
Combinatorics of explicit substitutions
λυ is an extension of the λcalculus which internalises the calculus of ...
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Towards a Semantic Measure of the Execution Time in CallbyValue lambdaCalculus (Long Version)
We investigate the possibility of a semantic account of the execution ti...
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Towards a Semantic Measure of the Execution Time in CallbyValue lambdaCalculus
We investigate the possibility of a semantic account of the execution ti...
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The Bang Calculus Revisited
CallbyPushValue (CBPV) is a programming paradigm subsuming both Call...
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Nonidempotent types for classical calculi in natural deduction style
In the first part of this paper, we define two resource aware typing sys...
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Reducing Lambda Terms with Traversals
We introduce a method to evaluate untyped lambda terms by combining the ...
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Resolution as Intersection Subtyping via Modus Ponens
Resolution and subtyping are two common mechanisms in programming langua...
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Towards the averagecase analysis of substitution resolution in λcalculus
Substitution resolution supports the computational character of βreduction, complementing its execution with a captureavoiding exchange of terms for bound variables. Alas, the metalevel definition of substitution, masking a nontrivial computation, turns βreduction into an atomic rewriting rule, despite its varying operational complexity. In the current paper we propose a somewhat indirect averagecase analysis of substitution resolution in the classic λcalculus, based on the quantitative analysis of substitution in λυ, an extension of λcalculus internalising the υcalculus of explicit substitutions. Within this framework, we show that for any fixed n ≥ 0, the probability that a uniformly random, conditioned on size, λυterm υnormalises in n normalorder (i.e. leftmostoutermost) 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 υnormalisable terms into classes of terms normalising in n normalorder 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 λυcalculus and combinatory logic.
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