## The expressive powers of logic programming semantics (1995)

Venue: | Abstract in Proc. PODS 90 |

Citations: | 84 - 5 self |

### BibTeX

@INPROCEEDINGS{Schlipf95theexpressive,

author = {John S. Schlipf},

title = {The expressive powers of logic programming semantics},

booktitle = {Abstract in Proc. PODS 90},

year = {1995},

pages = {196--204}

}

### Years of Citing Articles

### OpenURL

### Abstract

We study the expressive powers of two semantics for deductive databases and logic programming: the well-founded semantics and the stable semantics. We compare them especially to two older semantics, the two-valued and three-valued program completion semantics. We identify the expressive power of the stable semantics, and in fairly general circumstances that of the well-founded semantics. In particular, over infinite Herbrand universes, the four semantics all have the same expressive power. We discuss a feature of certain logic programming semantics, which we call the Principle of Stratification, a feature allowing a program to be built easily in modules. The three-valued program completion and well-founded semantics satisfy this principle. Over infinite Herbrand models, we consider a notion of translatability between the three-valued program completion and well-founded semantics which is in a sense uniform in the strata. In this sense of uniform translatability we show the well-founded semantics to be more expressive than the three-valued program completion. The proof is a corollary of our result that over non-Herbrand infinite models, the well-founded semantics is more expressive than the three-valued program completion semantics. 1

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Citation Context ...nt definition [VG93], devised jointly with Van Gelder to facilitate this work. 8 Van Gelder’s alternating fixed point definition was based upon the Gelfond-Lifschitz transformation of a logic program =-=[GL88]-=-. The definition below is Van Gelder’s minor variant of the original definition. Definition 3.4 Let P be a logic program, D be an EDB for P, and I a set of negative Dinstantiated literals of P. The Ge... |

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Citation Context ...th these contexts are commonly studied, we address expressive power in both. We shall see there are interesting contrasts between the contexts. But more general contexts have also been studied. Clark =-=[Cla78]-=- studied not just the Herbrand universe, but all preinterpretations satisfying a set of first order axioms called Clark Equality Theory. Kunen [Kun87] made an even broader domain assumption. Another v... |

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Citation Context ...negated atomic facts. Although all these semantics agree in inferring ¬mortal(Thor) above, they disagree on many other logic programs. We investigate two semantics here: the well-founded semantics of =-=[VGRS91]-=- and (a variant of) the stable semantics of [Gel87, GL88]. We compare them with two older semantics: the two-valued and three-valued program completion semantics of [Cla78, Fit85] Though logic program... |

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Citation Context ...ogy. So neither the two-valued program completion semantics nor the three-valued program completion semantics infers any literals. But P is a Horn-clause program, and the van Emden-Kowalski semantics =-=[VEK76]-=- infers ¬r(A). Thus the program completions semantics do not capture at least one well-accepted notion of negation as failure. By contrast, the well-founded and stable semantics will identify the posi... |

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Citation Context ... a positive EDB literal false in D, then the completion of P over D contains the formula α ↔ false. The original interpretation of the completion was as a theory in ordinary two-valued logic. Fitting =-=[Fit85]-=- observed that Clark’s completion has an especially nice interpretation in threevalued logic (truth values true, false, and undetermined), where the ↔ of the completion is given a ̷Lukasiewicz interpr... |

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Citation Context ...tial model is constructible in polynomial time (in the size of the EDB), can easily be converted into a proof that the well-founded partial model is uniformly inductively definable, using a result of =-=[Imm86]-=-: if a relation is inductively definable uniformly over all finite structures, then it is also coinductively definable uniformly over all finite structures. The stable semantics is more powerful. Mare... |

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Citation Context ...re general contexts have also been studied. Clark [Cla78] studied not just the Herbrand universe, but all preinterpretations satisfying a set of first order axioms called Clark Equality Theory. Kunen =-=[Kun87]-=- made an even broader domain assumption. Another variant is presented in [VGRS91]. By considering even broader collections of domains, in particular, infinite universes other than Herbrand universes, ... |

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Citation Context ...l-founded and stable semantics can more generally be related to inductive definability, but to inductive definability over an in general somewhat richer structure the the original EDB. Definition 4.2 =-=[Bar75]-=-. Let D be an EDB. Consider the elements of D (as anybody but a settheorist would) as being, not sets, but indivisible objects, or urelementen. The set HFD is the smallest set where D ⊆ HFD and, if X1... |

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Citation Context ...inductively definable uniformly over all finite structures, then it is also coinductively definable uniformly over all finite structures. The stable semantics is more powerful. Marek and Truszczynski =-=[MT91]-=- proved that deciding whether a propositional logic program P has a stable model is N P-complete. A minor modification of their proof gives that there is a function-free logic program P where, as a fu... |

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Citation Context ...t recursively saturated structures (again see [BS76, Sch78]) is that if D is recursively saturated, every inductive definition on D closes off in ≤ ω steps. It follows from the Boundedness Theorem of =-=[Mos74]-=- that if a relation r on a recursively saturated EDB is both inductively and coinductively definable, it is first order definable. Similarly, every relation on a countable recursively saturated struct... |

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Citation Context ...it will infer ¬r(A). The wellfounded semantics was originally proposed in [VGRS91]. Subsequently several equivalent definitions have been given, including well-known definitions such as in [VG93] and =-=[Prz89]-=-. We give here a variant of Van Gelder’s alternating fixed point definition [VG93], devised jointly with Van Gelder to facilitate this work. 8 Van Gelder’s alternating fixed point definition was based... |

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Citation Context ...It has also been investigated in several recent papers in relation to normal logic programming, for example in [AB90] (on the degrees of uncomputability of the perfect models of stratified programs), =-=[KP88]-=- (including a result on the computational complexity of Clark’s semantics over finite databases), [Kun88] (some results on the program completion semantics over finite extensional databases), and [MNR... |

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Citation Context ...l-founded semantics. Expressive power is a standard concern in database query languages. It has also been investigated in several recent papers in relation to normal logic programming, for example in =-=[AB90]-=- (on the degrees of uncomputability of the perfect models of stratified programs), [KP88] (including a result on the computational complexity of Clark’s semantics over finite databases), [Kun88] (some... |

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Citation Context ...inciple of Stratification under such variants. Also, we find it reasonable that one might require only the first part of the Principle of Stratification of a programming language semantics. Elsewhere =-=[Sch92]-=- we have referred to the first requirement of the Principle of Stratification alone as the Weak Principle of Stratification. There we also suggested a logic programming semantics which obeys the Weak ... |

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Citation Context ...ple in [AB90] (on the degrees of uncomputability of the perfect models of stratified programs), [KP88] (including a result on the computational complexity of Clark’s semantics over finite databases), =-=[Kun88]-=- (some results on the program completion semantics over finite extensional databases), and [MNR92] (an extensive recursion-theoretic discussion of the class of stable Herbrand models of programs with ... |

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Citation Context ...P88] (including a result on the computational complexity of Clark’s semantics over finite databases), [Kun88] (some results on the program completion semantics over finite extensional databases), and =-=[MNR92]-=- (an extensive recursion-theoretic discussion of the class of stable Herbrand models of programs with function symbols, which overlaps at one significant point with this paper). Of course, increased e... |

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Citation Context ...ductively definable on an EDB D is definable over D in the well-founded semantics. Corollary 3.12 Any relation Π 1 1 tics. 12 definable on an EDB D is definable over D in the stable semanTheorem 3.13 =-=[Sch91]-=- Over all EDB’s, the stable semantics is at least as expressive as the wellfounded semantics. Over all classes of EDB’s, the stable semantics is uniformly at least as expressive as the well-founded se... |

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