### Abstract

In order to verify semialgebraic programs, we automatize the Floyd/Naur/Hoare proof method. The main task is to automatically infer valid invariants and rank functions. First we express the program semantics in polynomial form. Then the unknown rank function and invariants are abstracted in parametric form. The implication in the Floyd/Naur/Hoare verification conditions is handled by abstraction into numerical constraints by Lagrangian relaxation. The remaining universal quantification is handled by semidefinite programming relaxation. Finally the parameters are computed using semidefinite programming solvers. This new approach exploits the recent progress in the numerical resolution of linear or bilinear matrix inequalities by semidefinite programming using efficient polynomial primal/dual interior point methods generalizing those well-known in linear programming to convex optimization. The framework is applied to invariance and termination proof of sequential, nondeterministic, concurrent, and fair parallel imperative polynomial programs and can easily be extended to other safety and liveness properties.

Original language | English (US) |
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Title of host publication | Lecture Notes in Computer Science |

Editors | R. Cousot |

Pages | 1-24 |

Number of pages | 24 |

Volume | 3385 |

State | Published - 2005 |

Event | 6th International Conference on Verification, Model Checking, and Abstract Interpretation, VMCAI 2005 - Paris, France Duration: Jan 17 2005 → Jan 19 2005 |

### Other

Other | 6th International Conference on Verification, Model Checking, and Abstract Interpretation, VMCAI 2005 |
---|---|

Country | France |

City | Paris |

Period | 1/17/05 → 1/19/05 |

### Fingerprint

### Keywords

- Bilinear matrix inequality (BMI)
- Convex optimization
- Invariance
- Lagrangian relaxation
- Linear matrix inequality (LMI)
- Liveness
- Parametric abstraction
- Polynomial optimization
- Proof
- Rank function
- S-procedure
- Safety
- Semidefinite programming

### ASJC Scopus subject areas

- Computer Science (miscellaneous)

### Cite this

*Lecture Notes in Computer Science*(Vol. 3385, pp. 1-24)

**Proving program invariance and termination by parametric abstraction, lagrangian relaxation and semidefinite programming.** / Cousot, Patrick.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science.*vol. 3385, pp. 1-24, 6th International Conference on Verification, Model Checking, and Abstract Interpretation, VMCAI 2005, Paris, France, 1/17/05.

}

TY - GEN

T1 - Proving program invariance and termination by parametric abstraction, lagrangian relaxation and semidefinite programming

AU - Cousot, Patrick

PY - 2005

Y1 - 2005

N2 - In order to verify semialgebraic programs, we automatize the Floyd/Naur/Hoare proof method. The main task is to automatically infer valid invariants and rank functions. First we express the program semantics in polynomial form. Then the unknown rank function and invariants are abstracted in parametric form. The implication in the Floyd/Naur/Hoare verification conditions is handled by abstraction into numerical constraints by Lagrangian relaxation. The remaining universal quantification is handled by semidefinite programming relaxation. Finally the parameters are computed using semidefinite programming solvers. This new approach exploits the recent progress in the numerical resolution of linear or bilinear matrix inequalities by semidefinite programming using efficient polynomial primal/dual interior point methods generalizing those well-known in linear programming to convex optimization. The framework is applied to invariance and termination proof of sequential, nondeterministic, concurrent, and fair parallel imperative polynomial programs and can easily be extended to other safety and liveness properties.

AB - In order to verify semialgebraic programs, we automatize the Floyd/Naur/Hoare proof method. The main task is to automatically infer valid invariants and rank functions. First we express the program semantics in polynomial form. Then the unknown rank function and invariants are abstracted in parametric form. The implication in the Floyd/Naur/Hoare verification conditions is handled by abstraction into numerical constraints by Lagrangian relaxation. The remaining universal quantification is handled by semidefinite programming relaxation. Finally the parameters are computed using semidefinite programming solvers. This new approach exploits the recent progress in the numerical resolution of linear or bilinear matrix inequalities by semidefinite programming using efficient polynomial primal/dual interior point methods generalizing those well-known in linear programming to convex optimization. The framework is applied to invariance and termination proof of sequential, nondeterministic, concurrent, and fair parallel imperative polynomial programs and can easily be extended to other safety and liveness properties.

KW - Bilinear matrix inequality (BMI)

KW - Convex optimization

KW - Invariance

KW - Lagrangian relaxation

KW - Linear matrix inequality (LMI)

KW - Liveness

KW - Parametric abstraction

KW - Polynomial optimization

KW - Proof

KW - Rank function

KW - S-procedure

KW - Safety

KW - Semidefinite programming

UR - http://www.scopus.com/inward/record.url?scp=24144488686&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=24144488686&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:24144488686

VL - 3385

SP - 1

EP - 24

BT - Lecture Notes in Computer Science

A2 - Cousot, R.

ER -