77 research outputs found
On the confluence of lambda-calculus with conditional rewriting
The confluence of untyped \lambda-calculus with unconditional rewriting is
now well un- derstood. In this paper, we investigate the confluence of
\lambda-calculus with conditional rewriting and provide general results in two
directions. First, when conditional rules are algebraic. This extends results
of M\"uller and Dougherty for unconditional rewriting. Two cases are
considered, whether \beta-reduction is allowed or not in the evaluation of
conditions. Moreover, Dougherty's result is improved from the assumption of
strongly normalizing \beta-reduction to weakly normalizing \beta-reduction. We
also provide examples showing that outside these conditions, modularity of
confluence is difficult to achieve. Second, we go beyond the algebraic
framework and get new confluence results using a restricted notion of
orthogonality that takes advantage of the conditional part of rewrite rules
Substitution, jumps, and algebraic effects
Contains fulltext :
129931.pdf (author's version ) (Open Access
Handling Fibred Algebraic Effects
International audienceWe study algebraic computational effects and their handlers in the dependently typed setting. We describecomputational effects using a generalisation of Plotkin and Pretnar’s effect theories, whose dependentlytyped operations allow us to capture precise notions of computation, e.g., state with location-dependent storetypes and dependently typed update monads. Our treatment of handlers is based on an observation that theirconventional term-level definition leads to unsound program equivalences being derivable in languages thatinclude a notion of homomorphism. We solve this problem by giving handlers a novel type-based treatmentvia a new computation type, the user-defined algebra type, which pairs a value type (the carrier) with a set ofvalue terms (the operations), capturing Plotkin and Pretnar’s insight that effect handlers denote algebras. Wethen show that the conventional presentation of handlers can be routinely derived, and demonstrate that thistype-based treatment of handlers provides a useful mechanism for reasoning about effectful computations.We also equip the resulting language with a sound denotational semantics based on families fibrations
Initiality for Typed Syntax and Semantics
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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