Modules over Monads and Linearity

Hirschowitz, Andrè; Maggesi, Marco

doi:10.1007/978-3-540-73445-1_16

Cited by 25 publications

(71 citation statements)

References 17 publications

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“…9.3], Hur suggests the use of preorders, or more generally, arbitrary relations to model inequational systems. Hirschowitz and Maggesi [9] prove initiality of the set of lambda terms modulo beta and eta conversion in a category of exponential monads. In an unpublished paper [10], they define a notion of half-equation and equation to express congruence between terms.…”

Section: Discussionmentioning

confidence: 99%

Initiality for Typed Syntax and Semantics

Ahrens¹

2012

Logic, Language, Information and Computation

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We give an algebraic characterization of the syntax and semantics of a class of simply-typed languages, such as the language PCF: we characterize simply-typed binding syntax equipped with reduction rules via a universal property, namely as the initial object of some category. For this purpose, we employ techniques developed in two previous works: in [2], we model syntactic translations between languages over different sets of types as initial morphisms in a category of models. In [1], we characterize untyped syntax with reduction rules as initial object in a category of models. In the present work, we show that those techniques are modular enough to be combined: we thus characterize simply-typed syntax with reduction rules as initial object in a category. The universal property yields an operator which allows to specify translations - that are semantically faithful by construction - between languages over possibly different sets of types. We specify a language by a 2-signature, that is, a signature on two levels: the syntactic level specifies the types and terms of the language, and associates a type to each term. The semantic level specifies, through inequations, reduction rules on the terms of the language. To any given 2-signature we associate a category of models. We prove that this category has an initial object, which integrates the types and terms freely generated by the 2-signature, and the reduction relation on those terms generated by the given inequations. We call this object the (programming) language generated by the 2-signature. [1] Ahrens, B.: Modules over relative monads for syntax and semantics (2011), arXiv:1107.5252, to be published in Math. Struct. in Comp. Science [2] Ahrens, B.: Extended Initiality for Typed Abstract Syntax. Logical Methods in Computer Science 8(2), 1-35 (2012)Comment: presented at WoLLIC 2012, 15 page

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Section: Discussionmentioning

confidence: 99%

Initiality for Typed Syntax and Semantics

Ahrens¹

2012

Logic, Language, Information and Computation

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“…Hirschowitz and Maggesi [17,18] equip the term monad of a signature S with a universal property by considering a category of representations of a given signature. A representation of S is given by any monad T and a family of module morphisms of suitable type over T .…”

Section: Initiality For the Constructed Monadmentioning

confidence: 99%

From Signatures to Monads in UniMath

2018

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The term UniMath refers both to a formal system for mathematics, as well as a computerchecked library of mathematics formalized in that system. The UniMath system is a core dependent type theory, augmented by the univalence axiom. The system is kept as small as possible in order to ease verification of it-in particular, general inductive types are not part of the system.In this work, we partially remedy the lack of inductive types by constructing some datatypes and their associated induction principles from other type constructors. This involves a formalization of a category-theoretic result on the construction of initial algebras, as well as a mechanism to conveniently use the datatypes obtained. We also connect this construction to a previous formalization of substitution for languages with variable binding. Altogether, we construct a framework that allows us to concisely specify, via a simple notion of binding signature, a language with variable binding. From such a specification we obtain the datatype of terms of that language, equipped with a certified monadic substitution operation and a suitable recursion scheme. Using this we formalize the untyped lambda calculus and the raw syntax of Martin-Löf type theory.

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“…For more details we refer to our paper [13] (Theorems 1 and 2). We give below a more general result (Theorem 7.8).…”

Section: Now We Give Two Examples Of Morphisms Of Aritiesmentioning

confidence: 99%

“…Although this problem may ultimately concern typed languages and their operational semantics, it already concerns untyped languages. In this work, we extend to new kinds of constructions our treatment of higher-order abstract syntax [13], based on modules and linearity.…”

Section: Introductionmentioning

confidence: 99%

Initial Semantics for Strengthened Signatures

Hirschowitz¹,

Maggesi

2012

Electron. Proc. Theor. Comput. Sci.

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We give a new general definition of arity, yielding the companion notions of signature and associated syntax. This setting is modular in the sense requested by [12]: merging two extensions of syntax corresponds to building an amalgamated sum. These signatures are too general in the sense that we are not able to prove the existence of an associated syntax in this general context. So we have to select arities and signatures for which there exists the desired initial monad. For this, we follow a track opened by Matthes and Uustalu [16]: we introduce a notion of strengthened arity and prove that the corresponding signatures have initial semantics (i.e. associated syntax). Our strengthened arities admit colimits, which allows the treatment of the λ-calculus with explicit substitution in the spirit of [12].

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Modules over Monads and Linearity

Cited by 25 publications

References 17 publications

Initiality for Typed Syntax and Semantics

Initiality for Typed Syntax and Semantics

From Signatures to Monads in UniMath

Initial Semantics for Strengthened Signatures

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