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In a first-order theory |$\varTheta $|⁠ , the decision problem for a class of formulae |$\varPhi $| is solvable if there is an algorithmic procedure that can assess whether or not the existential closure |$\varphi ^{\exists }$| of |$\varphi $| belongs to |$\varTheta $|⁠ , for any |$\varphi \in \varPhi $|⁠. In 1988, Parlamento and Policriti already showed how to tailor arguments à la Gödel to a very weak axiomatic set theory, referring them to the class of |$\varSigma _{1}$| -formulae with |$(\forall \exists \forall)_{0}$| -matrix, i.e. existential closures of formulae that contain just restricted quantifiers of the forms |$(\forall x \in y)$| and |$(\exists x \in y)$| and are writable in prenex form with at most two alternations of restricted quantifiers (the outermost quantifier being a ' |$\forall $| '). While revisiting their work, we show slightly less weak theories under which incompleteness for recursively axiomatizable extensions holds with respect to existential closures of |$(\forall \exists)_{0}$| -matrices, namely formulae with at most one alternation of restricted quantifiers. [ABSTRACT FROM AUTHOR]

We report on an investigation aimed at identifying small fragments of set theory (typically, sublanguages of Multi-Level Syllogistic) endowed with polynomial-time satisfiability decision tests, potentially useful for automated proof verification. Leaving out of consideration the membership relator ∈ for the time being, in this paper we provide a complete taxonomy of the polynomial and the NP-complete fragments involving, besides variables intended to range over the von Neumann set-universe, the Boolean operators ∪ ∩ \, the Boolean relators ⊆, ⊈,=, ≠, and the predicates '• = Ø' and 'Disj(•, •)', meaning 'the argument set is empty' and 'the arguments are disjoint sets', along with their opposites '• ≠ Ø and '¬Disj(•, •)'. We also examine in detail how to test for satisfiability the formulae of six sample fragments: three sample problems are shown to be NP-complete, two to admit quadratic-time decision algorithms, and one to be solvable in linear time. [ABSTRACT FROM AUTHOR]

In this paper we present a methodology for introducing customizable metalogic features in logic-based knowledge representation and reasoning languages. The proposed approach is based on concepts of introspection and reflection previously introduced and discussed by various authors in relevant literature. This allows a knowledge engineer to specify enhanced reasoning engines by defining properties and meta-properties of relations as expressible for instance in OWL. We employ meta-level axiom schemata based upon a naming (reification) device. We propose general principles for extending the semantics of "host" formalisms accordingly. Consequently, suitable pre-defined libraries of properties can be made available, while user-defined new schemata are also allowed. We make the specific cases of Answer Set Programming (ASP) and Datalog±, where such features may be part of software engineering toolkits for these programming paradigms. On the one hand, concerning ASP, we extend the programming principles and practice to accommodate the proposed methodology, so as to perform meta-reasoning within the plain ASP semantics. The computational complexity of the resulting framework does not change. On the other hand, we show how metalogic features can significantly enrich Datalog± with minor changes to its operational semantics (provided in terms of "chase") and, also in this case, no additional complexity burden. [ABSTRACT FROM AUTHOR]

In this paper, we study an information flow security property for systems specified as terms of a quantitative Markovian process algebra, namely the Performance Evaluation Process Algebra (PEPA). We propose a quantitative extension of the Non-Interference property used to secure systems from the functional point view by assuming that the observers are able to measure also the timing properties of the system, e.g., the response time of certain actions or its throughput. We introduce the notion of Persistent Stochastic Non-Interference (PSNI) based on the idea that every state reachable by a process satisfies a basic Stochastic Non-Interference (SNI) property. The structural operational semantics of PEPA allows us to give two characterizations of PSNI: one based on a bisimulation-like equivalence relation inducing a lumping on the underlying Markov chain, and another one based on unwinding conditions which demand properties of individual actions. These two different characterizations naturally lead to efficient methods for the verification and construction of secure systems. A decision algorithm for PSNI is presented and an application of PSNI to a queueing system is discussed. [ABSTRACT FROM AUTHOR]

The axiom of elementary states the existence of a void set and the availability of an unordered pair formation operation. This axiom states that distinct sets cannot have precisely the same elements immediately after the extensionality axiom.

This paper considers ∃ ∀ prenex sentences of pure first-order predicate calculus with equality. This is the set of formulas which Ramsey's treated in a famous article of 1930. We demonstrate that the satisfiability problem and the problem of existence of arbitrarily large models for these formulas can be reduced to the satisfiability problem for ∃ ∀ prenex sentences of Set Theory (in the relators ∈, =).We present two satisfiability-preserving (in a broad sense) translations Φ ↦ $\dot{\Phi}$ and Φ ↦ Φσ of ∃ ∀ sentences from pure logic to well-founded Set Theory, so that if $\dot{\Phi}$ is satisfiable (in the domain of Set Theory) then so is Φ, and if Φσ is satisfiable (again, in the domain of Set Theory) then Φ can be satisfied in arbitrarily large finite structures of pure logic.It turns out that |$\dot{\Phi}$| = $\mathcal{O}$(|Φ|) and |Φσ| = $\mathcal{O}$(|Φ|2).Our main result makes use of the fact that ∃ ∀ sentences, even though constituting a decidable fragment of Set Theory, offer ways to describe infinite sets. Such a possibility is exploited to glue together infinitely many models of increasing cardinalities of a given ∃ ∀ logical formula, within a single pair of infinite sets. [ABSTRACT FROM AUTHOR]

We introduce and prove the basic properties of encodings that generalize to non-wellfounded hereditarily finite sets the bijection defined by Ackermann in 1937 between hereditarily finite sets and natural numbers. [ABSTRACT FROM AUTHOR]

An undirected graph is commonly represented as a set of vertices and a set of vertex doubletons; but one can also represent each vertex by a finite set so as to ensure that membership mimics, over these vertex-representing sets, the edge relation of the graph. This alternative modelling, applied to connected claw-free graphs, recently gave crucial clues for obtaining simpler proofs of some of their properties (e.g. Hamiltonicity of the square of the graph). This article adds a computer-checked contribution. On the one hand, we discuss our development, by means of the Ref verifier, of two theorems on representing graphs by families of finite sets: a weaker theorem pertains to general graphs, and a stronger one to connected claw-free graphs. Before proving those theorems, we must show that every graph admits an acyclic, weakly extensional orientation, which becomes fully extensional when connectivity and claw-freeness are met. This preliminary work enables injective decoration, à la Mostowski, of the vertices by the sought-for finite sets. By this new scenario, we complement our earlier formalization with Ref of two classical properties of connected claw-free graphs. On the other hand, our present work provides another example of the ease with which graph-theoretic results are proved with the Ref verifier. For example, we managed to define and exploit the notion of connected graph without resorting to the notion of path. [ABSTRACT FROM AUTHOR]

It is shown that within the language of Set Theory, if membership is assumed to be non-well-founded à la Aczel, then one can state the existence of infinite sets by means of an ∃∃∀∀ prenex sentence. Somewhat surprisingly, this statement of infinity is essentially the one which was proposed in 1988 for well-founded sets, and it is satisfied exclusively by well-founded sets. Stating infinity inside the BSR (Bernays–Schönfinkel–Ramsey) class of the ∃*∀*-sentences becomes more challenging if no commitment is taken as whether membership is well-founded or not: for this case, we produce an ∃∃∀∀∀ -sentence, thus lowering the complexity of the quantificational prefix with respect to earlier prenex formulations of infinity. We also show that no prenex specification of infinity can have a prefix simpler than ∃∃∀∀.The problem of determining whether a BSR-sentence involving an uninterpreted predicate symbol and = can be satisfied over a large domain is then reduced to the satisfiability problem for the set theoretic class BSR subject to the ill-foundedness assumption. Envisaged enhancements of this reduction, cleverly exploiting the expressive power of the set theoretic BSR-class, add to the motivation for tackling the satisfaction problem for this class, which appears to be anything but unchallenging. [ABSTRACT FROM AUTHOR]

Formal set theory is traditionally concerned with pure sets; consequently, the satisfiability problem for fragments of set theory was most often addressed (and in many cases positively solved) in the pure framework. In practical applications, however, it is common to assume the existence of a number of primitive objects (sometimes called atoms) that can be members of sets but behave differently from them. If these entities are assumed to be devoid of members, the standard extensionality axiom must be revised; then decidability results can sometimes be achieved via reduction to the pure case and sometimes can be based on direct goal-driven algorithms. An alternative approach to modeling atoms that allows one to retain the original formulation of extensionality was proposed by Quine: atoms are self-singletons. In this article we adopt this approach in coping with the satisfiability problem: We show the decidability of this problem relativized to ∃*∀-sentences, and develop a goal-driven unification algorithm. [ABSTRACT FROM AUTHOR]

The article presents a study on the decidability of ∃*∀ class of sentences using a general decision method in diversified axiomatic set theories. It illustrates a decision method in finding the solution for the ∃*∀ sentences with single universal quantifier. The authors also achieve sub theory of complete Zermelo-Fraenkel and presents a technique in restricting the universal quantifier.

In this paper we present a methodology for introducing customizable metalogic features in logic-based knowledge representation and reasoning languages. The proposed approach is based on concepts of introspection and reflection previously introduced and discussed by various authors in relevant literature. This allows a knowledge engineer to specify enhanced reasoning engines by defining properties and meta-properties of relations as expressible for instance in OWL. We employ meta-level axiom schemata based upon a naming (reification) device. We propose general principles for extending the semantics of "host" formalisms accordingly. Consequently, suitable pre-defined libraries of properties can be made available, while user-defined new schemata are also allowed. We make the specific cases of Answer Set Programming (ASP) and Datalog±, where such features may be part of software engineering toolkits for these programming paradigms. On the one hand, concerning ASP, we extend the programming principles and practice to accommodate the proposed methodology, so as to perform meta-reasoning within the plain ASP semantics. The computational complexity of the resulting framework does not change. On the other hand, we show how metalogic features can significantly enrich Datalog± with minor changes to its operational semantics (provided in terms of "chase") and, also in this case, no additional complexity burden. [ABSTRACT FROM AUTHOR]

We report on an investigation aimed at identifying small fragments of set theory (typically, sublanguages of Multi-Level Syllogistic) endowed with polynomial-time satisfiability decision tests, potentially useful for automated proof verification. Leaving out of consideration the membership relator ∈ for the time being, in this paper we provide a complete taxonomy of the polynomial and the NP-complete fragments involving, besides variables intended to range over the von Neumann set-universe, the Boolean operators ∪ ∩ \, the Boolean relators ⊆, ⊈,=, ≠, and the predicates '• = Ø' and 'Disj(•, •)', meaning 'the argument set is empty' and 'the arguments are disjoint sets', along with their opposites '• ≠ Ø and '¬Disj(•, •)'. We also examine in detail how to test for satisfiability the formulae of six sample fragments: three sample problems are shown to be NP-complete, two to admit quadratic-time decision algorithms, and one to be solvable in linear time. [ABSTRACT FROM AUTHOR]