A New Arithmetically Incomplete First- Order Extension of Gl All Theorems of Which Have Cut Free Proofs

Reference [12] introduced a novel formula to formula translation tool (“formulators”) that enables syntactic metatheoretical investigations of first-order modal logics, bypassing a need to convert them first into Gentzen style logics in order to rely on cut elimination and the subformula property. In fact, the formulator tool, as was already demonstrated in loc. cit., is applicable even to the metatheoretical study of logics such as QGL, where cut elimination is (provably, [2]) unavailable. This paper applies the formulator approach to show the independence of the axiom schema _A ! _8xA of the logics M3 and ML3 of [17, 18, 11, 13]. This leads to the conclusion that the two logics obtained by removing this axiom are incomplete, both with respect to their natural Kripke structures and to arithmetical interpretations. In particular, the so modified ML3 is, similarly to QGL, an arithmetically incomplete first-order extension of GL, but, unlike QGL, all its theorems have cut free proofs. We also establish here, via formulators, a stronger version of the disjunction property for GL and QGL without going through Gentzen versions of these logics (compare with the more complex proofs in [2, 8]).This research was partially supported by NSERC grant No. 8250

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later

On the Invariance of G\"odel's Second Theorem with regard to Numberings

The prevalent interpretation of G\"odel's Second Theorem states that a sufficiently adequate and consistent theory does not prove its consistency. It is however not entirely clear how to justify this informal reading, as the formulation of the underlying mathematical theorem depends on several arbitrary formalisation choices. In this paper I examine the theorem's dependency regarding G\"odel numberings. I introduce deviant numberings, yielding provability predicates satisfying L\"ob's conditions, which result in provable consistency sentences. According to the main result of this paper however, these "counterexamples" do not refute the theorem's prevalent interpretation, since once a natural class of admissible numberings is singled out, invariance is maintained.Comment: Forthcoming in The Review of Symbolic Logi

Model Checking Linear Logic Specifications

The overall goal of this paper is to investigate the theoretical foundations of algorithmic verification techniques for first order linear logic specifications. The fragment of linear logic we consider in this paper is based on the linear logic programming language called LO enriched with universally quantified goal formulas. Although LO was originally introduced as a theoretical foundation for extensions of logic programming languages, it can also be viewed as a very general language to specify a wide range of infinite-state concurrent systems. Our approach is based on the relation between backward reachability and provability highlighted in our previous work on propositional LO programs. Following this line of research, we define here a general framework for the bottom-up evaluation of first order linear logic specifications. The evaluation procedure is based on an effective fixpoint operator working on a symbolic representation of infinite collections of first order linear logic formulas. The theory of well quasi-orderings can be used to provide sufficient conditions for the termination of the evaluation of non trivial fragments of first order linear logic.Comment: 53 pages, 12 figures "Under consideration for publication in Theory and Practice of Logic Programming

An Effective Fixpoint Semantics for Linear Logic Programs

In this paper we investigate the theoretical foundation of a new bottom-up semantics for linear logic programs, and more precisely for the fragment of LinLog that consists of the language LO enriched with the constant 1. We use constraints to symbolically and finitely represent possibly infinite collections of provable goals. We define a fixpoint semantics based on a new operator in the style of Tp working over constraints. An application of the fixpoint operator can be computed algorithmically. As sufficient conditions for termination, we show that the fixpoint computation is guaranteed to converge for propositional LO. To our knowledge, this is the first attempt to define an effective fixpoint semantics for linear logic programs. As an application of our framework, we also present a formal investigation of the relations between LO and Disjunctive Logic Programming. Using an approach based on abstract interpretation, we show that DLP fixpoint semantics can be viewed as an abstraction of our semantics for LO. We prove that the resulting abstraction is correct and complete for an interesting class of LO programs encoding Petri Nets.Comment: 39 pages, 5 figures. To appear in Theory and Practice of Logic Programmin

The ILLTP Library for Intuitionistic Linear Logic

Benchmarking automated theorem proving (ATP) systems using standardized problem sets is a well-established method for measuring their performance. However, the availability of such libraries for non-classical logics is very limited. In this work we propose a library for benchmarking Girard's (propositional) intuitionistic linear logic. For a quick bootstrapping of the collection of problems, and for discussing the selection of relevant problems and understanding their meaning as linear logic theorems, we use translations of the collection of Kleene's intuitionistic theorems in the traditional monograph "Introduction to Metamathematics". We analyze four different translations of intuitionistic logic into linear logic and compare their proofs using a linear logic based prover with focusing. In order to enhance the set of problems in our library, we apply the three provability-preserving translations to the propositional benchmarks in the ILTP Library. Finally, we generate a comprehensive set of reachability problems for Petri nets and encode such problems as linear logic sequents, thus enlarging our collection of problems