#### DMCA

## Proving termination starting from the end (2013)

Citations: | 6 - 0 self |

### Citations

2315 |
Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints
- Cousot, Cousot
- 1977
(Show Context)
Citation Context ... ′ with i < j we have (si, s j) ∈ W`. This contradicts that W` is well-founded and we are done. uunionsq Remark 1. When fixpoints are not computable, they can be approximated from above or from below =-=[14]-=-. It is routine to check that the results of Lemmas 3 and 4 remain valid when replacing G = gfp λY.W ∩ g̃(Y) in Def. 1 with G′ ⊆ gfp λY.W ∩ g̃(Y). Therefore we have that, even when approximating gfp λ... |

1422 |
Introduction to Lattices and Order
- Davey, Priestley
- 2002
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Citation Context ...ion R ⊆ Z2 × Z2 such that R = {〈(x, y), (x′, y′)〉 | x ≥ 0 ∧ x′ = x − y ∧ y′ = y}. Finally, we briefly recall classical results of lattice theory and refer to the classical book of Davey and Priestley =-=[15]-=- for further information. Let f be a function over a partially ordered set (L,v). A fixpoint of f is an element l ∈ L such that f (l) = l. We denote by lfp f and gfp f , respectively, the least and th... |

742 | Construction of abstract state graphs with PVS
- Graf, Saidi
- 1997
(Show Context)
Citation Context ...an underapproximation of gfp λY.W ∩ g̃(Y) therefore complying with the requirement on G. As far as abstract interpretation is concerned, our implementation uses a combination of predicate abstraction =-=[18]-=- and trace partitioning [22]. The set of predicates is given by a finite set of atomic linear constraints and is also closed under negation, e.g., if x+y ≥ 0 is a predicate then x+y ≤ −1 is also a pre... |

473 |
On a problem of formal logic
- Ramsey
- 1930
(Show Context)
Citation Context ...n where each W` is well-founded. For any si, s j ∈ S with i < j it holds that (si, s j) ∈ RG ◦ R∗, and thus, according to Lem. 3, we also have that (si, s j) ∈ W` for some 1 ≤ ` ≤ n. Ramsey’s theorem =-=[25]-=- guarantees the existence of an infinite subsequence S ′ = s j1 , s j2 , . . . of S , and a single W`, such that for all si, s j ∈ S ′ with i < j we have (si, s j) ∈ W`. This contradicts that W` is we... |

203 | The size-change principle for program termination.
- Lee, Jones, et al.
- 2001
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Citation Context ...](X), the set of non-terminating states, is computable. It is worth “terminating” by mentioning that several formulations, of the termination problem, similar to the check R+ ⊆ W have appeared before =-=[8,21,16]-=-. They have also led to practical tools for corresponding programming paradigms. The relation between these approaches was recently studied [19]. Works based on these formulations, in particular those... |

174 | A.: Termination proofs for systems code
- Cook, Podelski, et al.
- 2006
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Citation Context .... This result has important practical consequences because it triggered the emergence of effective techniques, based on transition invariants, to solve the termination problem for real-world programs =-=[11,2,28,20]-=-. By replacing the well-foundedness problem of R with the equivalent disjunctive well-foundedness problem of R+, one allows for the incremental construction of W: when the inclusion of R+ into W fails... |

119 | Transition invariants.
- Podelski, Rybalchenko
- 2004
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Citation Context ... paper, is a result showing that the well-foundedness problem of a given relation R is equivalent to the problem of asking whether the transitive closure of R, noted R+, is disjunctively well-founded =-=[24]-=-. That is whether R+ is included in some W (in which case W is called a transition invariant) such that W = W1 ∪ · · · ∪Wn, n ∈ N and each Wi is well-founded (in which case W is said to be disjunctive... |

106 | A semantic basis for the termination analysis of logic programs.
- Codish, Taboch
- 1999
(Show Context)
Citation Context ...](X), the set of non-terminating states, is computable. It is worth “terminating” by mentioning that several formulations, of the termination problem, similar to the check R+ ⊆ W have appeared before =-=[8,21,16]-=-. They have also led to practical tools for corresponding programming paradigms. The relation between these approaches was recently studied [19]. Works based on these formulations, in particular those... |

96 |
M'ethodes it'eratives de construction et d'approximation de points fixes d'op'erateurs monotones sur un treillis, analyse s'emantique de programmes
- Cousot
- 1978
(Show Context)
Citation Context ...where Acc denotes (an overapproximation of) the reachable states in the system. Formally, Acc is given by the least fixpoint lfp λX.I ∪ post[R](X). The second option is inspired by the work of Cousot =-=[12]-=- where he mixes backward and forward reasoning. We give here some intuitions and preliminary development. Recall that the greatest fixpoint gfp λY.W ∩ g̃(Y) of line 3 is best understood as the result ... |

85 | Trace partitioning in abstract interpretation based static analyzers
- Mauborgne, Rival
- 2005
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Citation Context ... λY.W ∩ g̃(Y) therefore complying with the requirement on G. As far as abstract interpretation is concerned, our implementation uses a combination of predicate abstraction [18] and trace partitioning =-=[22]-=-. The set of predicates is given by a finite set of atomic linear constraints and is also closed under negation, e.g., if x+y ≥ 0 is a predicate then x+y ≤ −1 is also a predicate. Abstract values are ... |

65 | Abstraction refinement for termination.
- Cook, Podelski, et al.
- 2005
(Show Context)
Citation Context ... states into account. In a future work, we will formally develop those two options and evaluate their benefit. Related Works. As for termination, our work is mostly related to the work of Cook et al. =-=[10,11]-=- where the inclusion check R+ ⊆ W [24] is put to work by incrementally constructing W. Our approach, being based on the dual check R ⊆ W−, adds a new dimension of modularity/incrementality in which R ... |

58 |
Fixpoint induction and proofs of program properties.
- Park
- 1969
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Citation Context ...dering ¬G instead, what we are looking for is an overapproximation of ¬(gfp λY.W ∩ g̃(Y)). Next we recall Park’s theorem replacing the above expression by a least fixpoint expression. Theorem 3 (From =-=[23]-=-). Let 〈L,v,d,⊔,>,⊥,¬〉 be a complete Boolean algebra and let f ∈ L → L be an order-preserving function then f ′ = λX.¬( f (¬X)) is an orderpreserving function on L and ¬(gfp f ) = lfp f ′. Park’s theo... |

50 | The polyranking principle.
- Bradley, Manna, et al.
- 2005
(Show Context)
Citation Context ...junction of linear constraints over the variables x̄ and x̄′. In this context, the set Q of states is equal to Zn where n is the number of variables in x̄. This is a classical setting for termination =-=[4,6,24]-=-. Internally, we represent sets of states and relations over them as DNF formulas where the atoms are linear constraints. In what follows, we explain sufficient implementation details so that our expe... |

45 | A general framework for automatic termination analysis of logic programs
- DERSHOWITZ, LINDENSTRAUSS, et al.
- 2001
(Show Context)
Citation Context ...](X), the set of non-terminating states, is computable. It is worth “terminating” by mentioning that several formulations, of the termination problem, similar to the check R+ ⊆ W have appeared before =-=[8,21,16]-=-. They have also led to practical tools for corresponding programming paradigms. The relation between these approaches was recently studied [19]. Works based on these formulations, in particular those... |

35 | Costa: Design and implementation of a cost and termination analyzer for java bytecode
- ALBERT, ARENAS, et al.
(Show Context)
Citation Context .... This result has important practical consequences because it triggered the emergence of effective techniques, based on transition invariants, to solve the termination problem for real-world programs =-=[11,2,28,20]-=-. By replacing the well-foundedness problem of R with the equivalent disjunctive well-foundedness problem of R+, one allows for the incremental construction of W: when the inclusion of R+ into W fails... |

23 | Partial completeness of abstract fixpoint checking
- Cousot
- 2000
(Show Context)
Citation Context ... is thus reduced to finding some W and prove that the inclusion hold. In the above inclusion check, R+ coincides with the least fixpoint of the function λY.R ∪ g(Y) where g def= λY.Y ◦ R. It is known =-=[13]-=- that if we can find an adjoint function g̃ to g such that g(X) ⊆ Y iff X ⊆ g̃(Y) for all X,Y then there exists an equivalent inclusion check to R+ ⊆ W. This equivalent check, denoted R ⊆ W− in the in... |

22 | Size-change termination, monotonicity constraints and ranking functions
- Ben-Amram
(Show Context)
Citation Context ...s that that every infinite trace in R must have a suffix that exclusively consists of transitions from R(2)B . Recursively applying Acabar on R (2) B does not yield any further partitioning, that is R=-=(3)-=-B = R (2) B . The reason being that no potential ranking function is automatically inferred. Thus, Acabar fails to prove well-foundedness of R, which is indeed not well-founded. However, due to the ab... |

22 | Proving conditional termination
- Cook, Gulwani, et al.
- 2008
(Show Context)
Citation Context ...ion problem, and the problem of inferring a precondition that guarantees termination. 2 Example In this section, we informally overview our proposed techniques on an example taken from the literature =-=[9]-=-. Consider the following loop: while ( x>0 ) { x:=x+y; y:=y+z; } represented by the transition relation R = {x > 0, x′ = x + y, y′ = y + z, z′ = z}, where the primed variables represent the values of ... |

22 | Termination analysis with compositional transition invariants.
- Kroening, Sharygina, et al.
- 2010
(Show Context)
Citation Context .... This result has important practical consequences because it triggered the emergence of effective techniques, based on transition invariants, to solve the termination problem for real-world programs =-=[11,2,28,20]-=-. By replacing the well-foundedness problem of R with the equivalent disjunctive well-foundedness problem of R+, one allows for the incremental construction of W: when the inclusion of R+ into W fails... |

16 | On the linear ranking problem for integer linear-constraint loops
- Ben-Amram, Genaim
- 2013
(Show Context)
Citation Context ...junction of linear constraints over the variables x̄ and x̄′. In this context, the set Q of states is equal to Zn where n is the number of variables in x̄. This is a classical setting for termination =-=[4,6,24]-=-. Internally, we represent sets of states and relations over them as DNF formulas where the atoms are linear constraints. In what follows, we explain sufficient implementation details so that our expe... |

15 | Deciding conditional termination
- Bozga, Iosif, et al.
- 2012
(Show Context)
Citation Context ... ending in Z; and (iii) compute P as the complement of V. Formally, (i) is given by a greatest fixpoint expression gfp λX. pre[RB](X). This expression is directly inspired by the work of Bozga et al. =-=[5]-=- on deciding conditional termination. This greatest fixpoint is the largest set Z of states each of which has an RB-successor inZ. Because of this property, every infinite RB-trace visits only states ... |

15 |
A termination analyzer for java bytecode based on path-length
- Spoto, Mesnard, et al.
- 2010
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Citation Context |

7 | Size-change termination and transition invariants
- Heizmann, Jones, et al.
- 2010
(Show Context)
Citation Context ..., similar to the check R+ ⊆ W have appeared before [8,21,16]. They have also led to practical tools for corresponding programming paradigms. The relation between these approaches was recently studied =-=[19]-=-. Works based on these formulations, in particular those that construct global ranking functions for R [3], might serve as a starting point to understand some (completeness) properties of our approach... |

6 |
S.: Termination proofs for linear simple loops
- Chen, Flur, et al.
- 2012
(Show Context)
Citation Context ..., using a combination of predicate abstraction and trace partitioning. 7 Experiments We have evaluated our prototype implementation against a set of benchmarks collected from publications in the area =-=[9,7]-=-. In what follows, we present the results of our implementation for those loops, and compare them to existing tools for proving termination [26,7,6] as well as tools for inferring preconditions for te... |

4 |
The Fixpoint Checking Problem: An Abstraction Refinement Perspective
- Ganty
- 2007
(Show Context)
Citation Context ...lk of A. Podelski at ETAPS ’11 who remarked that the inclusion check R+ ⊆ W is equivalently formulated as a safety verification problem where states are made of pairs. Back to late 2007, a PhD thesis =-=[17]-=- proposed a new approach to the safety verification problem in which the author shows how to leverage the equivalent backward and forward formulations of the inclusion check. Those two events planted ... |