## Termination of Linear Programs (2004)

Venue: | In CAV’2004: Computer Aided Verification, volume 3114 of LNCS |

Citations: | 42 - 0 self |

### BibTeX

@INPROCEEDINGS{Tiwari04terminationof,

author = {Ashish Tiwari},

title = {Termination of Linear Programs},

booktitle = {In CAV’2004: Computer Aided Verification, volume 3114 of LNCS},

year = {2004},

pages = {70--82},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

We show that termination of a class of linear loop programs is decidable. Linear loop programs are discrete-time linear systems with a loop condition governing termination, that is, a while loop with linear assignments. We relate the termination of such a simple loop, on all initial values, to the eigenvectors corresponding to only the positive real eigenvalues of the matrix defining the loop assignments. This characterization of termination is reminiscent of the famous stability theorems in control theory that characterize stability in terms of eigenvalues.

### Citations

1399 |
A Discipline of Programming
- Dijkstra
- 1976
(Show Context)
Citation Context ...e interpreted over the set of integers, is decidable. 6 The General Case We consider the termination of a set of nondeterministic linear conditional assignments, written in a guarded command language =-=[8] as, P7 : [ B1x -=-> b1 −→ x := A1x + c1 [] B2x > b2 −→ x := A2x + c2 . . . [] Bkx > bk −→ x := Akx + ck ] which we will write in shorthand as P7: [] k i=1 (Bix > bi −→ x := Aix + ci)sCounter machines ca... |

112 | A survey of computational complexity results in systems and control
- Blondel, Tsitsiklis
(Show Context)
Citation Context ...ollability, are decidable for linear systems [10, 16], they soon become undecidable even when a “little” nonlinearity is introduced [16]. In particular, this is also true for piecewise affine syst=-=ems [3, 4]-=-, see also Section 6. In this context, it is interesting to note that termination is decidable for linear loop programs. Techniques to prove termination of programs have attracted renewed attention la... |

112 |
A complete method for the synthesis of linear ranking functions
- Podelski, Rybalchenko
- 2004
(Show Context)
Citation Context ... also Section 6. In this context, it is interesting to note that termination is decidable for linear loop programs. Techniques to prove termination of programs have attracted renewed attention lately =-=[7, 5, 6, 13]-=-. The popular approach to prove termination is through the synthesis of a ranking function, a mapping from the state space to a well-founded domain, whose value monotonically decreases as the system m... |

107 |
Dynamical Systems
- Arrowsmith, Place
- 1992
(Show Context)
Citation Context ...rminating. Example 3. Consider the program: while (x − y > 0) { x := −x + y; y := y }. The matrix A = [−1, 1; 0, 1] has two eigenvalues, 1 and −1. The eigenvector corresponding to the eigenval=-=ue 1 is [1; 2] and-=- we note that 1 − 2 �> 0. Hence, it follows from Corollary 2 that the above loop is terminating. 2.2 Two Variable Case Theorem 1 and Theorem 2 show that nonterminating linear loops almost always h... |

89 | Transition Invariants
- Podelski, Rybalchenko
- 2004
(Show Context)
Citation Context ... research has focused mostly on generating linear ranking functions – some effective heuristics have been proposed [5, 6] and recently a complete method was presented in [13] for a model motivated b=-=y [14]-=-. This paper investigates termination at a more basic theoretical level. The main result establishes the decidability of the termination problem for programs of the form (in matrix notation) while (Bx... |

64 | Symbolic reachability computation for families of linear vector fields
- Lafferriere, Pappas, et al.
(Show Context)
Citation Context .... In recent years, computer scientists have begun to apply the rich mathematical knowledge that has been developed in systems theory for analyzing such systems for safety properties, see for instance =-=[17, 12, 11]-=-. One of the most basic results in the theory of linear systems, both discretetime and continuous-time, is the characterization of the stability of linear systems in terms of the eigenvalues of the co... |

57 |
Commutation, transformation, and termination
- Bachmair, Dershowitz
- 1986
(Show Context)
Citation Context ...rminating. Example 3. Consider the program: while (x − y > 0) { x := −x + y; y := y }. The matrix A = [−1, 1; 0, 1] has two eigenvalues, 1 and −1. The eigenvector corresponding to the eigenval=-=ue 1 is [1; 2] and-=- we note that 1 − 2 �> 0. Hence, it follows from Corollary 2 that the above loop is terminating. 2.2 Two Variable Case Theorem 1 and Theorem 2 show that nonterminating linear loops almost always h... |

57 | Practical methods for proving program termination
- Colón, Sipma
- 2002
(Show Context)
Citation Context ... also Section 6. In this context, it is interesting to note that termination is decidable for linear loop programs. Techniques to prove termination of programs have attracted renewed attention lately =-=[7, 5, 6, 13]-=-. The popular approach to prove termination is through the synthesis of a ranking function, a mapping from the state space to a well-founded domain, whose value monotonically decreases as the system m... |

57 |
Fixed Point Theorems
- Smart
- 1974
(Show Context)
Citation Context ... x and y, written as x ∼ y, that are nonzero scalar multiples of each other (x = λy), then the resulting set (NT ′ / ∼) is closed and bounded (as a subset of ℜ n−1 ). By Brouwer’s fixed p=-=oint theorem [15], it follo-=-ws that there is an eigenvector v (with positive eigenvalue) of A in NT ′ . For all points u ∈ NT , we know c T u > 0. By continuity, for all points u ∈ NT ′ , we have c T u ≥ 0. If, in fact... |

50 |
Synthesis of linear ranking functions
- Colón, Sipma
(Show Context)
Citation Context ... also Section 6. In this context, it is interesting to note that termination is decidable for linear loop programs. Techniques to prove termination of programs have attracted renewed attention lately =-=[7, 5, 6, 13]-=-. The popular approach to prove termination is through the synthesis of a ranking function, a mapping from the state space to a well-founded domain, whose value monotonically decreases as the system m... |

39 |
Approximate reachability for linear systems
- Tiwari
- 2003
(Show Context)
Citation Context .... In recent years, computer scientists have begun to apply the rich mathematical knowledge that has been developed in systems theory for analyzing such systems for safety properties, see for instance =-=[17, 12, 11]-=-. One of the most basic results in the theory of linear systems, both discretetime and continuous-time, is the characterization of the stability of linear systems in terms of the eigenvalues of the co... |

23 | From linear to nonlinear: some complexity comparisons
- Sontag
- 1995
(Show Context)
Citation Context ...zed piecewise affine systems, which themselves are special kinds of nonlinear systems. While several properties, such as reachability, stability, and controllability, are decidable for linear systems =-=[10, 16], th-=-ey soon become undecidable even when a “little” nonlinearity is introduced [16]. In particular, this is also true for piecewise affine systems [3, 4], see also Section 6. In this context, it is in... |

22 | Deciding stability and mortality of piecewise affine systems
- Blondel, Bournez, et al.
- 2001
(Show Context)
Citation Context ...ollability, are decidable for linear systems [10, 16], they soon become undecidable even when a “little” nonlinearity is introduced [16]. In particular, this is also true for piecewise affine syst=-=ems [3, 4]-=-, see also Section 6. In this context, it is interesting to note that termination is decidable for linear loop programs. Techniques to prove termination of programs have attracted renewed attention la... |

18 |
Polynomial-Time Algorithm for the Orbit Problem
- Kannan, Lipton
- 1986
(Show Context)
Citation Context ...zed piecewise affine systems, which themselves are special kinds of nonlinear systems. While several properties, such as reachability, stability, and controllability, are decidable for linear systems =-=[10, 16], th-=-ey soon become undecidable even when a “little” nonlinearity is introduced [16]. In particular, this is also true for piecewise affine systems [3, 4], see also Section 6. In this context, it is in... |

12 | A heuristic for the automatic generation of ranking functions
- Dams, Gerth, et al.
(Show Context)
Citation Context |

1 |
Computing metatransitions for linear transition systems
- Musset, Rusinowitch
- 2003
(Show Context)
Citation Context .... In recent years, computer scientists have begun to apply the rich mathematical knowledge that has been developed in systems theory for analyzing such systems for safety properties, see for instance =-=[17, 12, 11]-=-. One of the most basic results in the theory of linear systems, both discretetime and continuous-time, is the characterization of the stability of linear systems in terms of the eigenvalues of the co... |