## Confluence results for the pure strong categorical logic CCL. $$\lambda$$- calculi as subsystems of CCL.(English)Zbl 0707.03012

Author’s summary: “The Strong Categorical Combinatory Logic (CCL, CCL$$\beta\eta$$ SP), developed by Curien (1986) is, when typed and augmented with a rule defining a terminal object, a presentation of Cartesian Closed Categories. Furthermore, it is equationally equivalent to the Lambda-calculus with explicit couples and Surjective Pairing. Here we study the confluence properties of (CCL, CCL$$\beta\eta$$ SP) and of several of its subsystems, and the relationship between untyped Lambda- calculi and (CCL, CCL$$\beta\eta$$ SP) as rewriting systems. We prove that there exists a subset $${\mathcal D}$$ of CCL, and a subsystem SL$$\beta$$ of CCL$$\beta\eta$$ SP confluent on $${\mathcal D}$$, a very simple isomorphism between $$\Lambda$$, the classical Lambda-calculus, and a subset $${\mathcal S}{\mathcal D}_{\lambda}$$ of $${\mathcal D}$$, which is extended between $$\beta$$- derivations of $$\Lambda$$ and a class of derivations of SL$$\beta$$. Substitution, which is a one-step operation belonging to the metalanguage of $$\Lambda$$, is now described by rewritings with SL$$\beta$$ and calculations between several substitutions launched at the same time may be performed by SL$$\beta$$. This point is a real increase in the calculation capacities of Lambda-calculus (same results for $${\mathcal D}).$$
The same result holds for the Lambda-calculus with couples and projection rules (without Surjective Pairing).
The locally confluent subsystem CCL$$\beta$$ SP (that is $$SL\beta +(SP))$$ is not confluent. This result is obtained by firstly designing a new counter-example (different from J. W. Klop’s one) for confluence of the Lambda-calculus with couples and Surjective Pairing and then translating it into CCL. However, CCL$$\beta$$ SP is shown to be confluent on the set derived from $${\mathcal S}{\mathcal D}_{\lambda}.$$
These results cannot be obtained with classical methods of confluence and we designed a new method called Interpretation Method based on this trick: a given relation R is confluent on a set X if and only if a relation $${\mathcal E}(R)$$ induced by R on a set of regularized terms $${\mathcal E}(X)$$ is confluent.”
Reviewer: C.Berline

### MSC:

 03B40 Combinatory logic and lambda calculus 68Q42 Grammars and rewriting systems 18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.) 03G30 Categorical logic, topoi
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### References:

 [1] Barendregt, H.P., The lambda-calculus, (1984), North-Holland Amsterdam · Zbl 0549.03012 [2] Barendregt, H.P., Pairing without conventional restraints, Z. math. logik grundlag. math., 20, 289-306, (1974) · Zbl 0299.02030 [3] de Brujin, N., Lambda-calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the church – rosser theorem, Indag. math., 34, 381-392, (1972) · Zbl 0253.68007 [4] de Brujin, N., Lambda-calculus notation with namefree formulas involving symbols that represent reference transforming mappings, Indag. math., 40, 348-356, (1978) · Zbl 0393.03009 [5] Church, A., The calculi of lambda-conversion, Annals of mathematics studies, Vol. 6, (1941), Princeton Univ. Press · JFM 67.0041.01 [6] Curien, P.L., Categorical combinators, sequential algorithms and functional programming, Research notes in theoretical computer science, (1986), Pitman London · Zbl 0643.68004 [7] Curry, H.B., Combinatory logic, (1958), North-Holland Amsterdam · Zbl 0158.24703 [8] Curry, H.B.; Hindley, J.R.; Seldin, J.P., Combinatory logic, (1972), North-Holland Amsterdam · Zbl 0269.02005 [9] Hardin, T., Résultats de confluence pour LES règles fortes de la logique combinatoire catégorique et liens avec LES lambda-calculs, Thèse de doctorat, 7, (1987), Université de Paris [10] Hardin, T.; Laville, A., Proof of termination of the rewriting system SUBST on CCL, Theoret. comput. sci., 46, 305-312, (1986) · Zbl 0618.68031 [11] Hindley, R.; Seldin, J., Introduction to combinators and λ-calculus, London mathematical society student texts, Vol. 1, (1986), Cambridge University Press · Zbl 0614.03014 [12] Klop, J.W., Combinatory reduction systems, Dissertation, (1982), Mathematisch Centrum Amsterdam · Zbl 0466.03006 [13] Klop, J.W.; de Vrijer, R., Unique normal forms for λ-calculus with surjective pairing tech. report., 87-03, (1987), Centre of Mathematics and Computer Science Amsterdam [14] Knuth, D.; Bendix, P., Simple word problems in universal algebras, (), 263-297 · Zbl 0188.04902 [15] Lambek, J.; Scott, P.J., Introduction to higher order categorical logic, (1987), Cambridge Univ. Press · Zbl 0596.03002 [16] Levy, J.J., Réductions correctes and optimales dans le lambda-calcul, Thèse d’etat, 7, (1978), Université de Paris [17] Mauny, M., Compilation des langages fonctionnels dans LES combinateurs categoriques, Thèse de troisième cycle, 7, (1985), Université de Paris [18] Turner, D., A new implementation technique for applicative languages, Software practice experience, 9, 31-49, (1979) · Zbl 0386.68009 [19] R. de Vrijer, Surjective pairing and strong normalisation: two themes in λ-calculus, Dissertation, University of Amsterdam. [20] H. Yokouchi, Relationship between λ-calculus and rewriting systems for categorical combinators, Preprint, IBM Research Center, Tokyo. · Zbl 0672.03006
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