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**Published in:** LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)

Substitution resolution supports the computational character of beta-reduction, complementing its execution with a capture-avoiding exchange of terms for bound variables. Alas, the meta-level definition of substitution, masking a non-trivial computation, turns beta-reduction into an atomic rewriting rule, despite its varying operational complexity. In the current paper we propose a somewhat indirect average-case analysis of substitution resolution in the classic lambda-calculus, based on the quantitative analysis of substitution in lambda-upsilon, an extension of lambda-calculus internalising the upsilon-calculus of explicit substitutions. Within this framework, we show that for any fixed n >= 0, the probability that a uniformly random, conditioned on size, lambda-upsilon-term upsilon-normalises in n normal-order (i.e. leftmost-outermost) reduction steps tends to a computable limit as the term size tends to infinity. For that purpose, we establish an effective hierarchy (G_n)_n of regular tree grammars partitioning upsilon-normalisable terms into classes of terms normalising in n normal-order rewriting steps. The main technical ingredient in our construction is an inductive approach to the construction of G_{n+1} out of G_n based, in turn, on the algorithmic construction of finite intersection partitions, inspired by Robinson’s unification algorithm. Finally, we briefly discuss applications of our approach to other term rewriting systems, focusing on two closely related formalisms, i.e. the full lambda-upsilon-calculus and combinatory logic.

Maciej Bendkowski. Towards the Average-Case Analysis of Substitution Resolution in Lambda-Calculus. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 7:1-7:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bendkowski:LIPIcs.FSCD.2019.7, author = {Bendkowski, Maciej}, title = {{Towards the Average-Case Analysis of Substitution Resolution in Lambda-Calculus}}, booktitle = {4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)}, pages = {7:1--7:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-107-8}, ISSN = {1868-8969}, year = {2019}, volume = {131}, editor = {Geuvers, Herman}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.7}, URN = {urn:nbn:de:0030-drops-105144}, doi = {10.4230/LIPIcs.FSCD.2019.7}, annote = {Keywords: lambda calculus, explicit substitutions, complexity, combinatorics} }

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**Published in:** LIPIcs, Volume 108, 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)

Environments and closures are two of the main ingredients of evaluation in lambda-calculus. A closure is a pair consisting of a lambda-term and an environment, whereas an environment is a list of lambda-terms assigned to free variables. In this paper we investigate some dynamic aspects of evaluation in lambda-calculus considering the quantitative, combinatorial properties of environments and closures. Focusing on two classes of environments and closures, namely the so-called plain and closed ones, we consider the problem of their asymptotic counting and effective random generation. We provide an asymptotic approximation of the number of both plain environments and closures of size n. Using the associated generating functions, we construct effective samplers for both classes of combinatorial structures. Finally, we discuss the related problem of asymptotic counting and random generation of closed environments and closures.

Maciej Bendkowski and Pierre Lescanne. Counting Environments and Closures. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bendkowski_et_al:LIPIcs.FSCD.2018.11, author = {Bendkowski, Maciej and Lescanne, Pierre}, title = {{Counting Environments and Closures}}, booktitle = {3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)}, pages = {11:1--11:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-077-4}, ISSN = {1868-8969}, year = {2018}, volume = {108}, editor = {Kirchner, H\'{e}l\`{e}ne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2018.11}, URN = {urn:nbn:de:0030-drops-91817}, doi = {10.4230/LIPIcs.FSCD.2018.11}, annote = {Keywords: lambda-calculus, combinatorics, functional programming, mathematical analysis, complexity} }