Endomorphisms for Non-trivial Non-linear Loop Invariant Generation

Present approaches for non-linear loop invariant generation are limited to linear (affine) systems, or they relay on non scalable methods which have high complexity. Moreover, for programs with nested loops and conditional statements that describe multivariate polynomials or multivariate fractional systems, no applicable method is known to lend itself to non-trivialnon-linear invariants generation. We demonstrate a powerful computational completemethod to solve this problem. Our approach avoids first-order quantifier elimination, cylindrical algebraic decomposition and Grobner bases computation, hereby circumventing difficulties met by recent methods.

[1]  Enric Rodríguez-Carbonell,et al.  Generating all polynomial invariants in simple loops , 2007, J. Symb. Comput..

[2]  Robert W. Floyd,et al.  Assigning Meanings to Programs , 1993 .

[3]  Laura Kovács,et al.  Reasoning Algebraically About P-Solvable Loops , 2008, TACAS.

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

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

[6]  Zohar Manna,et al.  Temporal verification of reactive systems - safety , 1995 .

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

[8]  Yinghua Chen,et al.  Generating Polynomial Invariants with DISCOVERER and QEPCAD , 2007, Formal Methods and Hybrid Real-Time Systems.

[9]  Markus Müller-Olm,et al.  Polynomial Constants Are Decidable , 2002, SAS.

[10]  Patrick Cousot,et al.  Abstract Interpretation and Application to Logic Programs , 1992, J. Log. Program..

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

[12]  Enric Rodríguez-Carbonell,et al.  Automatic generation of polynomial invariants of bounded degree using abstract interpretation , 2007, Sci. Comput. Program..

[13]  Zohar Manna,et al.  Temporal Verification of Reactive Systems , 1995, Springer New York.

[14]  Deepak Kapur.  Automatically Generating Loop Invariants Using Quantifier Elimination , 2005, Deduction and Applications.

[15]  Yassine Lakhnech,et al.  A Transformational Approach for Generating Non-linear Invariants , 2000, SAS.

[16]  Ashish Tiwari,et al.  A Technique for Invariant Generation , 2001, TACAS.

[17]  Hassen Saïdi,et al.  Powerful Techniques for the Automatic Generation of Invariants , 1996, CAV.

[18]  Henny B. Sipma,et al.  Non-linear loop invariant generation using Gröbner bases , 2004, POPL.

[19]  Volker Weispfenning,et al.  Quantifier Elimination for Real Algebra — the Quadratic Case and Beyond , 1997, Applicable Algebra in Engineering, Communication and Computing.