Proof Systems For Logics Research Articles

An introduction to INCAPS system

INCAPS, a subsystem of XYZ system, is an INteractive Computer-Assisted Proving System, The primary targets to develop it range from proving temporal logic formal theorem to verifying XYZ/SE program's correctness which are supported respectively by the mechanized logics—FOTL logic and Hoare-like proof system. This paper discusses five main topics concerning INCAPS system: the rules, implementation, tactics, forward proof and backward proof. It also gives several typical examples for demonstration of INCAPS' working principle. The achievement to data is that we have now accomplished successfully the verification of the hierarchical specification of AB protocol and the correctness of XYZ/SE program.

A resolution framework for finitely-valued first-order logics

In this paper we propose a resolution proof framework on the basis of which automated proof systems for finitely-valued first-order logics ( FFO logics) can be introduced and studied. We define the notion of a first-order resolution proof system and we show that for every disjunctive FFO logic a refutationally complete resolution proof system can be constructed. Moreover, we discuss two theorem proving strategies, the polarity and set of support strategies, and we prove their completeness.

Mechanical proof systems for logic: Reaching consensus by groups of intelligent agents

Reasoning about reaching consensus by groups of intelligent agents has been discussed and formalized by Rasiowa and Marek within the framework of a modified logic for approximation reasoning. Two kinds of knowledge operators have been applied: perception operators and epistemic operators. The present paper is restricted to the fragment with perception operators and presents theorem-proving systems of Gentzen style and of the resolution style. They are based on systems formulated by Rasiowa, Sikorski; and Robinson for classical predicate calculi and on those given by Orlowska for Post predicate calculi.

The power of temporal proofs

Some methods for reasoning about concurrent programs and hardware devices have been based on proof systems for temporal logic. Unfortunately, all effective proof systems for temporal logic are incomplete for the standard semantics, in the sense that some formulas hold in every intended model but cannot be proved. We evaluate and compare the power of several proof systems for temporal logic. Specifically, we relate temporal systems to classical systems with explicit time parameters. A typical temporal system turns out to be incomplete in a strong sense; we exhibit a short, valid formula it fails to prove. We suggest the addition of new rules to define auxiliary predicates. With these rules, we obtain nonstandard soundness and completeness results. In particular, one of the simple temporal systems we describe is as powerful as Peano Arithmetic.

Arithmetical completeness versus relative completeness

In this paper we study the status of the arithmetical completeness of dynamic logic. We prove that for finitistic proof systems for dynamic logic results beyond arithmetical completeness are very unlikely. The role of the set of natural numbers is carefully analyzed.

Model existence theorems for modal and intuitionistic logics

In classical logic a collection of sets of statements (or equivalently, a property of sets of statements) is called a consistency property if it meets certain simple closure conditions (a definition is given in §2). The simplest example of a consistency property is the collection of all consistent sets in some formal system for classical logic. The Model Existence Theorem then says that any member of a consistency property is satisfiable in a countable domain. From this theorem many basic results of classical logic follow rather simply: completeness theorems, the compactness theorem, the Lowenheim-Skolem theorem, and the Craig interpolation lemma among others. The central position of the theorem in classical logic is obvious. For the infinitary logic the Model Existence Theorem is even more basic as the compactness theorem is not available; [8] is largely based on it.In this paper we define appropriate notions of consistency properties for the first-order modal logics S4, T and K (without the Barcan formula) and for intuitionistic logic. Indeed we define two versions for intuitionistic logic, one deriving from the work of Gentzen, one from Beth; both have their uses. Model Existence Theorems are proved, from which the usual known basic results follow. We remark that Craig interpolation lemmas have been proved model theoretically for these logics by Gabbay ([5], [6]) using ultraproducts. The existence of both ultra-product and consistency property proofs of the same result is a common phenomena in classical and infinitary logic. We also present extremely simple tableau proof systems for S4, T, K and intuitionistic logics, systems whose completeness is an easy consequence of the Model Existence Theorems.