Second-Order and Dependently-Sorted Abstract Syntax

Fiore, Marcelo

doi:10.1109/lics.2008.38

Cited by 42 publications

(85 citation statements)

References 14 publications

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“…One important observation related to our development is that calculi and algebras for programming languages are often described naturally as second-order algebraic theories [Fiore and Hur 2010;Fiore and Mahmoud 2010]. Second-order algebraic theories are founded on the mathematical theory of second-order abstract syntax [Fiore 2008;Fiore et al 1999;Hamana 2004]. Staton has shown that second-order algebraic theories are a useful framework that models various important notions of programming languages, such as logic programming [Staton 2013a], algebraic effects [Staton 2013b], and quantum computation [Staton 2015].…”

Section: Proof By Rewritingmentioning

confidence: 99%

How to prove your calculus is decidable: practical applications of second-order algebraic theories and computation

Hamana

2017

Proc. ACM Program. Lang.

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We present a general methodology of proving the decidability of equational theory of programming language concepts in the framework of second-order algebraic theories. We propose a Haskell-based analysis tool SOL, Second-Order Laboratory, which assists the proofs of confluence and strong normalisation of computation rules derived from second-order algebraic theories.To cover various examples in programming language theory, we combine and extend both syntactical and semantical results of second-order computation in a non-trivial manner. We demonstrate how to prove decidability of various algebraic theories in the literature. It includes the equational theories of monad and λ-calculi, Plotkin and Power's theory of states, and Stark's theory of π -calculus.

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Section: Proof By Rewritingmentioning

confidence: 99%

How to prove your calculus is decidable: practical applications of second-order algebraic theories and computation

Hamana

2017

Proc. ACM Program. Lang.

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“…The free substitution algebra construction S yields a strong monad on the category of context-indexed-sets ( [12],[42, Cor. 1]): for any context-indexed-sets P and Q there is a context-indexedfunction >>= : SP × (SQ) P → SQ, satisfying the monad laws.…”

Section: Substitution Algebrasmentioning

confidence: 99%

Substitution, jumps, and algebraic effects

Fiore

Staton

2014

Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Nin

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Algebraic structures abound in programming languages. The starting point for this paper is the following theorem: (first-order) algebraic signatures can themselves be described as free algebras for a (second-order) algebraic theory of substitution. Transporting this to the realm of programming languages, we investigate a computational metalanguage based on the theory of substitution, demonstrating that substituting corresponds to jumping in an abstract machine. We use the theorem to give an interpretation of a programming language with arbitrary algebraic effects into the metalanguage with substitution/jumps.

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“…The view (2) was explored in [Ham04,Ham05,Fio08,FH10]. The presentation of axioms using metavariables is certainly more economical than (1), but technically more involved.…”

Section: On Substitutions and Future Workmentioning

confidence: 99%

Polymorphic Abstract Syntax via Grothendieck Construction

Hamana

2011

Foundations of Software Science and Computational Structures

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Abstract. Abstract syntax with variable binding is known to be characterised as an initial algebra in a presheaf category. This paper extends it to the case of polymorphic typed abstract syntax with binding. We consider two variations, secondorder and higher-order polymorphic syntax. The central idea is to apply Fiore's initial algebra characterisation of typed abstract syntax with binding repeatedly, i.e. first to the type structure and secondly to the term structure of polymorphic system. In this process, we use the Grothendieck construction to combine differently staged categories of polymorphic contexts.

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Second-Order and Dependently-Sorted Abstract Syntax

Cited by 42 publications

References 14 publications

How to prove your calculus is decidable: practical applications of second-order algebraic theories and computation

How to prove your calculus is decidable: practical applications of second-order algebraic theories and computation

Substitution, jumps, and algebraic effects

Polymorphic Abstract Syntax via Grothendieck Construction

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