Towards Automated Reasoning in Herbrand Structures

Herbrand structures have the advantage, computationally speaking, of being guided by the definability of all elements in them. A salient feature of the logics induced by them is that they internally exhibit the induction scheme, thus providing a congenial, computationally-oriented framework for formal inductive reasoning. Nonetheless, their enhanced expressivity renders any effective proof system for them incomplete. Furthermore, the fact that they are not compact poses yet another prooftheoretic challenge. This paper offers several layers for coping with the inherent incompleteness and non-compactness of these logics. First, two types of infinitary proof system are introduced—one of infinite width and one of infinite height—which manipulate infinite sequents and are sound and complete for the intended semantics. The restriction of these systems to finite sequents induces a completeness result for finite entailments. Then, in search of effectiveness, two finite approximations of these systems are presented and explored. Interestingly, the approximation of the infinite-width system via an explicit induction scheme turns out to be weaker than the effective cyclic fragment of the infinite-height system

A Linear Logic Based Approach to Timed Petri Nets

1.1 Relationship between Petri net and linear logic Petri nets were first introduced by Petri in his seminal Ph.D. thesis, and both the theory and the applications of his model have flourished in concurrency theory (Reisig & Rozenberg, 1998a; Reisig & Rozenberg, 1998b)

Proof Analysis in Temporal Logic

The logic of time is one of the most interesting modal logics, and its importance is widely acknowledged both for philosophical and formal reasons. In this thesis, we apply the method of internalisation of Kripke-style semantics into the syntax of sequent calculus to the proof-theoretical analysis of temporal logics. Sequent systems for different flows of time are obtained as modular extensions of a basic temporal calculus, through the addition of appropriate mathematical rules that correspond to the properties of temporal frames: a general and uniform treatment is thus achieved for a wide range of temporal logics. All the calculi enjoy remarkable structural properties, in particular are contraction and cut free. Linear discrete time is analysed by means of two infinitary calculi. The first is obtained by means of a rule with infinitely many premises, and the second through a new definition of provability which admits, under certain conditions, derivation trees with infinite branches. The first calculus enjoys the desired structural properties, but the presence of an infinitary rule is harmful for proof analysis. Two finitary systems are identified by replacing the infinitary rule with a weaker finitary rule, and by bounding the number of its premises, respectively. Corresponding, somehow complementary, conservativity results are proved with respect to adequate fragments of the original calculus. The second calculus stems from a closure algorithm which exploits the fixed-point equations for temporal operators and gives saturated sets of closure formulas from a given formula. Finitisation is obtained in the form of an upper bound to the proof-search procedure, and decidability follows as a major consequence

Cyclic proof systems for modal fixpoint logics

This thesis is about cyclic and ill-founded proof systems for modal fixpoint logics, with and without explicit fixpoint quantifiers.Cyclic and ill-founded proof-theory allow proofs with infinite branches or paths, as long as they satisfy some correctness conditions ensuring the validity of the conclusion. In this dissertation we design a few cyclic and ill-founded systems: a cyclic one for the weak Grzegorczyk modal logic K4Grz, based on our explanation of the phenomenon of cyclic companionship; and ill-founded and cyclic ones for the full computation tree logic CTL* and the intuitionistic linear-time temporal logic iLTL. All systems are cut-free, and the cyclic ones for K4Grz and iLTL have fully finitary correctness conditions.Lastly, we use a cyclic system for the modal mu-calculus to obtain a proof of the uniform interpolation property for the logic which differs from the original, automata-based one

Real-time and Probabilistic Temporal Logics: An Overview

Over the last two decades, there has been an extensive study on logical formalisms for specifying and verifying real-time systems. Temporal logics have been an important research subject within this direction. Although numerous logics have been introduced for the formal specification of real-time and complex systems, an up to date comprehensive analysis of these logics does not exist in the literature. In this paper we analyse real-time and probabilistic temporal logics which have been widely used in this field. We extrapolate the notions of decidability, axiomatizability, expressiveness, model checking, etc. for each logic analysed. We also provide a comparison of features of the temporal logics discussed

Encoding logical theories of programs

Nowadays, in many critical situations (such as on-board software), it is manda-tory to certify programs and systems, that is, to prove formally that they meet their specifications. To this end, many logics and formal systems have been proposed for rea-soning rigorously on properties of programs and systems. Their usage on non-trivial cases, however, is often cumbersome and error-prone; hence, a computerized proof assistant is required. This thesis is a contribution to the field of computer-aided formal reasoning. In recent years, Logical Frameworks (LF's) have been proposed as general metalan-guages for the description (encoding) of formal systems. LF's streamline the implementa-tion of proof systems on a machine; moreover, they allow for conceptual clarification of the object logics. The encoding methodology of LF's (based on the judgement as types, proofs as \u3bb-terms paradigm) has been successfully applied to many logics; however, the encoding of the many peculiarities presented by formal systems for program logics is problematic. In this thesis we propose a general methodology for adequately encoding formal systems for reasoning on programs. We consider Structured and Natural Operational Semantics, Modal Logics, Dynamic Logics, and the \ub5-calculus. Each of these systems presents distinc-tive problematic features; in each case, we propose, discuss and prove correct, alternative solutions. In many cases, we introduce new presentations of these systems, in Natural Deduction style, which are suggested by the metalogical analysis induced by the method-ology. At the metalogical level, we generalize and combine the concept of consequence relation by Avron and Aczel, in order to handle schematic and multiple consequences. We focus on a particular Logical Framework, namely the Calculus of Inductive Con-structions, originated by Coquand and Huet, and its implementation, Coq. Our inves-tigation shows that this framework is particularly flexible and suited for reasoning on properties of programs and systems. Our work could serve as a guide and reference to future users of Logical Frameworks

Strong Normalization of a Typed Lambda Calculus for Intuitionistic Bounded Linear-time Temporal Logic

Linear-time temporal logics (LTLs) are known to be useful for verifying concurrent systems, and a simple natural deduction framework for LTLs has been required to obtain a good computational interpretation. In this paper, a typed -calculus B[l] with a Curry-Howard correspondence is introduced for an in-tuitionistic bounded linear-time temporal logic B[l], of which the time domain is bounded by a fixed positive integer l. The strong normalization theorem for B[l] is proved as a main result. The base logic B[l] is defined as a Gentzen-type sequent calculus, and despite the restriction on the time domain, B[l] can derive almost all the typical temporal axioms of LTLs. The proposed frame-work allows us to obtain a uniform and simple proof-theoretical treatment of both natural deduction and sequent calculus, i.e., the equivalence between them, the cut-elimination theorem, the decidability theorem, the Curry-Howard correspondence and the strong normalization theorem can be obtained uniformly

Towards a tableau-based procedure for PLTL based on a multi-conclusion rule and logical optimizations

We present an ongoing work on a proof-search procedure for Propositional Linear Temporal Logic (PLTL) based on a one-pass tableau calculus with a multiple-conclusion rule. The procedure exploits logical optimization rules to reduce the proof-search space. We also discuss the performances of a Prolog prototype of our procedure

A Non-wellfounded, Labelled Proof System for Propositional Dynamic Logic

We define a infinitary labelled sequent calculus for PDL, G3PDL^{\infty}. A finitarily representable cyclic system, G3PDL^{\omega}, is then given. We show that both are sound and complete with respect to standard models of PDL and, further, that G3PDL^{\infty} is cut-free complete. We additionally investigate proof-search strategies in the cyclic system for the fragment of PDL without tests