179 research outputs found
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
On Graph Refutation for Relational Inclusions
We introduce a graphical refutation calculus for relational inclusions: it
reduces establishing a relational inclusion to establishing that a graph
constructed from it has empty extension. This sound and complete calculus is
conceptually simpler and easier to use than the usual ones.Comment: In Proceedings LSFA 2011, arXiv:1203.542
Mendler-style Iso-(Co)inductive predicates: a strongly normalizing approach
We present an extension of the second-order logic AF2 with iso-style
inductive and coinductive definitions specifically designed to extract programs
from proofs a la Krivine-Parigot by means of primitive (co)recursion
principles. Our logic includes primitive constructors of least and greatest
fixed points of predicate transformers, but contrary to the common approach, we
do not restrict ourselves to positive operators to ensure monotonicity, instead
we use the Mendler-style, motivated here by the concept of monotonization of an
arbitrary operator on a complete lattice. We prove an adequacy theorem with
respect to a realizability semantics based on saturated sets and
saturated-valued functions and as a consequence we obtain the strong
normalization property for the proof-term reduction, an important feature which
is absent in previous related work.Comment: In Proceedings LSFA 2011, arXiv:1203.542
An ecumenical view of proof-theoretic semantics
Debates concerning philosophical grounds for the validity of classical and
intuitionistic logics often have the very nature of logical proofs as one of
the main points of controversy. The intuitionist advocates for a strict notion
of constructive proof, while the classical logician advocates for a notion
which allows non-construtive proofs through \textit{reductio ad absurdum}. A
great deal of controversy still subsists to this day on the matter, as there is
no agreement between disputants on the precise standing of non-constructive
methods.
Two very distinct approaches to logic are currently providing interesting
contributions to this debate. The first, oftentimes called logical ecumenism,
aims to provide a unified framework in which two "rival" logics may peacefully
coexist, thus providing some sort of neutral ground for the contestants. The
second, proof-theoretic semantics, aims not only to elucidate the meaning of a
logical proof, but also to provide means for its use as a basic concept of
semantic analysis. Logical ecumenism thus provides a medium in which meaningful
interactions may occur between classical and intuitionistic logic, whilst
proof-theoretic semantics provides a way of clarifying what is at stake when
one accepts or denies reductio ad absurdum as a meaningful proof method.
In this paper we show how to coherently combine both approaches by providing
not only a medium in which classical and intuitionistic logics may coexist, but
also one in which classical and intuitionistic notions of proof may coexist.Comment: Submitted to TABLEAUX 202
Lazy Strong Normalization
AbstractAmong all the reduction strategies for the untyped λ-calculus, the so called lazy β-evaluation is of particular interest due to its large applicability to functional programming languages (e.g. Haskell [Bird, R., “Introduction to Functional Programming using Haskell,” Series in Computer Science (2nd edition), Prentice Hall, (1998)]). This strategy reduces only redexes not inside a lambda abstraction.The lazy strongly β- normalizing terms are the λ-terms that don't have infinite lazy β-reduction sequences.This paper presents a logical characterization of lazy strongly β-normalizing terms using intersection types. This characterization, besides being interesting by itself, allows an interesting connection between call-by-name and call-by-value λ-calculus.In fact, it turns out that the class of lazy strongly β-normalizing terms coincides with that of call-by-value potentially valuable terms. This last class is of particular interest since it is a key notion for characterizing solvability in the call-by-value setting
Strong normalization from an unusual point of view
AbstractA new complete characterization of β-strong normalization is given, both in the classical and in the lazy λ-calculus, through the notion of potential valuability inside two suitable parametric calculi
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