Abstract
We present an algebraic theory for a fragment of predicate logic. The fragment has disjunction, existential quantification and equality. It is not an algebraic theory in the classical sense, but rather within a new framework that we call ‘parameterized algebraic theories’.
We demonstrate the relevance of this algebraic presentation to computer science by identifying a programming language in which every type carries a model of the algebraic theory. The result is a simple functional logic programming language.
We provide a syntax-free representation theorem which places terms in bijection with sieves, a concept from category theory.
We study presentation-invariance for general parameterized algebraic theories by providing a theory of clones. We show that parameterized algebraic theories characterize a class of enriched monads.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Aczel, P.: A general Church-Rosser theorem (1978)
Adámek, J., Borceux, F., Lack, S., Rosický, J.: A classification of accessible categories. J. Pure Appl. Algebra 175(1-3), 7–30 (2002)
Adámek, J., Rosický, J.: On sifted colimits and generalized varieties. Theory Appl. Categ. 8(3), 33–53 (2001)
Adams, R.: Lambda-free logical frameworks. Ann. Pure Appl. Logic (to appear)
Altenkirch, T., Chapman, J., Uustalu, T.: Monads Need Not Be Endofunctors. In: Ong, L. (ed.) FOSSACS 2010. LNCS, vol. 6014, pp. 297–311. Springer, Heidelberg (2010)
Amato, G., Lipton, J., McGrail, R.: On the algebraic structure of declarative programming languages. Theor. Comput. Sci. 410(46), 4626–4671 (2009)
Antoy, S., Hanus, M.: Functional logic programming. C. ACM 53(4), 74–85 (2010)
Asperti, A., Martini, S.: Projections instead of variables: A category theoretic interpretation of logic programs. In: Proc. ICLP 1989 (1989)
Bauer, A., Pretnar, M.: Programming with algebraic effects and handlers. arXiv:1203.1539v1
Berger, C., Melliès, P.-A., Weber, M.: Monads with arities and their associated theories. J. Pure Appl. Algebra 216(8-9), 2029–2048 (2012)
Braßel, B., Fischer, S., Hanus, M., Reck, F.: Transforming Functional Logic Programs into Monadic Functional Programs. In: Mariño, J. (ed.) WFLP 2010. LNCS, vol. 6559, pp. 30–47. Springer, Heidelberg (2011)
Bronsard, F., Reddy, U.S.: Axiomatization of a Functional Logic Language. In: Kirchner, H., Wechler, W. (eds.) ALP 1990. LNCS, vol. 463, pp. 101–116. Springer, Heidelberg (1990)
Clouston, R.A., Pitts, A.M.: Nominal equational logic. In: Computation, Meaning, and Logic. Elsevier (2007)
Cohn, P.M.: Universal algebra, 2nd edn. D Reidel (1981)
Curien, P.-L.: Operads, clones and distributive laws. In: Operads and Universal Algebra. World Scientific (2012)
Finkelstein, S.E., Freyd, P.J., Lipton, J.: Logic Programming in Tau Categories. In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933, pp. 249–263. Springer, Heidelberg (1995)
Fiore, M., Plotkin, G., Turi, D.: Abstract syntax and variable binding. In: Proc. LICS 1999 (1999)
Fiore, M., Hur, C.-K.: Second-Order Equational Logic (Extended Abstract). In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 320–335. Springer, Heidelberg (2010)
Fiore, M., Mahmoud, O.: Second-Order Algebraic Theories. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 368–380. Springer, Heidelberg (2010)
Gabbay, M.J., Mathijssen, A.: One and a halfth order logic. J. Logic Comput. 18 (2008)
Jagadeesan, R., Panangaden, P., Pingali, K.: A fully abstract semantics for a functional language with logic variables. In: LICS 1989 (1989)
Johann, P., Simpson, A., Voigtländer, J.: A generic operational metatheory for algebraic effects. In: LICS 2010 (2010)
Johnstone, P.T.: Sketches of an Elephant. OUP (2002)
Kammar, O., Plotkin, G.D.: Algebraic foundations for effect-dependent optimisations. In: Proc. POPL 2012 (2012)
Kelly, G.M., Power, A.J.: Adjunctions whose counits are coequalisers. J. Pure Appl. Algebra 89, 163–179 (1993)
Kinoshita, Y., Power, A.J.: A fibrational Semantics for Logic Programs. In: Herre, H., Dyckhoff, R., Schroeder-Heister, P. (eds.) ELP 1996. LNCS, vol. 1050, pp. 177–191. Springer, Heidelberg (1996)
Komendantskaya, E., Power, J.: Coalgebraic Semantics for Derivations in Logic Programming. In: Corradini, A., Klin, B., Cîrstea, C. (eds.) CALCO 2011. LNCS, vol. 6859, pp. 268–282. Springer, Heidelberg (2011)
Kurz, A., Petrişan, D.: Presenting functors on many-sorted varieties and applications. Inform. Comput. 208(12), 1421–1446 (2010)
Lack, S., Rosický, J.: Notions of Lawvere theory. Appl. Categ. Structures 19(1) (2011)
Melliès, P.-A.: Segal condition meets computational effects. In: Proc. LICS 2010 (2010)
Møgelberg, R.E., Staton, S.: Linearly-Used State in Models of Call-by-Value. In: Corradini, A., Klin, B., Cîrstea, C. (eds.) CALCO 2011. LNCS, vol. 6859, pp. 298–313. Springer, Heidelberg (2011)
Moggi, E.: Notions of computation and monads. Inform. Comput. 93(1) (1991)
Moreno-Navarro, J.J., Rodríguez-Artalejo, M.: Logic programming with functions and predicates. J. Log. Program 12(3&4), 191–223 (1992)
Plotkin, G.: Some Varieties of Equational Logic. In: Futatsugi, K., Jouannaud, J.-P., Meseguer, J. (eds.) Goguen Festschrift. LNCS, vol. 4060, pp. 150–156. Springer, Heidelberg (2006)
Plotkin, G., Power, J.: Notions of Computation Determine Monads. In: Nielsen, M., Engberg, U. (eds.) FOSSACS 2002. LNCS, vol. 2303, pp. 342–356. Springer, Heidelberg (2002)
Plotkin, G.D., Power, J.: Algebraic operations and generic effects. Appl. Categ. Structures 11(1), 69–94 (2003)
Plotkin, G., Pretnar, M.: Handlers of Algebraic Effects. In: Castagna, G. (ed.) ESOP 2009. LNCS, vol. 5502, pp. 80–94. Springer, Heidelberg (2009)
Reddy, U.S.: Functional Logic Languages, Part I. In: Fasel, J.H., Keller, R.M. (eds.) Graph Reduction 1986. LNCS, vol. 279, pp. 401–425. Springer, Heidelberg (1987)
Saraswat, V.A., Rinard, M.C., Panangaden, P.: Semantic foundations of concurrent constraint programming. In: Proc. POPL 1991, pp. 333–352 (1991)
Schrijvers, T., Stuckey, P.J., Wadler, P.: Monadic constraint programming. J. Funct. Program. 19(6) (2009)
Staton, S.: Relating Coalgebraic Notions of Bisimulation. In: Kurz, A., Lenisa, M., Tarlecki, A. (eds.) CALCO 2009. LNCS, vol. 5728, pp. 191–205. Springer, Heidelberg (2009)
Velebil, J., Kurz, A.: Equational presentations of functors and monads. Math. Struct. in Comp. Science 21 (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Staton, S. (2013). An Algebraic Presentation of Predicate Logic. In: Pfenning, F. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2013. Lecture Notes in Computer Science, vol 7794. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37075-5_26
Download citation
DOI: https://doi.org/10.1007/978-3-642-37075-5_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-37074-8
Online ISBN: 978-3-642-37075-5
eBook Packages: Computer ScienceComputer Science (R0)