## Ground reducibility is EXPTIME-complete (1997)

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Venue: | Information and Computation |

Citations: | 19 - 9 self |

### BibTeX

@ARTICLE{Comon97groundreducibility,

author = {Hubert Comon and Florent Jacquemard},

title = {Ground reducibility is EXPTIME-complete},

journal = {Information and Computation},

year = {1997},

volume = {187},

pages = {2003}

}

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### Abstract

Abstract. We prove that ground reducibility is EXPTIME-complete in the general case. EXP-TIME-hardness is proved by encoding the emptiness problem for the intersection of recognizable tree languages. It is more difficult to show that ground reducibility belongs to DEXPTIME. We associate first an automaton with disequality constraints AR,t to a rewrite system R and a term t. This automaton is deterministic and accepts at least one term iff t is not ground reducible by R. The number of states of AR,t is O(2 �R��t � ) and the size of its constraints is polynomial in the size of R, t. Then we prove some new pumping lemmas, using a total ordering on the computations of the automaton. Thanks to these lemmas, we can show that emptiness for an automaton with disequality constraints can be decided in a time which is polynomial in the number of states and exponential in the size of the constraints. Altogether, we get a simply exponential time deterministic algorithm for ground reducibility decision. 1

### Citations

751 | Rewrite systems
- Dershowitz, Jouannaud
- 1992
(Show Context)
Citation Context ...we write t|p for the subterm of t at position p and t[s]p for the term obtained by replacing t|p by s (at position p) int. We assume the reader familiar with (constrained) term rewriting systems (see =-=[6]-=- for a survey). Let us only recall that a term t is ground reducible by a rewrite system R iff all the ground instances of t are reducible by R. The rewriting relation associated to a rewrite system R... |

95 |
E.: Logic programs as types for logic programs
- Frühwirth, Shapiro, et al.
- 1991
(Show Context)
Citation Context ...bound was proved by Kapur et al [10]. We give here a simple proof of EXPTIME-hardness. It is known that the emptiness problem for the intersection of n recognizable languages is EXPTIME-complete, see =-=[7, 14]-=-. We show here that this problem is reducible to ground reducibility in polynomial time. In section 2, we recall the definition of automata with disequality constraints. In section 3, we show how to c... |

78 |
Deciding equivalence of finite tree automata
- Seidl
- 1990
(Show Context)
Citation Context ...ion lemma relies on these properties. 6 Lower bound Theorem 22 Ground reducibility is EXPTIME-hard, for linear rewrite systems R and linear terms t. The proof is very much similar to H. Seidl's proof =-=[15]-=- that inclusion of tree languages is EXPTIME-hard. We encode computation trees of an alternating Turing machine M which works on a polynomially bounded space as a term in T (F). (This encoding is poly... |

44 |
Reduction properties and automata with constraints
- Dauchet, Caron, et al.
- 1995
(Show Context)
Citation Context ...m of ground reducibility as an emptiness problem for some tree language [3]. He also gave a decision proof whose complexity is even worse than the former ones. A.-C Caron, J.-L. Coquid and M. Dauchet =-=[2, 5]-=- proved a very beautiful result in 1993, enlighting the pumping properties and their difficulty. They actually show a more general result: the first-order theory of unary encompassment predicates is d... |

42 | overloading is dexptime-complete
- Seidl, Haskell
- 1994
(Show Context)
Citation Context ...bound was proved by Kapur et al [10]. We give here a simple proof of EXPTIME-hardness. It is known that the emptiness problem for the intersection of n recognizable languages is EXPTIME-complete, see =-=[7, 14]-=-. We show here that this problem is reducible to ground reducibility in polynomial time. In section 2, we recall the definition of automata with disequality constraints. In section 3, we show how to c... |

36 |
Automatic proofs by induction in theories without constructors
- Jouannaud, Kounalis
- 1989
(Show Context)
Citation Context ...em R expresses that all ground instances (instances without variables) of t are reducible by R. This property is fundamental in automating inductive proofs in equational theories without constructors =-=[9]-=-. It is also related to sufficient completeness in algebraic specifications (see e.g. [11]). Roughly, it expresses that all cases have been covered by R and that t will be reducible for any inputs. Ma... |

36 |
Semantic confluence tests and completion methods
- Plaisted
- 1985
(Show Context)
Citation Context ... devoted to decision of ground reducibility. Let us report a brief history of the milestones, starting only in 1985 with the general case. Ground reducibility was first shown decidable by D. Plaisted =-=[13]-=-. The algorithm is however quite complex: a tower of 9 exponentials though there is no explicit complexity analysis in the paper. D. Kapur et al. [11] gave another decidability proof which is conceptu... |

22 |
On sufficient completeness and related properties of term rewriting systems
- Kapur, Narendran, et al.
- 1987
(Show Context)
Citation Context ... by R. This property is fundamental in automating inductive proofs in equational theories without constructors [9]. It is also related to sufficient completeness in algebraic specifications (see e.g. =-=[11]-=-). Roughly, it expresses that all cases have been covered by R and that t will be reducible for any inputs. Many papers have been devoted to decision of ground reducibility. Let us report a brief hist... |

13 |
J.L.: Automata for reduction properties solving
- Dauchet, Caron, et al.
- 1995
(Show Context)
Citation Context ...m of ground reducibility as an emptiness problem for some tree language [3]. He also gave a decision proof whose complexity is even worse than the former ones. A.-C Caron, J.-L. Coquid and M. Dauchet =-=[2, 5]-=- proved a very beautiful result in 1993, enlighting the pumping properties and their difficulty. They actually show a more general result: the first-order theory of unary encompassment predicates is d... |

11 | F.: Pumping, cleaning and symbolic constraints solving
- Caron, Comon, et al.
- 1994
(Show Context)
Citation Context ...ives any hint on how to implement a ground reducibility test. As we have seen, it is not tractable in general. A possible way to implement these techniques as efficiently as possible was suggested in =-=[1]-=-. In the average, some algorithms may behave well. In any case, we claim that tree automata help both in theory and in practice. Acknowledgments The authors wish to thank Bruce Reed for his proof of t... |

11 |
Sufficient completeness, ground reducibility and their complexity
- Kapur, Narendran, et al.
- 1991
(Show Context)
Citation Context ...orithm for ground reducibility. This result is optimal since ground reducibility is EXPTIME-hard, already for linear rewrite systems and linear t. AO(2 n log n ) lower bound was proved by Kapur et al =-=[10]-=-. We give here a simple proof of EXPTIME-hardness. It is known that the emptiness problem for the intersection of n recognizable languages is EXPTIME-complete, see [7, 14]. We show here that this prob... |

8 | Ground reducibility and automata with disequality constraints
- Comon, Jacquemard
- 1994
(Show Context)
Citation Context ...lexity which results from their proof is not better than Plaisted’s bound. We tried to specialize the tree automata technique for ground reducibility and we got in this way a triple exponential bound =-=[4]-=-. This is better than previous methods, but still far from the lower bound. The problem in all works about ground reducibility is that they give a bound on the depth of a minimal irreducible instance ... |

8 |
Reductions in tree replacement systems
- Gallier, Book
- 1985
(Show Context)
Citation Context ...guage. In support of this idea, when both t and the left members of R are linear, i.e. each variable appears only once, then the set of reducible instances of t is accepted by a finite tree automaton =-=[8]-=-. Hence the set of irreducible ground instances is also accepted by a trees2 automaton, by complement. This easily gives a simply exponential algorithm in the linear case. (As we will see this algorit... |

6 |
An effective method for handling initial algebras
- Comon
- 1988
(Show Context)
Citation Context ...simply exponential algorithm in the linear case. (As we will see this algorithm is optimal). H. Comon expressed first the problem of ground reducibility as an emptiness problem for some tree language =-=[3]-=-. He also gave a decision proof whose complexity is even worse than the former ones. A.-C Caron, J.-L. Coquid and M. Dauchet [2, 5] proved a very beautiful result in 1993, enlighting the pumping prope... |

4 |
Testing for the ground (co)-reducibility in term rewriting systems
- Kounalis
- 1992
(Show Context)
Citation Context ...can be reduced to checking the reducibility of all ground instances of t of depth smaller than N(R) where N(R) is a tower of 5 exponentials in the size of R. A third proof was proposed by E. Kounalis =-=[12]-=-. The result is generalized to co-ground reducibility and the expected complexity is 5 exponentials, though there is no explicit complexity analysis in the paper. These three algorithms use combinator... |

3 |
Automates d’Arbres et Réécriture de Termes
- Jacquemard
- 1996
(Show Context)
Citation Context ...6. Some proofs are long and technical, hence not included in the present abstract. Except for a small combinatorial argument due to B. Reed, the complete proofs can be found in F. Jacquemard's thesis =-=[9] (in-=- French). A complete version of the paper is also available on the Web (http://www.mpi-sb.de/��florent/art/). 2 Automata with disequality constraints F will always be a fixed finite set of functio... |