Axioms and decidability for type isomorphism in the presence of sums

Ilik, Danko

doi:10.1145/2603088.2603115

Cited by 5 publications

(6 citation statements)

References 24 publications

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“…Equation validity (and consequently formula isomorphism) poses interesting meta-theoretic problems that go back to Tarski's High-School Identity Problem (cf. [4,8,15]). In particular, validity (and isomorphism) is not finitely axiomatizable: there is no finite set of equality axioms Ax that is sufficient to derive every valid equation (isomorphism), i.e., such that N + F = G implies Ax F = G.…”

Section: Proof Rules As Equalitiesmentioning

confidence: 99%

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An intuitionistic formula hierarchy based on high‐school identities

Brock-Nannestad

Ilik

2019

Mathematical Logic Qtrly

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To the memory of Kosta Došen and Roy DyckhoffWe revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal) propositional sequent calculi are formula (i.e., sequent) isomorphisms corresponding to the highschool identities, we show that one can obtain a more compact variant of a proof system, consisting of noninvertible proof rules only, and where the invertible proof rules have been replaced by a formula normalization procedure. Moreover, for certain proof systems such as the G4ip sequent calculus of Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the non-invertible proof rules as strict inequalities between exponential polynomials; a careful combinatorial treatment is given in order to establish this fact. Finally, we extend the exponential polynomial analogy to the first-order quantifiers, showing that it gives rise to an intuitionistic hierarchy of formulas, resembling the classical arithmetical hierarchy, and the first one that classifies formulas while preserving isomorphism.

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Section: Proof Rules As Equalitiesmentioning

confidence: 99%

“…I.e., when we close these axioms under appropriate equality and congruence rules (cf., e.g., [15]), we can talk about formal derivability of an equation, HSI F = G, for which we have:…”

Section: Proof Rules As Equalitiesmentioning

confidence: 99%

An intuitionistic formula hierarchy based on high‐school identities

Brock-Nannestad

Ilik

2019

Mathematical Logic Qtrly

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“…where χi are atomic types (or type variables), it is not always clear when two given types are essentially the same one. More precisely, it is not known how to decide whether two types are isomorphic (Ilik 2014). Although the notion of isomorphism can be treated abstractly in Category Theory, in bi-Cartesian closed categories, and without committing to a specific term calculus inhabiting the types, in the language of the standard syntax and equational theory of lambda calculus with sums ( Figure 1), the types τ and σ are isomorphic when there exist coercing lambda terms M : σ → τ and N : τ → σ such that λx.M (N x) = βη λx.x and λy.N (M y) = βη λy.y.…”

Section: The Exp-log Normal Form Of Typesmentioning

confidence: 99%

“…Recognizing isomorphic types If we leave aside the problems of canonicity of and equality between terms, there is a further problem at the level of types that makes it hard to determine whether two type signatures are essentially the same one. Namely, although for each of the type languages {→, ×} and {→, +} there is a very simple algorithm for deciding type isomorphism, for the whole of the language {→, +, ×} it is only known that type isomorphism is decidable when types are to be interpreted as finite structures, and that without a practically implementable algorithm in sight (Ilik 2014).…”

Section: Introductionmentioning

confidence: 99%

The exp-log normal form of types: decomposing extensional equality and representing terms compactly

Ilik¹

2017

Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages

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Lambda calculi with algebraic data types lie at the core of functional programming languages and proof assistants, but conceal at least two fundamental theoretical problems already in the presence of the simplest non-trivial data type, the sum type. First, we do not know of an explicit and implemented algorithm for deciding the beta-eta-equality of terms-and this in spite of the first decidability results proven two decades ago. Second, it is not clear how to decide when two types are essentially the same, i.e. isomorphic, in spite of the meta-theoretic results on decidability of the isomorphism.In this paper, we present the exp-log normal form of typesderived from the representation of exponential polynomials via the unary exponential and logarithmic functions-that any type built from arrows, products, and sums, can be isomorphically mapped to. The type normal form can be used as a simple heuristic for deciding type isomorphism, thanks to the fact that it is a systematic application of the high-school identities.We then show that the type normal form allows to reduce the standard beta-eta equational theory of the lambda calculus to a specialized version of itself, while preserving the completeness of equality on terms.We end by describing an alternative representation of normal terms of the lambda calculus with sums, together with a Coqimplemented converter into/from our new term calculus. The difference with the only other previously implemented heuristic for deciding interesting instances of eta-equality by Balat, Di Cosmo, and Fiore, is that we exploit the type information of terms substantially and this often allows us to obtain a canonical representation of terms without performing sophisticated term analyses.

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“…This observation is the basis of the promising unpublished work of Ahmad, Licata, and Harper (2010), also strongly relying on (higher-order) focusing. Finiteness hypotheses also play an important role in Ilik (2014), where they are used to reason on type isomorphisms in presence of sums. Our own work does not handle 1 or 0; the latter at least is a notorious source of difficulties for equivalence, but is also seldom necessary in practical programming applications.…”

Section: Equivalence Of Terms In Presence Of Sumsmentioning

confidence: 99%

Which simple types have a unique inhabitant?

Scherer

Rémy

2015

SIGPLAN Not.

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We study the question of whether a given type has a unique inhabitant modulo program equivalence. In the setting of simplytyped lambda-calculus with sums, equipped with the strong βηequivalence, we show that uniqueness is decidable. We present a saturating focused logic that introduces irreducible cuts on positive types "as soon as possible". Backward search in this logic gives an effective algorithm that returns either zero, one or two distinct inhabitants for any given type. Preliminary application studies show that such a feature can be useful in strongly-typed programs, inferring the code of highly-polymorphic library functions, or "glue code" inside more complex terms.

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Axioms and decidability for type isomorphism in the presence of sums

Cited by 5 publications

References 24 publications

An intuitionistic formula hierarchy based on high‐school identities

An intuitionistic formula hierarchy based on high‐school identities

The exp-log normal form of types: decomposing extensional equality and representing terms compactly

Which simple types have a unique inhabitant?

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