4 research outputs found
A calculus of multiary sequent terms
Multiary sequent terms were originally introduced as a tool for
proving termination of permutative conversions in cut-free sequent
calculus. This work develops the language of multiary sequent terms
into a term calculus for the computational (Curry-Howard)
interpretation of a fragment of sequent calculus with cuts and
cut-elimination rules. The system, named generalised multiary
lambda-calculus, is a rich extension of the lambda-calculus
where the computational content of the sequent calculus format is
explained through an enlarged form of the application constructor.
Such constructor exhibits the features of multiarity (the ability of
forming lists of arguments) and generality (the ability of
prescribing a kind of continuation). The system integrates in a
modular way the multiary lambda-calculus and an isomorphic copy
of the lambda-calculus with generalised application LambdaJ
(in particular, natural deduction is captured internally up to
isomorphism). In addition, the system: (i) comes with permutative
conversion rules, whose role is to eliminate the new features of
application;
(ii) is equipped with reduction rules --- either the mu-rule,
typical of the multiary setting, or rules for cut-elimination,
which enlarge the ordinary beta-rule.
This paper establishes the meta-theory of the system, with emphasis
on the role of the mu-rule, and including a study of the
interaction of reduction and permutative conversions.Fundação para a Ciência e a Tecnologia (FCT
Permutability in proof terms for intuitionistic sequent calculus with cuts
This paper gives a comprehensive and coherent view on permutability in the intuitionistic sequent calculus with cuts. Specifically we show that, once permutability is packaged into appropriate global reduction procedures, it organizes the internal structure of the system and determines fragments with computational interest, both for the computation-as-proof-normalization and the computation-as-proof-search paradigms. The vehicle of the study is a lambda-calculus of multiary proof terms with generalized application, previously developed by the authors (the paper argues this system represents the simplest fragment of ordinary sequent calculus that does not fall into mere natural deduction). We start by adapting to our setting the concept of normal proof, developed by Mints, Dyckhoff, and Pinto, and by defining natural proofs, so that a proof is normal iff it is natural and cut-free. Natural proofs form a subsystem with a transparent Curry-Howard interpretation (a kind of formal vector notation for lambda-terms with vectors consisting of lists of lists of arguments), while searching for normal proofs corresponds to a slight relaxation of focusing (in the sense of LJT). Next, we define a process of permutative conversion to natural form, and show that its combination with cut elimination gives a concept of normalization for the sequent calculus. We derive a systematic picture of the full system comprehending a rich set of reduction procedures (cut elimination, flattening, permutative conversion, normalization, focalization), organizing the relevant subsystems and the important subclasses of cut-free, normal, and focused proofs.Partially financed by FCT through project UID/MAT/00013/2013, and by COST action CA15123 EUTYPES. The first and the last authors were partially financed by Fundação para
a Ciência e a Tecnologia (FCT) through project UID/MAT/00013/2013. The first author got
financial support by the COST action CA15123 EUTYPES.info:eu-repo/semantics/publishedVersio