Proving Divide and Conquer Complexities in Isabelle/HOL

The Akra–Bazzi method (Akra and Bazzi in Comput Optim Appl 10(2):195–210,  1998 . doi: 10.1023/A:1018373005182 ), a generalisation of the well-known Master Theorem, is a useful tool for analysing the complexity of Divide and Conquer algorithms. This work describes a formalisation of the Akra–Bazzi m...

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Vydáno v:Journal of automated reasoning Ročník 58; číslo 4; s. 483 - 508
Hlavní autor: Eberl, Manuel
Médium: Journal Article
Jazyk:angličtina
Vydáno: Dordrecht Springer Netherlands 01.04.2017
Témata:
Algorithms
Artificial Intelligence
Automated reasoning
Automation
Complexity
Computer Science
Derivation
Mathematical Logic and Formal Languages
Mathematical Logic and Foundations
Symbolic and Algebraic Manipulation
Texts
Theorem proving
Theorems
Divide and Conquer algorithms
Isabelle/HOL
Akra–Bazzi
Master Theorem
Landau symbols
Recurrences
Complexity
ISSN:0168-7433, 1573-0670
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Shrnutí:The Akra–Bazzi method (Akra and Bazzi in Comput Optim Appl 10(2):195–210,  1998 . doi: 10.1023/A:1018373005182 ), a generalisation of the well-known Master Theorem, is a useful tool for analysing the complexity of Divide and Conquer algorithms. This work describes a formalisation of the Akra–Bazzi method (as generalised by Leighton in Notes on better Master theorems for divide-and-conquer recurrences,  1996 . http://courses.csail.mit.edu/6.046/spring04/handouts/akrabazzi.pdf ) in the interactive theorem prover Isabelle/HOL and the derivation of a generalised version of the Master Theorem from it. We also provide some automated proof methods that facilitate the application of this Master Theorem and allow mostly automatic verification of Θ -bounds for these Divide and Conquer recurrences. To our knowledge, this is the first formalisation of theorems for the analysis of such recurrences.
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ISSN:0168-7433
1573-0670
DOI:10.1007/s10817-016-9378-0