Fragments Of 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.

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.

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 ...

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.

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.

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].

Graph properties checkable in linear time in the number of vertices

This paper originates from the observation that many classical NP graph problems, including some NP-complete problems, are actually of very low nondeterministic time complexity. In order to formalize this observation, we define the complexity class vertexNLIN, which collects the graph problems computable on a nondeterministic RAM in time O( n), where n is the number of vertices of the input graph G=( V, E), rather than its usual size | V|+| E|. It appears that this class is robust (it is defined by a natural restrictive computational device; it is logically characterized by several simple fragments of existential second-order logic; it is closed under various combinatorial operators, including some restrictions of transitive closure) and meaningful (it contains many natural NP problems: connectivity, hamiltonicity, non-planarity, etc.). Furthermore, the very restrictive definition of vertexNLIN seems to have beneficial effects on our ability to answer difficult questions about complexity lower bounds or separation between determinism and nondeterminism. For instance, we prove that vertexNLIN strictly contains its deterministic counterpart, vertexDLIN, and even that it does not coincide with its complementary class, co-vertexNLIN. Also, we prove that several famous graph problems (e.g. planarity, 2-colourability) do not belong to vertexNLIN, although they are computable in deterministic time O(| V|+| E|).

Counterexamples of the 0-1 Law for Fragments of Existential Second-Order Logic: an Overview

AbstractWe propose an original use of techniques from random graph theory to find a Monadic(Minimal Scott without equality) sentence without an asymptotic probability. Our result implies that the 0-1 law fails for the logics(FO2) and](Minimal Gödel without equality). Therefore we complete the classification of first-order prefix classes with or without equality, according to the existence of the 0-1 law for the correspondingfragment. In addition, our counterexample can be viewed as a single explanation of the failure of the 0-1 law of all the fragments of existential second-order logic for which the failure is already known.

0–1 Laws and decision problems for fragments of second-order logic

The probability of a property on the collection of all finite relational structures is the limit as n → ∞ of the fraction of structures with n elements satisfying the property, provided the limit exists. It is known that a 0–1 law holds for any property expressible in first-order logic; i.e., the probability of any such property exists and is either 0 or 1. Moreover, the associated decision problem for the probabilities in PSPACE-complete We investigate here fragments of existential second-order logic in which we restric the patterns of first-order quantifiers. We focus on the class Σ11 (Ackermann) of existential second-order sentences in which the first-order part belongs to the Ackermann class; i.e., it contains at most one universal first-order quantifier. All properties expressible by Σ11 (Ackermann) sentences are NP-computable and there are natural NP-complete properties, such as SATISFIABILITY, that are expressible by such sentences. We establish that the 0–1 law holds for the class Σ11 (Ackermann) and identify the complexity of the associated decision problem. We also show that the 0–1 law fails for other fragments of existential second-order logic in which the first-order part belongs to certain prefix classes with an unsolvable decision problem. Thus, the emerging picture is that the classifications of prefix classes according to the solvability of the satisfiability problem and according to the 0–1 law for the corresponding Σ11 fragment are identical.