The power of temporal proofs

The main goal of this paper is to provide an abstract framework for constructing proof systems for various many-valued logics. Using the framework it is possible to generate strongly complete proof systems with respect to any finitely valued deterministic and non-deterministic logic. I provide a couple of examples of proof systems for well-known many-valued logics and prove the completeness of proof systems generated by the framework.

In this paper we discuss the problem of termination of the deductive process in resolution proof systems for non-classical logics.

We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof.A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by “nearly linear” is meant the ratio of proof lengths is O(α(n)) where α is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O(n . α(n)). Also we show that a Frege proof of n lines can be transformed into a tree-like Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O(n log n).

Focused proofs are sequent calculus proofs that group inference rules into alternating positive and negative phases. These phases can then be used to define macro-level inference rules Gentzen’s original and tiny introduction and structural rules. We show here that the inference rules of labeled proof systems for modal logics can similarly be described as pairs of such phases within the LKF focused proof system for first-order classical logic. We consider the system $${ G3K} $$ of Negri for the modal logic $$K$$ and define a translation from labeled modal formulas into first-order polarized formulas and show a strict correspondence between derivations in the two systems, i.e., each rule application in $${ G3K} $$ corresponds to a bipole—a pair of a positive and a negative phases—in $${ LKF} $$ . Since geometric axioms (when properly polarized) induce bipoles, this strong correspondence holds for all modal logics whose Kripke frames are characterized by geometric properties. We extend these results to present a focused labeled proof system for this same class of modal logics and show its soundness and completeness. The resulting proof system allows one to define a rich set of normal forms of modal logic proofs.

A methodology for developing sound proof systems for program verification is demonstrated by the development of a proof system for reasoning about concurrent programs based on the unity logic of K.M. Chandy and J. Misra (1988). This proof system has been validated by an automated theorem prover, a computer program that checks the correctness of proofs. The Boyer-Moore logic in which the proof system is formalized and the Boyer-Moore theorem prover which mechanizes this logic are described. The motivation behind the unity logic and the way in which it may be used to prove the correctness of concurrent programs are examined. A proof system for concurrent programs based on the unity logic system is provided as an example. An operational semantics of concurrency is formalized in Boyer-Moore logic by use of the transition system model. Unity's proof rules are proved as theorems about this operational semantics. The proofs of these theorems are mechanically checked. The entire proof system has been verified by the Boyer-Moore prover, making it possible to prove mechanically the consequence of other concurrent programs.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency (1979). Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What George Boolos does is to show how the concepts, techniques, and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self-referential' sentences constructed in their proof. The book explores the effects of reinterpreting the notions of necessity and possibility to mean provability and consistency.

A uniform semantic proof for cut-elimination and completeness of various first and higher order logics

A complete proof system for propositional projection temporal logic

Many important parallel computer programs exhibit ongoing behaviour that is characterized naturally in terms of infinite execution traces, which can be organized into “branching” trees, and which reflect the high degree of nondeterminism inherent in parallel computation. In this paper, we give a systematic account of Branching Time Temporal Logics, which provide a formal system for describing and reasoning about the correct behaviour of such programs. Several systems of branching time temporal logic that have appeared in the literature are presented, and significant related issues such as their axiomatizations, and decision procedures for their satisfiability and model checking problems are discussed. The applicability of their axiomatizations to formulate deductive systems of these temporal logics to reason about the correctness of concurrent programs is then described as is that of their decision procedures to the tasks of mechanical synthesis and verification. A comparison of the relative expressive power of these systems of branching time temporal logic is also presented, and their ability to specify important correctness properties of programs, including those that involve fairness, is discussed. Moreover, their expressiveness is related to both, the expressiveness of corresponding linear time temporal logics, as well as that of other standard formalisms such as the monadic second-order theory of many successors and finite-state tree automata. Finally, a comparison is undertaken between branching time temporal logics and linear time ones, particularly with respect to their adequacy for such applications as specifying and reasoning about the correctness of concurrent programs.

Modalities for model checking: branching time logic strikes back

This paper presents a sound and complete logical system whose atomic sentences are the equalities of recursive terms involving sets. There are two interpretations of this language: one makes use of non-wellfounded sets with finite transitive closure, and the other uses pointed finite graphs modulo bisimulation. Our logical system is a sequent-style deduction system. The main axioms and inference rules come from the $\mbox{{\it FLR}$_0$}$ -proof system from [6], including the Recursion Inference Rule (but an additional axiom is needed), and also axioms corresponding to the extensionality axiom of set theory.

CL (Clausal Language) is a computer programming language with mathematical syntax and a proof system based on Peano arithmetic which we have repeatedly used in the teaching of three (first and second year) undergraduate courses covering respectively declarative programming, program verification, and program and abstract data specification.

Transitive closure logic is a known extension of first-order logic obtained by introducing a transitive closure operator. While other extensions of first-order logic with inductive definitions are a priori parametrized by a set of inductive definitions, the addition of the transitive closure operator uniformly captures all finitary inductive definitions. In this paper we present an infinitary proof system for transitive closure logic which is an infinite descent-style counterpart to the existing (explicit induction) proof system for the logic. We show that, as for similar systems for first-order logic with inductive definitions, our infinitary system is complete for the standard semantics and subsumes the explicit system. Moreover, the uniformity of the transitive closure operator allows semantically meaningful complete restrictions to be defined using simple syntactic criteria. Consequently, the restriction to regular infinitary (i.e. cyclic) proofs provides the basis for an effective system for automating inductive reasoning.

The chilled water system of central air conditioning is a typical hybrid system. The dynamic adjustment of cooling capacity based on hybrid system can achieve accurate temperature control and real-time energy saving. Mixed logical dynamical (MLD) systems have advantage for solving constrained optimization problems of this type of case by numerical methods. This paper proposes a novel modified type of MLD system, which enhances the model applicability and improves the switching flexibility and control effect for the framework. In order to meet the needs of cooling capacity and energy saving, the optimal control problem is transformed as MIQP problem by defined performance index. As a numeral example of application, the model and control method is used in pumps group control for variable water volume in chilled water system of central air conditioning. At last, the dynamic and energy-saving effects of the system are simulated, which shows the ideal control results.

The generation of too much information prohibits efficient resolution proof search in classical logics. Subsumption is used to discard redundant information and strategies have been developed to guide the proof search avoiding irrelevant information. The extension of the resolution method to temporal logics, further magnifies this problem. Here we develop an algorithm for the removal of irrelevant information during resolution proofs for a particular resolution proof system for propositional linear-time temporal logic. The resolution system operates on formulae in a normal form. Following a phase of classical style resolution, temporal resolution is carried out by detecting sets of formulae that together represent an □-formula to be resolved with a ◊-formula. It is before the application of the temporal resolution rule that the new algorithm is applied to efficiently remove irrelevant information. The completeness of the new algorithm is discussed and its complexity considered. Its practical efficiency is demonstrated by giving timings for a set of examples.