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27
The expressive powers of logic programming semantics
 Abstract in Proc. PODS 90
, 1995
"... We study the expressive powers of two semantics for deductive databases and logic programming: the wellfounded semantics and the stable semantics. We compare them especially to two older semantics, the twovalued and threevalued program completion semantics. We identify the expressive power of the ..."
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Cited by 89 (6 self)
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We study the expressive powers of two semantics for deductive databases and logic programming: the wellfounded semantics and the stable semantics. We compare them especially to two older semantics, the twovalued and threevalued program completion semantics. We identify the expressive power of the stable semantics, and in fairly general circumstances that of the wellfounded 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 threevalued program completion and wellfounded semantics satisfy this principle. Over infinite Herbrand models, we consider a notion of translatability between the threevalued program completion and wellfounded semantics which is in a sense uniform in the strata. In this sense of uniform translatability we show the wellfounded semantics to be more expressive than the threevalued program completion. The proof is a corollary of our result that over nonHerbrand infinite models, the wellfounded semantics is more expressive than the threevalued program completion semantics. 1
A natural model of the multiverse axioms
 Notre Dame J. Form. Log
, 2010
"... Abstract. If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms of [Hama]. 1. ..."
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Abstract. If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms of [Hama]. 1.
The automorphism group of a countable recursively saturated structure
 Proceedings of the London Mathematical Society, Series 3 , 65:225244
, 1992
"... The automorphism groups of K0categorical structures have been studied extensively by both permutation group theorists and model theorists, and this collaboration has turned out to be very fruitful. (See, for example, [10,6,2].) The notion of a recursively saturated structure generalizes that of a ( ..."
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The automorphism groups of K0categorical structures have been studied extensively by both permutation group theorists and model theorists, and this collaboration has turned out to be very fruitful. (See, for example, [10,6,2].) The notion of a recursively saturated structure generalizes that of a (countable)
Axiomatizing first order consequences in dependence logic
, 2012
"... Dependence logic, introduced in [8], cannot be axiomatized. However, firstorder consequences of dependence logic sentences can be axiomatized, and this is what we shall do in this paper. We give an explicit axiomatization and prove the respective Completeness Theorem. 1 ..."
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Dependence logic, introduced in [8], cannot be axiomatized. However, firstorder consequences of dependence logic sentences can be axiomatized, and this is what we shall do in this paper. We give an explicit axiomatization and prove the respective Completeness Theorem. 1
P # NP over the nonstandard reals implies P # NP over W, Theoret. Comput. Sci
 J. Symbolic Logic
, 1994
"... ..."
Real closures of models of weak arithmetic
, 2011
"... D’Aquino et al. (J. Symb. Log. 75(1)(2010)) have recently shown that every realclosed field with an integer part satisfying the arithmetic theory IΣ4 is recursively saturated, and that this theorem fails if IΣ4 is replaced by I∆0. We prove that the theorem holds if IΣ4 is replaced by weak subtheori ..."
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Cited by 2 (0 self)
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D’Aquino et al. (J. Symb. Log. 75(1)(2010)) have recently shown that every realclosed field with an integer part satisfying the arithmetic theory IΣ4 is recursively saturated, and that this theorem fails if IΣ4 is replaced by I∆0. We prove that the theorem holds if IΣ4 is replaced by weak subtheories of Buss ’ bounded arithmetic: PV or Σ b 1IND xk. It also holds for I∆0 (and even its subtheory IE 2) under a rather mild assumption on cofinality. On the other hand, it fails for the extension of IOpen by an axiom expressing the Bézout property, even under the same assumption on cofinality. A discretely ordered subring A of a realclosed field (henceforth often: rcf) R is an integer part of R if for every r ∈ R there exists a ∈ A such that a ≤ r < a + 1. It is wellknown that every rcf has an integer part [MR93], which is then a model of the weak arithmetic theory IOpen (induction for quantifierfree formulas in the language of ordered rings). On the other hand, every model of IOpen is an integer part of its real closure (or, more precisely, the real closure of its fraction field). Recently, d’Aquino et al. [DKS10] studied the question which rcfs have integer parts satisfying more arithmetic, e.g. Peano Arithmetic. It turns out
Resplendent models and Σ 1 1definability with an oracle
 SUBMITTED TO ARCHIVE FOR MATHEMATICAL LOGIC
, 2005
"... In this article we find some sufficient and some necessary Σ 1 1conditions with oracles for a model to be resplendent or chronically resplendent. The main tool of our proofs is internal arguments, that is analogues of classical theorems and modeltheoretic constructions conducted inside a model of ..."
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In this article we find some sufficient and some necessary Σ 1 1conditions with oracles for a model to be resplendent or chronically resplendent. The main tool of our proofs is internal arguments, that is analogues of classical theorems and modeltheoretic constructions conducted inside a model of firstorder Peano Arithmetic: arithmetised backandforth constructions and versions of the arithmetised completeness theorem, namely constructions of recursively saturated and resplendent models from the point of view of a model of arithmetic. These internal arguments are used in conjunction with Pabion’s theorem that ensures that certain oracles are coded in a sufficiently saturated model of arithmetic. Examples of applications are provided for the theories DLO (of dense linear orders) and DIS (of discrete linear orders). These results are then generalised to other ωcategorical theories and theories with a unique countable recursively saturated model.