Abstract
Every variable-free logic program induces a P f P f -coalgebra 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 variable-free logic program with a P f P f -coalgebra on Set and showed that, if C(P f P f ) is the cofree comonad on P f P f , then given a logic program P qua P f P f -coalgebra, the corresponding C(P f P f )-coalgebra structure describes the parallel and-or 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 Poset-valued 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.
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References
Adámek, J., Milius, S., Velebil, J.: Semantics of higher-order recursion schemes. In: CoRR, abs/1101.4929 (2011)
Amato, G., Lipton, J., McGrail, R.: On the algebraic structure of declarative programming languages. Theor. Comput. Sci. 410(46), 4626–4671 (2009)
Asperti, A., Martini, S.: Projections instead of variables: A category theoretic interpretation of logic programs. In: ICLP, pp. 337–352 (1989)
Bonchi, F., Montanari, U.: Reactive systems (semi-)saturated semantics and coalgebras on presheaves. Theor. Comput. Sci. 410(41), 4044–4066 (2009)
Bruni, R., Montanari, U., Rossi, F.: An interactive semantics of logic programming. TPLP 1(6), 647–690 (2001)
Comini, M., Levi, G., Meo, M.C.: A theory of observables for logic programs. Inf. Comput. 169(1), 23–80 (2001)
Corradini, A., Montanari, U.: An algebraic semantics of logic programs as structured transition systems. In: Proc. NACLP 1990. MIT Press, Cambridge (1990)
Corradini, A., Montanari, U.: An algebraic semantics for structured transition systems and its application to logic programs. TCS 103, 51–106 (1992)
Costa, V.S., Warren, D.H.D., Yang, R.: Andorra-I: A parallel prolog system that transparently exploits both and-and or-parallelism. In: PPOPP, pp. 83–93 (1991)
Girard, J.-Y.: Linear logic. Theor. Comput. Sci. 50, 1–102 (1987)
Gupta, G., Bansal, A., Min, R., Simon, L., Mallya, A.: Coinductive logic programming and its applications. In: Dahl, V., Niemelä, I. (eds.) ICLP 2007. LNCS, vol. 4670, pp. 27–44. Springer, Heidelberg (2007)
Gupta, G., Costa, V.S.: Optimal implementation of and-or parallel prolog. In: Conference proceedings on PARLE 1992, pp. 71–92. Elsevier North-Holland, Inc., New York (1994)
Kelly, G.M.: Coherence theorems for lax algebras and for distributive laws. In: Category Seminar. LNM, vol. 420, pp. 281–375 (1974)
Kinoshita, Y., Power, A.J.: A fibrational semantics for logic programs. In: Proceedings of the Fifth International Workshop on Extensions of Logic Programming. LNCS (LNAI), vol. 1050. Springer, Heidelberg (1996)
Komendantskaya, E., McCusker, G., Power, J.: Coalgebraic semantics for parallel derivation strategies in logic programming. In: Johnson, M., Pavlovic, D. (eds.) AMAST 2010. LNCS, vol. 6486, pp. 111–127. Springer, Heidelberg (2011)
Komendantskaya, E., Power, J.: Fibrational semantics for many-valued logic programs: Grounds for non-groundness. In: Hölldobler, S., Lutz, C., Wansing, H. (eds.) JELIA 2008. LNCS (LNAI), vol. 5293, pp. 258–271. Springer, Heidelberg (2008)
Lawvere, W.: Functional semantics of algebraic theories. PhD thesis, Columbia University (1963)
Lloyd, J.: Foundations of Logic Programming, 2nd edn. Springer, Heidelberg (1987)
Miller, D., Nadathur, G.: Higher-order logic programming. In: ICLP, pp. 448–462 (1986)
Pym, D.: The Semantics and Proof Theory of the Logic of Bunched Implications. Applied Logic Series, vol. 26. Kluwer Academic Publishers, Dordrecht (2002)
Rydeheard, D., Burstall, R.: Computational Category theory. Prentice Hall, Englewood Cliffs (1988)
Simon, L., Bansal, A., Mallya, A., Gupta, G.: Co-logic programming: Extending logic programming with coinduction. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 472–483. Springer, Heidelberg (2007)
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Komendantskaya, E., Power, J. (2011). Coalgebraic Semantics for Derivations in Logic Programming. In: Corradini, A., Klin, B., Cîrstea, C. (eds) Algebra and Coalgebra in Computer Science. CALCO 2011. Lecture Notes in Computer Science, vol 6859. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22944-2_19
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DOI: https://doi.org/10.1007/978-3-642-22944-2_19
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