## Simplifying logic programs under uniform and strong equivalence (2004)

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Venue: | In LPNMR’04 |

Citations: | 37 - 22 self |

### BibTeX

@INPROCEEDINGS{Eiter04simplifyinglogic,

author = {Thomas Eiter and Michael Fink and Hans Tompits and Stefan Woltran},

title = {Simplifying logic programs under uniform and strong equivalence},

booktitle = {In LPNMR’04},

year = {2004},

pages = {87--99},

publisher = {Springer}

}

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### Abstract

Abstract. We consider the simplification of logic programs under the stablemodel semantics, with respect to the notions of strong and uniform equivalence between logic programs, respectively. Both notions have recently been considered for nonmonotonic logic programs (the latter dates back to the 1980s, though) and provide semantic foundations for optimizing programs with input. Extending previous work, we investigate syntactic and semantic rules for program transformation, based on proper notions of consequence. We furthermore provide encodings of these notions in answer-set programming, and give characterizations of programs which are semantically equivalent to positive and Horn programs, respectively. Finally, we investigate the complexity of program simplification and determining semantical equivalence, showing that the problems range between coNP and Π P 2 complexity, and we present some tractable cases. 1

### Citations

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Citation Context ...P is head-cycle free (HCF) in P if no distinct atoms a; b 2 H(r) mutually positively depend on each other. A DLP P is HCF [1] if each r 2 P is HCF in P . We recall the stable-model semantics for DLPs =-=[8, 19]-=-. Let I be an interpretation, i.e., a subset of At . Then, an atom a is true under I , symbolically I j= a, iff a 2 I , and false under I otherwise. For a rule r, I j= H(r) iff some a 2 H(r) is true u... |

322 | Extending and implementing the stable model semantics
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Citation Context ...ical equivalence, showing that the problems range between coNP and P 2 complexity, and we present some tractable cases. 1 Introduction Implementations of answer-set solvers such as DLV [4], Smodels [=-=21]-=-, or ASSAT [13] led to the consideration of practical applications of nonmonotonic logic programs in the last years, but also renewed interest in the study of foundational properties. In particular, s... |

213 | Disjunctive Datalog
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Citation Context ...t B (r); not H(r 0 ); r 2 Q; (4) v 00 isv 0 i ; v 00 isv i ; not v 00 i ; v 00 isnot v 00 i ; 1 i n; (5) wsnot v 00 i ; v i ; ssv 00 i ; not v 0 i ; 1 i n; (6)snot w; not u;snot s; not u; (7)sB + (r 00 ); not B (r); not H(r 00 ); not u; r 2 Q: The intuition behind C U P;Q is as follows. (1) computes all UE-models (X; Y ) of P via atoms V 0 and V (characterizing X 0 resp. Y ). We must chec... |

213 | ASSAT: Computing answer sets of a logic program by SAT solvers
- Lin, Zhao
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Citation Context ...e, showing that the problems range between coNP and P 2 complexity, and we present some tractable cases. 1 Introduction Implementations of answer-set solvers such as DLV [4], Smodels [21], or ASSAT [=-=13]-=- led to the consideration of practical applications of nonmonotonic logic programs in the last years, but also renewed interest in the study of foundational properties. In particular, semantical notio... |

205 | Strongly equivalent logic programs
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Citation Context ...ast years, but also renewed interest in the study of foundational properties. In particular, semantical notions of equivalence between logic programs such as strong equivalence have been studied (cf. =-=[11, 22, 23, 17, 12, 3]-=-): Programs P 1 and P 2 are strongly equivalent, if, for any set R of rules, the programs P 1 [R and P 2 [R are equivalent under the stable semantics, i.e., have the same set of stable models. This ca... |

160 | Stable semantics for disjunctive programs
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Citation Context ...P is head-cycle free (HCF) in P if no distinct atoms a; b 2 H(r) mutually positively depend on each other. A DLP P is HCF [1] if each r 2 P is HCF in P . We recall the stable-model semantics for DLPs =-=[8, 19]-=-. Let I be an interpretation, i.e., a subset of At . Then, an atom a is true under I , symbolically I j= a, iff a 2 I , and false under I otherwise. For a rule r, I j= H(r) iff some a 2 H(r) is true u... |

147 | Propositional semantics for disjunctive logic programs
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Citation Context ... P , and either b 2 B + (r), or some c 2 B + (r) positively depends on b; r 2 P is head-cycle free (HCF) in P if no distinct atoms a; b 2 H(r) mutually positively depend on each other. A DLP P is HCF =-=[1]-=- if each r 2 P is HCF in P . We recall the stable-model semantics for DLPs [8, 19]. Let I be an interpretation, i.e., a subset of At . Then, an atom a is true under I , symbolically I j= a, iff a 2 I ... |

91 |
Optimizing DATALOG Programs
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Citation Context ...6]: if a subprogram Q of P is strongly equivalent to a (simpler) program Q 0 , then we can replace Q by Q 0 . Since strong equivalence is rather strict, the more liberal notion of uniform equivalences=-=[20, 14]-=- has been considered in [5, 18], where R is restricted to sets of facts. A hierarchical component C within a program P may be replaced by a uniformly equivalent set of rules C 0 , providing the global... |

80 |
Equivalences of logic programs
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Citation Context ...6]: if a subprogram Q of P is strongly equivalent to a (simpler) program Q 0 , then we can replace Q by Q 0 . Since strong equivalence is rather strict, the more liberal notion of uniform equivalences=-=[20, 14]-=- has been considered in [5, 18], where R is restricted to sets of facts. A hierarchical component C within a program P may be replaced by a uniformly equivalent set of rules C 0 , providing the global... |

64 | Declarative problem-solving using the DLV system
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Citation Context ...mining semantical equivalence, showing that the problems range between coNP and P 2 complexity, and we present some tractable cases. 1 Introduction Implementations of answer-set solvers such as DLV [=-=4]-=-, Smodels [21], or ASSAT [13] led to the consideration of practical applications of nonmonotonic logic programs in the last years, but also renewed interest in the study of foundational properties. In... |

62 | Strong equivalence made easy: Nested expressions and weight constraints. Theory and Practice of Logic Programming - Turner |

58 |
Reducing strong equivalence of logic programs to entailment in classical propositional logic
- Lin
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Citation Context ...ast years, but also renewed interest in the study of foundational properties. In particular, semantical notions of equivalence between logic programs such as strong equivalence have been studied (cf. =-=[11, 22, 23, 17, 12, 3]-=-): Programs P 1 and P 2 are strongly equivalent, if, for any set R of rules, the programs P 1 [R and P 2 [R are equivalent under the stable semantics, i.e., have the same set of stable models. This ca... |

57 | Semantics of (disjunctive) logic programs based on partial evaluation
- Brass, Dix
- 1999
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Citation Context ...l both with syntactic and semantic transformation rules. Syntactic transformation rules for strong equivalence have previously been considered by Osorio et al. [16]. Besides rules from there and from =-=[2]-=-, we also examine the recent notion of s-implication [24], as well as a new transformation rule for head-cycle free rules, called local shifting. Both preserve uniform equivalence, and the former also... |

51 | Uniform equivalence of logic programs under the stable model semantics
- Eiter, Fink
- 2003
(Show Context)
Citation Context ...rongly equivalent to a (simpler) program Q 0 , then we can replace Q by Q 0 . Since strong equivalence is rather strict, the more liberal notion of uniform equivalences[20, 14] has been considered in =-=[5, 18]-=-, where R is restricted to sets of facts. A hierarchical component C within a program P may be replaced by a uniformly equivalent set of rules C 0 , providing the global hierarchical component structu... |

48 | Encodings for equilibrium logic and logic programs with nested expressions - Pearce, Tompits, et al. - 2001 |

27 | Strong equivalence for logic programs and default theories (made easy
- Turner
(Show Context)
Citation Context ...ast years, but also renewed interest in the study of foundational properties. In particular, semantical notions of equivalence between logic programs such as strong equivalence have been studied (cf. =-=[11, 22, 23, 17, 12, 3]-=-): Programs P 1 and P 2 are strongly equivalent, if, for any set R of rules, the programs P 1 [R and P 2 [R are equivalent under the stable semantics, i.e., have the same set of stable models. This ca... |

23 | Characterization of strongly equivalent logic programs in intermediate logics. http://turing.wins.uva.nl/ lhendrik
- Jongh, Hendriks
- 2001
(Show Context)
Citation Context |

22 | Equivalence in Answer Set Programming
- Osorio, Navarro, et al.
- 2001
(Show Context)
Citation Context ...ivalent, if, for any set R of rules, the programs P 1 [R and P 2 [R are equivalent under the stable semantics, i.e., have the same set of stable models. This can be used to simplify a logic program P =-=[23, 16]-=-: if a subprogram Q of P is strongly equivalent to a (simpler) program Q 0 , then we can replace Q by Q 0 . Since strong equivalence is rather strict, the more liberal notion of uniform equivalences[2... |

14 | Comparisons and computation of well-founded semantics for disjunctive logic programs
- Wang, Zhou
- 2005
(Show Context)
Citation Context ... Syntactic transformation rules for strong equivalence have previously been considered by Osorio et al. [16]. Besides rules from there and from [2], we also examine the recent notion of s-implication =-=[24]-=-, as well as a new transformation rule for head-cycle free rules, called local shifting. Both preserve uniform equivalence, and the former also strong equivalence. The semantic transformation rules em... |

12 | Testing the equivalence of logic programs under stable model semantics
- Janhunen, Oikarinen
- 2002
(Show Context)
Citation Context ...his problem, whilst deciding P j= s r is "only" coNP-complete. We remark that Pearce and Valverde [18] have provided a tableau system for deciding uniform equivalence, and that Janhunen and =-=Oikarinen [9, 10]-=- provided an alternative method in terms of ASP for testing strong and ordinary equivalence between normal logic programs. (3) Beyond local transformations, we present general conditions under which p... |

9 | lpeq and dlpeq – translators for automated equivalence testing of logic programs
- Janhunen, Oikarinen
- 2004
(Show Context)
Citation Context ...his problem, whilst deciding P j= s r is "only" coNP-complete. We remark that Pearce and Valverde [18] have provided a tableau system for deciding uniform equivalence, and that Janhunen and =-=Oikarinen [9, 10]-=- provided an alternative method in terms of ASP for testing strong and ordinary equivalence between normal logic programs. (3) Beyond local transformations, we present general conditions under which p... |

4 |
Some Types of Equivalence for Logic Programs and Equilibrium Logic
- Pearce, Valverde
(Show Context)
Citation Context ...rongly equivalent to a (simpler) program Q 0 , then we can replace Q by Q 0 . Since strong equivalence is rather strict, the more liberal notion of uniform equivalences[20, 14] has been considered in =-=[5, 18]-=-, where R is restricted to sets of facts. A hierarchical component C within a program P may be replaced by a uniformly equivalent set of rules C 0 , providing the global hierarchical component structu... |

2 | Eliminating Disjunction from Propositional Logic Programs under Stable Model Preservation
- Eiter, Fink, et al.
- 2003
(Show Context)
Citation Context ... giving characterizations of programs which are semantically equivalent to positive and Horn programs. Similar characterizations for equivalence to normal programs are considered in a companion paper =-=[6]-=-. Future work concerns implementing the methods presented, as well as extending the results to programs with two kinds of negation and to the case of programs with variables. The current results provi... |