Automating Algebraic Methods in Isabelle

We implement a large Isabelle/HOL repository of algebras for application in modelling computing systems. They subsume computational logics such as dynamic and Hoare logics and form a basis for various software development methods. Isabelle has recently been extended by automated theorem provers and SMT solvers. We use these integrated tools for automatically proving several rather intricate refinement and termination theorems. We also automate a modal correspondence result and soundness and relative completeness proofs of propositional Hoare logic. These results show, for the first time, that Isabelle's tool integration makes automated algebraic reasoning particularly simple. This is a step towards increasing the automation of formal methods.

[1]  Norbert Schirmer,et al.  Verification of sequential imperative programs in Isabelle-HOL , 2006 .

[2]  Maribel Fernández,et al.  Curry-Style Types for Nominal Terms , 2006, TYPES.

[3]  Norbert Schirmer,et al.  A Verification Environment for Sequential Imperative Programs in Isabelle/HOL , 2005, LPAR.

[4]  Simon Foster,et al.  Automated Engineering of Relational and Algebraic Methods in Isabelle/HOL - (Invited Tutorial) , 2011, RAMiCS.

[5]  Georg Struth,et al.  On Automating the Calculus of Relations , 2008, IJCAR.

[6]  Georg Struth,et al.  Internal axioms for domain semirings , 2011, Sci. Comput. Program..

[7]  Tobias Nipkow,et al.  Proof Pearl: Regular Expression Equivalence and Relation Algebra , 2011, Journal of Automated Reasoning.

[8]  Jörg H. Siekmann,et al.  8th International Conference on Automated Deduction , 1986, Lecture Notes in Computer Science.

[9]  Dexter Kozen.  A Completeness Theorem for Kleene Algebras and the Algebra of Regular Events , 1994, Inf. Comput..

[10]  Roger D. Maddux,et al.  Relation-Algebraic Semantics , 1996, Theor. Comput. Sci..

[11]  Georg Struth,et al.  Automated Reasoning in Kleene Algebra , 2007, CADE.

[12]  Nikolaj Bjørner,et al.  Automated Deduction - CADE-23 - 23rd International Conference on Automated Deduction, Wroclaw, Poland, July 31 - August 5, 2011. Proceedings , 2011, CADE.

[13]  J. Conway.  Regular algebra and finite machines , 1971 .

[14]  Ernst-Rüdiger Olderog,et al.  Verification of Sequential and Concurrent Programs , 1997, Graduate Texts in Computer Science.

[15]  Nachum Dershowitz,et al.  Commutation, Transformation, and Termination , 1986, CADE.

[16]  Markus Wenzel,et al.  Local Theory Specifications in Isabelle/Isar , 2009, TYPES.

[17]  Edsger W. Dijkstra,et al.  A Discipline of Programming , 1976 .

[18]  Stephen A. Cook,et al.  Soundness and Completeness of an Axiom System for Program Verification , 1978, SIAM J. Comput..

[19]  Yde Venema,et al.  Dynamic Logic by David Harel, Dexter Kozen and Jerzy Tiuryn. The MIT Press, Cambridge, Massachusetts. Hardback: ISBN 0–262–08289–6, $50, xv + 459 pages , 2002, Theory and Practice of Logic Programming.

[20]  Jerzy Tiuryn,et al.  Dynamic logic , 2001, SIGA.

[21]  C. A. R. Hoare,et al.  Laws of programming , 1987, CACM.

[22]  T. Nipkow.  Hoare Logics in Isabelle/HOL , 2002 .

[23]  Dexter Kozen.  On Hoare logic and Kleene algebra with tests , 2000, TOCL.

[24]  Georg Struth,et al.  Kleene algebra with domain , 2003, TOCL.

[25]  Lawrence C. Paulson,et al.  Extending Sledgehammer with SMT Solvers , 2011, CADE.

[26]  Sascha Böhme,et al.  Fast LCF-Style Proof Reconstruction for Z3 , 2010, ITP.

[27]  H. Schwichtenberg,et al.  Proof and System-Reliability , 2002 .

[28]  Georg Struth,et al.  Automated verification of refinement laws , 2009, Annals of Mathematics and Artificial Intelligence.

[29]  Nick Bezhanishvili,et al.  Finitely generated free Heyting algebras via Birkhoff duality and coalgebra , 2011, Log. Methods Comput. Sci..

[30]  Larry Wos,et al.  What Is Automated Reasoning? , 1987, J. Autom. Reason..

[31]  C. A. R. Hoare,et al.  Unifying theories of programming , 1998, RelMiCS.

[32]  Georg Struth,et al.  Algebras of modal operators and partial correctness , 2006, Theor. Comput. Sci..

[33]  Georg Struth,et al.  Algebraic Notions of Termination , 2010, Log. Methods Comput. Sci..

[34]  Marcel Jackson,et al.  SEMIGROUPS WITH if-then-else AND HALTING PROGRAMS , 2009, Int. J. Algebra Comput..

[35]  Roland Carl Backhouse,et al.  A Calculational Approach to Mathematical Induction , 1997, Theor. Comput. Sci..

[36]  Frank Pfenning,et al.  Automated Deduction - CADE-21, 21st International Conference on Automated Deduction, Bremen, Germany, July 17-20, 2007, Proceedings , 2007, CADE.

[37]  Ernie Cohen,et al.  Separation and Reduction , 2000, MPC.