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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Published in:	Journal of automated reasoning Vol. 58; no. 4; pp. 483 - 508
Main Author:	Eberl, Manuel
Format:	Journal Article
Language:	English
Published:	Dordrecht Springer Netherlands 01.04.2017
Subjects:	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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Abstract	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.
AbstractList	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. (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).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 ...-bounds for these Divide and Conquer recurrences. To our knowledge, this is the first formalisation of theorems for the analysis of such recurrences.
Author	Eberl, Manuel
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Keywords	Divide and Conquer algorithms Isabelle/HOL Akra–Bazzi Master Theorem Landau symbols Recurrences Complexity
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References_xml	– start-page: 103 year: 2010 end-page: 117 ident: CR11 article-title: Code generation via higher-order rewrite systems publication-title: Functional and Logic Programming. doi: 10.1007/978-3-642-12251-4_9 – year: 2009 ident: CR7 publication-title: Introduction to Algorithms, 4th printing – volume: 60 start-page: 16:1 issue: 3 year: 2013 end-page: 16:49 ident: CR8 article-title: A Master theorem for discrete divide and conquer recurrences publication-title: J. ACM doi: 10.1145/2487241.2487242 – volume: 39 start-page: 1546 issue: 5 year: 1993 end-page: 1554 ident: CR6 article-title: Block arithmetic coding for source compression publication-title: IEEE Trans. Inf. Theory doi: 10.1109/18.259639 – ident: CR3 – ident: CR14 – volume: 36 start-page: 41 issue: 1 year: 2003 end-page: 57 ident: CR5 article-title: The solution of linear probabilistic recurrence relations publication-title: Algorithmica doi: 10.1007/s00453-002-1003-4 – ident: CR12 – ident: CR13 – ident: CR10 – ident: CR9 – volume: 52 start-page: 123 issue: 2 year: 2014 end-page: 153 ident: CR4 article-title: Locales: a module system for mathematical theories publication-title: J. Autom. Reason. doi: 10.1007/s10817-013-9284-7 – start-page: 357 year: 2004 end-page: 371 ident: CR2 article-title: Formalizing notation in Isabelle/HOL publication-title: Automated Reasoning, – volume: 10 start-page: 195 issue: 2 year: 1998 end-page: 210 ident: CR1 article-title: On the solution of linear recurrence equations publication-title: Comput. Optim. Appl. doi: 10.1023/A:1018373005182 – ident: 9378_CR3 – volume: 36 start-page: 41 issue: 1 year: 2003 ident: 9378_CR5 publication-title: Algorithmica doi: 10.1007/s00453-002-1003-4 – start-page: 357 volume-title: Automated Reasoning, Lecture Notes in Computer Science year: 2004 ident: 9378_CR2 doi: 10.1007/978-3-540-25984-8_27 – start-page: 103 volume-title: Functional and Logic Programming. Lecture Notes in Computer Science year: 2010 ident: 9378_CR11 doi: 10.1007/978-3-642-12251-4_9 – volume: 60 start-page: 16:1 issue: 3 year: 2013 ident: 9378_CR8 publication-title: J. ACM doi: 10.1145/2487241.2487242 – volume: 39 start-page: 1546 issue: 5 year: 1993 ident: 9378_CR6 publication-title: IEEE Trans. Inf. Theory doi: 10.1109/18.259639 – ident: 9378_CR10 – ident: 9378_CR13 – ident: 9378_CR12 – volume: 52 start-page: 123 issue: 2 year: 2014 ident: 9378_CR4 publication-title: J. Autom. Reason. doi: 10.1007/s10817-013-9284-7 – ident: 9378_CR14 – volume: 10 start-page: 195 issue: 2 year: 1998 ident: 9378_CR1 publication-title: Comput. Optim. Appl. doi: 10.1023/A:1018373005182 – ident: 9378_CR9 – volume-title: Introduction to Algorithms, 4th printing year: 2009 ident: 9378_CR7
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Snippet	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... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).The Akra-Bazzi method (Akra and Bazzi in Comput Optim Appl 10(2):195-210, 1998....
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SubjectTerms	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
Title	Proving Divide and Conquer Complexities in Isabelle/HOL
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