## Sat solving for argument filterings (2006)

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Venue: | In Logic for Programming, Artificial Intelligence and Reasoning (LPAR |

Citations: | 15 - 8 self |

### BibTeX

@INPROCEEDINGS{Codish06satsolving,

author = {Michael Codish and Vitaly Lagoon and René Thiemann and Jürgen Giesl},

title = {Sat solving for argument filterings},

booktitle = {In Logic for Programming, Artificial Intelligence and Reasoning (LPAR},

year = {2006},

pages = {30--44},

publisher = {Springer}

}

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### Abstract

Abstract. This paper introduces a propositional encoding for lexicographic path orders in connection with dependency pairs. This facilitates the application of SAT solvers for termination analysis of term rewrite systems based on the dependency pair method. We address two main inter-related issues and encode them as satisfiability problems of propositional formulas that can be efficiently handled by SAT solving: (1) the combined search for a lexicographic path order together with an argument filtering to orient a set of inequalities; and (2) how the choice of the argument filtering influences the set of inequalities that have to be oriented. We have implemented our contributions in the termination prover AProVE. Extensive experiments show that by our encoding and the application of SAT solvers one obtains speedups in orders of magnitude as well as increased termination proving power. 1

### Citations

1006 |
Term Rewriting and All That
- Baader, Nipkow
- 1998
(Show Context)
Citation Context ...n are not very powerful for proving termination. Example 2. Consider the following TRS R for division on natural numbers [2]. minus(x, 0) → x (1) minus(s(x), s(y)) → minus(x, y) (2) quot(0, s(y)) → 0 =-=(3)-=- quot(s(x), s(y)) → s(quot(minus(x, y), s(y))) (4) Rules (1) - (3) can easily be oriented using an LPO, but rule (4) cannot. To see this, observe that if we instantiate y by s(x), we obtain quot(s(x),... |

513 |
An extensible sat-solver
- Eén, Sörensson
(Show Context)
Citation Context ...lowing [5] and for argument filtering constraints (ca. 300 lines). (c) Interfaces to several SAT solvers (ca. 300 lines). For the scope of this paper all results are obtained using the MiniSAT solver =-=[8]-=-. For the translation to conjunctive normal form (CNF) we used the implementation of Tseitin’s algorithm [21] offered by SAT4J [19] - a freely available Java implementation of MiniSAT. Our implementat... |

469 | Termination of rewriting
- Dershowitz
- 1987
(Show Context)
Citation Context ...reduction order is an order which is well-founded, monotonic, and stable (closed under contexts and substitutions). In practice, most reduction orders amenable to automation are simplification orders =-=[7]-=-, i.e., they contain the embedding relation ≻emb. The lexicographic path order is one of the most prominent simplification orders and raises the associated decision problem: For terms s and t, does th... |

314 |
On the complexity of derivations in the propositional calculus
- Tseitin
- 1968
(Show Context)
Citation Context .... 300 lines). For the scope of this paper all results are obtained using the MiniSAT solver [8]. For the translation to conjunctive normal form (CNF) we used the implementation of Tseitin’s algorithm =-=[21]-=- offered by SAT4J [19] - a freely available Java implementation of MiniSAT. Our implementation uses several optimizations to minimize encoding size: 1. We apply basic simplification axioms for true an... |

229 | Termination of term rewriting using dependency pairs
- Arts, Giesl
- 2000
(Show Context)
Citation Context ...ermination problems and hence LPO is typically combined with more sophisticated termination proving techniques. One of the most popular and powerful such techniques is the dependency pair (DP) method =-=[2]-=-. Essentially, for any TRS the DP method generates a set of inequalities between terms. If one can find a well-founded order satisfying these inequalities, then termination is proved. A main advantage... |

70 | Automating the dependency pair method
- Hirokawa, Middeldorp
- 2005
(Show Context)
Citation Context ...sgemeinschaft DFG under grant GI 274/5-1.sof the DP method [9, 20], the choice of π also influences the set of usable rules which contribute to the inequalities that have to be oriented. As stated in =-=[14]-=-, “the dependency pairs method derives much of its power from the ability to use argument filterings to simplify constraints”. However, argument filterings represent a severe bottleneck for the automa... |

63 | The dependency pair framework: Combining techniques for automated termination proofs
- Giesl, Thiemann, et al.
- 2005
(Show Context)
Citation Context ...4GHz dual-CPU platform. The AProVE analyzer and our new SAT-based analyzer are run on an AMD Athlon 64 at 2.2 GHz. Apart from the reduction pair processor, we also used the dependency graph processor =-=[2, 10, 14]-=-, which is the other main processor of the dependency pair framework. This processor is used to split up dependency pair problems into 12ssmaller ones. As AProVE and TTT use slightly different techniq... |

48 |
Two generalizations of the recursive path ordering
- Kamin, Levy
- 1980
(Show Context)
Citation Context ...ymbols. Observe that if >F is strict then ≈F and ∼ are the identity of symbols and terms respectively. Each precedence >F on the symbols induces a lexicographic path order on terms. Definition 1 (LPO =-=[16]-=-). The lexicographic path order ≻LP O on terms induced by the partial order >F is defined as s = f(s1, . . . , sn) ≻LP O t if and only if one of the following holds: 1. t = g(t1, . . . , tm) and s ≻LP... |

46 | Proving and disproving termination of higher-order functions
- Giesl, Thiemann, et al.
(Show Context)
Citation Context ...x, 0) � π LP O true) ∧ τ(ge(0, s(y)) � π LP O false) ∧ τ(ge(s(x), s(y)) � π LP O ge(x, y)))) 4 The definition of ω can easily be adapted to more advanced definitions of usable rules as well, cf. e.g. =-=[2, 9, 11]-=-. 11s5 Implementation and Experiments The propositional encodings for LPO with argument filterings and for the reduction pair processors described in Sect. 3 and 4 have been fully implemented and inte... |

42 | Argument filtering transformation
- Kusakari, Nakamura, et al.
- 1999
(Show Context)
Citation Context ...air processor. Additional processors are described in [10]. For a DP problem (P, R), the reduction pair processor generates inequality constraints which should be satisfied by a reduction pair (�, ≻) =-=[18]-=- where � is reflexive, transitive, monotonic, and stable and ≻ is a stable well-founded order compatible with � (i.e., � ◦ ≻ ⊆ ≻ or ≻ ◦ � ⊆ ≻). However, ≻ need not be monotonic. A typical choice for a... |

28 | Solving partial order constraints for LPO termination
- Codish, Lagoon, et al.
- 2006
(Show Context)
Citation Context ...sive experiments show that by our encoding and the application of SAT solvers one obtains speedups in orders of magnitude as well as increased termination proving power. 1 Introduction In recent work =-=[5]-=-, Codish et al. introduce a propositional encoding of lexicographic path orders (LPO) and demonstrate that SAT solving can drastically speedup the solving of LPO termination problems. The key idea is ... |

25 | Improved Modular Termination Proofs Using Dependency Pairs
- Thiemann, Schneider-Kamp
- 2004
(Show Context)
Citation Context ...pecifies which parts of a term f(. . .) may be eliminated before comparing terms. In recent refinements ⋆ Supported by the Deutsche Forschungsgemeinschaft DFG under grant GI 274/5-1.sof the DP method =-=[9, 20]-=-, the choice of π also influences the set of usable rules which contribute to the inequalities that have to be oriented. As stated in [14], “the dependency pairs method derives much of its power from ... |

21 | S.Falke, Improving Dependency Pairs
- Giesl, Schneider-Kamp
- 2003
(Show Context)
Citation Context ...pecifies which parts of a term f(. . .) may be eliminated before comparing terms. In recent refinements ⋆ Supported by the Deutsche Forschungsgemeinschaft DFG under grant GI 274/5-1.sof the DP method =-=[9, 20]-=-, the choice of π also influences the set of usable rules which contribute to the inequalities that have to be oriented. As stated in [14], “the dependency pairs method derives much of its power from ... |

21 |
A.: Tyrolean termination tool
- Hirokawa, Middeldorp
- 2005
(Show Context)
Citation Context ...) are usable. Since the right-hand sides of the minus-rules do not contain additional symbols, these are in fact all of the usable rules. Hence, the quot-rules (3) and (4) are not usable. As shown in =-=[15, 20]-=-, under certain conditions on the reduction pair, Restriction (b) ensures that in chains s1 → t1, s2 → t2, . . . with tiσi →∗ R si+1σi+1, we have tiσi � si+1σi+1. The required conditions hold in parti... |

19 | Tsukuba termination tool
- Hirokawa, Middeldorp
- 2003
(Show Context)
Citation Context ...nd t, does there exist a precedence >F such that s ≻LP O t holds? This decision problem comes in two flavours: “strict-LPO” and “quasi-LPO” depending on whether >F is required to be strict or not. In =-=[13]-=-, the authors observe that finding >F such that s ≻LP O t is tantamount to solving a constraint obtained by unfolding the definition of s ≻LP O t. As an example, let F = {−, +, ∗}. Then there exists a... |

17 |
Ecient BDD encodings for partial order constraints with application to expert systems in software veri
- Kurihara, Kondo
- 2004
(Show Context)
Citation Context ...ups in orders of magnitude. We conclude in Sect. 6. 2 Preliminaries This section briefly describes the starting points for the rest of the paper: propositional encodings for lexicographic path orders =-=[5, 17]-=- and the dependency pair framework [2, 10, 14]. We refer to [3] for further details on term rewriting. We assume an algebra of terms constructed over given sets of symbols F and variables V. Let >F de... |

16 | Decision problems in ordered rewriting
- Comon, Narendran, et al.
- 1998
(Show Context)
Citation Context ...O” and “quasi-LPO” depending on whether >F is required to be strict or not. Finding >F such that s ≻LP O t is tantamount to solving a constraint obtained by unfolding the definition of s ≻LP O t, cf. =-=[7, 15]-=-. As an example, let F = {−, +, ∗}. Then there exists a strict precedence such that −(x + y) ≻LP O (−x) ∗ (−y) if and only if the partial order constraint (− >F ∗) ∨ ((+ >F ∗) ∧ (+ >F −)) has a soluti... |

13 |
Ordering-based methods for proving termination automatically
- Borralleras
- 2003
(Show Context)
Citation Context ...e the dependency pair method. The dependency pair framework [10] is a modular reformulation and improvement of Arts and Giesl’s dependency pair approach [2] which was also inspired by related work in =-=[4, 14]-=-. To ease readability, the following presentation is slightly simplified yet sufficient to state the contributions of this paper. For further details on the dependency pair framework see [10]. For a t... |

11 |
AProVE 1.2: Automatic termination proofs in the DP framework
- Giesl, Schneider-Kamp, et al.
(Show Context)
Citation Context ... extend this encoding to take into account the influence of an argument filtering on the set of usable rules. In Sect. 5 we describe the implementation of our results in the termination prover AProVE =-=[12]-=- and provide extensive experimental evidence which indicates speedups in orders of magnitude. We conclude in Sect. 6. 2 Preliminaries This section briefly describes the starting points for the rest of... |

7 | Termination proofs using gpo ordering constraints - Genet, Gnaedig - 1997 |

2 | Solving gpo ordering constraints with shared term data structure
- Genet, Gnaedig
- 1996
(Show Context)
Citation Context ...owing inequality constraints. minus(x, 0) � x minus(s(x), s(y)) � minus(x, y) MINUS(s(x), s(y)) � MINUS(x, y) ( ) (8) QUOT(s(x), s(y)) � MINUS(x, y) ( ) (9) QUOT(s(x), s(y)) � QUOT(minus(x, y), s(y)) =-=(10)-=- ( ) By Thm. 5, all dependency pairs corresponding to strictly decreasing inequalities (8) - (10) can be removed. To solve the inequalities we may take (� π LP O , ≻π LP O ) where π(minus)=1, π(s)=π(M... |