Fragment Of Second-order Logic Research Articles

Alternating (In)Dependence-Friendly Logic

Hintikka and Sandu originally proposed Independence Friendly Logic (▪) as a first-order logic of imperfect information to describe game-theoretic phenomena underlying the semantics of natural language. The logic allows for expressing independence constraints among quantified variables, in a similar vein to Henkin quantifiers, and has a nice game-theoretic semantics in terms of imperfect information games. However, the ▪ semantics exhibits some limitations, at least from a purely logical perspective. It treats the players asymmetrically, considering only one of the two players as having imperfect information when evaluating truth, resp., falsity, of a sentence. In addition, truth and falsity of sentences coincide with the existence of a uniform winning strategy for one of the two players in the semantic imperfect information game. As a consequence, ▪ does admit undetermined sentences, which are neither true nor false, thus failing the law of excluded middle. These idiosyncrasies limit its expressive power to the existential fragment of Second Order Logic (▪). In this paper, we investigate an extension of ▪, called Alternating Dependence/Independence Friendly Logic (▪), tailored to overcome these limitations. To this end, we introduce a novel compositional semantics, generalising the one based on trumps proposed by Hodges for ▪. The new semantics (i) allows for meaningfully restricting both players at the same time, (ii) enjoys the property of game-theoretic determinacy, (iii) recovers the law of excluded middle for sentences, and (iv) grants ▪ the full descriptive power of ▪. We also provide an equivalent Herbrand-Skolem semantics and a game-theoretic semantics for the prenex fragment of ▪, the latter being defined in terms of a determined infinite-duration game that precisely captures the other two semantics on finite structures.

The spectrum problem for Abelian $$\ell $$-groups and MV-algebras

This paper deals with the problem of characterizing those topological spaces which are homeomorphic to the prime spectra of MV-algebras or Abelian $$\ell $$ -groups. As a first main result, we show that a topological space X is the prime spectrum of an MV-algebra if and only if X is spectral, and the lattice K(X) of compact open subsets of X is a closed epimorphic image of the lattice of “cylinder rational polyhedra” (a natural generalization of rational polyhedra) of $$[0,1]^Y$$ for some set Y. As a second main result we extend our results to Abelian $$\ell $$ -groups. That is, a topological space X is the prime spectrum of an Abelian $$\ell $$ -group if and only if X is generalized spectral, and the lattice K(X) is a closed epimorphic image of the lattice of “cylinder rational cones” (a generalization of rational cones) in $${{\mathbb {R}}}^Y$$ for some set Y. Finally, we axiomatize, in monadic second order logic, the Belluce lattices of free MV-algebras (equivalently, the lattice of cylinder rational polyhedra) of dimension 1, 2 and infinite, and we study the problem of describing Belluce lattices in certain fragments of second order logic.

On the Expressive Power of Logics on Constraint Databases with Complex Objects

We extend the constraint data model to allow complex objects and study the expressive power of various query languages over this sort of constraint databases. The tools we use come in the form of collapse results which are well established in the context of first-order logic. We show that the natural-active collapse with a condition and the activegeneric collapse carry over to the second-order logic for structures with o-minimality property and any signature in the complex value relations. The expressiveness results for more powerful logics including monadic second-order logic, monadic second-order logic with fix-point operators, and fragments of second-order logic are investigated in the paper. We discuss the data complexity for second-order logics over constraint databases. The main results are that the complexity upper bounds for three theories, $$ \mathcal{MSO} $$ + Lin, $$ \mathcal{MSO} $$ + Poly, and Inflationary Datalog $$ {}_{\mathrm{act}}^{\mathrm{cv},\neg } $$ (SC, $$ \mathfrak{M} $$ ) without powerset operator are $$ {\cup}_i{\sum}_i^{N{C}^1},\mathbf{NCH}={\cup}_i{\sum}_i^{NC} $$ , and AC0/poly, respectively. We also consider the problem of query closure property in the context of embedded finite models and constraint databases with complex objects and the issue of how to determine safe constraint queries.

Expressiveness of Logic Programs under the General Stable Model Semantics

Stable model semantics had been recently generalized to non-Herbrand structures by several works, which provides a unified framework and solid logical foundations for answer set programming. This article focuses on the expressiveness of normal and disjunctive logic programs under general stable model semantics. A translation from disjunctive logic programs to normal logic programs is proposed for infinite structures. Over finite structures, some disjunctive logic programs are proved to be intranslatable to normal logic programs if the arities of auxiliary predicates and functions are bounded in a certain way. The equivalence of the expressiveness of normal logic programs and disjunctive logic programs over arbitrary structures is also shown to coincide with that over finite structures and coincide with whether the complexity class NP is closed under complement. Moreover, to obtain a more explicit picture of the expressiveness, some intertranslatability results between logic program classes, and fragments of second-order logic are established.

SOME FRAGMENTS OF SECOND-ORDER LOGIC OVER THE REALS FOR WHICH SATISFIABILITY AND EQUIVALENCE ARE (UN)DECIDABLE

We consider the 10-fragment of second-order logic over the vocabulary h+;; 0; 1; <; S1; :::; Ski, interpreted over the reals, where the predicate symbols Si are interpreted as semialgebraic sets. We show that, in this context, satisability of formulas is decidable for the rst-order 9-quantier fragment and undecidable for the 98- and 8-fragments. We also show that for these three fragments the same (un)decidability results hold for containment and equivalence of formulas.

On formalism freeness: Implementing Gödel's 1946 Princeton bicentennial lecture

In this paper we isolate a notion that we call “formalism freeness” from Gödel's 1946 Princeton Bicentennial Lecture, which asks for a transfer of the Turing analysis of computability to the cases of definability and provability. We suggest an implementation of Gödel's idea in the case of definability, via versions of the constructible hierarchy based on fragments of second order logic. We also trace the notion of formalism freeness in the very wide context of developments in mathematical logic in the 20th century.

On Formalism Freeness: Implementing Gödel's 1946 Princeton Bicentennial Lecture

AbstractIn this paper we isolate a notion that we call “formalism freeness” from Gödel's 1946 Princeton Bicentennial Lecture, which asks for a transfer of the Turing analysis of computability to the cases of definability and provability. We suggest an implementation of Gödel's idea in the case of definability, via versions of the constructible hierarchy based on fragments of second order logic. We also trace the notion of formalism freeness in the very wide context of developments in mathematical logic in the 20th century.

Characterising equilibrium logic and nested logic programs: Reductions and complexity,

AbstractEquilibrium logic is an approach to non-monotonic reasoning that extends the stable-model and answer-set semantics for logic programs. In particular, it includes the general case ofnested logic programs, where arbitrary Boolean combinations are permitted in heads and bodies of rules, as special kinds of theories. In this paper, we present polynomial reductions of the main reasoning tasks associated with equilibrium logic and nested logic programs intoquantified propositional logic, an extension of classical propositional logic where quantifications over atomic formulas are permitted. Thus, quantified propositional logic is a fragment of second-order logic, and its formulas are usually referred to asquantified Boolean formulas(QBFs). We provide reductions not only for decision problems, but also for the central semantical concepts of equilibrium logic and nested logic programs. In particular, our encodings map a given decision problem into some QBF such that the latter is valid precisely in case the former holds. The basic tasks we deal with here are theconsistency problem,brave reasoningandskeptical reasoning. Additionally, we also provide encodings for testing equivalence of theories or programs under different notions of equivalence, viz.ordinary,stronganduniform equivalence. For all considered reasoning tasks, we analyse their computational complexity and give strict complexity bounds. Hereby, our encodings yield upper bounds in a direct manner. Besides this useful feature, our approach has the following benefits: First, our encodings yield auniform axiomatisationfor a variety of problems in a common language. Second, extant solvers for QBFs can be used as back-end inference engines to realise implementations of the encoded task in a rapid prototyping manner. Third, our axiomatisations also allow us to straightforwardly relate equilibrium logic with circumscription.

Zero-one law and definability of linear order

§1. Introduction. A logic ℒ has a limit law, if the asymptotic probability of every query definable in ℒ converges. It has a 0–1-law if the probability converges to 0 or 1. The 0–1-law for first-order logic on relational vocabularies was independently found by Glebski et al. [6] and Fagin [5]. Later it has been shown for many other logics, for instance for fragments of second order logic [12], for finite variable logic [13] and for FO extended with the rigidity quantifier [3]. Lynch [14] has shown a limit law for first-order logic on vocabularies with unary functions.We say that two formulas or two logics are almost everywhere equivalent, if they are equivalent on a class of structures whose asymptotic probability measure is one [7]. A 0–1-law is usually proved by showing that every quantifier of the logic has almost everywhere quantifier elimination, i.e., every formula with just one quantifier in front of it is almost everywhere equivalent to a quantifier-free formula. Besides proving 0–1-law, this implies that the logic is (weakly) almost everywhere equivalent to first-order logic.The aim of this paper is to study, whether a logic with a 0–1-law can have greater expressive power than FO in the almost everywhere sense and to what extent. In particular, we are interested on the definability of linear order. Because a 0–1-law determines the almost everywhere expressive power of the sentences of the logic completely, but does not say anything about formulas explicitly, we have to assume some regularity on logics. We will therefore mostly consider extensions of first-order logic with generalized quantifiers.

Back and forth between guarded and modal logics

Guarded fixed-point logic μGF extends the guarded fragment by means of least and greatest fixed points, and thus plays the same role within the domain of guarded logics as the modal μ-calculus plays within the modal domain. We provide a semantic characterization of μGF within an appropriate fragment of second-order logic, in terms of invariance under guarded bisimulation. The corresponding characterization of the modal μ-calculus, due to Janin and Walukiewicz, is lifted from the modal to the guarded domain by means of model theoretic translations. Guarded second-order logic, the fragment of second-order logic which is introduced in the context of our characterization theorem, captures a natural and robust level of expressiveness with several equivalent characterizations. For a wide range of issues in guarded logics it may take up a role similar to that of monadic second-order in relation to modal logics. At the more general methodological level, the translations between the guarded and modal domains make the intuitive analogy between guarded and modal logics available as a tool in the further analysis of the model theory of guarded logics.

Comment On Eklund And Kolak

I appreciate greatly the occasion that Eklund and Kolak (referred to in the sequel as 'the authors') have provided me to reflect on the nature of independence-friendly (IF) logic. Naturally I also welcome their sup port of my position. However, the issues they raise cut even deeper than the authors bring out. In order to help our readers to put their paper in perspective, I therefore propose to offer a few comments on it. The most general comment concerns the framework the authors are using. They consider different logics the received first-order logic IF first-order logic, the E{ fragment of second-order logic, the whole second order logic, and so on, as different 'systems' of logic, each with their own 'commitments'. This way of talking and thinking clearly presupposes some idea of a system of logic as an independent unit of inquiry, some thing like a structure of formulas held together by inferential relations are to be captured by suitable axioms and inference rules. Now G?del-type incompleteness results show that one cannot conceive of mathematical theories as such formal systems. Even arithmetical truth can only be char acterized model-theoretically, not formally. However, the necessity of the same shift away from a syntactical and deductive viewpoint to a model theoretical (semantical) one in the case of systems of logic has not reached the consciousness of philosophers. The culprit here is historically speaking G?del's completeness proof for ordinary first-order logic which had the disastrous effect of leading philosophers and philosophical logicians to believe that logical theories (branches of logic) can be thought of as formal systems, that is, thought of in axiomatic and deductive terms. One of the most important impacts of IF logic on the philosophy of logic is to destroy this belief for good. In spite of its crystal clear semantical meaning, IF first order logic is not axiomatizable, that is to say, representable as a formal system. It can only be characterized semantically (model-theoretically). And comparisons with other kinds of logic can a fortiori be carried out only in model-theoretical terms.

SO(∀∃*) Sentences and Their Asymptotic Probabilities

We prove a 0-1 law for the fragment of second order logic SO(∀∃*) over parametric classes of finite structures which allow only one unary atomic type. This completes the investigation of 0-1 laws for fragments of second order logic defined in terms of first order quantifier prefixes over, e.g., simple graphs and tournaments. We also prove a low oscillation law, and establish the 0-1 law for Σ14(∀∃*) without any restriction on the number of unary types.

On the expressiveness of frame satisfiability and fragments of second-order logic

AbstractIt was conjectured by Halpern and Kapron (Annals of Pure and Applied Logic, vol. 69, 1994) that frame satisfiability of propositional modal formulas is incomparable in expressive power to both (Ackermann) and (Bernays-Schönfinkel). We prove this conjecture. Our results imply that (Ackermann) and (Bernays-Schönfinkel) are incomparable in expressive power, already on finite graphs. Moreover, we show that on ordered finite graphs, i.e., finite graphs with a successor, (Bernays-Schönfinkel) is strictly more expressive than (Ackermann).

Capturing complexity classes by fragments of second-order logic

We investigate the expressive power of certain fragments of second-order logic on finite structures. The fragments are second-order Horn logic, second-order Krom logic as well as a symmetric and a deterministic version of the latter. It is shown that all these logics collapse to their existential fragments. In the presence of successor relation they provide characterizations of polynomial time, deterministic and nondeterministic logspace and of the complement of symmetric logspace. Without successor relation these logics still can express certain problems that are complete in the corresponding complexity classes, but on the other hand they are strictly weaker than previously known logics for these classes and fail to express some very simple properties.