José Espírito Santo

An Isomorphism between a Fragment of Sequent Calculus and an Extension of Natural Deduction

Variants of Herbelins λ-calculus, here collectively named Herbelin calculi, have proved useful both in foundational studies and as internal languages for the efficient representation of λ-terms. An obvious requirement of both these two kinds of applications is a clear understanding of the relationship between cut-elimination in Herbelin calculi and normalisation in the λ-calculus. However,this understanding is not complete so far. Our previous work showed that λ is isomorphic to a Herbelin calculus,here named λP, only admitting cuts that are both left- and right-permuted. In this paper we consider a generalisation λPh admitting any kind of right-permuted cut. We show that there is a natural deduction system λNh which conservatively extends λ and is isomorphic to λPh. The idea is to build in the natural deduction system a distinction between applicative term and application, together with a distinction between head and tail application. This is suggested by examining how natural deduction proofs are mapped to sequent calculus derivations according to a translation due to Prawitz. In addition to β, λNh includes a reduction rule that mirrors left permutation of cuts, but without performing any append of lists/spines.

Characterising Strongly Normalising Intuitionistic Terms

This paper gives a characterisation, via intersection types, of the strongly normalising proof-terms of an intuitionistic sequent calculus where LJ easily embeds. The soundness of the typing system is reduced to that of a well known typing system with intersection types for the ordinary λ-calculus. The completeness of the typing system is obtained from subject expansion at root position. Next we use our result to analyze the characterisation of strong normalisability for three classes of intuitionistic terms: ordinary λ-terms, ΛJ-terms λ-terms with generalised application, and λx-terms λ-terms with explicit substitution. We explain via our system why the type systems in the natural deduction format for ΛJ and λx known from the literature contain extra, exceptional rules for typing generalised application or substitution; and we show a new characterisation of the β-strongly normalising λ-terms, as a corollary to a PSN-result, relating the λ-calculus and the intuitionistic sequent calculus. Finally, we obtain variants of our characterisation by restricting the set of assignable types to sub-classes of intersection types, notably strict types. In addition, the known characterisation of the β-strongly normalising λ-terms in terms of assignment of strict types follows as an easy corollary of our results.

The λ -Calculus and the Unity of Structural Proof Theory

In the context of intuitionistic implicational logic, we achieve a perfect correspondence (technically an isomorphism) between sequent calculus and natural deduction, based on perfect correspondences between left-introduction and elimination, cut and substitution, and cut-elimination and normalisation. This requires an enlarged system of natural deduction that refines von Plato’s calculus. It is a calculus with modus ponens and primitive substitution; it is also a “coercion calculus”, in the sense of Cervesato and Pfenning. Both sequent calculus and natural deduction are presented as typing systems for appropriate extensions of the λ-calculus. The whole difference between the two calculi is reduced to the associativity of applicative terms (sequent calculus = right associative, natural deduction = left associative), and in fact the achieved isomorphism may be described as the mere inversion of that associativity.The novel natural deduction system is a “multiary” calculus, because “applicative terms” may exhibit a list of several arguments. But the combination of “multiarity” and left-associativity seems simply wrong, leading necessarily to non-local reduction rules (reason: normalisation, like cut-elimination, acts at the head of applicative terms, but natural deduction focuses at the tail of such terms). A solution is to extend natural deduction even further to a calculus that unifies sequent calculus and natural deduction, based on the unification of cut and substitution. In the unified calculus, a sequent term behaves like in the sequent calculus, whereas the reduction steps of a natural deduction term are interleaved with explicit steps for bringing heads to focus. A variant of the calculus has the symmetric role of improving sequent calculus in dealing with tail-active permutative conversions.In the context of intuitionistic implicational logic, we achieve a perfect correspondence (technically an isomorphism) between sequent calculus and natural deduction, based on perfect correspondences between left-introduction and elimination, cut and substitution, and cut-elimination and normalisation. This requires an enlarged system of natural deduction that refines von Plato’s calculus. It is a calculus with modus ponens and primitive substitution; it is also a “coercion calculus”, in the sense of Cervesato and Pfenning. Both sequent calculus and natural deduction are presented as typing systems for appropriate extensions of the λ-calculus. The whole difference between the two calculi is reduced to the associativity of applicative terms (sequent calculus = right associative, natural deduction = left associative), and in fact the achieved isomorphism may be described as the mere inversion of that associativity. The novel natural deduction system is a “multiary” calculus, because “applicative terms” may exhibit a list of several arguments. But the combination of “multiarity” and left-associativity seems simply wrong, leading necessarily to non-local reduction rules (reason: normalisation, like cut-elimination, acts at the head of applicative terms, but natural deduction focuses at the tail of such terms). A solution is to extend natural deduction even further to a calculus that unifies sequent calculus and natural deduction, based on the unification of cut and substitution. In the unified calculus, a sequent term behaves like in the sequent calculus, whereas the reduction steps of a natural deduction term are interleaved with explicit steps for bringing heads to focus. A variant of the calculus has the symmetric role of improving sequent calculus in dealing with tail-active permutative conversions.

Confluence and Strong Normalisation of the Generalised Multiary λ-Calculus

In a previous work we introduced the generalised multiaryλ-calculus λ J m , an extension of the λ-calculus where functions can be applied to lists of arguments (a feature which we call “multiarity”) and encompassing “generalised” eliminations of von Plato. In this paper we prove confluence and strong normalisation of the reduction relations of λ J m . Proofs of these results lift corresponding ones obtained by Joachimski and Matthes for the system Λ J. Such lifting requires the study of how multiarity and some forms of generality can express each other. This study identifies a variant of λJ, and another system isomorphic to it, as being the subsystems of λ J m with, respectively, minimal and maximal use of multiarity. We argue then that λ J m is the system with the right use of multiarity.

Monadic translation of classical sequent calculus

We study monadic translations of the call-by-name (cbn) and call-by-value (cbv) fragments of the classical sequent calculus λμ˜ due to Curien and Herbelin, and give modular and syntactic proofs of strong normalisation. The target of the translations is a new meta-language for classical logic, named monadic λμ. This language is a monadic reworking of Parigot’s λμ-calculus, where the monadic binding is confined to commands, thus integrating the monad with the classical features. Also, its μ-reduction rule is replaced by a rule expressing the interaction between monadic binding and μ-abstraction. Our monadic translations produce very tight simulations of the respective fragments of λμ˜ within monadic λμ, with reduction steps of λμ˜ being translated in a 1–1 fashion, except for β steps, which require two steps. The monad of monadic λμ can be instantiated to the continuations monad so as to ensure strict simulation of monadic λμ within simply typed λ-calculus with β -a ndη-reduction. Through strict simulation, the strong normalisation of simply typed λ -calculus is inherited by monadicλμ, and then by cbn and cbv λμ˜, thus reproving strong normalisation in an elementary syntactical way for these fragments of λμ˜, and establishing it for our new calculus. These results extend to second-order logic, with polymorphic λ-calculus as the target, giving new strong normalisation results for classical second-order logic in sequent calculus style. CPS translations of cbn and cbv λμ˜ with the strict simulation property are obtained by composing our monadic translations with the continuations-monad instantiation. In an appendix to the paper, we investigate several refinements of the continuations-monad instantiation in order to obtain in a modular way improvements of the CPS translations enjoying extra properties like simulation by cbv β-reduction or reduction of administrative redexes at compile time.

Towards a canonical classical natural deduction system

Abstract This paper studies a new classical natural deduction system, presented as a typed calculus named λ μ let . It is designed to be isomorphic to Curien and Herbelinʼs λ ¯ μ μ ˜ -calculus, both at the level of proofs and reduction, and the isomorphism is based on the correct correspondence between cut (resp. left-introduction) in sequent calculus, and substitution (resp. elimination) in natural deduction. It is a combination of Parigotʼs λμ-calculus with the idea of “coercion calculus” due to Cervesato and Pfenning, accommodating let-expressions in a surprising way: they expand Parigotʼs syntactic class of named terms. This calculus and the mentioned isomorphism Θ offer three missing components of the proof theory of classical logic: a canonical natural deduction system; a robust process of “read-back” of calculi in the sequent calculus format into natural deduction syntax; a formalization of the usual semantics of the λ ¯ μ μ ˜ -calculus, that explains co-terms and cuts as, respectively, contexts and hole-filling instructions. λ μ let is not yet another classical calculus, but rather a canonical reflection in natural deduction of the impeccable treatment of classical logic by sequent calculus; and Θ provides the “read-back” map and the formalized semantics, based on the precise notions of context and “hole-expression” provided by λ μ let . We use “read-back” to achieve a precise connection with Parigotʼs λμ, and to derive λ-calculi for call-by-value combining control and let-expressions in a logically founded way. Finally, the semantics Θ, when fully developed, can be inverted at each syntactic category. This development gives us license to see sequent calculus as the semantics of natural deduction; and uncovers a new syntactic concept in λ ¯ μ μ ˜ (“co-context”), with which one can give a new definition of η-reduction.

The polarized λ-calculus

This research was financed by Portuguese Funds through FCT Fundac¸ao para a Ci ˜ encia ˆ e a Tecnologia, within the Project UID/MAT/00013/2013.

A note on strong normalization in classical natural deduction

This research was financed by Portuguese Funds through FCT Fundacao para a Ciencia e a Tecnologia, within the Project UID/MAT/00013/2013.

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, called generalized multiary λ-calculus, is a rich extension of the λ-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 to form lists of arguments) and generality (the ability to prescribe a kind of continuation). The system integrates in a modular way the multiary λ-calculus and an isomorphic copy of the λ-calculus with generalized application, Λ J (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 μ-rule, typical of the multiary setting, or rules for cut-elimination, which enlarge the ordinary β-rule. This article establishes the metatheory of the system, with emphasis on the role of the μ-rule, and including a study of the interaction of reduction and permutative conversions.