740 research outputs found
Correspondences between Classical, Intuitionistic and Uniform Provability
Based on an analysis of the inference rules used, we provide a
characterization of the situations in which classical provability entails
intuitionistic provability. We then examine the relationship of these
derivability notions to uniform provability, a restriction of intuitionistic
provability that embodies a special form of goal-directedness. We determine,
first, the circumstances in which the former relations imply the latter. Using
this result, we identify the richest versions of the so-called abstract logic
programming languages in classical and intuitionistic logic. We then study the
reduction of classical and, derivatively, intuitionistic provability to uniform
provability via the addition to the assumption set of the negation of the
formula to be proved. Our focus here is on understanding the situations in
which this reduction is achieved. However, our discussions indicate the
structure of a proof procedure based on the reduction, a matter also considered
explicitly elsewhere.Comment: 31 page
Hilbert's Program Then and Now
Hilbert's program was an ambitious and wide-ranging project in the philosophy
and foundations of mathematics. In order to "dispose of the foundational
questions in mathematics once and for all, "Hilbert proposed a two-pronged
approach in 1921: first, classical mathematics should be formalized in
axiomatic systems; second, using only restricted, "finitary" means, one should
give proofs of the consistency of these axiomatic systems. Although Godel's
incompleteness theorems show that the program as originally conceived cannot be
carried out, it had many partial successes, and generated important advances in
logical theory and meta-theory, both at the time and since. The article
discusses the historical background and development of Hilbert's program, its
philosophical underpinnings and consequences, and its subsequent development
and influences since the 1930s.Comment: 43 page
Proof search in constructive logics
We present an overview of some sequent calculi organised not for
"theorem-proving" but for proof search, where the proofs themselves
(and the avoidance of known proofs on backtracking) are objects of
interest. The main calculus discussed is that of Herbelin [1994] for
intuitionistic logic, which extends methods used in hereditary
Harrop logic programming; we give a brief discussion of similar
calculi for other logics. We also point out to some related work on
permutations in intuitionistic Gentzen sequent calculi that
clarifies the relationship between such calculi and natural
deduction.Centro de MatemĂĄtica da Universidade do Minho (CMAT).UniĂŁo Europeia (UE) - Programa ESPRIT - BRA 7232 Gentzen
Proceedings of the Workshop on Linear Logic and Logic Programming
Declarative programming languages often fail to effectively address many aspects of control and resource management. Linear logic provides a framework for increasing the strength of declarative programming languages to embrace these aspects. Linear logic has been used to provide new analyses of Prolog\u27s operational semantics, including left-to-right/depth-first search and negation-as-failure. It has also been used to design new logic programming languages for handling concurrency and for viewing program clauses as (possibly) limited resources. Such logic programming languages have proved useful in areas such as databases, object-oriented programming, theorem proving, and natural language parsing.
This workshop is intended to bring together researchers involved in all aspects of relating linear logic and logic programming. The proceedings includes two high-level overviews of linear logic, and six contributed papers.
Workshop organizers: Jean-Yves Girard (CNRS and University of Paris VII), Dale Miller (chair, University of Pennsylvania, Philadelphia), and Remo Pareschi, (ECRC, Munich)
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
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