## Path Rewriting and Combined Word Problems (2000)

Citations: | 1 - 0 self |

### BibTeX

@TECHREPORT{Fiorentini00pathrewriting,

author = {Camillo Fiorentini and Silvio Ghilardi},

title = {Path Rewriting and Combined Word Problems},

institution = {},

year = {2000}

}

### OpenURL

### Abstract

. We give an algorithm solving combined word problems (over non necessarily disjoint signatures) based on rewriting of equivalence classes of terms. The canonical rewriting system we introduce consists of few transparent rules and is obtained by applying Knuth-Bendix completion procedure to presentations of pushouts among categories with products. It applies to pairs of theories which are both constructible over their common reduct (on which we do not make any special assumption) . Lavoro svolto nell'ambito del progetto MURST \Logica". 1 1 Introduction An essential problem in automated deduction consists in integrating theorem provers which are able to perform separated tasks. In the eld of equational logic, this leads in particular to the following question: suppose you are able to solve word problems for theories T 1 ; T 2 ; can you solve word problem for T 1 [ T 2 ? Better, can you design an algorithm taking as input two arbitrary algorithms for word problems for T 1 and T 2...

### Citations

950 |
Term Rewriting and all That
- Baader, Nipkow
- 1998
(Show Context)
Citation Context ...and less interested in their proofs). This technical report is fully detailed and self-contained. We only assume a certain familiarity with rewriting (for some unexplained notions readers may consult [2]). 2 A short summary in functorial semantics We recall that a category withsnite products C is a category in which for everysnite list of objects X 1 ; : : : ; X n (n 0) there is an object X 1 ... |

67 |
Categories of continuous functors
- Freyd, Kelly
(Show Context)
Citation Context ...sociated `to the left' and arrows in the second class will be associated `to the right', as in the case in which 0 is empty. There is a standard notion of factorization system in category theory (see =-=[6]-=-), however such a notion is too strong in the present context, so that we weaken it. Let C be any category; by a weak factorization system in C, we mean a pair of classes of arrows (E ; M) from C such... |

29 |
Lawvere. Functorial semantics of algebraic theories
- William
- 1963
(Show Context)
Citation Context ...we shall put such complex combinatorics under the control framework provided by the categorical approach to equational logic: such approach goes back to the classical pioneering paper of F.W. Lawvere =-=[9-=-] in functorial semantics. 2 Basically, equational theories are identied with categories with products, so that in our situation we need to manipulate presentations of pushouts among such categories. ... |

27 |
Combining matching algorithms: The regular case
- Nipkow
- 1991
(Show Context)
Citation Context ...lem for T 1 [ T 2 ? In case T 1 ; T 2 have disjoint signatures the positive answer was known from long time [12], although only more recently discovered within automated deduction community (see e.g. =-=[11]-=-). In the general case, combining decidable word problems may leads to undecidability, even if we suppose that T 1 ; T 2 are both conservative over their common reduct T 0 . To this aim, consider the ... |

25 | Combination techniques for non-disjoint equational theories
- Domenjoud, Klay, et al.
- 1994
(Show Context)
Citation Context ...terminology `they are both constructible over T 0 '). There are various denitions of constructors and depending on such denitions there are variable strength results. Main papers on the subject are [5] and [3, 4]: the second has a weaker denition and consequently a stronger result. Our denition is again weaker (see Section 10 for details) and, more important, it covers natural mathematical exampl... |

24 | Deciding the word problem in the union of equational theories sharing constructors
- Baader, Tinelli
- 1999
(Show Context)
Citation Context ...ogy `they are both constructible over T 0 '). There are various denitions of constructors and depending on such denitions there are variable strength results. Main papers on the subject are [5] and [3=-=, -=-4]: the second has a weaker denition and consequently a stronger result. Our denition is again weaker (see Section 10 for details) and, more important, it covers natural mathematical examples and does... |

20 | Monads and modular term rewriting
- Lüth, Ghani
- 1997
(Show Context)
Citation Context ...that there is another quite interesting category-theoretic approach to universal algebra, namely the monads approach (which has also been signicantly used in questions related to rewriting, see e.g. [=-=10]-=-). 3 Although R is a quite simply described system, the con uence proof requires long work, because all critical pairs must be examined. This leads to a large amount of details, all consisting of elem... |

14 |
The join of equational theories
- Pigozzi
- 1974
(Show Context)
Citation Context ...lgorithms for word problems for T 1 and T 2 and realizing a decision procedure for word problem for T 1 [ T 2 ? In case T 1 ; T 2 have disjoint signatures the positive answer was known from long time =-=[12]-=-, although only more recently discovered within automated deduction community (see e.g. [11]). In the general case, combining decidable word problems may leads to undecidability, even if we suppose th... |

10 |
An algebraic theory of normal forms
- Ghilardi
- 1995
(Show Context)
Citation Context ...s moved in third position. However, the term moved to last position for composition with w(x 1 ; x 2 ) (the arrow 1 sm of 38 These are well-known results. For a proof making use of normal forms, see [=-=7]-=-. 78 the Table of rules), requires a renaming away from x 1 and consequently it is the pair hx 1 ; u 0 (x 2 ; x 3 )i. Thus the products rule rewrite step produces X 3 - ht; u; x 1 ; x 3 i X 4 - hv; x ... |

3 |
Causes and remedies for undecidability in arrow logics and in multi-modal logics
- Kurucz, Nemeti, et al.
- 1996
(Show Context)
Citation Context ...f the generators), as well as T 1 . The union theory (which we better indicate with T 1 + T0 T 2 ) corresponds to the `distributive linear logic' of [8] and falls within the undecidability results of =-=[1]-=-. Clearly something must be assumed in order to have positive solution to combined word problems; in the literature it is usually assumed that T 1 ; T 2 share a set of constructors (we prefer the term... |

1 |
Modal logics with n-ary connectives
- Ghilardi, Meloni
- 1990
(Show Context)
Citation Context ...ord problem (free algebras aresnite sets of lists of the generators), as well as T 1 . The union theory (which we better indicate with T 1 + T0 T 2 ) corresponds to the `distributive linear logic' of =-=[8]-=- and falls within the undecidability results of [1]. Clearly something must be assumed in order to have positive solution to combined word problems; in the literature it is usually assumed that T 1 ; ... |