Weighted Model Counting in FO2 with Cardinality Constraints and Counting Quantifiers: A Closed Form Formula

Motivated by applications in software verification, we explore automated reasoning about the non-disjoint combination of theories of infinitely many finite structures, where the theories share set variables and set operations. We prove a combination theorem and apply it to show the decidability of the satisfiability problem for a class of formulas obtained by applying propositional connectives to formulas belonging to: 1) Boolean Algebra with Presburger Arithmetic (with quantifiers over sets and integers), 2) weak monadic second-order logic over trees (with monadic second-order quantifiers), 3) two-variable logic with counting quantifiers (ranging over elements), 4) the Bernays-Schonfinkel-Ramsey class of first-order logic with equality (with ∃*¬* quantifier prefix), and 5) the quantifier-free logic of multisets with cardinality constraints.

1 Introduction Program schemata (also called schemes, or abstract programs) are widely known as devices for treating properties of computer programs which are invariant across a range of interpretatio ns. Since the intent of schemata is to formalize a specific aspect of effective computation, the properties studied are usually those which have been prescribed in other models of computation: e.g., totality, equivalence, decision power. It is perhaps for this reason that the methodology for dealing with schemata has evolved around constructive mathematics and direct demonstration rather than continuous mathematics, transfinite mathematics and indirect proof. Recently, however, it has been pointed out (Kfoury [3], DeMillo [2]) that useful properties of schemata can be established using arguments which rest on nonconstructi ve foundations; in this case the model theory of first order logic. The outcome of this activity has been to recast some deep questions regarding, for instance, what can and cannot be proved about computer programs into a form which is similar to the corresponding logical questions, where some of the answers are known. In this note,* we present three such applications of model theory. We allow a free wheeling notion of schemata (allowing, for example, first order oracles of any sort desired) and show that direct limitations exist in the ability to draw inferences about schemata by examining finite entailment and axiomatizability. The third application shows that even the expansive concept of effectiveness allowed here does not admit more relative decision power than the standard models of computation. 2 Notation Logical notation follows such standard sources as Bell and Slomson [1]. In particular, such notions as first order language, similarity type, relational structure, and interpretation are assumed in this presentation. A given first order language always talks about

The satisfiability and finite satisfiability problems for the two-variable fragment of first-order logic with counting quantifiers are both in NEXPTIME, even when counting quantifiers are coded succinctly.

Data involving epistemic modals suggest that some classically valid argument forms, such as reductio, are invalid in natural language reasoning as they lead to modal collapses. We adduce further data showing that the classical argument forms governing the existential quantifier are similarly defective, as they lead to a de re–de dicto collapse. We observe a similar problem for disjunction. But if the classical argument forms for negation, disjunction and existential quantification are invalid, what are the correct forms that govern the use of these items? Our diagnosis is that epistemic modals interfere with hypothetical reasoning. We present a modal first-order logic and model theory that characterizes hypothetical reasoning with epistemic modals in a principled manner. One upshot is a sound and complete natural deduction system for reasoning with epistemic modals in first-order logic.

Two graphs with adjacency matrices A and B are isomorphic if there exists a permutation matrix P for which the identity PTAP = B holds. Multiplying through by P and relaxing the permutation matrix to a doubly stochastic matrix leads to the linear programming relaxation known as fractional isomorphism. We show that the levels of the Sherali-Adams (SA) hierarchy of linear programming relaxations applied to fractional isomorphism interleave in power with the levels of a well-known color-refinement heuristic for graph isomorphism called the Weisfeiler-Lehman algorithm, or equivalently, with the levels of indistinguishability in a logic with counting quantifiers and a bounded number of variables. This tight connection has quite striking consequences. For example, it follows immediately from a deep result of Grohe in the context of logics with counting quantifiers, that a fixed number of levels of SA suffice to determine isomorphism of planar and minor-free graphs. We also offer applications both in finite model theory and polyhedral combinatorics. First, we show that certain properties of graphs, such as that of having a flow-circulation of a prescribed value, are definable in the infinitary logic with counting with a bounded number of variables. Second, we exploit a lower bound construction due to Cai, Fürer and Immerman in the context of counting logics to give simple explicit instances that show that the SA relaxations of the vertex-cover and cut polytopes do not reach their integer hulls for up to Ω(n) levels, where n is the number of vertices in the graph.

We now discuss the model theory of (classical) first-order logic together with partial interpretations. As in the propositional case, the syntax of partial firstorder logic is essentially the same as in the classical case. The main difference from classical first-order logic is the model theory, and the resulting formal axiomatics. The reader is referred e.g. to Smullyan [94] for a comprehensive discussion of classical first-order logic.

Pebble games are a powerful tool in the study of finite model theory, constraint satisfaction and database theory. Monads and comonads are basic notions of category theory which are widely used in semantics of computation and in modern functional programming. We show that existential k-pebble games have a natural comonadic formulation. Winning strategies for Duplicator in the k-pebble game for structures A and B are equivalent to morphisms from A to B in the coKleisli category for this comonad. This leads on to comonadic characterisations of a number of central concepts in Finite Model Theory: - Isomorphism in the co-Kleisli category characterises elementary equivalence in the k-variable logic with counting quantifiers. - Symmetric games corresponding to equivalence in full k-variable logic are also characterized. - The treewidth of a structure A is characterised in terms of its coalgebra number: the least k for which there is a coalgebra structure on A for the k-pebbling comonad. - Co-Kleisli morphisms are used to characterize strong consistency, and to give an account of a Cai-F\"urer-Immerman construction. - The k-pebbling comonad is also used to give semantics to a novel modal operator. These results lay the basis for some new and promising connections between two areas within logic in computer science which have largely been disjoint: (1) finite and algorithmic model theory, and (2) semantics and categorical structures of computation.

Enriching logic formalisms with counting capabilities is an important task in view of the needs of many application areas, ranging from database theory to formal verification. In this paper, we consider a very expressive language obtained by enriching linear integer arithmetic with free function symbols and cardinality constraints for interpreted sets. We obtain positive results for a flat fragment via a reduction to decidability of Presburger arithmetic with unary counting quantifiers (Schweikhart in Arithmetic, first-order logic, and counting quantifiers, ACM TOCL, New York, 2004). We isolate also an easier simple flat subfragment, whose satisfiability is in NP, and we show that this subfragment is adequate to formalize problems arising in the area of the verification of fault-tolerant distributed algorithms. We finally discuss our first implementation, the related experimental results, as well as further algorithmic problems suggested by model-checking applications.

In this chapter we present a method for obtaining lower bounds on the computational complexity of logical theories, and give several illustrations of its use. This method is an extension of widely used procedures for proving the recursive undecidability of logical theories. (See Rabin [1965] and Eršov et al. [1965].) One important aspect of this method is that it is based on a family of inseparability results for certain logical problems, closely related to the well-known inseparability result of Trakhtenbrot (as refined by Vaught), that no recursive set separates the logically valid sentences from those which are false in some finite model, as long as the underlying language has at least one non-unary relation symbol. By using these inseparability results as a foundation, we are able to obtain hereditary lower bounds, i.e., bounds which apply uniformly to all subtheories of the theory. The second important aspect of this method is that we use interpretations to transfer lower bounds from one theory to another. By doing this we eliminate the need to code machine computations into the models of the theory being studied. (The coding of computations is done once and for all in proving the inseparability results.) By using interpretations, attention is centred on much simpler definability considerations, viz., what kinds of binary relations on large finite sets can be defined using short formulas in models of the theory. This is conceptually much simpler than other approaches that have been proposed for obtaining lower bounds, such as the method of bounded concatenations of Fleischmann et al. [1977]. We will deal primarily with theories in first-order logic and monadic second-order logic.

Data Analytics Using Ontologies of Management Theories: Towards Implementing 'From Theory to Practice'

It is well known that the hypothesis that all real numbers are constructible in the sense of Gödel [1] implies the existence of a Σ21well-ordering of the Baire space [1, p. 67]. We are concerned with the converse to this theorem. From the assumption of the existence of a Σ21well-ordering with total domain, we derive various consequences which in the presence of a nonconstructible real seem highly pathological. However, while several of these consequences are obviously absurd, none have as yet been disproven. Indeed some of the stranger consistency proofs of Jensen seem to indicate that there is a possibility that they may be consistent. I am referring to such results as existence of a model for ZF set theory in which the degrees of constructibility form a sequence of typeω+ 1 with the (ω+ 1)st degree being the only one which contains a nonconstructible real, and that degree being the degree of a Δ31nonconstructible real.In what follows we shall use the small Greek lettersα, β, γto range over the Baire spaceNNwhereNis the set of natural numbers. We assume that the reader is familiar with the use of trees to encode closed subsets of this space. The class Σ21is the collection of all those subsets of the Baire space which can be defined by a formula which is Σ21in a constructible parameter. Correspondingly Π21will be taken to mean Π11in a constructible parameter. The proof of the first lemma is left as an exercise. It involves nothing more than coding perfect sets by trees and then counting quantifiers.

Let G be a locally compact abelian group with Haar measure dx, and let ω be a symmetric Beurling weight function on G (Reiter, 1968). In this paper, using the relations between pi and qi, where 1 < pi, qi < ∞, pi ≠ qi(i = 1, 2), we show that the space of multipliers from to the space , the space of multipliers from to and the space of multipliers to .

An original boundary integral formulation is proposed for the problem of a semi-infinite crack at the interface between two dissimilar elastic materials in the presence of heat flows and mass diffusion. Symmetric and skew-symmetric weight function matrices are used together with a generalized Betti's reciprocity theorem in order to derive a system of integral equations that relate the applied loading, the temperature and mass concentration fields, the heat and mass fluxes on the fracture surfaces and the resulting crack opening. The obtained integral identities can have many relevant applications, such as for the modelling of crack and damage processes at the interface between different components in electrochemical energy devices characterized by multi-layered structures (solid oxide fuel cells and lithium ions batteries).

On the expressive power of counting

Abstract. Despite being widely regarded as a gloss on first-order logic and set theory, Z has not been found to be very supportive of proof. This paper attempts to distinguish between the different philosophies of proof in Z. It discusses some of the issues which must be addressed in creating a proof technology for Z, namely schemas, undefinedness, and what kind of logic to use.