The call-by-value λ-calculus: a semantic investigation

ALBERTO PRAVATO, SIMONA RONCHI della ROCCA, LUCA ROVERSI
<span title="">1999</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jdpukxfslnfmradqb65llq43na" style="color: black;">Mathematical Structures in Computer Science</a> </i> &nbsp;
This paper is about a categorical approach for modeling the pure (i.e., without constants) call-by-value λ-calculus, defined by Plotkin as a restriction of the call-by-name λ-calculus. In particular, the properties a category Cbv must enjoy to describe a model of call-by-value λ-calculus are given. The category Cbv is general enough to catch models in Scott Domains and Coherence Spaces. the intuition, a model for the λβ-calculus is not necessarily a model for the λβ v -calculus (see Remark
more &raquo; ... Moreover, we study the problem of modeling the call-by-value extensionality. Syntactically, the call-by-value extensionality is expressed by the η v -rule, which is a restriction of the classical η-rule. We define a semantic notion of extensionality, suitably restricting the analogous notion for the λβ-calculus. Namely, a model for the λβ v -calculus is extensional if the equality relation between its elements reflects their extensional functional behavior. However, the elements of the model are not seen as total functions. They are considered as partial functions, having the set of semantic values as domain. The unexpected consequence is that, unlike the λβ-calculus, a model of the λβ v -calculus can be extensional without modeling the β v η v -equality. As evidence for this, we show that the Coherence Space which is the least solution of D ≈ !(D =⇒ D) satisfies the β v η v -equality, while not being extensional. Roughly speaking, to model the β v η v -equality it is sufficient that only the elements of the models which are interpretation of valuable terms have an extensional behavior. The class Cbv is not a complete characterization of the models for the λβ v -calculus, at least with respect to those with an extensional theory. We prove that all Cbv-models, having a β v η v -theory, satisfy the equality IM = M , where I is the identity term λx.x, and M is any term. This equality, which is correct with respect to the operational semantics of the λβ v -calculus, does not belong to all βη v -theories. For example, it is not in the term model induced by the β v η v -theory, and it is not in the model of (Honsell and Lenisa, 1993) . The equality IM = M reflects the substitution property of the Intuitionistic Linear Logic which we choose for modeling the typed version of the λβ v -calculus. We leave as an open problem to say whether the class of Cbv-models not having an extensional theory is complete or not. Index. In Section 2 the λβ v -calculus, and its notion of model are recalled. In Section 3, starting from some logical argumentation, the categorical structure needed for modeling the λβ v -calculus is defined. This categorical structure is used in Section 4, and in Section 5 to define a categorical model for the λβ v -calculus. Section 6 is about extensionality. Section 7 proves the incompleteness of the subclass of Cbv-models with an extensional theory. In Section 8 two instances of the categorical model are introduced. In Section 9, we discuss the relation between Cbv, and the models of the λβ v -calculus, given in (Moggi, 1991) . Finally, A recalls some of the categorical concepts used in the paper. However, we assume a basic knowledge about Category Theory, Scott Domains, and Coherence Spaces. An earlier, and partial version of this paper was in (Pravato et al., 1995) . Modeling the call-by-value λ-calculus The call-by-value lambda calculus, or λβ v -calculus, is a restriction of the classical one, based on the concept of value. In particular, the restriction concerns the evaluation rule, namely the β-rule, which is replaced by the β v -rule.
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