Permutative Conversions Research Articles

The bang calculus revisited

Call-by-Push-Value (CBPV) is a programming paradigm subsuming both Call-by-Name (CBN) and Call-by-Value (CBV) semantics. The essence of this paradigm is captured by the Bang Calculus, a term language connecting CBPV and Linear Logic.This paper presents a revisited version of the Bang Calculus, called λ!, enjoying some important properties missing in the original formulation. Indeed, the new calculus integrates permutative conversions to unblock value redexes while preserving confluence. A second contribution is related to non-idempotent types. We provide a quantitative type system for our λ!-calculus, giving upper bounds to the length of the reduction to normal form plus its size. We also explore the properties of this type system with respect to CBN/CBV translations. Last but not least, the quantitative system is refined to a tight one, which transforms the previous upper bound into two independent exact measures for the reduction length and the normal form size respectively.

The Naturality of Natural Deduction

Developing a suggestion by Russell, Prawitz showed how the usual natural deduction inference rules for disjunction, conjunction and absurdity can be derived using those for implication and the second order quantifier in propositional intuitionistic second order logic NI $$^2$$ . It is however well known that the translation does not preserve the relations of identity among derivations induced by the permutative conversions and immediate expansions for the definable connectives, at least when the equational theory of NI $$^2$$ is assumed to consist only of $$\beta $$ - and $$\eta $$ -equations. On the basis of the categorial interpretation of NI $$^2$$ , we introduce a new class of equations expressing what in categorial terms is a naturality condition satisfied by the transformations interpreting NI $$^2$$ -derivations. We show that the Russell–Prawitz translation does preserve identity of proof with respect to the enriched system by highlighting the fact that naturality corresponds to a generalized permutation principle. Finally we sketch how these results could be used to investigate the properties of connectives definable in the framework of higher-level rules.

Compositional Z: Confluence Proofs for Permutative Conversion

This paper gives new confluence proofs for several lambda calculi with permutation-like reduction, including lambda calculi corresponding to intuitionistic and classical natural deduction with disjunction and permutative conversions, and a lambda calculus with explicit substitutions. For lambda calculi with permutative conversion, naive parallel reduction technique does not work, and (if we consider untyped terms, and hence we do not use strong normalization) traditional notion of residuals is required as Ando pointed out. This paper shows that the difficulties can be avoided by extending the technique proposed by Dehornoy and van Oostrom, called the Z theorem: existence of a mapping on terms with the Z property concludes the confluence. Since it is still hard to directly define a mapping with the Z property for the lambda calculi with permutative conversions, this paper extends the Z theorem to compositional functions, called compositional Z, and shows that we can adopt it to the calculi.

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.

Strong normalization results by translation

We prove the strong normalization of full classical natural deduction (i.e. with conjunction, disjunction and permutative conversions) by using a translation into the simply typed λ μ -calculus. We also extend Mendler’s result on recursive equations to this system.

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.

Strong normalization of classical natural deduction with disjunctions

This paper proves the strong normalization of classical natural deduction with disjunction and permutative conversions, by using CPS-translation and augmentations. Using them, this paper also proves the strong normalization of classical natural deduction with general elimination rules for implication and conjunction, and their permutative conversions. This paper also proves that natural deduction can be embedded into natural deduction with general elimination rules, strictly preserving proof normalization.

Strong normalization proofs by CPS-translations

This paper proposes a new proof method for strong normalization of classical natural deduction and calculi with control operators. For this purpose, we introduce a new CPS-translation, continuation and garbage passing style ( CGPS ) translation. We show that this CGPS-translation method gives simple proofs of strong normalization of λ μ → ∧ ∨ ⊥ , which is introduced in [P. de Groote, Strong normalization of classical natural deduction with disjunction, in: S. Abramsky (Ed.), Typed Lambda Calculi and Applications, 5th International Conference, TLCA 2001, in: Lecture Notes in Comput. Sci., vol. 2044, Springer, Berlin, 2001, pp. 182–196] by de Groote and corresponds to the classical natural deduction with disjunctions and permutative conversions.

Second order permutative conversions with Prawitz's strong validity

A clear and complete proof of strong normalization of second order natural deduction with permutative conversions is given by using Prawitz’s strong validity. This paper completes Prawitz’s original proof.

A simple proof of second-order strong normalization with permutative conversions

A simple and complete proof of strong normalization for first- and second-order intuitionistic natural deduction including disjunction, first-order existence and permutative conversions is given. The paper follows the Tait–Girard approach via computability predicates (reducibility) and saturated sets. Strong normalization is first established for a set of conversions of a new kind, then deduced for the standard conversions. Difficulties arising for disjunction are resolved using a new logic where disjunction is restricted to atomic formulas.

Continuous normalization for the lambda-calculus and Gödel’s [formula omitted

Building on previous work by Mints, Buchholz and Schwichtenberg, a simplified version of continuous normalization for the untyped λ -calculus and Gödel’s T is presented and analysed in the coalgebraic framework of non-wellfounded terms with so-called repetition constructors. The primitive recursive normalization function is uniformly continuous w.r.t. the natural metric on non-wellfounded terms. Furthermore, the number of necessary repetition constructors is locally related to the number of reduction steps needed to reach the normal form (as represented by the Böhm tree) and its size. It is also shown how continuous normal forms relate to derivations of strong normalizability in the typed λ -calculus and how this leads to new bounds for the sum of the height of the reduction tree and the size of the normal form. Finally, the methods are extended to an infinitary λ -calculus with ω -rule and permutative conversions and this is used to derive a strong form of normalization for an iterative version of Gödel’s system T , leading to a value table semantics for number-theoretic functions.

Short proofs of normalization for the simply- typed λ-calculus, permutative conversions and Gödel's T

Inductive characterizations of the sets of terms, the subset of strongly normalizing terms and normal forms are studied in order to reprove weak and strong normalization for the simply-typed λ-calculus and for an extension by sum types with permutative conversions. The analogous treatment of a new system with generalized applications inspired by generalized elimination rules in natural deduction, advocated by von Plato, shows the flexibility of the approach which does not use the strong computability/candidate style a la Tait and Girard. It is also shown that the extension of the system with permutative conversions by η-rules is still strongly normalizing, and likewise for an extension of the system of generalized applications by a rule of ``immediate simplification''. By introducing an infinitely branching inductive rule the method even extends to Godel's T.

Proof Reflection in Coq

We formalize natural deduction for first-order logic in the proof assistant Coq, using de Bruijn indices for variable binding. The main judgment we model is of the form Γvd [:] φ, stating that d is a proof term of formula φ under hypotheses Γ; it can be viewed as a typing relation by the Curry–Howard isomorphism. This relation is proved sound with respect to Coq's native logic and is amenable to the manipulation of formulas and of derivations. As an illustration, we define a reduction relation on proof terms with permutative conversions and prove the property of subject reduction.

Natural deduction with general elimination rules

The structure of derivations in natural deduction is analyzed through isomorphism with a suitable sequent calculus, with twelve hidden convertibilities revealed in usual natural deduction. A general formulation of conjunction and implication elimination rules is given, analogous to disjunction elimination. Normalization through permutative conversions now applies in all cases. Derivations in normal form have all major premisses of elimination rules as assumptions. Conversion in any order terminates.

Termination of permutative conversions in intuitionistic Gentzen calculi

It is shown that permutative conversions terminate for the cut-free intuitionistic Gentzen (i.e. sequent) calculus; this proves a conjecture by Dyckhoff and Pinto. The main technical tool is a term notation for derivations in Gentzen calculi. These terms may be seen as λ-terms with explicit substitution, where the latter corresponds to the left introduction rules.