## Constraint-based linearrelations analysis (2004)

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Venue: | In Proc. SAS, LNCS 3148 |

Citations: | 29 - 2 self |

### BibTeX

@INPROCEEDINGS{Sankaranarayanan04constraint-basedlinearrelations,

author = {Sriram Sankaranarayanan and Henny B. Sipma and Zohar Manna},

title = {Constraint-based linearrelations analysis},

booktitle = {In Proc. SAS, LNCS 3148},

year = {2004},

pages = {53--68},

publisher = {Springer}

}

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### Abstract

1 Introduction Linear-relations analysis discovers linear relationships among the variables of aprogram that hold in all the reachable program states. Such relationships are called linear invariants. Invariants are useful in the verification of both safetyand liveness properties. Many existing techniques rely on the presence of these invariants to prove properties of interest. Some types of analysis, e.g., variable-bounds analysis, can be viewed as specializations of linear-relations analysis. Traditionally, this analysis is framed as an abstract interpretation in the domainof polyhedra [6, 7]. The analysis is carried out using a propagation-based technique, wherein increasingly accurate polyhedral iterates, converging towards thefinal result, are computed. This convergence is ensured through the use of widening, or extrapolation, operators. Such techniques are popular in the domains ofdiscrete and hybrid programs, motivating tools like

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2004 |
Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints
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Citation Context ...analysis, e.g., variablebounds analysis, can be viewed as specializations of linear-relations analysis. Traditionally, this analysis is framed as an abstract interpretation in the domain of polyhedra =-=[6, 7]-=-. The analysis is carried out using a propagation-based technique, wherein increasingly accurate polyhedral iterates, converging towards the final result, are computed. This convergence is ensured thr... |

1546 |
Theory of Linear and Integer Programming
- Schrijver
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Citation Context ...eal-valued variables. Constant reals are denoted by a, b with subscripts, and unknown coefficients by c, d with subscripts. Further details about linear assertions can be obtained from standard texts =-=[14]-=-.sDefinition 1 (Linear Assertions) A linear expression is of the form a1x1 + · · · + anxn + b. The expression is homogeneous iff b = 0, or else it is inhomogeneous. A linear inequality is of the form ... |

603 | Automatic Discovery of Linear Restraints among Variables of a
- Cousot, Halbwachs
- 1978
(Show Context)
Citation Context ...analysis, e.g., variablebounds analysis, can be viewed as specializations of linear-relations analysis. Traditionally, this analysis is framed as an abstract interpretation in the domain of polyhedra =-=[6, 7]-=-. The analysis is carried out using a propagation-based technique, wherein increasingly accurate polyhedral iterates, converging towards the final result, are computed. This convergence is ensured thr... |

594 |
A Decision Method for Elementary Algebra and Geometry
- Tarski
- 1951
(Show Context)
Citation Context ...straints themselves are non-linear with existentially quantified parameters. Nevertheless, an exact elimination is possible in theory through quantifier elimination techniques for the theory of reals =-=[15, 4, 16]-=-. In practice, however, the technique using exact quantifier elimination does not scale to systems with more than five variables. Fortunately, the constraints obtained in this process, though non-line... |

402 |
Temporal verification of reactive systems: Safety
- Manna, Pnueli
- 1995
(Show Context)
Citation Context ... section, we define linear transition systems and linear invariants. Our presentation concentrates only on linear systems. The reader is referred to standard textbooks for a more general presentation =-=[12]-=-. Definition 2 (Linear Transition Systems) Let V = {x1, . . . , xn} be a set of system variables. A linear transition system over V is a tuple 〈L, T , ℓ0, Θ〉, where L is a set of locations, T is a set... |

388 |
Quantifier elimination for real closed fields by cylindrical algebraic decomposition
- Collins
(Show Context)
Citation Context ...straints themselves are non-linear with existentially quantified parameters. Nevertheless, an exact elimination is possible in theory through quantifier elimination techniques for the theory of reals =-=[15, 4, 16]-=-. In practice, however, the technique using exact quantifier elimination does not scale to systems with more than five variables. Fortunately, the constraints obtained in this process, though non-line... |

116 | Verification of real-time systems using linear relation analysis,” CAV’93: Computer Aided Verification
- Halbwachs, Proy, et al.
- 1997
(Show Context)
Citation Context ... of widening, or extrapolation, operators. Such techniques are popular in the domains of discrete and hybrid programs, motivating tools like HyTech [11] and improved widening operators over polyhedra =-=[10, 1]-=-. ⋆ This research was supported in part by NSF grants CCR-01-21403, CCR-02-20134 and CCR-02-09237, by ARO grant DAAD19-01-1-0723, by ARPA/AF contracts F33615-00-C-1693 and F33615-99-C-3014, and by NAV... |

84 |
The complexity of linear problems in fields
- Weispfenning
- 1988
(Show Context)
Citation Context ...straints themselves are non-linear with existentially quantified parameters. Nevertheless, an exact elimination is possible in theory through quantifier elimination techniques for the theory of reals =-=[15, 4, 16]-=-. In practice, however, the technique using exact quantifier elimination does not scale to systems with more than five variables. Fortunately, the constraints obtained in this process, though non-line... |

83 | HYTECH : The Cornell HYbrid TECHnology Tool
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- 1995
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Citation Context ...omputed. This convergence is ensured through the use of widening, or extrapolation, operators. Such techniques are popular in the domains of discrete and hybrid programs, motivating tools like HyTech =-=[11]-=- and improved widening operators over polyhedra [10, 1]. ⋆ This research was supported in part by NSF grants CCR-01-21403, CCR-02-20134 and CCR-02-09237, by ARO grant DAAD19-01-1-0723, by ARPA/AF cont... |

81 | The double description method revisited
- Fukuda, Prodon
- 1996
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Citation Context ....32 57 27 0 0 0 0.01 0.01 0 − − lifo(7) 0.88 58 51 0.29 0.27 0.02 0.32 0.29 0.03 �= �= lifo-nat(7) 10.13 1191 593 0.27 0.25 0.02 0.32 0.27 0.04 + + cars-midpt(7) 0.1 17 8 32.8 5 27.8 > 3600 + ? barber=-=(8)-=- 1.68 125 84 0.18 0.17 0.01 20.41 0.18 20.23 + + swim(9) 0.42 36 22 0.08 0.06 0.02 0.61 0.06 0.55 − − swim1(9) 0.88 65 32 0.07 0.06 0.01 0.59 0.06 0.53 = = Fig. 3. Result on Low-dimensional Systems. #... |

79 | Linear invariant generation using non-linear constraint solving
- Colón, Sankaranarayanan, et al.
- 2003
(Show Context)
Citation Context ...ns arising from abstract interpretation may be posed explicitly and solved without relying directly on iteration or widening. This is achieved through applications of Farkas Lemma in our earlier work =-=[5]-=-. Given a template inequality with unknown coefficients, our technique computes constraints on the values of the coefficients, such that substituting any solution back into the template yields a valid... |

72 | FAST: Fast Acceleration of Symbolic Transition systems
- Bardin, Finkel, et al.
- 2003
(Show Context)
Citation Context ...nger invariants. 4.1 Low Dimensional Systems Figure 3 shows the experimental results for some small to medium sized examples from the related work and some benchmarks from analysis tools such as fast =-=[3]-=-. The number of variables for each program is shown in the second column. The table shows for each program the time (in seconds) of our (constraint-based) approach, and the time taken by the ch79 and ... |

66 | Possibly not closed convex polyhedra and the Parma Polyhedra Library
- Bagnara, Ricci, et al.
- 2002
(Show Context)
Citation Context ...cient and simple, especially if some invariants are to be generated in as short a time as possible. 4 Experimental Results Our prototype implementation uses the library ppl for manipulating polyhedra =-=[2]-=-, and our own implementation of some of the rules in Figure 2, discussed below. PPL provides implementations of many widening operators [7, 1]. We consider the standard ch79 widening and a refinement ... |

44 | Precise widening operators for convex polyhedra
- Bagnara, Hill, et al.
(Show Context)
Citation Context ... of widening, or extrapolation, operators. Such techniques are popular in the domains of discrete and hybrid programs, motivating tools like HyTech [11] and improved widening operators over polyhedra =-=[10, 1]-=-. ⋆ This research was supported in part by NSF grants CCR-01-21403, CCR-02-20134 and CCR-02-09237, by ARO grant DAAD19-01-1-0723, by ARPA/AF contracts F33615-00-C-1693 and F33615-99-C-3014, and by NAV... |

18 | Polyhedral analysis for synchronous languages
- Besson, Jensen, et al.
- 1999
(Show Context)
Citation Context ... ϕ that contains (roughly) all the inequalities in ϕ1 that are satisfied by ϕ2. The details along with key mathematical properties of widening are described in [7, 6], and enhanced versions appear in =-=[10, 3, 1]-=-. As mentioned earlier the analysis begins with an initial assertion map defined by η0(ℓ0) = Θ, and η0(ℓ) = ∅ for ℓ �= ℓ0. At each step, the map ηi is updated to map ηi+1 as follows: ⎡ ηi+1(ℓ) = ηi(ℓ)... |

7 | Cartesian factoring of polyhedra in linear relation analysis
- Halbwachs, D, et al.
- 2003
(Show Context)
Citation Context ...0.27 0.02 0.32 0.29 0.03 �= �= lifo-nat(7) 10.13 1191 593 0.27 0.25 0.02 0.32 0.27 0.04 + + cars-midpt(7) 0.1 17 8 32.8 5 27.8 > 3600 + ? barber(8) 1.68 125 84 0.18 0.17 0.01 20.41 0.18 20.23 + + swim=-=(9)-=- 0.42 36 22 0.08 0.06 0.02 0.61 0.06 0.55 − − swim1(9) 0.88 65 32 0.07 0.06 0.01 0.59 0.06 0.53 = = Fig. 3. Result on Low-dimensional Systems. #br represents the number of branches, #sub denotes the n... |

7 | Petri Net Analysis Using Invariant Generation
- Sankaranarayanan, Sipma, et al.
- 2004
(Show Context)
Citation Context ...eries of simplifications resolves the constraints into a linear system. For instance, whenever the underlying transition system is a Petri net, the system of constraints resolves into a linear system =-=[13]-=-. This has led us to verify transition systems derived from Petri Nets with as many as 40 dimensions and 50 transitions. The use of quantifier elimination is clearly inefficient in such situations. In... |