1,813 research outputs found
Constructive Provability Logic
We present constructive provability logic, an intuitionstic modal logic that
validates the L\"ob rule of G\"odel and L\"ob's provability logic by permitting
logical reflection over provability. Two distinct variants of this logic, CPL
and CPL*, are presented in natural deduction and sequent calculus forms which
are then shown to be equivalent. In addition, we discuss the use of
constructive provability logic to justify stratified negation in logic
programming within an intuitionstic and structural proof theory.Comment: Extended version of IMLA 2011 submission of the same titl
Computational reverse mathematics and foundational analysis
Reverse mathematics studies which subsystems of second order arithmetic are
equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main
philosophical application of reverse mathematics proposed thus far is
foundational analysis, which explores the limits of different foundations for
mathematics in a formally precise manner. This paper gives a detailed account
of the motivations and methodology of foundational analysis, which have
heretofore been largely left implicit in the practice. It then shows how this
account can be fruitfully applied in the evaluation of major foundational
approaches by a careful examination of two case studies: a partial realization
of Hilbert's program due to Simpson [1988], and predicativism in the extended
form due to Feferman and Sch\"{u}tte.
Shore [2010, 2013] proposes that equivalences in reverse mathematics be
proved in the same way as inequivalences, namely by considering only
-models of the systems in question. Shore refers to this approach as
computational reverse mathematics. This paper shows that despite some
attractive features, computational reverse mathematics is inappropriate for
foundational analysis, for two major reasons. Firstly, the computable
entailment relation employed in computational reverse mathematics does not
preserve justification for the foundational programs above. Secondly,
computable entailment is a complete relation, and hence employing it
commits one to theoretical resources which outstrip those available within any
foundational approach that is proof-theoretically weaker than
.Comment: Submitted. 41 page
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
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
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
- …