## Linear invariant generation using non-linear constraint solving (2003)

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Venue: | IN COMPUTER AIDED VERIFICATION |

Citations: | 79 - 12 self |

### BibTeX

@INPROCEEDINGS{Colón03linearinvariant,

author = {Michael A. Colón and Sriram Sankaranarayanan and Henny B. Sipma},

title = {Linear invariant generation using non-linear constraint solving},

booktitle = {IN COMPUTER AIDED VERIFICATION},

year = {2003},

pages = {420--432},

publisher = {Springer Verlag}

}

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### Abstract

We present a new method for the generation of linear invariants which reduces the problem to a non-linear constraint solving problem. Our method, based on Farkas' Lemma, synthesizes linear invariants by extracting non-linear constraints on the coefficients of a target invariant from a program. These constraints guarantee that the linear invariant is inductive. We then apply existing techniques, including specialized quantifier elimination methods over the reals, to solve these non-linear constraints. Our method has the advantage of being complete for inductive invariants. To our knowledge, this is the first sound and complete technique for generating inductive invariants of this form. We illustrate the practicality of our method on several examples, including cases in which traditional methods based on abstract interpretation with widening fail to generate sufficiently strong invariants.

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Citation Context ...ssertion that strengthens it [16]. Thus invariant generation methods are, normally, methods for generating inductive assertions. The dominant invariant generation technique is abstract interpretation =-=[6]-=-. The main idea behind this approach is to perform an approximate symbolic ⋆ This research was supported in part by NSF(ITR) grant CCR-01-21403, by NSF grant CCR-99-00984-001, by ARO grant DAAD19-01-1... |

1546 |
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Citation Context ...rator often causes the technique to produce weak invariants. The design of a widening operator with some guarantee of completeness remains a key challenge for abstract interpretation based techniques =-=[7, 1]-=-. In fact, tools like HyTech [12] have given up widening in favor of extrapolation heuristics with no convergence guarantees and have reported good results. In this paper, we generate linear invariant... |

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Citation Context ...ful both as a direct means of checking program specifications and as an indirect means of obtaining intermediate assertions that can be used as lemmas for proving other safety and liveness properties =-=[16]-=-. An assertion is said to be inductive at a program location if it holds the first time the location is reached and is preserved under every cycle back to the location. It has been established that al... |

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Citation Context ...ial root solving. The original breakthrough in quantifier elimination was due to Tarski [18]. However, it was not computationally feasible until Collins introduced Cylindrical Algebraic Decomposition =-=[5]-=-. Recently, there have been many practical implementations based on this technique. Notable among them is qepcad, which incorporates many improvements to the original CAD algorithm [4]. The time compl... |

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Citation Context ...c Decomposition [5]. Recently, there have been many practical implementations based on this technique. Notable among them is qepcad, which incorporates many improvements to the original CAD algorithm =-=[4]-=-. The time complexity of the algorithm is exponential in the size of the formula. However, the running time can be brought down significantly for low degree polynomials using the elimination at test p... |

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Citation Context ...lation heuristics with no convergence guarantees and have reported good results. In this paper, we generate linear invariants for linear transition systems, a class of programs that is widely studied =-=[7, 1, 11]-=-. A large number of reactive systems may be modeled directly or approximately as linear transition systems [13]. Rather than perform a least fixed point computation by iteration, we solve constraints ... |

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Citation Context ...f additional, independent variables weakens the invariants generated using abstract interpretation, can be observed in more realistic programs, e.g. the implementation of mergesort presented by Wirth =-=[20]-=-. Heapsort We applied our method to heapsort, shown in Figure 3 and taken from [7]. Arrays and operations involving arrays were not modeled in the transition system, and branches involving array condi... |

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Citation Context ...is to compute the generators of the polyhedron and then project these generators on to the free variables. These generators can be computed by the simplex method [17] or the double description method =-=[9]-=-. In our examples, presented in the next section, we use an implementation of the double description method called PolKa [10]. Alternatively, projection and the computation of generators can be achiev... |

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Citation Context ...rithm is exponential in the size of the formula. However, the running time can be brought down significantly for low degree polynomials using the elimination at test points method due to Weispfenning =-=[19]-=-. After quantifier elimination, simplification is carried out using factorization and Gröbner Bases. This method has been implemented in redlog [8]. Another approach is to use linear programming and d... |

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Citation Context ...rator often causes the technique to produce weak invariants. The design of a widening operator with some guarantee of completeness remains a key challenge for abstract interpretation based techniques =-=[7, 1]-=-. In fact, tools like HyTech [12] have given up widening in favor of extrapolation heuristics with no convergence guarantees and have reported good results. In this paper, we generate linear invariant... |

18 |
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Citation Context ... non-linear assertion is called a semi-algebraic set. Problems such as satisfiability, projection, intersection and union, though still decidable, have a higher complexity than for linear constraints =-=[2]-=-. We say that a transition system is linear if its initial assertion Θ is linear over V and its transition relations ρτ are linear assertions over V ∪ V ′ . Consequently, for every simple path π, the ... |

13 |
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Citation Context ...aint solving. Thus, non-linear constraint solvers have been incorporated into the clp paradigm. For instance,sthe risc-clp(r) system uses quantifier elimination in the background to solve constraints =-=[14]-=-. Heuristics The constraints we obtain are of a low degree and hence, redlog is the most suitable tool. A disadvantage of using quantifier elimination is the size of the result after elimination. In g... |

12 |
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4 |
Model-checking strategies for hybrid systems. Presented at the
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Citation Context ... produce weak invariants. The design of a widening operator with some guarantee of completeness remains a key challenge for abstract interpretation based techniques [7, 1]. In fact, tools like HyTech =-=[12]-=- have given up widening in favor of extrapolation heuristics with no convergence guarantees and have reported good results. In this paper, we generate linear invariants for linear transition systems, ... |

2 |
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Citation Context ...can be computed by the simplex method [17] or the double description method [9]. In our examples, presented in the next section, we use an implementation of the double description method called PolKa =-=[10]-=-. Alternatively, projection and the computation of generators can be achieved through a quantifier elimination method called Fourier’s elimination, which eliminates variables from the system of constr... |