## Canonical Inference for Implicational Systems ⋆

### Cached

### Download Links

Citations: | 2 - 2 self |

### BibTeX

@MISC{Bonacina_canonicalinference,

author = {Maria Paola Bonacina},

title = {Canonical Inference for Implicational Systems ⋆},

year = {}

}

### OpenURL

### Abstract

Abstract. Completion is a general paradigm for applying inferences to generate a canonical presentation of a logical theory, or to semi-decide the validity of theorems, or to answer queries. We investigate what canonicity means for implicational systems that are axiomatizations of Moore families – or, equivalently, of propositional Horn theories. We build a correspondence between implicational systems and associative-commutative rewrite systems, give deduction mechanisms for both, and show how their respective inferences correspond. Thus, we exhibit completion procedures designed to generate canonical systems that are “optimal ” for forward chaining, to compute minimal models, and to generate canonical systems that are rewrite-optimal. Rewrite-optimality is a new notion of “optimality ” for implicational systems, one that takes contraction by simplification into account. 1

### Citations

310 | Linear-time algorithms for testing the satisfiability of propositional horn formulae - Dowling, Gallier - 1984 |

76 |
On sentences which are true of direct unions of algebras
- Horn
- 1956
(Show Context)
Citation Context ...a semantic correspondence between Horn theories and Moore families, since Horn theories are those theories whose models are closed under intersection, a fact observed first by McKinsey [16] (see also =-=[15]-=-). A (one-step) deduction mechanism ❀ is a binary relation over presentations. A deduction step Q ❀ Q∪Q ′ is an expansion provided Q ′ ⊆ Th Q, where Th Q is the set of theorems of Q. A deduction step ... |

44 |
The decision problem for some classes of sentences without quantifiers
- McKinsey
- 1943
(Show Context)
Citation Context ... is matched by a semantic correspondence between Horn theories and Moore families, since Horn theories are those theories whose models are closed under intersection, a fact observed first by McKinsey =-=[12]-=-. A (one-step) deduction mechanism ❀ is a binary relation over presentations. A deduction step Q ❀ Q ∪ Q ′ is an expansion provided Q ′ ⊆ Th Q, where Th Q is the set of theorems of Q. A deduction step... |

34 |
Computing with Rewrite Systems
- Dershowitz
- 1985
(Show Context)
Citation Context ...ated into a rewrite rule a1 · · · anc1 · · · cm → a1 · · · an, where juxtaposition stands for associative-commutative-idempotent conjunction, and the arrow → signifies logical equivalence (see, e.g., =-=[9, 5]-=-). A positive literal c is translated into a rule c → true, where true is a new constant. We will be making use of a well-founded ordering ≻ on V ∪ {true}, wherein true is minimal. Conjunctions of pro... |

30 | Equational inference, canonical proofs, and proof orderings
- Bachmair, Dershowitz
- 1994
(Show Context)
Citation Context ..., and it is unique for the assumed ordering, a property first noticed by Mike Ballantyne (see [12]). Otherwise, completion semi-decides validity by working refutationally on E and ũ ̸≃ ˜v (see, e.g., =-=[11,1, 6]-=- for basic definitions and more references). More generally, the notion of canonicity can be articulated into three properties of increasing strength (e.g., [4]): a presentation is complete if it affo... |

30 |
J.-P.: Rewrite systems
- Dershowitz, Jouannaud
- 1990
(Show Context)
Citation Context ..., and it is unique for the assumed ordering, a property first noticed by Mike Ballantyne (see [12]). Otherwise, completion semi-decides validity by working refutationally on E and ũ ̸≃ ˜v (see, e.g., =-=[11,1,6]-=- for basic definitions and more references). More generally, the notion of canonicity can be articulated into three properties of increasing strength (e.g., [4]): a presentation is complete if it affo... |

20 |
and Nachum Dershowitz. Equational inference, canonical proofs and proof orderings
- Bachmair
- 1994
(Show Context)
Citation Context ...♯ is finite, it serves as decision procedure, because validity can be decided by “blind” rewriting. Otherwise, completion semi-decides validity by working refutationally on E and �u �� �v (see, e.g., =-=[1, 6]-=-, also for more references). More generally, the notion of canonicity can be articulated into three properties of increasing strength (e.g., [4]): a presentation is complete, if it affords a normal-fo... |

18 | Towards a foundation of completion procedures as semidecision procedures
- Bonacina, Hsiang
- 1995
(Show Context)
Citation Context ..., and it is unique for the assumed ordering, a property first noticed by Mike Ballantyne (see [12]). Otherwise, completion semi-decides validity by working refutationally on E and ũ ̸≃ ˜v (see, e.g., =-=[11,1,6]-=- for basic definitions and more references). More generally, the notion of canonicity can be articulated into three properties of increasing strength (e.g., [4]): a presentation is complete if it affo... |

17 | Abstract canonical inference - Bonacina, Dershowitz |

14 |
Existence, uniqueness, and construction of rewrite systems
- Dershowitz, Marcus, et al.
- 1988
(Show Context)
Citation Context ...dure, because validity can be decided by “blind” rewriting. If E ♯ is also reduced, it is called canonical, and it is unique for the assumed ordering, a property first noticed by Mike Ballantyne (see =-=[12]-=-). Otherwise, completion semi-decides validity by working refutationally on E and ũ ̸≃ ˜v (see, e.g., [11,1,6] for basic definitions and more references). More generally, the notion of canonicity can ... |

12 |
On rewrite programs: semantics and relationship with Prolog
- Bonacina, Hsiang
- 1992
(Show Context)
Citation Context ...be translated into a rewrite rule a1 ···anc1 ···cm → a1 ···an, where juxtaposition stands for associative-commutative-idempotent conjunction, and the arrow → signifies logical equivalence (see, e.g., =-=[9,5]-=-). A positive literal c is translated into a rule c → true, wheretrue is a special constant. We will be making use of a well-founded ordering ≻ on V ∪{true}, whereintrue is minimal. Conjunctions of pr... |

11 |
The On-Line Encyclopedia
- Sloane, Plouffe
(Show Context)
Citation Context ... future work include generalizing this analysis beyond propositional Horn theories, studying enumerations of Moore families and related structures (see [10] and Sequences A102894–7 and A108798–801 in =-=[18]-=-), and exploring connections between canonical systems and decision procedures, or the rôle of canonicity of presentations in specific contexts where Moore families occur, such as in the abstract inte... |

8 |
B.: The lattices of Moore families and closure operators on a finite set: a survey
- Caspard, Monjardet
- 2003
(Show Context)
Citation Context ...ed a rôle in a variety of fields in computer science, including relational databases, data mining, artificial intelligence, logic programming, lattice theory and abstract interpretations. We refer to =-=[7]-=- and [2] for surveys, including applications, related formalisms and historical notes. An implicational systems can be regarded as a Horn presentation of its Moore family. Since a Moore family may be ... |

7 | Bonacina and Nachum Dershowitz. Abstract canonical inference - Paola - 2007 |

6 |
Bonacina and Jieh Hsiang. Towards a foundation of completion procedures as semidecision procedures
- Paola
(Show Context)
Citation Context ...♯ is finite, it serves as decision procedure, because validity can be decided by “blind” rewriting. Otherwise, completion semi-decides validity by working refutationally on E and �u �� �v (see, e.g., =-=[1, 6]-=-, also for more references). More generally, the notion of canonicity can be articulated into three properties of increasing strength (e.g., [4]): a presentation is complete, if it affords a normal-fo... |

5 | M.: Efficient algorithms on the Moore family associated to an implicational system
- Bertet, Nebut
- 2004
(Show Context)
Citation Context ... This paper studies canonicity for implicational systems. An implicational system is a set of implications, whose family of models is a Moore family, meaning that it is closed under intersection (see =-=[3, 2]-=-). A Moore family defines a closure operator that associates with any set the least element of the Moore family that includes it. Moore families, closure operators and implicational systems have playe... |

4 |
Andrzej Tarlecki. Existence, uniqueness, and construction of rewrite systems
- Dershowitz, Marcus
- 1988
(Show Context)
Citation Context ...dure, because validity can be decided by “blind” rewriting. If E ♯ is also reduced, it is called canonical, and it is unique for the assumed ordering, a property first noticed by Mike Ballantyne (see =-=[12]-=-). Otherwise, completion semi-decides validity by working refutationally on E and ũ ̸≃ ˜v (see, e.g., [11,1, 6] for basic definitions and more references). More generally, the notion of canonicity can... |

3 |
The multiple facets of the canonical direct implicational basis. Cahiers de la
- Bertet, Monjardet
(Show Context)
Citation Context ... This paper studies canonicity for implicational systems. An implicational system is a set of implications, whose family of models is a Moore family, meaning that it is closed under intersection (see =-=[3, 2]-=-). A Moore family defines a closure operator that associates with any set the least element of the Moore family that includes it. Moore families, closure operators and implicational systems have playe... |

3 |
Bonacina and Jieh Hsiang. On rewrite programs: Semantics and relationship with Prolog
- Paola
(Show Context)
Citation Context ...ated into a rewrite rule a1 · · · anc1 · · · cm → a1 · · · an, where juxtaposition stands for associative-commutative-idempotent conjunction, and the arrow → signifies logical equivalence (see, e.g., =-=[9, 5]-=-). A positive literal c is translated into a rule c → true, where true is a new constant. We will be making use of a well-founded ordering ≻ on V ∪ {true}, wherein true is minimal. Conjunctions of pro... |

2 | Enumeration problems related to ground Horn theories. Available at http://arxiv.org/pdf/ cs.LO/0610054
- Dershowitz, Huang, et al.
(Show Context)
Citation Context ...d by completion up to redundancy. Directions for future work include generalizing this analysis beyond propositional Horn theories, studying enumerations of Moore families and related structures (see =-=[10]-=- and Sequences A102894–7 and A108798–801 in [14]), and exploring connections between canonical systems and decision procedures, or the rôle of canonicity of presentations in specific contexts where Mo... |

2 | Knowledge compilation for description logics
- Furbach, Obermaier
- 2007
(Show Context)
Citation Context ...t takes contraction by simplification into account. 1 Introduction Knowledge compilation is the transformation of a knowledge base into a canonical form that makes efficient reasoning possible (e.g., =-=[13, 8, 11]-=-). In automated reasoning the knowledge base is often the “presentation” of a theory, where we use “presentation” to mean a set of formulæ, reserving “theory” for a presentation with all its theorems.... |

2 | P.: Exact knowledge compilation in predicate calculus: the partial achievement case
- Roussel, Mathieu
- 1997
(Show Context)
Citation Context ...t takes contraction by simplification into account. 1 Introduction Knowledge compilation is the transformation of a knowledge base into a canonical form that makes efficient reasoning possible (e.g., =-=[13, 8, 11]-=-). In automated reasoning the knowledge base is often the “presentation” of a theory, where we use “presentation” to mean a set of formulæ, reserving “theory” for a presentation with all its theorems.... |

1 |
Searching while keeping a trace: the evolution from satisfiability to knowledge compilation
- Darwiche
- 2006
(Show Context)
Citation Context ...t takes contraction by simplification into account. 1 Introduction Knowledge compilation is the transformation of a knowledge base into a canonical form that makes efficient reasoning possible (e.g., =-=[13, 8, 11]-=-). In automated reasoning the knowledge base is often the “presentation” of a theory, where we use “presentation” to mean a set of formulæ, reserving “theory” for a presentation with all its theorems.... |

1 | The On-Line Encyclopedia of Integer Sequences. 1996–2006. Available at http://www.research.att.com/ ∼ njas/sequences
- Sloane
(Show Context)
Citation Context ... future work include generalizing this analysis beyond propositional Horn theories, studying enumerations of Moore families and related structures (see [10] and Sequences A102894–7 and A108798–801 in =-=[14]-=-), and exploring connections between canonical systems and decision procedures, or the rôle of canonicity of presentations in specific contexts where Moore families occur, such as in the abstract inte... |

1 | M.A.: Enumeration problems related
- Dershowitz, Huang, et al.
(Show Context)
Citation Context ...d by completion up to redundancy. Directions for future work include generalizing this analysis beyond propositional Horn theories, studying enumerations of Moore families and related structures (see =-=[10]-=- and Sequences A102894–7 and A108798–801 in [18]), and exploring connections between canonical systems and decision procedures, or the rôle of canonicity of presentations in specific contexts where Mo... |