A generalization of the Chen-Wu duality into quantale-valued setting

Document Type: Research Paper

Authors

1 Department of Physics, Hebei University of Science and Technology, Shijiazhuang 050018, P.R. China

2 Department of Physics, Hebei University of Science and Technol- ogy, Shijiazhuang 050018, P.R. China

3 Department of Physics, Hebei University of Science and Technology, Shi- jiazhuang 050018, P.R. China

4 Department of Physics, Hebei University of Science and Tech- nology, Shijiazhuang 050018, P.R. China

Abstract

With the unit interval [0,1] as the truth value table, Chen and Wu
presented the concept of  possibility computation over dcpos.
Indeed, every possibility computation can be considered as a
[0,1]-valued Scott open set on a dcpo. The aim of this paper is to
study Chen-Wu's duality on quantale-valued setting. For clarity,
with a commutative unital quantale $L$ as the truth value table, we
introduce a concept of fuzzy possibility computations over fuzzy
dcpos and then establish an equivalence between their denotational
semantics and their logical semantics.

Keywords


[1] R. Belohlavek, Fuzzy relational systems: foundations and principles, Kluwer Academic/
Plenum Publishers, New York, 2002.
[2] R. Belohlavek, Concept lattices and order in fuzzy logic, Annals of Pure and Applied Logic,
128(1-3) (2004), 227-298.
[3] Y. X. Chen and H. Y. Wu, Domain semantics of possibility computations, Information Sciences,
178(2) (2008), 2661-2679.
[4] Y. X. Chen and A. Jung, An introduction to fuzzy predicate transformers, The Invited Talk
at the Third International Symposium on Domain Theory, Shaanxi Normal University, Xian,
China, 2004.
[5] P. W. Chen, H. Lai and D. Zhang, Core
ective hull of nite strong L-topological spaces,
Fuzzy Sets and Systems, 182(1) (2011), 79-92.
[6] L. Fan, A new approach to quantitative domain theory, Electronic Notes in Theroretical
Computer Science, 45 (2001), 77-87.
[7] J. A. Goguen, L-fuzzy sets, Journal of Mathematial Analysis and Applications, 18(1) (1967),
145-174.
[8] C. A. R. Hoare, Communicating sequential process, Communications of the ACM, 21(8)
(1978), 666-677.
[9] C. Jones, Probabilistic non-determinism, PhD thesis, Department of Computer Science, University
of Edinburgh, Edinburgh, 1990.
[10] C. Jones and G. Plotkin, A probabilistic powerdomain of evaluations, In Proceedings of the
Fourth Annual Symposium on Logic in Computer Science, (1989), 186-195.
[11] H. Lai and D. Zhang, Complete and directed complete
-categories, Theoretical Computer
Science, 388(1-3) (2007), 1-25.
[12] G. D. Plotkin, A powerdomain construction, SIAM Journal on Computing, 5(3) (1976),
452-487.
[13] G. D. Plotkin, A powerdomain for countable non-determinism, In M. Nielsen and E. M.
Schmidt (editors), Automata, Languages and programming, Lecture Notes in Computer Science,
EATCS, Springer-Verlag, 140 (1982), 412-428.
[14] G. D. Plotkin, Probabilistic powerdomains, In Proceedings CAAP, (1982), 271-287.
[15] K. I. Rosenthal, Quantales and their applications, Longman Scienti c and Technical, New
York, 1990.
[16] S. E. Rodabaugh, Powerset operator foundations for poslat fuzzy set theories and topologies,
pp. 91-116, Chapter 2 in U. Hohle and S. E. Rodabaugh, eds, Mathematics of Fuzzy Sets:
Topology, and Measure Theory, The handbooks of Fuzzy Sets Series, Volume 3 (1999), Kluwer
Academic Publishers (Boston/ Dordrecht/London).
[17] S. E. Rodabaugh, Relationship of algebraic theories to powerset theories and fuzzy topo-
logical theories for lattice-valued mathematics, International Journal of Mathematics and
Mathematical Sciences, vol. 2007, Article ID 43645, 71 pages, 2007. doi:10.1155/2007/43645
[18] D. S. Scott, A type theoretical alternative to ISWIM, CUCH, OWHY, Theoretical Computer
Science, 121(1-2) (1993), 411-440.
[19] D. S. Scott, Continuous lattices, In: E. Lawvere (Ed.), Toposes, Algebraic Geometry and
Logic, Lecture Notes in Mathematics, Springer-Verlag, 274 (1972), 97-136.
[20] M. B. Smyth, Powerdomains, Journal of Computer and Systems Sciences, 16(1) (1978),
23-36.
[21] N. Saheb-Djahromi, CPOs of measures for non-determinism, Theoretical Computer Science,
12(1) (1980), 19-37.
[22] R. Tix, K. Keimel and G. Plotkin, Semantic domains for combining probability and non-
determinism, Electronic Notes in Theoretical Computer Science, 222 (2009), 3-99.
[23] K. R. Wagner, Liminf convergence in
-categories, Theoretical Computer Science, 184(1-2)
(1997), 61-104.
[24] W. Yao and L. X. Lu, Fuzzy Galois connections on fuzzy posets, Mathmatical Logic Quarterly,
55(1) (2009), 105-112.
[25] W. Yao, Quantitative domains via fuzzy sets: Part I: Continuity of fuzzy directed complete
posets, Fuzzy Sets and Systems, 161(7) (2010), 937-987.
[26] W. Yao and F. G. Shi, Quantitative domains via fuzzy sets: Part II: fuzzy Scott topology on
fuzzy directed complete posets, Fuzzy Sets and Systems, 173(1) (2011), 1-21.
[27] W. Yao, A survey of fuzzi cations of frames, the Papert-Papert-Isbell adjunction and sobri-
ety, Fuzzy Sets and Systems, 190(1) (2012), 63-81.
[28] W. Yao, A categorical isomorphism between injective fuzzy T0-spaces and fuzzy continuous
lattices, IEEE Transactions on Fuzzy Systems, Article in press.
[29] W. Yao, A more general truth valued table for lattice-valued convergence spaces, Preprint.
[30] Q. Y. Zhang and L. Fan, Continuity in quantitative domains, Fuzzy Sets and Systems, 154(1)
(2005), 118-131.
[31] Q. Y. Zhang and W. X. Xie, Section-retraction-pairs between fuzzy domains, Fuzzy Sets and
Systems, 158(1) (2007), 99-114.