Policy Iteration-Based Conditional Termination and Ranking Functions

Termination analyzers generally synthesize ranking functions or relations, which represent checkable proofs of their results. In [23], we proposed an approach for conditional termination analysis based on abstract fixpoint computation by policy iteration. This method is not based on ranking functions and does not directly provide a ranking relation, which makes the comparison with existing approaches difficult. In this paper we study the relationships between our approach and ranking functions and relations, focusing on extensions of linear ranking functions. We show that it can work on programs admitting a specific kind of segmented ranking functions, and that the results can be checked by the construction of a disjunctive ranking relation. Experimental results show the interest of this approach.

[1]  Rajeev Alur,et al.  A Temporal Logic of Nested Calls and Returns , 2004, TACAS.

[2]  Patrick Cousot,et al.  Méthodes itératives de construction et d'approximation de points fixes d'opérateurs monotones sur un treillis, analyse sémantique des programmes , 1978 .

[3]  Daniel Kroening,et al.  Termination Analysis with Compositional Transition Invariants , 2010, CAV.

[4]  Maurizio Parton,et al.  Deriving Numerical Abstract Domains via Principal Component Analysis , 2010, SAS.

[5]  Henny B. Sipma,et al.  Scalable Analysis of Linear Systems Using Mathematical Programming , 2005, VMCAI.

[6]  Marius Bozga,et al.  Deciding Conditional Termination , 2012, TACAS.

[7]  Ankur Taly,et al.  Static Analysis by Policy Iteration on Relational Domains , 2007, ESOP.

[8]  Paul Feautrier,et al.  Multi-dimensional Rankings, Program Termination, and Complexity Bounds of Flowchart Programs , 2010, SAS.

[9]  Andreas Podelski,et al.  Abstraction Refinement for Termination , 2005, SAS.

[10]  Caterina Urban,et al.  The Abstract Domain of Segmented Ranking Functions , 2013, SAS.

[11]  Helmut Seidl,et al.  Precise Fixpoint Computation Through Strategy Iteration , 2007, ESOP.

[12]  Sumit Gulwani,et al.  Proving Conditional Termination , 2008, CAV.

[13]  Bertrand Jeannet,et al.  Apron: A Library of Numerical Abstract Domains for Static Analysis , 2009, CAV.

[14]  Sriram Sankaranarayanan,et al.  Generalizing the Template Polyhedral Domain , 2011, ESOP.

[15]  A. Rybalchenko,et al.  Transition invariants , 2004, LICS 2004.

[16]  Yassamine Seladji,et al.  Fixpoint Computation in the Polyhedra Abstract Domain Using Convex and Numerical Analysis Tools , 2013, VMCAI.

[17]  Rocco De Nicola.  Programming Languages and Systems, 16th European Symposium on Programming, ESOP 2007, Held as Part of the Joint European Conferences on Theory and Practics of Software, ETAPS 2007, Braga, Portugal, March 24 - April 1, 2007, Proceedings , 2007, ESOP.

[18]  Meenakshi D'Souza,et al.  Theoretical Aspects of Computing – ICTAC 2012 , 2012, Lecture Notes in Computer Science.

[19]  Patrick Cousot,et al.  An abstract interpretation framework for termination , 2012, POPL '12.

[20]  Agostino Cortesi,et al.  Verification, Model Checking, and Abstract Interpretation , 2002, Lecture Notes in Computer Science.

[21]  L. Hogben.  Handbook of Linear Algebra , 2006 .

[22]  Supratik Mukhopadhyay,et al.  Termination Proofs for Linear Simple Loops , 2012, SAS.

[23]  Kenneth H. Rosen,et al.  Discrete Mathematics and its applications , 2000 .

[24]  Brian Campbell,et al.  Amortised Memory Analysis Using the Depth of Data Structures , 2009, ESOP.

[25]  Eric Goubault,et al.  A Policy Iteration Algorithm for Computing Fixed Points in Static Analysis of Programs , 2005, CAV.

[26]  Patrick Cousot,et al.  Proving Program Invariance and Termination by Parametric Abstraction, Lagrangian Relaxation and Semidefinite Programming , 2005, VMCAI.

[27]  Andreas Podelski,et al.  A Complete Method for the Synthesis of Linear Ranking Functions , 2004, VMCAI.

[28]  Marius Bozga,et al.  Deciding Conditional Termination , 2012, TACAS.

[29]  P. Cousot.  Thesis: These d'Etat es sciences mathematiques: Methodes iteratives de construction et d'approximation de points fixes d'operateurs monotones sur un treillis, analyse semantique de programmes (in French) , 1978 .

[30]  Kousha Etessami,et al.  Analysis of Recursive Game Graphs Using Data Flow Equations , 2004, VMCAI.

[31]  Ashish Tiwari,et al.  Termination of Linear Programs , 2004, CAV.

[32]  Liyun Dai,et al.  Non-termination Sets of Simple Linear Loops , 2012, ICTAC.

[33]  Eric Goubault,et al.  Coupling policy iteration with semi-definite relaxation to compute accurate numerical invariants in static analysis , 2010, Log. Methods Comput. Sci..

[34]  Florian Zuleger,et al.  Ramsey vs. Lexicographic Termination Proving , 2013, TACAS.

[35]  Xavier Rival,et al.  The trace partitioning abstract domain , 2007, TOPL.

[36]  Damien Massé,et al.  Proving Termination by Policy Iteration , 2012, NSAD@SAS.

[37]  Matthieu Martel,et al.  Static Analysis - 17th International Symposium, SAS 2010, Perpignan, France, September 14-16, 2010. Proceedings , 2010, SAS.