Abstract
We extend the \(\lambda \)-calculus with constructs suitable for relational and functional–logic programming: non-deterministic choice, fresh variable introduction, and unification of expressions. In order to be able to unify \(\lambda \)-expressions and still obtain a confluent theory, we depart from related approaches, such as \(\lambda \)Prolog, in that we do not attempt to solve higher-order unification. Instead, abstractions are decorated with a location, which intuitively may be understood as its memory address, and we impose a simple coherence invariant: abstractions in the same location must be equal. This allows us to formulate a confluent small-step operational semantics which only performs first-order unification and does not require strong evaluation (below lambdas). We study a simply typed version of the system. Moreover, a denotational semantics for the calculus is proposed and reduction is shown to be sound with respect to the denotational semantics.
Work partially supported by project grants ECOS Sud A17C01, PUNQ 1346/17, and UBACyT 20020170100086BA.
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Notes
- 1.
- 2.
Key in our proof of confluence is the fact that if \(\sigma \) and \(\sigma '\) are most general unifiers for unification problems \(\mathsf {G}\) and \(\mathsf {G}'\) respectively, then the most general unifier for \((\mathsf {G}\cup \mathsf {G}')\) is an instance of both \(\sigma \) and \(\sigma '\). See Example 4.5.
- 3.
E.g. \(\lambda ^{\ell } x.\,x \overset{\bullet }{=}\lambda ^{\ell } x.\,x\) succeeds but \(\lambda ^{\ell } x.\,x \overset{\bullet }{=}\lambda ^{\ell '} x.\,x\) fails.
- 4.
We expect that a less naive semantics should be stateful, involving a memory, in such a way that abstractions (\(\lambda x.\,P\)) allocate a memory cell and store a closure, whereas allocated abstractions (\(\lambda ^{\ell } x.\,P\)) denote a memory location in which a closure is already stored.
- 5.
Precisely, \(\underline{\mathbf {t}}(n) = \{f_n\}\) with \(f_n(m) = \{(n,m)\}\).
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Acknowledgements
To Alejandro Díaz-Caro for supporting our interactions. To Eduardo Bonelli, Delia Kesner, and the anonymous reviewers for their feedback and suggestions.
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Barenbaum, P., Lochbaum, F., Milicich, M. (2020). Semantics of a Relational \(\lambda \)-Calculus. In: Pun, V.K.I., Stolz, V., Simao, A. (eds) Theoretical Aspects of Computing – ICTAC 2020. ICTAC 2020. Lecture Notes in Computer Science(), vol 12545. Springer, Cham. https://doi.org/10.1007/978-3-030-64276-1_13
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