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Серия «Математика»
2023. Т. 44. С. 108—115
Онлайн-доступ к журналу: http://mathizv.isu.ru
Research article
УДК 510.665, 510.643 MSC 03B45, 03H05
DOI https://doi.org/10.26516/1997-7670.2023.44.108
Formulas Expressing Totally Nonstable Truth Values of Formulas
Vladimir V. Rybakov1, Vladimir R. Kiyatkin1^, Konstantin V. Grekovich1
1 Siberian Federal University, Krasnoyarsk, Russian Federation И [email protected].
Abstract. In this short paper we consider problem of nonstable truth values of formulas. We investigate temporal-modal logic for description of reliability information. The logic itself is generated by rather standard models on linear time. We consider intervals of nonstable truth values of formulas. This mean that the formula permanently and for a reasonably big time change its truth from true to false and vice versa. We aim to construct some technique and to find an algorithm which may recognize permanently nonstable formulas.
Keywords: modal logic, frame and model Kripke, satisfiability problem, decidability of logic
Acknowledgements: The research was financially supported by the Russian Scientific Foundation (Project No.23-21-00213 ).
For citation: Rybakov V. V., Kiyatkin V. R., Grekovich K. V. Formulas Expressing Totally Nonstable Truth Values of Formulas. The Bulletin of Irkutsk State University. Series Mathematics, 2023, vol. 44, pp. 108-115. https://doi.org/10.26516/1997-7670.2023.44.108
Научная статья
Формулы, выражающие тотальную нестабильность истинностных значений формул
В.В.Рыбаков1, В. Р. Кияткин1и, К. В. Грекович1
1 Сибирский федеральный университет, Красноярск, Российская Федерация И [email protected],
Аннотация. Рассматривается проблема нестабильности истинностных значений формул. Исследуется временная модальная логика на предмет описания надёжности информации. Логика сама по себе порождена стандартными моделями линейного времени. Рассматриваются интервалы нестабильности истинности формул в этих моделях. Это означает, что формула перманентно и в разумно большое время меняет свою истинность с истины на ложь и наоборот. Создается специальная техника и находится алгоритм, который позволит распознавать перманентную нестабильность истинности формул.
Ключевые слова: модальная логика, фрейм и модель Крипке, выполнимость формул, разрешимость логики
Благодарности: Работа выполнена при финансовой поддержке РНФ (проект 2321-00213).
Ссылка для цитирования: Rybakov V. V., Kiyatkin V. R., Grekovich K. V. Formulas Expressing Totally Nonstable Truth Values of Formulas // Известия Иркутского государственного университета. Серия Математика. 2023. Т. 44. C. 108-115. https://doi.org/10.26516/1997-7670.2023.44.108
1. Introduction
Usage of modal, temporal and other non-classical logics has a long and fruitful history. Origin of such many-valued logics may be dated to Lukasiewicz (1917) and his three-valued and many-valued propositional calculi, as well as to Goedel (1932), who refuted the finite — validness of intuitionistic logic. Later A. Tarski (1951) and S. Kripke (1960th) suggested semantical models for the studies of modal and temporal logics such as topological boolean algebras and relational models (Kripke-Hintikka models).
Conception of knowledge may be dated to the end of 1950. At 1962 Hintikka printed the manuscript: Knowledge and Belief, that was about the first book-length work involving modalities to represent the semantics of knowledge. That book contributed a good first addendum for the subject area, but a great deal of research has taken place since that time. One of logics in that line of research was temporal logic (cf. for historical outlook for reasonably close days Gabbay, Hodkinson, Reynolds [2; 3], Goldblat [4], Goranko [5], van Benthem [16], Yde Venema [19]).
Among various extensions of temporal logic, the linear temporal logic CTC with operation U - until - introduced by Amir Pnueli - was especially popular for applications and due to interesting mathematical base. Besides, it is a good idea to mention automaton technique for solution satisfiability in this logic developed by Vardi [17; 18]. From reasonably modern results concerning this logic we would mention the solution for admissibility problem for CTC in Rybakov [6; 7], the basis for admissible rules of CTC was obtained in Babenyshev and Rybakov [1]. The unification problem for CTC
was solved in [11]. Concerning applications of logical methods in AI and CS, the tools around temporal logic work well for analysis in multi-agent environment (cf. eg. [8;9]).
Another popular direction in Information Sciences studies representation of knowledge via multi-agent environment using usually many agent modal logics. That concerns diverse subjects of multi-agent environment - interaction and autonomy, effects of cooperation etc. For example tools for representation agents' interaction for the logic CTC of linear time were developed in Rybakov [8; 9]. In current time this logic was investigated from many viewpoints, in particular extensions of CTC for the case of non-transitive models, were studied in Rybakov [12; 15] for the case of the interval versions of the logic. Also modelling multi-agent reasoning via temporal models was applied in Rybakov [10; 13; 14] for the versions of liner logic.
In this our short paper we consider reasonably new logical problem in information sciences. We attempt to formalize what means that information (presumably written by modal formulas) is not stable, permanently not stable. We suggest in approach using Kripke-like linear models and technique of realizers sets of formulas. The aim is to find an algorithm which would recognize permanently nonstable formulas. The algorithm is constructed and we prove that it solves the pointed task.
2. Satisfiability in Logic
Formulas of our logic will be defined as the set of special formu-
las, which are true at states of certain relational Kripke-like model.
Alphabet for the language of our logic is defined in a standard
way and consists of denumerable set of propositional letters (variables), parenthesises, logical Boolean operations, and modal operations □ and ♦ and also special time operation Next (unary operator << NextTime >>).
We remind, that every modal operation □ can be defined by means of modal operation ♦ as follows □ = Now we give inductive definition
of the formulas in the language of our logic ).
1) Any propositional variable p £ Prop is formula.
2) If A is formula, then —A is formula also.
3) If A and В are formulas, then (А Л В), (A V В) and (A ^ В) are formulas as well.
4) If A is formula, then □ A is a formula also.
5) If A is formula, then MA is formula as well.
There is no other formulas in the language of logic £(MN). Now we turn to the definition of our Kripke-like models. Let we have not empty set W, binary relation R on this set, R C W2, and the set Prop of propositional variables.
Relational model is a model:
M = {W, R, V),
where the valuation V of any propositional variable p from a fixed chosen set of variables is defined as a some chosen subset V (p) from W.
We will first consider models where W is the set of all natural numbers N. So, the relation Next is binary relation where aNextb if and only if b = a + 1. It is convenient to write Next(a) = b. Now we precisely define the truth value of formulas in arbitrary model M as follows.
For any a,b,c € M the truth relations are as follows, let ^ is the standard linear order on N and
Vp € Prop : a IKy p ^^ a € V(p), a \Kv —P ^^ a ^v P, a \Kv (p A ^^ a \Kv P and a IKy ^, a IKy Mp ^^ Vb [(aNextb) ^ b IKy , a IKy □ p ^^ V b [(a < b) ^ (b IKy p)j, a IKy ♦ p ^^ 3 b [(a < b) A (b IKy p)j.
So, in accordance with our notation, a linear relational model Mn is a model:
Mn = {N, Next, V).
The set of all formulas written in the language of model MN and which are true in it, is called a logic, generated by model MN. Notice that this definition differs from standard definition of logics, because the our one not to be compulsory closed w.r.t. substitutions.
Formula p is said to be satisfiable in the logic £(MN), if there can be find a state a € MN such as a IKy p. Formula p is called not satisfiable in the logic £(Mn), if there is no a € MN such as a IKy p. Formula p is said to be refutable in the logic £(MN), if there exist a state a € MN such as a p. Respectively a formula p is said to be true in the logic £(MN) if it is true at any state a from Mn. Recall that formula p is a theorem of the logic £(Mn) if and if only — p is not satisfiable formula in that logic.
Construction of the model Mi
We need now some axillary technique. Let us consider the formula
A(a) = □♦p(a) A □♦p(—a)
(1)
where
р(а) = (а Л N (а Л N (а Л • • • Л N (a)...))^j, (fi(—a) = (—а Л N (—а Л N (—а Л • • • Л N (—a)...))^j.
Here every big parentheses contain exactly к formulas a. In what follows, always the formula a does not contain temporal operation M that is a is pure modal formula (we will speak about permanent non-stability only such formulas).
Let for some x £ N formula A(a) is true, so it is satisfiable in the state x. Common picture representing this event, may be depicted like this:
.. .x ... [.. .a... ]... [... —a... ]... [.. .a... ]...
Here every interval include exactly к states where formula a or —a to be true.
Without loss of generality, keeping intuition, we can express such a-intervals precisely in these sequences. If formula A(a) is satisfiable in a state of a model then formula a is called permanently unstable formula, (or information).
To solve the question of the satisfiability of formula A(a) we need to construct some finite models M2 and show, that this formula is satisfiable in some model if and only if it is satisfiable in some this model M2 with effectively computable number of states.
At first we shell consider some auxiliary considerations. So, let we assume that the formula A(a) = □♦^(a) Л □♦<£(—a) is true at some sate b from our model.
Lemma 1.
Vb, Ус, ((с ^ b) ^ 3S(S с Sub(a)) Л £ S) ^ (c lhy £)]л
ЛУР £ Sub(a)((b lly p) (P £ S))). (2)
Proof follows by simple direct evaluation of possible truth values of formulas from Sub(a).
The state b is called the realiser of the subset S, and S is realised in the state b. Equivalence (2) does mean that for any given natural number b only some realizers from some finite set of subsets S of the set of all sub-formulas from Sub (^(a)) are accessible.
In ascending order from any b, the variety of subsets realizers S decreases and therefore there exists the state с such that in all states у ^ с only some subsets S from the fixed finite set {Si,S2,..., Sn} are exactly to be
all possible realizers (starting from the state c) and these realizers to be infinitely many times repeated in the future. That means that for any x ^ c, for any S € {S1,S2,...,Sn} there is some y > x such that y is realizer for S. Also it is clear that S as a realizer is unique. We need now to construct certain intermediate model M1 with some desirable properties.
Basic set of the new model M1 will begin from the state c0 ^ c, where Co the earliest state after c where formula A(a) is true, that is c0 IKy A(a).
Then, in the basic set, we leave finite increasing set of all fixed states-realizers for all sets from {S1,S2,..., Sn} situated between first interval
Int1 := [.. .a... j
after c0 where a is true and next interval
Int2 := [... —a... j, where —a is true; then we take the first interval
Int3 := [.. .a ... j
after Int2 = [... —a... j where a is true; next we delete all other states starting from c0 and direct the state x the final one in Int3 to the state 2 which is next one in the obtained model after the final state w in the interval Int1. In the resulting model the relation ^ and valuation V remain to be as before. We can then observe now following picture:
... [co .. .a... j ... [... —a... j... [...a... j...
Lemma 2. For any natural number b ^ c0, where b € M1, and any subformulas ft € Sub(a) the next statement holds:
(Mn b) IKy ft ^ (M1, b), IKy ft
Proof. The proof follows by induction on the length I of subformulas ft € Sub(a).
1. I = 0. For the propositional variables it is evident.
2. Let the statement is true for VI < r. We shell prove for 1 = r. Inductive steps for the Boolean logical operations are evident. In the
case, when ft = O7 or ft = ^7, the inductive steps easy follow from the presence of all possible realizers. Lemma is proved. □
If a formula
A(a) = □♦<p(a) A □♦<p(—a)
is satisfiable in a model M1, it is also satisfiable in some usual model (it easy follows by standard unravelling technique).
As we may see, the model M\ has a finite size computable from the length of the formulas
A(a) = □ ♦<p(a) A □ ♦<p(-a).
Therefore from Lemma 1 and Lemma 2 we obtain our main result:
Theorem 1. The problem of satisfiability formulas A(a) is decidable and therefore the problem of recognizing permanent unstable formulas is solvable.
3. Conclusion
In this paper we considered problem of nonstable truth values of formulas. We investigated temporal modal logic C(^N) for description of reliability information. The logic C(^N) itself is generated by standard model = (N1, Next, V} on linear time. We considered intervals of nonstable truth values of formulas, when formula permanently changes its truth from true to false and vice versa. We constructed certain technique and offered algorithm which may recognize permanently unstable formulas.
References
1. Babenyshev S., Rybakov V. Linear Temporal Logic LTL: Basis for Admissible Rules. J. Logic and Computation, 2011, vol. 21, no. 2, pp. 157-177. https//doi.org/10.1-093/logcom/exq020
2. Gabbay D.M., Hodkinson I.M. An axiomatization of the temporal logic with Until and Since over the real numbers. Journal of Logic and Computation, 1990, vol. 1, no. 1, pp. 229-260.
3. Gabbay D.M., Hodkinson I.M., Reynolds M.A. Temporal Logic: Mathematical foundations and computational aspects. Vol. 1. Oxford, Clarendon Press, 1994, 653 p.
4. Goldblatt R. Logics of Time and Computations. CSLI pubblications, Stanford University, 1992, Chapter 6.
5. Goranko V. Hierarchies of Modal and Temporal Logics with Reference Pointers. Journal of Logic, Language and Information, 1996, vol. 5. no. 1, pp. 1-24.
6. Rybakov V. Logical Consecutions in Discrete Linear Temporal Logic. J. of Symbolic Logic, 2005, vol. 70, no. 4, pp. 1137-1149.
7. Rybakov V. Linear temporal logic with until and next, logical consecutions. Annals of Pure and Applied Logic, 2008, no. 155, pp. 32-45.
8. Rybakov V. Logic of knowledge and discovery via interacting agents decision algorithm for true and satisfiable statements. Information Sciences, 2009, vol. 179, no. 11, pp. 1608-1614.
9. Rybakov V. Linear Temporal Logic LTLk„ extended by Multi-Agent Logic K„ with Interacting Agents. Journal of logic and Computation, 2009, vol. 19, no. 6, pp. 989-1017.
10. Rybakov V. Logical Analysis for Chance Discovery in Multi-Agents' Environment. KES-2012, Conference Proceedings, Springer, 2012, pp. 1593-1601.
11. Rybakov V. Writing out unifiers in linear temporal logic. Journal of Logic and Computation, 2012, vol. 22, no. 5, pp. 1199-1206.
12. Rybakov V. Non-transitive linear temporal logic and logical knowledge operations. J. Logic and Computation, 2016, vol. 26, no. 3, pp. 945-958.
13. Rybakov V. Multiagent temporal logics with multivaluations. Siberian Mathematical Journal, Pleiades Publishing (USA), 2018, vol. 59, no. 4, pp. 710-720.
14. Rybakov V. Temporal multi-valued logic with lost worlds in the past. Siberian Electronic mathematical Reports, 2018, vol. 15, no. 1, pp. 436-449.
15. Rybakov V. Branching time agents' logic, satisfiability problem by rules in reduced form. Siberian Electronic mathematical Reports, 2019, vol. 16, no. 1, pp. 1158-1170. https://doi.org/10.33048/semi.201916.079
16. van Benthem J. Tense logic and time. Notre Dame J. Formal Logic, 1984, vol. 25, no. 1, pp. 1-16.
17. Vardi M.Y. An automata-theoretic approach to linear temporal logic. Y. Banff Higher Order Workshop, 1995, pp. 238-266.
18. Vardi M.Y. Reasoning about the past with two-way automata. Larsen K.G., Skyum S., Winskel G., ICALP, LNCS, Springer, 1998, no. 1443, pp. 628-641.
19. Venema Yde. Temporal Logic. Lou Goble (ed.) Blackwell Guide on Philosophical Logic, Blackwell Publishers, 2001, pp. 203-211.
Об авторах Рыбаков Владимир Владимирович, д-р физ.-мат. наук, проф., Институт математики, Сибирский федеральный университет, Красноярск, 660041, Российская Федерация, [email protected]
About the authors Vladimir V. Rybakov, Dr. Sci.
(Phys.-Math.), Prof., Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, [email protected]
Кияткин Владимир Ростиславович, канд. физ.-мат.
наук, доц., Институт математики, Сибирский федеральный университет, Красноярск, 660041, Российская Федерация, [email protected]
Грекович Константин Викентьевич, Институт математики, Сибирский федеральный университет, Красноярск, 660041, Российская Федерация, [email protected]
Vladimir R. Kiyatkin, Cand. Sci. (Phys.-Math), Assoc. Prof. Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, [email protected]
Konstantin V. Grekovich,
Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, [email protected]
Поступила в 'редакцию / Received 21.01.2023 Поступила после рецензирования / Revised 14.03.2021 Принята к публикации / Accepted 20.03.2023
