We develop techniques based on van Oostrom's decreasing diagrams that reduce confluence proofs to the checking of critical pairs for higher-order rewrite rules extending beta-reduction on pure lambda-terms. We show that confluence is preserved for a large subset of terms that contains all pure lambda terms. Our results are applied to famous Klop's examples of non-confluent behaviours in presence of convergent rewrite rules and to fragments of various encodings, in a dependent type theory with rewrite rules, of the Calculus of Constructions with polymorphic universes.

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To understand how evolutionary algorithms optimize the simple class of monotonic functions, Jansen (FOGA 2007) introduced the partially-ordered evolutionary algorithm (PO-EA) model and analyzed its runtime. The PO-EA is a pessimistic model of the true optimization process, hence performance guarantees for it immediately take over to the true optimization process.Based on the observation that Jansen's model leads to a process more pessimistic than what any monotonic function would, we extend his model by parameterizing the degree of pessimism. For all degrees of pessimism, and all mutation rates c/n, we give a precise runtime analysis of this process. For all degrees of pessimism lower than that of Jansen, we observe a Θ(n log n) runtime for the standard mutation probability of 1/n. However, we also observe a strange double-jump behavior in terms of the mutation probability. For all non-zero degrees of pessimism, there is a threshold c ∈ R such that (i) for mutation rates c /n with c < c we have a Θ(n log n) runtime, (ii) for the mutation rate c/n we have a runtime of Θ(n 3/2 ), and (iii) for mutation rates c /n with c > c we have an exponential runtime.To overcome the complicated interplay of mutation and selection in the PO-EA, by define artificial algorithms which provably (via a coupling argument) have the same asymptotic runtime, but allow a much easier computation of the drift towards the optimum.

User-defined higher-order rewrite rules are becoming a standard in proof assistants based on intuitionistic type theory. This raises the question of proving that they preserve the properties of beta-reductions for the corresponding type systems. In a series of papers, we develop techniques based on van Oostrom’s decreasing diagrams that reduce confluence proofs to the checking of various forms of critical pairs for higher-order rewrite rules extending beta-reduction on pure lambda-terms. As shown in a previous paper of the two middle authors, confluence of a terminating set of left-linear rewrite rules is obtained when their critical pairs are joinable, beta-rewrite steps being disallowed. The present paper concentrates on the case where arbitrary beta-rewrite steps are allowed for joining critical pairs. The rewrite relation used for analyzing confluence may rewrite arbitrarily many non-overlapping redexes in a single step. This relation gives rise to critical pairs that overlap both horizontally, as with parallel rewriting, but also vertically, forming chains of successive overlaps. Practical examples of use of this technique are analyzed.