## Connecting many-sorted theories (2007)

Venue: | The Journal of Symbolic Logic |

Citations: | 18 - 5 self |

### BibTeX

@INPROCEEDINGS{Baader07connectingmany-sorted,

author = {Franz Baader and Silvio Ghilardi},

title = {Connecting many-sorted theories},

booktitle = {The Journal of Symbolic Logic},

year = {2007},

pages = {278--294},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Basically, the connection of two many-sorted theories is obtained by taking their disjoint union, and then connecting the two parts through connection functions that must behave like homomorphisms on the shared signature. We determine conditions under which decidability of the validity of universal formulae in the component theories transfers to their connection. In addition, we consider variants of the basic connection scheme. 1

### Citations

392 | Simplification by cooperating decision procedures
- Nelson, Oppen
- 1982
(Show Context)
Citation Context ...e question is whether decidability transfers from T1, T2 to their combination T . One way of combining the theories T1, T2 is to build their union T1 ∪ T2. Both the Nelson-Oppen combination procedure =-=[16, 15]-=- and combination procedures for the word problem [19, 17, 5] address this type of combination, but for different types of formulae to be decided. Whereas the original combination procedures were restr... |

151 |
Logic for Computer Science. Foundation of Automatic Theorem Proving
- Gallier
- 1986
(Show Context)
Citation Context ...n this section, we fix the notation and give some important definitions, in particular a formal definition of the connection of two theories. We use standard many-sorted first-order logic (see, e.g., =-=[10]-=-), but try to avoid the notational overhead caused by the presence of sorts as much as possible. Thus, a signature Ω consists of a non-empty set of sorts S together with a set of function symbols F an... |

127 |
Complexity of Modal Logics
- Spaan
- 1993
(Show Context)
Citation Context ...es not transfer. Similar combination problems have also been investigated in modal logic, where one asks whether decidability of (relativized) validity transfers from two modal logics to their fusion =-=[12, 20, 23, 4]-=-. The approaches in [11, 3] actually generalize these results from equational theories induced by modal logics to more general first-order theories satisfying certain model-theoretic restrictions: the... |

95 | E-connections of abstract description systems
- Kutz, Lutz, et al.
(Show Context)
Citation Context ...ed models are finite). The theory Ti is compatible with the shared theory T0 iff (i) T0 ⊆ Ti; (ii) T0 has a model completion T ∗ 0 ; and (iii) every model of Ti embeds into a model of Ti ∪ T ∗ 0 . In =-=[13]-=-, a new combination scheme for modal logics, called E-connection, was introduced, for which decidability transfer is much simpler to show than in thescase of the fusion. Intuitively, the difference be... |

83 | Properties of independently axiomatizable bimodal logics - Kracht, Wolter - 1991 |

51 | Fusions of Description Logics and Abstract Description Systems
- Baader, Lutz, et al.
(Show Context)
Citation Context ...es not transfer. Similar combination problems have also been investigated in modal logic, where one asks whether decidability of (relativized) validity transfers from two modal logics to their fusion =-=[12, 20, 23, 4]-=-. The approaches in [11, 3] actually generalize these results from equational theories induced by modal logics to more general first-order theories satisfying certain model-theoretic restrictions: the... |

44 | Fusions of modal logics revisited
- Wolter
(Show Context)
Citation Context ...es not transfer. Similar combination problems have also been investigated in modal logic, where one asks whether decidability of (relativized) validity transfers from two modal logics to their fusion =-=[12, 20, 23, 4]-=-. The approaches in [11, 3] actually generalize these results from equational theories induced by modal logics to more general first-order theories satisfying certain model-theoretic restrictions: the... |

40 | Model theoretic methods in combined constraint satisfiability
- Ghilardi
- 2004
(Show Context)
Citation Context ...t types of formulae to be decided. Whereas the original combination procedures were restricted to the case of theories over disjoint signatures, there are now also solutions for the non-disjoint case =-=[8, 22, 6, 9, 11, 3]-=-, but they always require some additional restrictions since it is easy to see that in the unrestricted case decidability does not transfer. Similar combination problems have also been investigated in... |

37 |
Complexity, convexity and combinations of theories
- Oppen
- 1980
(Show Context)
Citation Context ...is worse then the known ExpTime-complexity [20] of the problem. The deterministic combination procedure described below overcomes this problem.sA deterministic combination procedure As pointed out in =-=[18]-=-, Nelson-Oppen style combination procedures can be made deterministic in the presence of a certain convexity condition. Let T be a theory over the signature Ω, and let Ω0 be a subsignature of Ω. Follo... |

36 | First-order Categorical Logic - Makkai, Reyes - 1977 |

34 | Unions of non-disjoint theories and combinations of satisfiability procedures
- Tinelli, Ringeissen
(Show Context)
Citation Context ...t types of formulae to be decided. Whereas the original combination procedures were restricted to the case of theories over disjoint signatures, there are now also solutions for the non-disjoint case =-=[8, 22, 6, 9, 11, 3]-=-, but they always require some additional restrictions since it is easy to see that in the unrestricted case decidability does not transfer. Similar combination problems have also been investigated in... |

27 |
Combining matching algorithms: The regular case
- Nipkow
- 1991
(Show Context)
Citation Context ...to their combination T . One way of combining the theories T1, T2 is to build their union T1 ∪ T2. Both the Nelson-Oppen combination procedure [16, 15] and combination procedures for the word problem =-=[19, 17, 5]-=- address this type of combination, but for different types of formulae to be decided. Whereas the original combination procedures were restricted to the case of theories over disjoint signatures, ther... |

24 | Deciding the word problem in the union of equational theories sharing constructors
- Baader, Tinelli
- 1999
(Show Context)
Citation Context ...t types of formulae to be decided. Whereas the original combination procedures were restricted to the case of theories over disjoint signatures, there are now also solutions for the non-disjoint case =-=[8, 22, 6, 9, 11, 3]-=-, but they always require some additional restrictions since it is easy to see that in the unrestricted case decidability does not transfer. Similar combination problems have also been investigated in... |

24 | Combination techniques for non-disjoint equational theories
- Domenjoud, Klay, et al.
- 1994
(Show Context)
Citation Context |

21 | A new approach for combining decision procedures for the word problem, and its connection to the nelson-oppen combination method
- Baader, Tinelli
- 1997
(Show Context)
Citation Context ...to their combination T . One way of combining the theories T1, T2 is to build their union T1 ∪ T2. Both the Nelson-Oppen combination procedure [16, 15] and combination procedures for the word problem =-=[19, 17, 5]-=- address this type of combination, but for different types of formulae to be decided. Whereas the original combination procedures were restricted to the case of theories over disjoint signatures, ther... |

14 |
The join of equational theories
- Pigozzi
- 1974
(Show Context)
Citation Context ...to their combination T . One way of combining the theories T1, T2 is to build their union T1 ∪ T2. Both the Nelson-Oppen combination procedure [16, 15] and combination procedures for the word problem =-=[19, 17, 5]-=- address this type of combination, but for different types of formulae to be decided. Whereas the original combination procedures were restricted to the case of theories over disjoint signatures, ther... |

13 | A new combination procedure for the word problem that generalizes fusion decidability results in modal logics
- Baader, Ghilardi, et al.
(Show Context)
Citation Context ...can show combination results that are analogous to Theorem 1: one just needs different compatibility conditions. To treat embeddings and isomorphisms, we use the compatibility condition introduced in =-=[11, 3]-=- for the case of unions of theories (see also the introduction of this paper). Following [11, 3], we call this condition T0-compatibility in the following. Theorem 4. Let T0, T1, T2 be theories over t... |

13 |
Combining satisfiability procedures by equality-sharing
- Nelson
- 1984
(Show Context)
Citation Context ...e question is whether decidability transfers from T1, T2 to their combination T . One way of combining the theories T1, T2 is to build their union T1 ∪ T2. Both the Nelson-Oppen combination procedure =-=[16, 15]-=- and combination procedures for the word problem [19, 17, 5] address this type of combination, but for different types of formulae to be decided. Whereas the original combination procedures were restr... |

13 | Cooperation of background reasoners in theory reasoning by residue sharing
- Tinelli
- 2003
(Show Context)
Citation Context ...son-Oppen style combination procedures can be made deterministic in the presence of a certain convexity condition. Let T be a theory over the signature Ω, and let Ω0 be a subsignature of Ω. Following =-=[21]-=-, we say that T is Ω0-convex iff every finite set of ground Ω X -literals (using additional free constants from X) T -entailing a disjunction of n > 1 ground Ω X 0 -atoms, already T -entails one of th... |

11 | Combining multisets with integers
- Zarba
- 2002
(Show Context)
Citation Context ...the false formula ⊥. We call the combined theory obtained this way the connection of T1 and T2. This kind of connection between theories has already been considered in automated deduction (see, e.g., =-=[1, 24]-=-), but only in very restricted cases where both T1 and T2 are fixed theories (e.g., the theory of sets and the theory of integers in [24]) and the connection functions have a fixed meaning (like yield... |

9 |
Combining word problems through rewriting in categories with products
- Fiorentini, Ghilardi
(Show Context)
Citation Context |

1 |
A modular framework for the combination of symbolic and built-in constraints
- Ajili, Kirchner
- 1997
(Show Context)
Citation Context ...the false formula ⊥. We call the combined theory obtained this way the connection of T1 and T2. This kind of connection between theories has already been considered in automated deduction (see, e.g., =-=[1, 24]-=-), but only in very restricted cases where both T1 and T2 are fixed theories (e.g., the theory of sets and the theory of integers in [24]) and the connection functions have a fixed meaning (like yield... |

1 |
Connecting many-sorted theories. LTCS-Report LTCS-05-04
- Baader, Ghilardi
- 2005
(Show Context)
Citation Context ...to the algebraic setting. The abstract description systems considered in [13], which cover all the usual modal and description logics, are closely related to to Boolean-based equational theories (see =-=[2]-=- for details). The theory E is called Boolean-based equational theory [3] iff its signature Σ has just one sort, equality is the only predicate symbol, the set of function symbols contains the Boolean... |