In this paper we introduce a term calculus ${\cal B}$ which adds to the affine $λ$-calculus with pairing a new construct allowing for a restricted form of contraction. We obtain a Curry-Howard correspondence between ${\cal B}$ and the sub-structural logical system which we call "minimal Łukasiewicz logic", also known in the literature as the logic of hoops (a generalisation of MV-algebras). This logic lies strictly in between affine minimal logic and standard minimal logic. We prove that ${\cal B}$ is strongly normalising and has the Church-Rosser property. We also give examples of terms in ${\cal B}$ corresponding to some important derivations from our work and the literature. Finally, we discuss the relation between normalisation in ${\cal B}$ and cut-elimination for a Gentzen-style formulation of minimal Łukasiewicz logic.
This is a survey of λ-calculi that, through the Curry-Howard isomorphism, correspond to constructive modal logics. We cover the prehistory of the subject and then concentrate on the developments that took place in the 1990s and early 2000s. We discuss logical aspects, modal λ-calculi, and their categorical semantics. The logics underlying the calculi surveyed are constructive versions of K, K4, S4, and LTL.
Dummett's logic LC is intuitionistic logic extended with Dummett's axiom: for every two statements the first implies the second or the second implies the first. We present a natural deduction and a Curry-Howard correspondence for first-order and second-order Dummett's logic. We add to the lambda calculus an operator which represents, from the viewpoint of programming, a mechanism for representing parallel computations and communication between them, and from the viewpoint of logic, Dummett's axiom. We prove that our typed calculus is normalizing and show that proof terms for existentially quan…
We present a variant of the calculus of deductive systems developed in (Lambek 1972, 1974), and give a generalization of the Curry-Howard-Lambek theorem giving an equivalence between the category of typed lambda-calculi and the category of cartesian closed categories and exponential-preserving morphisms that leverages the theory of generalized categories (Schoenbaum 2016). We discuss potential applications and extensions.
We add to intuitionistic logic infinitely many classical disjunctive tautologies and use the Curry--Howard correspondence to obtain typed concurrent $λ$-calculi; each of them features a specific communication mechanism, including broadcasting and cyclic message-exchange, and enhanced expressive power with respect to the $λ$-calculus. Moreover they all implement forms of code mobility. Our results provide a first concurrent computational interpretation for many propositional intermediate logics, classical logic included.
We show that an intuitionistic version of counting propositional logic corresponds, in the sense of Curry and Howard, to an expressive type system for the probabilistic event lambda-calculus, a vehicle calculus in which both call-by-name and call-by-value evaluation of discrete randomized functional programs can be simulated. Remarkably, proofs (respectively, types) do not only guarantee that validity (respectively, termination) holds, but also reveal the underlying probability. We finally show that by endowing the type system with an intersection operator, one obtains a system precisely captu…
Intuitionistic first-order logic extended with a restricted form of Markov's principle is constructive and admits a Curry-Howard correspondence, as shown by Herbelin. We provide a simpler proof of that result and then we study intuitionistic first-order logic extended with unrestricted Markov's principle. Starting from classical natural deduction, we restrict the excluded middle and we obtain a natural deduction system and a parallel Curry-Howard isomorphism for the logic. We show that proof terms for existentially quantified formulas reduce to a list of individual terms representing all possi…
This paper introduces a natural deduction calculus for intuitionistic logic of belief $\mathsf{IEL}^{-}$ which is easily turned into a modal $λ$-calculus giving a computational semantics for deductions in $\mathsf{IEL}^{-}$. By using that interpretation, it is also proved that $\mathsf{IEL}^{-}$ has good proof-theoretic properties. The correspondence between deductions and typed terms is then extended to a categorical semantics for identity of proofs in $\mathsf{IEL}^{-}$ showing the general structure of such a modality for belief in an intuitionistic framework.
The Curry-Howard Correspondence has a long history, and still is a topic of active research. Though there are extensive investigations into the subject, there doesn't seem to be a definitive formulation of this result in the level of generality that it deserves. In the current work, we introduce the formalism of p-institutions that could unify previous aproaches. We restate the tradicional correspondence between typed $λ$-calculi and propositional logics inside this formalism, and indicate possible directions in which it could foster new and more structured generalizations. Furthermore, we ind…
This reports introduces a novel sound and complete semantics for first order intuitionistic logic, in the framework of category theory and by the computational interpretation of the logic based on the so-called Curry-Howard isomorphism. Aside, a sound and complete semantics for the corresponding lambda-calculus is derived, too. This semantics extends, in a way, the more traditional meanings given by Heyting categories, the topos-theoretic interpretation, and Kripke models. The feature which justifies the introduction of this novel semantics is the fact that it is 'point-free', i.e., there is n…
Faculty of Technical Sciences, University of Novi Sad（诺维萨德大学技术科学学院）；Mathematical Institute of the Serbian Academy of Sciences and Arts（塞尔维亚科学院数学研究所）；School of Computer Science, University of Birmingham（伯明翰大学计算机学院）
The stack calculus is a functional language in which is in a Curry-Howard correspondence with classical logic. It enjoys confluence but, as well as Parigot's lambda-mu, does not admit the Bohm Theorem, typical of the lambda-calculus. We present a simple extension of stack calculus which is for the stack calculus what Saurin's Lambda-mu is for lambda-mu.
We offer a simple graphical representation for proofs of intuitionistic logic, which is inspired by proof nets and interaction nets (two formalisms originating in linear logic). This graphical calculus of proofs inherits good features from each, but is not constrained by them. By the Curry-Howard isomorphism, the representation applies equally to the lambda calculus, offering an alternative diagrammatic representation of functional computations.
Session type systems have been given logical foundations via Curry-Howard correspondences based on both intuitionistic and classical linear logic. The type systems derived from the two logics enforce communication correctness on the same class of pi-calculus processes, but they are significantly different. Caires, Pfenning and Toninho informally observed that, unlike the classical type system, the intuitionistic type system enforces locality for shared channels, i.e. received channels cannot be used for replicated input. In this paper, we revisit this observation from a formal standpoint. We d…
We introduce a Curry-Howard correspondence for a large class of intermediate logics characterized by intuitionistic proofs with non-nested applications of rules for classical disjunctive tautologies (1-depth intermediate proofs). The resulting calculus, we call it $λ_{\parallel}$, is a strongly normalizing parallel extension of the simply typed $λ$-calculus. Although simple, the $λ_{\parallel}$ reduction rules can model arbitrary process network topologies, and encode interesting parallel programs ranging from numeric computation to algorithms on graphs.
We introduce a functional calculus with simple syntax and operational semantics in which the calculi introduced so far in the Curry-Howard correspondence for Classical Logic can be faithfully encoded. Our calculus enjoys confluence without any restriction. Its type system enforces strong normalization of expressions and it is a sound and complete system for full implicational Classical Logic. We give a very simple denotational semantics which allows easy calculations of the interpretation of expressions.
The Curry-Howard correspondence is often described as relating proofs (in intutionistic natural deduction) to programs (terms in simply-typed lambda calculus). However this narrative is hardly a perfect fit, due to the computational content of cut-elimination and the logical origins of lambda calculus. We revisit Howard's work and interpret it as an isomorphism between a category of proofs in intuitionistic sequent calculus and a category of terms in simply-typed lambda calculus. In our telling of the story the fundamental duality is not between proofs and programs but between local (sequent c…
Beloved Curry--Howard correspondence tells that types are intuitionistic propositions, and in constructive math, a proof of proposition can be seen as some kind of a construction, or witness, conveying the information of the proposition. We demonstrate how useful this point of view is as the guiding principle for developing dependently-typed programs.
Propositional Gödel logic extends intuitionistic logic with the non-constructive principle of linearity $A\rightarrow B\ \lor\ B\rightarrow A$. We introduce a Curry-Howard correspondence for this logic and show that a particularly simple natural deduction calculus can be used as a typing system. The resulting functional language enriches the simply typed lambda calculus with a synchronous communication mechanism between parallel processes. Our normalization proof employs original termination arguments and sophisticated proof transformations with a meaningful computational reading. Our results …
This paper provides a call-by-name and a call-by-value term calculus, both of which have a Curry-Howard correspondence to the box fragment of the intuitionistic modal logic IK. The strong normalizability and the confluency of the calculi are shown. Moreover, we define a CPS transformation from the call-by-value calculus to the call-by-name calculus, and show its soundness and completeness.
Using the proof-program (Curry-Howard) correspondence, we give a new method to obtain models of ZF and relative consistency results in set theory. We show the relative consistency of ZF + DC + there exists a sequence of subsets of R the cardinals of which are strictly decreasing + other similar properties of R. These results seem not to have been previously obtained by forcing.
A system of session types is introduced as induced by a Curry Howard correspondence applied to Bounded Linear Logic, and then extending the thus obtained type system with probabilistic choices and ground types. The obtained system satisfies the expected properties, like subject reduction and progress, but also unexpected ones, like a polynomial bound on the time needed to reduce processes. This makes the system suitable for modelling experiments and proofs from the so-called computational model of cryptography.
We present a calculus providing a Curry-Howard correspondence to classical logic represented in the sequent calculus with explicit structural rules, namely weakening and contraction. These structural rules introduce explicit erasure and duplication of terms, respectively. We present a type system for which we prove the type-preservation under reduction. A mutual relation with classical calculus featuring implicit structural rules has been studied in detail. From this analysis we derive strong normalisation property.
University of North Texas（北德克萨斯大学）；Topos Institute（Topos研究所）
The problem we want to solve is how to generate all theorems of a given size in the implicational fragment of propositional intuitionistic linear logic. We start by filtering for linearity the proof terms associated by our Prolog-based theorem prover for Implicational Intuitionistic Logic. This works, but using for each formula a PSPACE-complete algorithm limits it to very small formulas. We take a few walks back and forth over the bridge between proof terms and theorems, provided by the Curry-Howard isomorphism, and derive step-by-step an efficient algorithm requiring a low polynomial effort …
We present an algorithm for deriving a spatial-behavioral type system from a formal presentation of a computational calculus. Given a 2-monad Calc: Catv$\to$ Cat for the free calculus on a category of terms and rewrites and a 2-monad BoolAlg for the free Boolean algebra on a category, we get a 2-monad Form = BoolAlg + Calc for the free category of formulae and proofs. We also get the 2-monad BoolAlg $\circ$ Calc for subsets of terms. The interpretation of formulae is a natural transformation $\interp{-}$: Form $\Rightarrow$ BoolAlg $\circ$ Calc defined by the units and multiplications of the m…
We present a generic framework that facilitates object level reasoning with logics that are encoded within the Higher Order Logic theorem proving environment of HOL Light. This involves proving statements in any logic using intuitive forward and backward chaining in a sequent calculus style. It is made possible by automated machinery that take care of the necessary structural reasoning and term matching automatically. Our framework can also handle type theoretic correspondences of proofs, effectively allowing the type checking and construction of computational processes via proof. We demonstra…
This paper establishes the normalisation of natural deduction or lambda calculus formulation of Intuitionistic Non Commutative Logic --- which involves both commutative and non commutative connectives. This calculus first introduced by de Groote and as opposed to the classical version by Abrusci and Ruet admits a full entropy which allow order to be relaxed into any suborder. Our result also includes, as a special case, the normalisation of natural deduction the Lambek calculus with product, which is unsurprising but yet unproved. Regarding Intuitionistic Non Commutative Logic with full entrop…
Let T be Goedel's system of primitive recursive functionals of finite type in the lambda formulation. We define by constructive means using recursion on nested multisets a multivalued function I from the set of terms of T into the set of natural numbers such that if a term a reduces to a term b and if a natural number I(a) is assigned to a then a natural number I(b) can be assigned to b such that I(a) is greater than I(b). The construction of I is based on Howard's 1970 ordinal assignment for T and Weiermann's 1996 treatment of T in the combinatory logic version. As a corollary we obtain an op…
Adding rewriting to a proof assistant based on the Curry-Howard isomorphism, such as Coq, may greatly improve usability of the tool. Unfortunately adding an arbitrary set of rewrite rules may render the underlying formal system undecidable and inconsistent. While ways to ensure termination and confluence, and hence decidability of type-checking, have already been studied to some extent, logical consistency has got little attention so far. In this paper we show that consistency is a consequence of canonicity, which in turn follows from the assumption that all functions defined by rewrite rules …
ML5 is a programming language for spatially distributed computing, based on a Curry-Howard correspondence with the modal logic S5. Despite being designed by a correspondence with S5 modal logic, the ML5 programming language differs from the logic in several ways. In this paper, we explain these discrepancies between ML5 and S5 by translating ML5 into a slightly different logic: intuitionistic S5 extended with a lax modality that encapsulates effectful computations in a monad. This translation both explains the existing ML5 design and suggests some simplifications and generalizations. We have f…
One leading question with respect to Bi-intuitionistic logic (BINT) is, what does BINT look like across the three arcs -- logic, typed $λ$-calculi, and category theory -- of the Curry-Howard-Lambek correspondence? Categorically, BINT can be seen as a mixing of two worlds: the first being intuitionistic logic (IL), which is modeled by a cartesian closed category, and the second being the dual to intuitionistic logic called cointuitionistic logic (coIL), which is modeled by a cocartesian coclosed category. Crolard showed that combining these two categories into the same category results in it de…
We propose a logic of interactive proofs as a framework for an intuitionistic foundation for interactive computation, which we construct via an interactive analog of the Goedel-McKinsey-Tarski-Artemov definition of Intuitionistic Logic as embedded into a classical modal logic of proofs, and of the Curry-Howard isomorphism between intuitionistic proofs and typed programs. Our interactive proofs effectuate a persistent epistemic impact in their intended communities of peer reviewers that consists in the induction of the (propositional) knowledge of their proof goal by means of the (individual) k…
In reductive proof search, proofs are naturally generalized by solutions, comprising all possibly infinite structures generated by locally correct, bottom-up application of inference rules. We propose an extension of the Curry-Howard paradigm of representation, from proofs to solutions: to represent solutions by possibly infinite terms of a dedicated lambda-calculus. This new, comprehensive approach to proof search is exemplified with the sequent calculus LJT for intuitionistic implication logic. A finitary representation is proposed, comprising lambda-terms extended with a formal greatest fix…
