Curry-Howard Research Articles

Comparing session type systems derived from linear logic

Session types are a typed approach to message-passing concurrency, where types describe sequences of intended exchanges over channels. Session type systems have been given strong logical foundations via Curry-Howard correspondences with linear logic, a resource-aware logic that naturally captures structured interactions. These logical foundations provide an elegant framework to specify and (statically) verify message-passing processes.In this paper, we rigorously compare different type systems for concurrency derived from the Curry-Howard correspondence between linear logic and session types. We address the main divide between these type systems: the classical and intuitionistic presentations of linear logic. Over the years, these presentations have given rise to separate research strands on logical foundations for concurrency; the differences between their derived type systems have only been addressed informally.To formally assess these differences, we develop πULL, a session type system that encompasses type systems derived from classical and intuitionistic interpretations of linear logic. Based on a fragment of Girard's Logic of Unity, πULL provides a basic reference framework: we compare existing session type systems by characterizing fragments of πULL that coincide with classical and intuitionistic formulations. We analyze the significance of our characterizations by considering the locality principle (enforced by intuitionistic interpretations but not by classical ones) and forms of process composition induced by the interpretations.

SIGLOG Monthly 251: July 2024

The 2024 Alonzo Church Award for Outstanding Contributions to Logic and Computation is presented jointly to Thomas Ehrhard and Laurent Regnier for giving a logical and computational account of differentiation, bringing Taylor expansion to the Curry-Howard correspondence, which had a major impact on programming language semantics.

Meaning and identity of proofs in a bilateralist setting: A two-sorted typed Lambda-calculus for proofs and refutations

Abstract In this paper, I will develop a $\lambda $-term calculus, $\lambda ^{2Int}$, for a bi-intuitionistic logic and discuss its implications for the notions of sense and denotation of derivations in a bilateralist setting. Thus, I will use the Curry–Howard correspondence, which has been well-established between the simply typed $\lambda $-calculus and natural deduction systems for intuitionistic logic, and apply it to a bilateralist proof system displaying two derivability relations, one for proving and one for refuting. The basis will be the natural deduction system of Wansing’s bi-intuitionistic logic 2Int, which I will turn into a term-annotated form. Therefore, we need a type theory that extends to a two-sorted typed $\lambda $-calculus. I will present such a term-annotated proof system for 2Int and prove a Dualization Theorem relating proofs and refutations in this system. On the basis of these formal results, I will argue that this gives us interesting insights into questions about sense and denotation as well as synonymy and identity of proofs from a bilateralist point of view.

Reification, Curry-Howard Correspondence, and Didactical Consequences

The roles of languages, processes, and objects in mathematical thinking have led to many theories, yet no consistent big picture has evolved from this. This paper puts forth the hypothesis that the Curry-Howard correspondence from computer science and the theories it is built on provide a unification framework. This correspondence asserts that (formal) proofs and programs (in functional programming languages) do not only have some similarities, but can, at least if formalized in an appropriate way, be mapped to each other by an isomorphism such that proofs are programs and vice versa. Moreover, objects can be realized as function evaluation strategies, and this provides a model of the reification process. The paper explores all this and discusses the didactical relevance; especially, the reification theories are revisited. Computer-based realizations of concepts are used as a tool to show the consistency of ideas and the practicability of concepts.

Limitations of and Lessons from the Learning of Large Language Models

It is argued that the Curry-Howard correspondence for classical logic implies limitations for logical reasoning that can be learned and performed by large language models. The correspondence establishes an isomorphism between proofs in logic and programs in functional typed lambda calculus. While intuitionistic logic maps to a version of lambda calculus that can be carried out in a local way, i.e., considering local parts of the code in isolation, the version of lambda calculus that corresponds to classical logic requires non-local relations – and this non-locality cannot be learned by large language models due to their restriction to investigate a relative short sequence of tokens. A possible way to go beyond this limitation is sketched as well. Implications for other areas are investigated as well.

UNDER LOCK AND KEY: A PROOF SYSTEM FOR A MULTIMODAL LOGIC

AbstractWe present a proof system for a multimode and multimodal logic, which is based on our previous work on modal Martin-Löf type theory. The specification of modes, modalities, and implications between them is given as a mode theory, i.e., a small 2-category. The logic is extended to a lambda calculus, establishing a Curry–Howard correspondence.

Natural deduction calculi for classical and intuitionistic S5

We propose an indexed natural deduction system for the modal logic , ideally following Wansing's previous work in the context of tableaux sequents. The system, given both in the classical and intuitionistic versions (called and respectively), is designed to match as much as possible the structure and properties of the standard system of natural deduction for first-order logic, exploiting the formal analogy between modalities and quantifiers. We study a (syntactical) normalization theorem for both and and its main consequences, the sub-formula principle and the consistency theorem. In particular, we propose an intuitionistic encoding of classical (via a suitable extension of the Gödel translation for first-order classical logic). Moreover, via the BHK interpretation of intuitionistic proofs, we propose a suitable Curry–Howard isomorphism for . By translation into the natural deduction system given by Galmiche and Salhi in [(2010b). Label-free proof systems for intuitionistic modal logic is5. In E. M. Clarke & A. Voronkov (Eds.), Logic for programming, artificial intelligence, and reasoning (pp. 255–271). Springer Berlin Heidelberg.], we prove the equivalence of w.r.t. an Hilbert-style axiomatization of . However, when considering the sheer provability of labelled formulas, our system is comparable to the one presented by Simpson in [(1993). The proof theory and semantics of intuitionistic modal logic [PhD thesis], University of Edinburgh, UK.]. Nevertheless, it remains uncertain whether it is feasible to establish a translation between the corresponding derivations.

Variations and interpretations of naturality in call-by-name lambda-calculi with generalized applications

In the context of intuitionistic sequent calculus, “naturality” means permutation-freeness (the terminology is essentially due to Mints). We study naturality in the context of the lambda-calculus with generalized applications and its multiary extension, to cover, under the Curry-Howard correspondence, proof systems ranging from natural deduction (with and without general elimination rules) to a fragment of sequent calculus with an iterable left-introduction rule, and which can still be recognized as a call-by-name lambda-calculus. In this context, naturality consists of a certain restricted use of generalized applications. We consider the further restriction obtained by the combination of naturality with normality w.r.t. the commutative conversion engendered by generalized applications. This combination sheds light on the interpretation of naturality as a vectorization mechanism, allowing a multitude of different ways of structuring lambda-terms, and the structuring of a multitude of interesting fragments of the systems under study. We also consider a relaxation of naturality, called weak naturality: this not only brings similar structural benefits, but also suggests a new “weak” system of natural deduction with generalized applications which is exempt from commutative conversions. In the end, we use all of this evidence as a stepping stone to propose a computational interpretation of generalized application (whether multiary or not, and without any restriction): it includes, alongside the argument(s) for the function, a general list – a new, very general, vectorization mechanism, that structures the continuation of the computation.

A bunch of sessions: a propositions-as-sessions interpretation of bunched implications in channel-based concurrency

The emergence of propositions-as-sessions, a Curry-Howard correspondence between propositions of Linear Logic and session types for concurrent processes, has settled the logical foundations of message-passing concurrency. Central to this approach is the resource consumption paradigm heralded by Linear Logic. In this paper, we investigate a new point in the design space of session type systems for message-passing concurrent programs. We identify O’Hearn and Pym’s Logic of Bunched Implications (BI) as a fruitful basis for an interpretation of the logic as a concurrent programming language. This leads to a treatment of non-linear resources that is radically different from existing approaches based on Linear Logic. We introduce a new π-calculus with sessions, called πBI; its most salient feature is a construct called spawn, which expresses new forms of sharing that are induced by structural principles in BI. We illustrate the expressiveness of πBI and lay out its fundamental theory: type preservation, deadlock-freedom, and weak normalization results for well-typed processes; an operationally sound and complete typed encoding of an affine λ-calculus; and a non-interference result for access of resources.

Normalization in the simply typed -calculus

AbstractIn this paper, in connection with the program of extending the Curry–Howard isomorphism to classical logic, we study the $\lambda \mu$ -calculus of Parigot emphasizing the difference between the original version of Parigot and the version of de Groote in terms of normalization properties. In order to talk about a satisfactory representation of the integers, besides the usual $\beta$ -, $\mu$ -, and $\mu '$ -reductions, we consider the $\lambda \mu$ -calculus augmented with the reduction rules $\rho$ , $\theta$ and $\varepsilon$ . We show that we need all of these rules for this purpose. Then we prove that, with the syntax of Parigot, the calculus enjoys the strong normalization property even when we add the rules $\rho$ , $\theta$ , and $\epsilon$ , while the $\lambda \mu$ -calculus presented with the more flexible de Groote-style syntax, in contrast, has only the weak normalization property. In particular, we present a normalization algorithm for the $\beta \mu \mu '\rho \theta \varepsilon$ -reduction in the de Groote-style calculus.

Multiparty GV: functional multiparty session types with certified deadlock freedom

Session types have recently been integrated with functional languages, bringing message-passing concurrency to functional programming. Channel endpoints then become first-class and can be stored in data structures, captured in closures, and sent along channels. Representatives of the GV (Wadler's "Good Variation") session type family are of particular appeal because they not only assert session fidelity but also deadlock freedom, inspired by a Curry-Howard correspondence to linear logic. A restriction of current versions of GV, however, is the focus on binary sessions, limiting concurrent interactions within a session to two participants. This paper introduces Multiparty GV (MPGV), a functional language with multiparty session types, allowing concurrent interactions among several participants. MPGV upholds the strong guarantees of its ancestor GV, including deadlock freedom, despite session interleaving and delegation. MPGV has a novel redirecting construct for modular programming with first-class endpoints, thanks to which we give a type-preserving translation from binary session types to MPGV to show that MPGV is strictly more general than binary GV. All results in this paper have been mechanized using the Coq proof assistant.

Automata theory approach to predicate intuitionistic logic

Abstract Predicate intuitionistic logic is a well-established fragment of dependent types. Proof construction in this logic, as the Curry–Howard isomorphism states, is the process of program synthesis. We present automata that can handle proof construction and program synthesis in full intuitionistic first-order logic. Given a formula, we can construct an automaton such that the formula is provable if and only if the automaton has an accepting run. As further research, this construction makes it possible to discuss formal languages of proofs or programs, the closure properties of the automata and their connections with the traditional logical connectives.

On Correspondence between Selective CPS Transformation and Selective Double Negation Translation

A double negation translation (DNT) embeds classical logic into intuitionistic logic. Such translations correspond to continuation passing style (CPS) transformations in programming languages via the Curry-Howard isomorphism. A selective CPS transformation uses a type and effect system to selectively translate only nontrivial expressions possibly with computational effects into CPS functions. In this paper, we review the conventional call-by-value (CBV) CPS transformation and its corresponding DNT, and provide a logical account of a CBV selective CPS transformation by defining a selective DNT via the Curry-Howard isomorphism. By using an annotated proof system derived from the corresponding type and effect system, our selective DNT translates classical proofs into equivalent intuitionistic proofs, which are smaller than those obtained by the usual DNTs. We believe that our work can serve as a reference point for further study on the Curry-Howard isomorphism between CPS transformations and DNTs.

Object-Level Reasoning with Logics Encoded in HOL Light

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 demonstrate our implementation using a simple propositional logic and its Curry-Howard correspondence to the lambda-calculus, and argue its use with linear logic and its various correspondences to session types.

Diamonds are not forever: liveness in reactive programming with guarded recursion

When designing languages for functional reactive programming (FRP) the main challenge is to provide the user with a simple, flexible interface for writing programs on a high level of abstraction while ensuring that all programs can be implemented efficiently in a low-level language. To meet this challenge, a new family of modal FRP languages has been proposed, in which variants of Nakano's guarded fixed point operator are used for writing recursive programs guaranteeing properties such as causality and productivity. As an apparent extension to this it has also been suggested to use Linear Temporal Logic (LTL) as a language for reactive programming through the Curry-Howard isomorphism, allowing properties such as termination, liveness and fairness to be encoded in types. However, these two ideas are in conflict with each other, since the fixed point operator introduces non-termination into the inductive types that are supposed to provide termination guarantees. In this paper we show that by regarding the modal time step operator of LTL a submodality of the one used for guarded recursion (rather than equating them), one can obtain a modal type system capable of expressing liveness properties while retaining the power of the guarded fixed point operator. We introduce the language Lively RaTT, a modal FRP language with a guarded fixed point operator and an `until' type constructor as in LTL, and show how to program with events and fair streams. Using a step-indexed Kripke logical relation we prove operational properties of Lively RaTT including productivity and causality as well as the termination and liveness properties expected of types from LTL. Finally, we prove that the type system of Lively RaTT guarantees the absence of implicit space leaks.

Deriving Theorems in Implicational Linear Logic, Declaratively

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 per generated theorem. The resulting Prolog program runs in O(N) space for terms of size N and generates in a few hours 7,566,084,686 theorems in the implicational fragment of Linear Intuitionistic Logic together with their proof terms in normal form. As applications, we generate datasets for correctness and scalability testing of linear logic theorem provers and training data for neural networks working on theorem proving challenges. The results in the paper, organized as a literate Prolog program, are fully replicable. Keywords: combinatorial generation of provable formulas of a given size, intuitionistic and linear logic theorem provers, theorems of the implicational fragment of propositional linear intuitionistic logic, Curry-Howard isomorphism, efficient generation of linear lambda terms in normal form, Prolog programs for lambda term generation and theorem proving.

Composition of Deductions within the Propositions-As-Types Paradigm

Kosta Dosen argued in his papers Inferential Semantics Dosen (in Inferential semantics, Springer, Berlin 2015) and On the Paths of Categories Dosen (in On the paths of categories, Springer, Cham 2016) that the propositions-as-types paradigm is less suited for general proof theory because—unlike proof theory based on category theory—it emphasizes categorical proofs over hypothetical inferences. One specific instance of this, Dosen points out, is that the Curry–Howard isomorphism makes the associativity of deduction composition invisible. We will show that this is not necessarily the case.

Computation focusing

Focusing is a technique from proof theory that exploits type information to prune inessential nondeterminism from proof search procedures. Viewed through the lens of the Curry-Howard correspondence, a focused typing derivation yields terms in normal form. This paper explores how to exploit focusing for reasoning about contextual equivalences and full abstraction. We present a focused polymorphic call-by-push-value calculus and prove a computational completeness result: for every well-typed term, there exists a focused term that is βη-equivalent to it. This completeness result yields a powerful way to refine the context lemmas for establishing contextual equivalences, cutting down the set that must be considered to just focused contexts. The paper demonstrates the application of focusing to establish program equivalences, including free theorems. It also uses focusing to prove full abstraction of a translation of the pure, total call-by-push-value language into a language with divergence and simple effect types, yielding a novel solution to a simple-to-state, but hitherto difficult to solve problem.

CUT ELIMINATION AND NORMALIZATION FOR GENERALIZED SINGLE AND MULTI-CONCLUSION SEQUENT AND NATURAL DEDUCTION CALCULI

AbstractAny set of truth-functional connectives has sequent calculus rules that can be generated systematically from the truth tables of the connectives. Such a sequent calculus gives rise to a multi-conclusion natural deduction system and to a version of Parigot’s free deduction. The elimination rules are “general,” but can be systematically simplified. Cut-elimination and normalization hold. Restriction to a single formula in the succedent yields intuitionistic versions of these systems. The rules also yield generalized lambda calculi providing proof terms for natural deduction proofs as in the Curry–Howard isomorphism. Addition of an indirect proof rule yields classical single-conclusion versions of these systems. Gentzen’s standard systems arise as special cases.

The placeholder view of assumptions and the Curry–Howard correspondence

Proofs from assumptions are amongst the most fundamental reasoning techniques. Yet the precise nature of assumptions is still an open topic. One of the most prominent conceptions is the placeholder view of assumptions generally associated with natural deduction for intuitionistic propositional logic. It views assumptions essentially as holes in proofs, either to be filled with closed proofs of the corresponding propositions via substitution or withdrawn as a side effect of some rule, thus in effect making them an auxiliary notion subservient to proper propositions. The Curry–Howard correspondence is typically viewed as a formal counterpart of this conception. I will argue against this position and show that even though the Curry–Howard correspondence typically accommodates the placeholder view of assumptions, it is rather a matter of choice, not a necessity, and that another more assumption-friendly view can be adopted.