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A Semantical Analysis of Focusing and Contraction in Intuitionistic Logic

Avellone A., Fiorentini C., Momigliano A.

Fundamenta Informaticae

2015

Vol. 140, nr 3/4

247--262

Focusing is a proof-theoretic device to structure proof search in the sequent calculus: it provides a normal form to cut-free proofs in which the application of invertible and non-invertible inference rules is structured in two separate and disjoint phases. Although stemming from proofsearch considerations, focusing has not been thoroughly investigated in actual theorem proving, in particular w.r.t. termination. We present a contraction-free (and hence terminating) focused multisuccedent sequent calculus for propositional intuitionistic logic, which refines the G4ip calculus in the tradition of Vorob’ev, Hudelmeier and Dyckhoff. We prove completeness of the calculus semantically and argue that this offers a viable alternative to other more syntactical means.

On the π-calculus and Co-intuitionistic Logic : Notes on Logic for Concurrency and λP Systems

Bellin G., Menti A.

Fundamenta Informaticae

2014

Vol. 130, nr 1

21--65

We reconsider work by Bellin and Scott in the 1990s on R. Milner and S. Abramsky’s encoding of linear logic in the π-calculus and give an account of efforts to establish a tight connection between the structure of proofs and of the cut elimination process in multiplicative linear logic, on one hand, and the input-output behaviour of the processes that represent them, on the other, resulting in a proof-theoretic account of (a variant of) Chu’s construction. But Milner’s encoding of the linear lambda calculus suggests consideration of multiplicative co-intuitionistic linear logic: we provide a term assignment for it, a calculus of coroutines which presents features of concurrent and distributed computing. Finally, as a test case of its adequacy as a logic for distributed computation, we represent our term assignment as a λP system. We argue that translations of typed functional languages in concurrent and distributed systems (such as π-calculi or λP systems) are best typed with co-intuitionistic logic, where some features of computations match the logical properties in a natural way.

Monteiro Spaces and Rough Sets Determined by Quasiorder Relations : Models for Nelson algebras

Järvinen J., Radeleczki S.

Fundamenta Informaticae

2014

Vol. 131, nr 2

205--215

The theory of rough sets provides a widely used modern tool, and in particular, rough sets induced by quasiorders are in the focus of the current interest, because they are strongly interrelated with the applications of preference relations and intuitionistic logic. In this paper, a structural characterisation of rough sets induced by quasiorders is given. These rough sets form Nelson algebras defined on algebraic lattices. We prove that any Nelson algebra can be represented as a subalgebra of an algebra defined on rough sets induced by a suitable quasiorder. We also show that Monteiro spaces, rough sets induced by quasiorders and Nelson algebras defined on T0-spaces that are Alexandrov topologies can be considered as equivalent structures, because they determine each other up to isomorphism.