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Coalgebraic semantics for derivations in logic programming
 In CALCO’11
, 2011
"... Abstract. Every variablefree logic program induces a PfPfcoalgebra on the set of atomic formulae in the program. The coalgebra p sends an atomic formula A to the set of the sets of atomic formulae in the antecedent of each clause for which A is the head. In an earlier paper, we identified a variab ..."
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Abstract. Every variablefree logic program induces a PfPfcoalgebra on the set of atomic formulae in the program. The coalgebra p sends an atomic formula A to the set of the sets of atomic formulae in the antecedent of each clause for which A is the head. In an earlier paper, we identified a variablefree logic program with a PfPfcoalgebra on Set and showed that, if C(PfPf) is the cofree comonad on PfPf, then given a logic program P qua PfPfcoalgebra, the corresponding C(PfPf)coalgebra structure describes the parallel andor derivation trees of P. In this paper, we extend that analysis to arbitrary logic programs. That requires a subtle analysis of lax natural transformations between Posetvalued functors on a Lawvere theory, of locally ordered endofunctors and comonads on locally ordered categories, and of coalgebras, oplax maps of coalgebras, and the relationships between such for locally ordered endofunctors and the cofree comonads on them.
Coalgebraic Derivations in Logic Programming ∗
"... Coalgebra may be used to provide semantics for SLDderivations, both finite and infinite. We first give such semantics to classical SLDderivations, proving results such as adequacy, soundness and completeness. Then, based upon coalgebraic semantics, we propose a new sound and complete algorithm for ..."
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Coalgebra may be used to provide semantics for SLDderivations, both finite and infinite. We first give such semantics to classical SLDderivations, proving results such as adequacy, soundness and completeness. Then, based upon coalgebraic semantics, we propose a new sound and complete algorithm for parallel derivations. We analyse this new algorithm in terms of the Theory of Observables, and we prove soundness, completeness, correctness and full abstraction results.
Saturated Semantics for Coalgebraic Logic Programming
"... Abstract. A series of recent papers introduces a coalgebraic semantics for logic programming, where the behavior of a goal is represented by a parallel model of computation called coinductive tree. This semantics fails to be compositional, in the sense that the coalgebra formalizing such behavior do ..."
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Abstract. A series of recent papers introduces a coalgebraic semantics for logic programming, where the behavior of a goal is represented by a parallel model of computation called coinductive tree. This semantics fails to be compositional, in the sense that the coalgebra formalizing such behavior does not commute with the substitutions that may apply to a goal. We suggest that this is an instance of a more general phenomenon, occurring in the setting of interactive systems (in particular, nominal process calculi), when one tries to model their semantics with coalgebrae on presheaves. In those cases, compositionality can be obtained through saturation. We apply the same approach to logic programming: the resulting semantics is compositional and enjoys an elegant formulation in terms of coalgebrae on presheaves and their right Kan extensions. 1
Coalgebraic Logic Programming: implicit versus explicit resource handling
"... Abstract. We compare approaches to implicit and explicit resource handling in coinductive and concurrent logic programming. We show various effects that implicit and explicit handling of resources have on implementation and semantics. In particular, we show that recently introduced coalgebraic logi ..."
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Abstract. We compare approaches to implicit and explicit resource handling in coinductive and concurrent logic programming. We show various effects that implicit and explicit handling of resources have on implementation and semantics. In particular, we show that recently introduced coalgebraic logic programming [17] is a paradigm in which, in contrast to many other alternative systems, the aspects of logic and control are intertwined, and computational resources are handled implicitly.