Logic Theorem Research Articles

Tool Building Requirements for an API to First-Order Solvers

Effective formal verification tools require that robust implementations of automatic procedures for first-order logic and satisfiability modulo theories be integrated into expressive interactive frameworks for logical deduction, such as higher-order logic theorem provers. This paper states some pragmatic requirements for implementations of decision procedures that make them well-suited to integration into such frameworks. The aim is to open a dialogue with the designers of decision procedure software that will lead to greater and easier uptake of their implementations by verification users.

On Löb algebras

We study the variety of Lob algebras , the algebraic structures associated with formal propositional calculus. Among other things, we prove a completeness theorem for formal propositional logic with respect to the variety of Lob algebras. We show that the variety of Lob algebras has the weak amalgamation property. Some interesting subclasses of the variety of Lob algebras, e.g. linear, faithful and strongly linear Lob algebras are introduced. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Weak cardinality theorems

AbstractKummer's Cardinality Theorem states that a language A must be recursive if a Turing machine can exclude for any n words , …, one of the n + 1 possibilities for the cardinality of {, …, }⋂ A. There was good reason to believe that this theorem is a peculiarity of recursion theory: neither the Cardinality Theorem nor weak forms of it hold for resource-bounded computational models like polynomial time. This belief may be flawed. In this paper it is shown that weak cardinality theorems hold for finite automata and also for other models. An explanation is proposed as to why recursion-theoretic and automata-theoretic weak cardinality theorems hold, but not corresponding 'middle-ground theorems': The recursion- and automata-theoretic weak cardinality theorems are instantiations of purely logical weak cardinality theorems. The logical theorems can be instantiated for logical structures characterizing recursive computations and finite automata computations. A corresponding structure characterizing polynomial time computations does not exist.

Fuzzy Modal Logics

In the paper we introduce formal calculi which are a generalization of propositional modal logics. These calculi are called fuzzy modal logics. We introduce the concept of a fuzzy Kripke model and consider a semantics of these calculi in the class of fuzzy Kripke models. The main result of the paper is the completeness theorem of a minimal fuzzy modal logic in the class of fuzzy Kripke models.

Composition of Semantic Web services using Linear Logic theorem proving

This paper introduces a method for automatic composition of Semantic Web services using Linear Logic (LL) theorem proving. The method uses a Semantic Web service language (DAML-S) for external presentation of Web services, while, internally, the services are presented by extralogical axioms and proofs in LL. LL, as a resource conscious logic, enables us to capture the concurrent features of Web services formally (including parameters, states and non-functional attributes). We use a process calculus to present the process model of the composite service. The process calculus is attached to the LL inference rules in the style of type theory. Thus, the process model for a composite service can be generated directly from the complete proof. We introduce a set of subtyping rules that defines a valid dataflow for composite services. The subtyping rules that are used for semantic reasoning are presented with LL inference figures. We propose a system architecture where the DAML-S Translator, LL Theorem Prover and Semantic Reasoner can operate together. This architecture has been implemented in Java.

A local normal form theorem for infinitary logic with unary quantifiers

We prove a local normal form theorem of the Gaifman type for the infinitary logic L∞ω(Qu)ω whose formulas involve arbitrary unary quantifiers but finite quantifier rank. We use a local Ehrenfeucht-Fraïssé type game similar to the one in [9]. A consequence is that every sentence of L∞ω(Qu)ω of quantifier rank n is equivalent to an infinite Boolean combination of sentences of the form (∃≥iy)ψ(y), where ψ(y) has counting quantifiers restricted to the (2n–1 – 1)-neighborhood of y. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Labelled transition systems as a Stone space

A fully abstract and universal domain model for modal transition systems and refinement is shown to be a maximal-points space model for the bisimulation quotient of labelled transition systems over a finite set of events. In this domain model we prove that this quotient is a Stone space whose compact, zero-dimensional, and ultra-metrizable Hausdorff topology measures the degree of bisimilarity such that image-finite labelled transition systems are dense. Using this compactness we show that the set of labelled transition systems that refine a modal transition system, its ''set of implementations'', is compact and derive a compactness theorem for Hennessy-Milner logic on such implementation sets. These results extend to systems that also have partially specified state propositions, unify existing denotational, operational, and metric semantics on partial processes, render robust consistency measures for modal transition systems, and yield an abstract interpretation of compact sets of labelled transition systems as Scott-closed sets of modal transition systems.

ファジィ理論を用いた大型ヘリカル装置超電導コイルシステムのクエンチ検出

We have analyzed the state of the superconducting coil system in a LHD at NIFS (National Institute of Fusion Science) using a fuzzy logic theorem to detect quenching at an early stage. In this method, the “warning coefficient" of the coil system is calculated. As for the fuzzy variables, “effective stored heat" in the coil is introduced in addition to the voltage signal in order to improve quench detection and state estimation. The “effective stored heat" is an integrated value of the heat generated in the coil on the assumption that instantaneous heat in the conductor is continuously cooled by liquid helium. Experiments conducted using the LHD coils confirmed that quench alarm signals can be issued with sufficient lead time before quenching. On the other hand, in the case of small local disturbances, the system shows only a small increase in the caution level.

Formal Verification of Pipelined Synthesized Designs by Exploiting Intermediary Rtls

We propose a formal methodology based on higher-order logic theorem proving for completely automated verification of Register Transfer Level (RTL) designs with pipelined iterative control constructs. We will show that the shortest path in verification of a pipelined design is often not a direct comparison and equivalence checking between the design and its behaviour specification, but through an intermediary design at the same level of abstraction. Viability of automating the entire verification process is a result of: (1) a two-step verification approach that relies on transforming the design into an equivalent non-pipelined RTL to serve as an intermediary between the design and its specification; and (2) identification of three correctness conditions for RTL designs with pipelined control flow, that allow for a concise proof decomposition based on the control and data properties of this class of designs. We use the theorem proving environment PVS (Prototype Verification System) [1], integrated with a synthesis tool, for conducting the verification task. This technique is implemented and integrated in a formal verification tool employed for verifying the correctness of RTL designs generated by a High-Level Synthesis (HLS) system. The technique has been successfully applied for verification of several benchmarks with different Data Introduction Intervals (DII) and clock frequencies.

Formal Verification of Pipelined Synthesized Designs by Exploiting Intermediary RTLs

We propose a formal methodology based on higher-order logic theorem proving for completely automated verification of Register Transfer Level (RTL) designs with pipelined iterative control constructs. We will show that the shortest path in verification of a pipelined design is often not a direct comparison and equivalence checking between the design and its behaviour specification, but through an intermediary design at the same level of abstraction. Viability of automating the entire verification process is a result of: (1) a two-step verification approach that relies on transforming the design into an equivalent non-pipelined RTL to serve as an intermediary between the design and its specification; and (2) identification of three correctness conditions for RTL designs with pipelined control flow, that allow for a concise proof decomposition based on the control and data properties of this class of designs. We use the theorem proving environment PVS (Prototype Verification System) [1], integrated with a synthesis tool, for conducting the verification task. This technique is implemented and integrated in a formal verification tool employed for verifying the correctness of RTL designs generated by a High-Level Synthesis (HLS) system. The technique has been successfully applied for verification of several benchmarks with different Data Introduction Intervals (DII) and clock frequencies.

Algebras of Intervals and a Logic of Conditional Assertions

Intervals in boolean algebras enter into the study of conditional assertions (or events) in two ways: directly, either from intuitive arguments or from Goodman, Nguyen and Walker's representation theorem, as suitable mathematical entities to bear conditional probabilities, or indirectly, via a representation theorem for the family of algebras associated with de Finetti's three-valued logic of conditional assertions/events. Further representation theorems forge a connection with rough sets. The representation theorems and an equivalent of the boolean prime ideal theorem yield an algebraic completeness theorem for the three-valued logic. This in turn leads to a Henkin-style completeness theorem. Adequacy with respect to a family of Kripke models for de Finetti's logic, Łukasiewicz's three-valued logic and Priest's Logic of Paradox is demonstrated. The extension to first-order yields a short proof of adequacy for Korner's logic of inexact predicates.

Effective completeness theorems for modal logic

We initiate the study of computable model theory of modal logic, by proving effective completeness theorems for a variety of first-order modal logics. We formulate a natural definition of a decidable Kripke model, and show how to construct such a decidable Kripke model of a given decidable theory. Our construction is inspired by the effective Henkin construction for classical logic. The Henkin construction, however, depends in an essential way on the Deduction Theorem. In its usual form the Deduction Theorem fails for modal logic. In our construction, the Deduction Theorem is replaced by a result about objects called finite Kripke diagrams. We argue that this result can be viewed as an analogue of the Deduction Theorem for modal logic.

A Sahlqvist theorem for distributive modal logic

In this paper we consider distributive modal logic, a setting in which we may add modalities, such as classical types of modalities as well as weak forms of negation, to the fragment of classical propositional logic given by conjunction, disjunction, true, and false. For these logics we define both algebraic semantics, in the form of distributive modal algebras, and relational semantics, in the form of ordered Kripke structures. The main contributions of this paper lie in extending the notion of Sahlqvist axioms to our generalized setting and proving both a correspondence and a canonicity result for distributive modal logics axiomatized by Sahlqvist axioms. Our proof of the correspondence result relies on a reduction to the classical case, but our canonicity proof departs from the traditional style and uses the newly extended algebraic theory of canonical extensions.

Modal and guarded characterisation theorems over finite transition systems

We explore the finite model theory of the characterisation theorems for modal and guarded fragments of first-order logic over transition systems and relational structures of width two. A new construction of locally acyclic bisimilar covers provides a useful analogue of the well known tree-like unravellings that can be used for the purposes of finite model theory. Together with various other finitary bisimulation respecting model transformations, and Ehrenfeucht–Fraı̈ssé game arguments, these covers allow us to upgrade finite approximations for full bisimulation equivalence towards approximations for elementary equivalence. These techniques are used to prove several ramifications of the van Benthem–Rosen characterisation theorem of basic modal logic for refinements of ordinary bisimulation equivalence, both in the sense of classical and of finite model theory.

Effective completeness theorems for modal logic

We initiate the study of computable model theory of modal logic, by proving effective completeness theorems for a variety of first-order modal logics. We formulate a natural definition of a decidable Kripke model, and show how to construct such a decidable Kripke model of a given decidable theory. Our construction is inspired by the effective Henkin construction for classical logic. The Henkin construction, however, depends in an essential way on the Deduction Theorem. In its usual form the Deduction Theorem fails for modal logic. In our construction, the Deduction Theorem is replaced by a result about objects called finite Kripke diagrams. We argue that this result can be viewed as an analogue of the Deduction Theorem for modal logic.

Boolos‐style proofs of limitative theorems

AbstractBoolos's proof of incompleteness is extended straightforwardly to yield simple “diagonalization‐free” proofs of some classical limitative theorems of logic. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Reasoning with Sets of Defaults in Default Logic

We present a general approach for representing and reasoning with sets of defaults in default logic, focusing on reasoning about preferences among sets of defaults. First, we consider how to control the application of a set of defaults so that either all apply (if possible) or none do (if not). From this, an approach to dealing with preferences among sets of default rules is developed. We begin with an ordered default theory, consisting of a standard default theory, but with possible preferences on sets of rules. This theory is transformed into a second, standard default theory wherein the preferences are respected. The approach differs from other work, in that we obtain standard default theories and do not rely on prioritized versions of default logic. In practical terms this means we can immediately use existing default logic theorem provers for an implementation. Also, we directly generate just those extensions containing the most preferred applied rules; in contrast, most previous approaches generate all extensions, then select the most preferred. In a major application of the approach, we show how semimonotonic default theories can be encoded so that reasoning can be carried out at the object level. With this, we can reason about default extensions from within the framework of a standard default logic. Hence one can encode notions such as skeptical and credulous conclusions, and can reason about such conclusions within a single extension.

Algorithmic uses of the Feferman–Vaught Theorem

The classical Feferman–Vaught Theorem for First Order Logic explains how to compute the truth value of a first order sentence in a generalized product of first order structures by reducing this computation to the computation of truth values of other first order sentences in the factors and evaluation of a monadic second order sentence in the index structure. This technique was later extended by Läuchli, Shelah and Gurevich to monadic second order logic. The technique has wide applications in decidability and definability theory. Here we give a unified presentation, including some new results, of how to use the Feferman–Vaught Theorem, and some new variations thereof, algorithmically in the case of Monadic Second Order Logic MSOL. We then extend the technique to graph polynomials where the range of the summation of the monomials is definable in MSOL. Here the Feferman–Vaught Theorem for these polynomials generalizes well known splitting theorems for graph polynomials. Again, these can be used algorithmically. Finally, we discuss extensions of MSOL for which the Feferman–Vaught Theorem holds as well.

An annotated logic theorem prover for an extended possibilistic logic

In this paper we present a theorem prover for possibilistic logic extended with fuzzy constants and fuzzily restricted quantifiers. First of all, we propose a representation for this logic in terms of Horn clauses. We then show how to transform this Horn clause formalism into the generalized annotated logic formalism proposed by Kifer and Subrahmanian in 1992. Specifically, the valuation in a possibilistic clause generates the annotation which will be attached to the head of the annotated clause. We also show how the inference rules of this possibilistic logic can be translated in terms of the mechanisms provided by this annotated logic. Finally, we discuss the implementation of a theorem prover for this possibilistic logic, now translated to annotated logic, in system KOMET, a large-purpose system which has generalized annotated logic as its underlying framework. In this paper we also present proofs relative to the theoretical issues and some examples implemented in KOMET.

Logic may be simple. Logic, congruence and algebra

This paper is an attempt to clear some philosophical questions about the nature of logic by setting up a mathematical framework. The notion of congruence in logic is defined. A logical structure in which there is no non-trivial congruence relation, like some paraconsistent logics, is called simple. The relations between simplicity, the replacement theorem and algebraization of logic are studied (including MacLane-Curry’s theorem and a discussion about Curry’s algebras). We also examine how these concepts are related to such notions as semantics, truth-functionality and bivalence. We argue that a logic, which is simple, can deserve the name logic and that the opposite view is connected with a reductionist perspective (reduction of logic to algebra).