Proving Divide and Conquer Complexities in Isabelle/HOL

A Fistful of Dollars: Formalizing Asymptotic Complexity Claims via Deductive Program Verification

“…We contend that, in order to define what it means for a ∈ A to "grow large", or "tend towards infinity", the type A must be equipped with a filter [6], that is, a quantifier Ua.P . (Eberl [13] does so as well.) We propose a library of lemmas and tactics that can prove nonnegativeness, monotonicity, and domination assertions (Sect.…”

“…Second, one can invoke Cormen et al's Master Theorem [12] or the more general Akra-Bazzi theorem [1,21]. Unfortunately, at present, these theorems are not available in Coq, although an Isabelle/HOL formalization exists [13].…”

“…Eberl [13] formalizes the Akra-Bazzi method [1,21], a generalization of the well-known Master Theorem [12], in Isabelle/HOL. He creates a library of Landau symbols specifically for this purpose.…”

Calcul Formel et Preuves Formelles

“…En fait, il s'agit plutôt d'une petite vitrine de preuves mathématiques par calcul symbolique certifié. Ainsi, on a écarté les formalisations de théorèmes liés au calcul formel, mais qui reposent sur des techniques moins spécifiques, comme par exemple les résultats de complexité de l'algorithmique du "diviser pour régner" [16], la décidabilité des corps réels clos [13] ou la théorie élémentaire des courbes elliptiques [1]. Enfin, ce choix privilégie des résultats récents, publiés au cours des cinq dernières années, et écarte ainsi des sujets pourtant pertinents comme les tests de primalité [19].…”

Verifying Asymptotic Time Complexity of Imperative Programs in Isabelle

“…Compared to [12], we go further in two directions. First, we incorporate the work of Eberl [11] on the Akra-Bazzi theorem to analyze several divide-and-conquer algorithms. Second, we extend the auto2 prover [21] to provide substantial automation in reasoning about separation logic with time credits, as well as deriving asymptotic behavior of functions.…”