Tense and dynamic algebras related to GFA

Document Type : Original Article


1 Department of Mathemathics, Shiraz Branch, Islamic Azad University, Shiraz, Iran

2 Department of Mathemathics, Graduate University of Advanced Technology, Kerman, Iran

3 Khatam Alanbia University of Technology, Behbahan, Iran


The present paper is an attempt to suggest and scrutinize tense operators in the dynamic logic $\textbf{B}$ which is regarded as a set of propositions about the general fuzzy automaton $ \tilde{F} $, in which its underlying structure has been a bounded poset. Here, the operators $ T_{\delta}, P_{\delta}, H_{\delta}$ and $ F_{\delta} $ are proposed regardless of what propositional connectives the logic comprises. For this purpose, the axiomatization of universal quantifiers is applied as a starting point and these axioms are modified. In this study, firstly, we demonstrate that the operators can be identified as modal operators and the pairs $ (T_{\delta},P_{\delta}) $ are examined as the so-called dynamic pairs. In addition, constructions of these operators are attained in the corresponding algebra and in the following a transition frame is suggested. Besides, the problem of finding a transition frame is solved in the case when the tense operators are given. Specifically, this study shows that the tense algebra $ \textbf{B} $ is representable in its Dedekind-MacNeille completion. Representation theorems for dynamic and tense algebra are explicated in details in the related given theorems.


[1] Kh. Abolpour, M.M. Zahedi, Isomorphism between two BL-general fuzzy automata, Soft Computing,
16 (2012), 729{736.
[2] Kh. Abolpour, M.M. Zahedi, BL-general fuzzy automata and accept behavior, Journal of Applied
Mathematics and Computing, 38 (2012), 103{118.
[3] Kh. Abolpour, M.M. Zahedi, General fuzzy automata based on complete residuated lattice-
valued, Iranian Journal of Fuzzy Systems, 14 (2017), 103{121.
[4] Kh. Abolpour, M.M. Zahedi, M. Shamsizadeh, BL-general fuzzy automata and minimal re-
alization: Based on the associated categories, Iranian Journal of Fuzzy Systems, 17 (2020),
[5] Kh. Abolpour, M.M. Zahedi, M. Shamsizadeh, New directions in general fuzzy automata: A
dynamic-logical view, AUT Journal of Mathematics and Computing, 1 (2020), 251{262.
[6] M. Botur, I. Chajda, R. Halas, M. Kolarik, Tense operators on basic algebras, International
Journal of Theoretical Physics, 50 (2011), 3737{3749.
[7] J. Burges, Basic tense logic, In: Handbook of Philosophical Logic, (D. M, Gabbay, F. Guntther,
eds), D. Reidel Publishing Company, 2 (1984), 89{139.
[8] G. Cattaneo, D. Ciucci, D. Dubois, Algebraic models of deviant modal operators based on De
Morgan and Kleene lattices, Information Sciences, 181 (2011), 4075{4100.
[9] I. Chajda, Algebraic axiomatization of tense intuitionistic logic, Central European Journal of
Mathematics, 9 (2012), 1185{1191.
[10] I. Chajda, M. Kolarik, Dynamic e ect algebras, Mathematica Slovaca, 62 (2012), 379{388.
[11] I. Chajda, J. Paseka, Dynamic e ect algebras and their Representations, Soft Computing, 16
(2012), 1733{1741.
[12] I. Chajada, J. Paseka, Transition operators assigned to physical systems, Reports on Mathematical
Physics, 78 (2016), 259{280.
[13] C. Chirita, Tense--valued Lukasiewicz-Moisil algebras, Journal of Multiple-Valued Logic and
Soft Computing, 17 (2011), 1{24.
[14] D. Diaconescu, G. Georgescu, Tense operators on MV-algebras and Lukasiewicz-Moisil alge-
bras, Fundamental Information, 81 (2007), 379{408.
[15] M. Doostfatemeh, S.C. Kremer, New directions in fuzzy automata, International Journal of
Approximate Reasoning, 38 (2005), 175{214.
[16] A. Dvurecenskij, S. Pulmannova, New trends in quantum structures, Kluwer Acad. Publ.,
Dordrecht/ Boston/ London, Lster Sci., Bratislava, (2000).
[17] W.B. Ewald, Intuitionistic tense and modal logic, Journal of Symbolic Logic, 51 (1986), 166{
[18] D.J. Foulis, M.K. Bennett, E ect algebras and unsharp quantum logics, Foundations of
Physics, 24 (1994), 1325{1346.
[19] J. Paseka, J. Janda, A dynamic e ect algebras with dual operation, Mathematics for Applications,
1 (2012), 79{89.
[20] M. Shamsizadeh, M.M. Zahedi, Kh. Abolpour, Admissible partition for BL-general fuzzy au-
tomaton, Iranian Journal of Fuzzy Systems, 15 (2018), 79{90.
[21] M. Shamsizadeh, M.M. Zahedi, Kh. Abolpour, Bisimulation for BL-general fuzzy automata,
Iranian Journal of Fuzzy Systems, 13 (2016), 35{50.
[22] D. Wijesekera, Constructive modal logics I, Annals of Pure and Applied Logic, (1990), 271{