Abstract
Stratified complete lattices are complete lattices equipped with a sequence of preorderings associated with the ordinals less than a given nonzero ordinal, typically a limit ordinal. They have been used to give semantics to recursive definitions involving nonmonotonic operations. We provide representation theorems for stratified complete lattices by inverse limits of complete lattices.
Partially supported by grant no. ANN 110883 from the National Foundation of Hungary for Scientific Research.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
The complete lattices and projections of an inverse system of [11] are continuous, and the ordinal \(\kappa \) is \(\omega \), the least infinite ordinal. Inverse systems of complete lattices over arbitrary directed partial orders are considered in [9], where following [11], the projections are usually assumed to be continuous as well.
References
Charalambidis, A., Ésik, Z., Rondogiannis, P.: Minimum model semantics for extensional higher-order logic programming with negation. Theor. Pract. Logic Programm. 14, 725–737 (2014)
Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, New York (2002)
Denecker, M., Marek, V.W., Truszczyński, M.: Approximations, stable operations, well-founded fixed points and applications in nonmonotonic reasoning. In: Minker, J. (ed.) Logic-Based Artificial Intelligence, pp. 127–144. Kluwer, Norwell (2000)
Ésik, Z.: Equational properties of stratified least fixed points (Extended Abstract). In: Paiva, V., Queiroz, R., Moss, L.S., Leivant, D., Oliveira, A.G. (eds.) WoLLIC 2015. LNCS, vol. 9160, pp. 174–188. Springer, Heidelberg (2015). doi:10.1007/978-3-662-47709-0_13
Ésik, Z.: A representation theorem for stratified complete lattices, arXiv:1503.05124 (2015)
Ésik, Z., Rondogiannis, P.: A fixed-point theorem for non-monotonic functions. Theoret. Comput. Sci. 574, 18–38 (2015)
Ésik, Z., Rondogiannis, P.: Theorems on pre-fixed points of non-monotonic functions with applications in logic programming and formal grammars. In: Kohlenbach, U., Barceló, P., Queiroz, R. (eds.) WoLLIC 2014. LNCS, vol. 8652, pp. 166–180. Springer, Heidelberg (2014). doi:10.1007/978-3-662-44145-9_12
Fitting, M.: Fixed point semantics for logic programming a survey. Theoret. Comput. Sci. 278, 25–51 (2002)
Gierz, G., Hoffman, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains. Cambridge University Press, Cambridge (2003)
Rondogiannis, R., Wadge, W.W.: Minimum model semantics for logic programs with negation. ACM Trans. Comput. Logic 6, 441–467 (2005)
Scott, D.: Continuous lattices. In: Lawvere, F.W. (ed.) Toposes, Algebraic Geometry and Logic. LNM, vol. 274, pp. 97–136. Springer, Heidelberg (1972). doi:10.1007/BFb0073967
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer-Verlag GmbH Germany
About this paper
Cite this paper
Ésik, Z. (2017). A Representation Theorem for Stratified Complete Lattices. In: Hansen, H., Murray, S., Sadrzadeh, M., Zeevat, H. (eds) Logic, Language, and Computation. TbiLLC 2015. Lecture Notes in Computer Science(), vol 10148. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-54332-0_15
Download citation
DOI: https://doi.org/10.1007/978-3-662-54332-0_15
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-54331-3
Online ISBN: 978-3-662-54332-0
eBook Packages: Computer ScienceComputer Science (R0)