasoning about probabilities and statistical independencies. Therefore, an extension from teams to multiteams (i.e. multisets of assignments) has been proposed by several authors. In this paper we aim at a systematic development of logics of dependence and independence based on multiteam semantics. We study atomic dependency properties of finite multiteams and discuss the appropriate meaning of logical operators to extend the atomic dependencies to full-fledged logics for reasoning about dependence properties in a multiteam setting. We explore properties and expressive power of a wide spectrum of different multiteam logics and compare them to second-order logic and to logics with team semantics. In many cases the results resemble what is known in team semantics, but there are also interesting differences. While in team semantics, the combination of inclusion and exclusion dependencies leads to a logic with the full power of both independence logic and existential second-order logic, independence properties of multiteams are not definable by any combination of properties that are downwards closed or union closed and thus are strictly more powerful than inclusion-exclusion logic. We also study the relationship of logics with multiteam semantics with existential second-order logic for a specific class of metafinite structures. It turns out that inclusion-exclusion logic can be characterised in a precise sense by the Presburger fragment of this logic, but for capturing independence, we need to go beyond it and add some form of multiplication. Finally, we also consider multiteams with weights in the reals and study the expressive power of formulae by means of topological properties.

On the Union Closed Fragment of Existential Second-Order Logic and Logics with Team Semantics

We present syntactic characterisations for the union closed fragments of existential second-order logic and of logics with team semantics. Since union closure is a semantical and undecidable property, the normal form we introduce enables the handling and provides a better understanding of this fragment. We also introduce inclusion-exclusion games that turn out to be precisely the corresponding model-checking games. These games are not only interesting in their own right, but they also are a key factor towards building a bridge between the semantic and syntactic fragments. On the level of logics with team semantics we additionally present restrictions of inclusion-exclusion logic to capture the union closed fragment. Moreover, we define a team based atom that when adding it to first-order logic also precisely captures the union closed fragment of existential second-order logic which answers an open question by Galliani and Hella.

Logics of dependence and independence: The local variants

Abstract Modern logics of dependence and independence are based on team semantics, which means that formulae are evaluated not on a single assignment of values to variables, but on a set of such assignments, called a team. This leads to high expressive power, on the level of existential second-order logic. As an alternative, Baltag and van Benthem have proposed a local variant of dependence logic, called logic of functional dependence ($ {\textsf {LFD}}$). While its semantics is also based on a team, the formulae are evaluated locally on just one of its assignments, and the team just serves as the supply of the possible assignments that are taken into account in the evaluation process. This logic thus relies on the modal perspective of generalized assignments semantics and can be seen as a fragment of first-order logic. For the variant of $ {\textsf {LFD}}$ without equality, the satisfiability problem is decidable. We extend the idea of localizing logics of dependence and independence in a systematic way, taking into account local variants of standard atomic dependency properties: besides dependence and independence, also inclusion, exclusion and anonymity. We study model-theoretic and algorithmic questions of the localized logics and also resolve some of the questions that had been left open by Baltag and van Benthem. In particular, we study decidability issues of the local logics and prove that satisfiability of $ {\textsf {LFD}}$ with equality is undecidable. Further, we establish characterization theorems via appropriate notions of bisimulation and study the complexity of model checking problems for these logics.

A Proof of the Algebraic Tractability Conjecture for Monotone Monadic SNP

The logic MMSNP is a restricted fragment of existential second-order logic which can express many interesting queries in graph theory and finite model theory. The logic was introduced by Feder and ...

On Existential MSO and Its Relation to ETH

Impagliazzo et al. proposed a framework, based on the logic fragment defining the complexity class SNP, to identify problems that are equivalent to k -CNF- Sat modulo subexponential-time reducibility (serf-reducibility). The subexponential-time solvability of any of these problems implies the failure of the Exponential Time Hypothesis (ETH). In this article, we extend the framework of Impagliazzo et al. and identify a larger set of problems that are equivalent to k -CNF- Sat modulo serf-reducibility. We propose a complexity class, referred to as Linear Monadic NP, that consists of all problems expressible in existential monadic second-order logic whose expressions have a linear measure in terms of a complexity parameter, which is usually the universe size of the problem. This research direction can be traced back to Fagin’s celebrated theorem stating that NP coincides with the class of problems expressible in existential second-order logic. Monadic NP, a well-studied class in the literature, is the restriction of the aforementioned logic fragment to existential monadic second-order logic. The proposed class Linear Monadic NP is then the restriction of Monadic NP to problems whose expressions have linear measure in the complexity parameter. We show that Linear Monadic NP includes many natural complete problems such as the satisfiability of linear-size circuits, dominating set, independent dominating set, and perfect code. Therefore, for any of these problems, its subexponential-time solvability is equivalent to the failure of ETH. We prove, using logic games, that the aforementioned problems are inexpressible in the monadic fragment of SNP, and hence, are not captured by the framework of Impagliazzo et al. Finally, we show that Feedback Vertex Set is inexpressible in existential monadic second-order logic, and hence is not in Linear Monadic NP, and investigate the existence of certain reductions between Feedback Vertex Set (and variants of it) and 3-CNF-Sat .

Polyteam semantics

Abstract Team semantics is the mathematical framework of modern logics of dependence and independence in which formulae are interpreted by sets of assignments (teams) instead of single assignments as in first-order logic. In order to deepen the fruitful interplay between team semantics and database dependency theory, we define Polyteam Semantics in which formulae are evaluated over a family of teams. We begin by defining a novel polyteam variant of dependence atoms and give a finite axiomatization for the associated implication problem. We relate polyteam semantics to team semantics and investigate in which cases logics over the former can be simulated by logics over the latter. We also characterize the expressive power of poly-dependence logic by properties of polyteams that are downwards closed and definable in existential second-order logic ($\textsf{ESO}$). The analogous result is shown to hold for poly-independence logic and all $\textsf{ESO}$-definable properties. We also relate poly-inclusion logic to greatest fixed point logic.

A logical approach to locality in pictures languages

This paper deals with descriptive complexity of picture languages of any dimension by fragments of existential second-order logic:1) We generalize to any dimension the characterization by Giammarresi et al. (1996) of the class of recognizable picture languages in existential monadic second-order logic.2) We state natural logical characterizations of the class of picture languages of any dimension d≥1 recognized in linear time on nondeterministic cellular automata, a robust complexity class that contains, for d=1, all the natural NP-complete problems.Our characterizations are essentially deduced from normalization results for first-order and existential second-order logics over pictures.

Existential second-order logic and modal logic with quantified accessibility relations

This article investigates the role of arity of second-order quantifiers in existential second-order logic, also known as Σ11. We identify fragments L of Σ11 where second-order quantification of relations of arity k>1 is (nontrivially) vacuous in the sense that each formula of L can be translated to a formula of (a fragment of) monadicΣ11. Let polyadic Boolean modal logic with identity (PBML=) be the logic obtained by extending standard polyadic multimodal logic with built-in identity modalities and with constructors that allow for the Boolean combination of accessibility relations. Let Σ11(PBML=) be the extension of PBML= with existential prenex quantification of accessibility relations and proposition symbols. The principal result of the article is that Σ11(PBML=) translates into monadicΣ11. As a corollary, we obtain a variety of decidability results for multimodal logic. The translation can also be seen as a step towards establishing whether every property of finite directed graphs expressible in Σ11(FO2) is also expressible in monadic Σ11. This question was left open in the 1999 paper of Grädel and Rosen in the 14th Annual IEEE Symposium on Logic in Computer Science.

Hierarchies in independence and inclusion logic with strict semantics

We study the expressive power of fragments of inclusion and independence logic defined by restricting the number k of universal quantifiers in formulas. Assuming the so-called strict semantics for these logics, we relate these fragments of inclusion and independence logic to sublogics ESOf(k∀) of existential second-order logic, which in turn are known to capture the complexity classes NTIMERAM(nk)⁠.

Independence friendly logic with classical negation via flattening is a second-order logic with weak dependencies

It is well-known that Independence Friendly (IF) logic is equivalent to existential second-order logic (Σ11) and, therefore, is not closed under classical negation. The Boolean closure of IF sentences, called Extended IF-logic, on the other hand, corresponds to a proper fragment of Δ21. In this article we consider SL(↓), IF-logic extended with Hodges' flattening operator ↓, which allows to define a classical negation. SL(↓) contains Extended IF-logic and hence it is at least as expressive as the Boolean closure of Σ11. We prove that SL(↓) corresponds to a weak syntactic fragment of SO which we show to be strictly contained in Δ21. The separation is derived almost trivially from the fact that Σn1 defines its own truth-predicate. We finally show that SL(↓) is equivalent to the logic of Henkin quantifiers, which shows, we argue, that Hodges' notion of negation is adequate.

Expressivity of Imperfect Information Logics without Identity

In this article we investigate the family of independence-friendly (IF) logics in the equality-free setting, concentrating on questions related to expressive power. Various natural equality-free fragments of logics in this family translate into existential second-order logic with prenex quantification of function symbols only and with the first-order parts of formulae equality-free. We study this fragment of existential second-order logic. Our principal technical result is that over finite models with a vocabulary consisting of unary relation symbols only, this fragment of second-order logic is weaker in expressive power than first-order logic (with equality). Results about the fragment could turn out useful for example in the study of independence-friendly modal logics. In addition to proving results of a technical nature, we address issues related to a perspective from which IF logic is regarded as a specification framework for games, and also discuss the general significance of understanding fragments of second-order logic in investigations related to non-classical logics.

Characterizing quantifier extensions of dependence logic

AbstractWe characterize the expressive power of extensions of Dependence Logic and Independence Logic by monotone generalized quantifiers in terms of quantifier extensions of existential second-order logic.

Hierarchies in Dependence Logic

We study fragments D ( k ∀) and D ( k -dep) of dependence logic defined either by restricting the number k of universal quantifiers or the width of dependence atoms in formulas. We find the sublogics of existential second-order logic corresponding to these fragments of dependence logic. We also show that, for any fixed signature, the fragments D ( k ∀) give rise to an infinite hierarchy with respect to expressive power. On the other hand, for the fragments D ( k -dep), a hierarchy theorem is otained only in the case the signature is also allowed to vary. For any fixed signature, this question is open and is related to the so-called Spectrum Arity Hierarchy Conjecture.

The Descriptive Complexity of the Deterministic Exponential Time Hierarchy

In Descriptive Complexity, we investigate the use of logics to characterize computational complexity classes. Since 1974, when Fagin proved that the class NP is captured by existential second-order logic, considered the first result in this area, other relations between logics and complexity classes have been established. Well-known results usually involve first-order logic and its extensions, and complexity classes in polynomial time or space. Some examples are that the first-order logic extended by the least fixed-point operator captures the class P and the second-order logic extended by the transitive closure operator captures the class PSPACE. In this paper, we will analyze the combined use of higher-order logics of order i, HOi, for i⩾2, extended by the least fixed-point operator, and we will prove that each level of this hierarchy captures each level of the deterministic exponential time hierarchy. As a corollary, we will prove that the hierarchy of HOi(LFP), for i⩾2, does not collapse, that is, HOi(LFP)⊂HOi+1(LFP).

Team Logic and Second-Order Logic

Team logic is a new logic, introduced by Vaananen [12], extending dependence logic by classical negation. Dependence logic adds to first-order logic atomic formulas expressing functional dependence of variables on each other. It is known that on the level of sentences dependence logic and team logic are equivalent with existential second-order logic and full second-order logic, respectively. In this article we show that, in a sense that we make explicit, team logic and second-order logic are also equivalent with respect to open formulas. A similar earlier result relating open formulas of dependence logic to the negative fragment of existential second-order logic was proved in [8].

Erratum to: On Definability in Dependence Logic

A correction is needed for the singular case X = ∅ in Theorems 4.9, 4.10, 5.1, and 5.2. In Kontinen and Vaananen (2009), the open formulas of Dependence Logic (D) were studied. Formulas of dependence logic express properties of sets of assignments (teams). It was shown in Vaananen (2007) that every formula of dependence logic can be represented in an equivalent form in existential second-order logic ( 1 1) with an extra predicate, occurring only negatively, interpreting the team. In Theorems 4.9 and 4.10 of Kontinen and Vaananen (2009) it was claimed that also the converse holds, i.e., that for every vocabulary L and sentence φ ∈ 1 1[L ∪ {R}], in which R is k-ary (for some k ≥ 1) and occurs only negatively, there is a formula ψ(y1, . . . , yk) ∈ D[L] such that for all models A and teams X with domain {y1, . . . , yk} A | X ψ ⇔ (A, rel(X)) | φ, (1)

Independence-friendly cylindric set algebras

Independence-friendly logic is a conservative extension of first-order logic that has the same expressive power as existential second-order logic. In her Ph.D. thesis, Dechesne introduces a variant of independence-friendly logic called IFG logic. We attempt to algebraize IFG logic in the same way that Boolean algebra is the algebra of propositional logic and cylindric algebra is the algebra of first-order logic. We define independence-friendly cylindric set algebras and prove two main results. First, every independence-friendly cylindric set algebra over a structure has an underlying Kleene algebra. Moreover, the class of such underlying Kleene algebras generates the variety of all Kleene algebras. Hence the equational theory of the class of Kleene algebras that underly an independence-friendly cylindric set algebra is finitely axiomatizable. Second, every one-dimensional independence-friendly cylindric set algebra over a structure has an underlying monadic Kleene algebra. However, the class of such underlying monadic Kleene algebras does not generate the variety of all monadic Kleene algebras. Finally, we offer a conjecture about which subvariety of monadic Kleene algebras the class of such monadic Kleene algebras does generate.

Revisiting constraint-directed search

In constraint-based local search the solutions are described declaratively by a conjunction of (often high-level) constraints. In this article we show that this opens up new ideas for constraint-directed search. For a constraint we introduce three neighbourhoods, where the penalty for that constraint alone is decreasing, increasing, or unchanged. We give specialised algorithms for common constraints that efficiently implement these neighbourhoods. Further, we give a general algorithm that implements these neighbourhoods from specifications of constraints in monadic existential second-order logic. Finally, we show how common constraint-directed local search algorithms are often easier to express using these neighbourhoods.

Generic Incremental Algorithms for Local Search

When a new (global) constraint is introduced in local search, measures for the penalty and variable conflicts of that constraint must be defined, and incremental algorithms for maintaining these measures must be implemented. These are complicated and time-consuming tasks, which clearly reduces the productivity of the local-search practitioner. We introduce a generic scheme that, from a description of a constraint in monadic existential second-order logic extended with counting, automatically gives penalty and variable-conflict measures for such a constraint, as well as incremental algorithms for maintaining these measures. We prove that our variable-conflict measure for a variable x is lower-bounded by the maximum penalty decrease that may be achieved by only changing the value of x, as well as upper bounded by the penalty measure. Without these properties, the local search performance may degrade. We also demonstrate the usefulness of the approach by replacing a built-in global constraint by a modelled version, while still obtaining competitive results in terms of runtime and robustness. This is especially attractive when a particular (global) constraint is not built in.