CC BY
5
DOI: 10.17516/1997-1397-2022-15-1-56-74 УДК 510.64, 510.65, 510.66, 004.82, 004.83
Satisfiability in a Temporal Multi-valueted Logic Based on Z
Vladimir R. Kiyatkin* Anna V. Kosheleva^
Siberian Federal University Krasnoyarsk, Russian Federation
Received 31.10.2020, received in revised form 21.02.2021, accepted 03.05.2021 Abstract. In this paper we continue the series of papers by V. V. Rybakov devoted to properties of multi-valueted logics and where he propose a new approach for modelling knowledge and reasoning of agents in a multi-agent system. We prove that the satisfiability problem is decidable in a temporal multi-valueted logic based on Z.
Keywords: temporal logic, multi-agent logic, epistemic modal logic, multi-valueted logic, satisfiability, decidability in logic, knowledge representation and reasoning, multi-agent systems.
Citation: V.R. Kiyatkin, A.V. Kosheleva, Satisfiability in a Temporal Multi-valueted Logic Based on Z, J. Sib. Fed. Univ. Math. Phys., 2022, 15(1), 56-74. DOI: 10.17516/1997-1397-2022-15-1-56-74.
Introduction
In our paper we continue the research of V. V. Rybakov devoted to properties of multi-valueted logics ( [17-20]). We consider a temporal multi-valueted logic L (MZ) and prove that the satisfiability problem in this logic is decidable.
Multi-valuation is a new approach developed by V. V. Rybakov for modelling and interpreting knowledge and reasoning of agents in a multi-agent system.
Multi-agent systems are one of the demanded directions in data science and artificial intelligence. They are used in online commerce, computer games, mobile and network technologies, geographic information systems, decision-making systems for emergency response, in medicine and the defense industry. Multi-agent systems have a complex architecture; a variety of software and mathematical tools are used in their design, including methods of mathematical logic. For example, using statements in various logical systems, one can set reasoning about agents, express their so-called mental properties: knowledge, beliefs, goals, etc.
Logical formalization of multi-agent systems is important since many of mental properties of agents should have a consistent description. At the beginning, first-order logic was used to formalize statements about agents. But many properties of agents are inexpressible in terms of the first-order predicate calculus, and first-order logics that underlie the description of properties of agents, are undecidable. The language of multimodal logics and logics of knowledge (which are a part of multimodal logics) turned out much more flexible and promising, especially the language of multimodal temporal logics since such logics have means for describing properties associated with real time, which is especially important in view of dynamic nature of an agent's
* [email protected] [email protected] © Siberian Federal University. All rights reserved
and a multi-agent system's functioning. Therefore, a large number of studies concerning the logical properties of multi-agent systems lies in the field of multimodal temporal logics. Since the 1960s and since the first major work of Jaakoo Hintikka "Knowledge and belief: an introduction to the logic of the two notions" [11] on logics of knowledge, many works have been published on modal, epistemological and temporal logics and their applications in multi-agent systems and on related to the theory of such systems - game theory ([1,4,8,9,13-16,21]). For example, the Nobel laureate in economics Robert Aumann applied a logic of knowledge in his research on the game theory for the analysis of economic systems, [1].
Now such scientists as M. Wooldridge and W. van der Hoek [12], F. Baader [2], R. Fagin, J.Halpern, Y.Moses and M.Vardi ([9,10,22,23]), A.Perea ([14,15]), J.van Benthem [7], V. V.Rybakov ([3,5,6,17-20]) and others are working in this direction.
In the language of logics of knowledge and the language of multimodal logics with several modal operators □i, in the language of temporal logics with several temporary operators of the same type, as well as in logics that simultaneously use logical operators of all three listed logics, it is possible to model reasoning about knowledge and behaviour of agents in multi-agent systems interpreting an index i of logical operators in a formula as knowledge of the i-th agent in a multiagent system.
V. V. Rybakov proposed in his papers another way of constructing reasoning of agents, namely, through formulas of multi-valueted logics.
The main difference between multi-valueted logics and logics, the language of which contains operators of logics of knowledge, multimodal logics and k operators of one type of temporary logics, is the interpretation of the i-th logical operator of each type in formulas of such logics. In multi-valueted logics, each index i is associated not with a binary relation Ri and (or) with one of several binary relations Nextj, etc. on Kripke frame, but with the i-th valuation Vi in a selected model.
Examples of formulas in multi-valueted logics and their interpretation from the point of view of agents can be found in the papers of V. V. Rybakov mentioned above. In this article, we also provide examples of agent's reasoning for our logic.
1. Logic L (Mz)
In Section 1 we introduce a temporal multi-valueted logic L (MZ), for which we investigate the satisfiability problem in this paper. The logic L (MZ) is defined as the set of formulas that are valid on certain relational multi-valueted models.
The alphabet of the language L (MZ) consists of a countable set of propositional variables Prop = {pi, ..., pi, ...}, brackets (, ), Boolean logical operators {-, A, V, modal operators {□1; ^2, ..., □k} ("it is necessary that ..."), and temporal operators {Ni, Si, Ui | i = 1, ..., k} (that is, unary operators "NextTime" or "Tomorrow", and binary operators "Since" and "Untill").
Recall, that each modal operator ("it is possible that ...") is defined by means of the modal operator □i in the following away: = -□i-, i = 1, ..., k.
Now we give an inductive definition of a formula in the language of L (MZ).
1. Any propositional variable p € Prop is a formula.
2. If A is a formula, then —A is a formula.
3. If A and B are formulas, then (A A B), (A V B) and (A —y B) are formulas.
4. If A is a formula, then Ui A are formulas, i = 1, ..., k.
5. If A is a formula, then Ni A is a formula, i = 1, ..., k.
6. If A and B are formulas, then (A Si B), (AUi B) are formulas, i = 1, ..., k.
There are no other formulas in the language of the logic L (MZ). We omit the outer parentheses of the formulas in what follows.
Let there be given a non-empty set W, binary relations R1, ..., Rk on this set: Ri C W2, and a set Prop of propositional variables.
Definition 1.1. A relational multi-valueted model is a model
M = {W,R1,...,Rk ,V1,...,Vt), (1)
where p € Prop: Vi(p) C W.
Definition 1.2. A relation Next is a binary relation such that
a Next b if and only if b = a + 1.
Further, for convenience, instead of a Next b we write Next(a) = b. Let < be a standard linear order on Z.
Definition 1.3. Let MN be a following relational linear multi-valueted model:
Mn = (N, Next, Vi, ..., Vk). (2)
Definition 1.4. Let MZ be a following relational linear multi-valueted model:
Mz = (Z, Next, Vi, ..., Vk). (3)
Denote by K (MN) a set of all possible models MN, and by K (MZ), consequently, a set of all possible models MZ.
If a, b € K (MN) or a, b € K (MZ), then the fact when a < b and a = b we denote by a <b. Let M € K (MN) or M € K (MZ). Define the truth of formulas in the model M.
Definition 1.5. For any points a, b, c € M the following holds.
Vp € Prop : a II v. p ^^ a € Vj(p), a Ibv. —ty ^^ a ¥Vj ty, a Iby. (ty A ^^ a Iby. ty u a Iby. ^, a IIV. Ni ty ^^ V b [(a Next b) ^ b IIV. ty] ,
a II Vj (ty Ui ^^ 3 b [(a < b) A (b II Vj AV c [(a < c<b) ^ (c II Vj ty)]] , a IIVj (ty Si ^^ 3 b [(b < a) A (b IIVj A V c [(b < c < a) ^ (c IIVj ty)]] , a II v. Ui ty ^ V b [(a < b) ^ (b 11*.. ty)] , a I V. ^ ty ^ 3 b [(a < b) A (b IIV. ty)\.
It is easy to see that Oi y = TUiy, that is, modal operators □i and Oi can be expressed by means the operator Ui in the considerable models.
Definition 1.6. A formula y in the language of the logic L (MZ) is called satisfiable in K (MZ) if there is a model MZ € K (MZ) and a point a € MZ such that a Iby. y for some j = 1, ..., k.
Definition 1.7. A formula y in the language of the logic L (MZ) is called refutable in K (MZ) if there is a model MZ € K (MZ) and a point a € MZ such that a y for some j = 1, ..., k.
Definition 1.8. A formula y in the language of the logic L (MZ) is called valid in a model MZ € K (Mz) if for any point a € MZ : a IbVj y for any j = 1, ..., k.
Definition 1.9. A formula y in the language of the logic L (MZ) is called unsatisfiable in this logic, if for any model MZ € K (MZ) and for any point a € MZ : a y for any j = 1, ..., k.
Definition 1.10. The multi-valueted logic L (MZ) is the set of formulas in the language of this logic that are valid in any model from the class K (MZ).
Formulas that belong to the logic L (MZ) are called theorems of this logic. Definitions 1.8 and 1.10 yield that
Theorem 1.11. A formula y is a theorem of the logic L (MZ) if and only if—y is an unsatisfiable formula in this logic.
In the paper by V. V. Rybakov [17] similarly it is defined the multi-valueted logic L (K). The language of L (K) is defined in the same away as the language of L (MZ), but without the temporal operators Si. The logic L (K) is defined as the set of formulas that are valid in any model from the class K (MN).
Thus, the fundamental difference between the logics L (MZ), L (K), and others multi-valueted logics from multi-modal logics with k modal operators □i, temporal logics with k temporal operators of the same type, as well as logics using both modal and temporal operators, is in the interpretation of the i-th modal or temporal operators of each type in formulas of such logics. In multi-valueted logics, every index i is connected not with a binary relation Ri on Kripke-frame and (or) with one of several binary relations Nexti, and etc. on a temporal frame, but with the i-th valuation on a selected model.
As we write in the introduction, in the language of multimodal logics with several modal operators □i and the language of temporal logic with several temporal operators of the same type, as well as in logics that use both modal and temporal operators, one can model reasoning about the knowledge and behaviour of agents in multi-agent systems, interpreting an index i of modal and temporal operators in a formula as knowledge of the i-th agent in a multi-agent system.
In the papers by V.V. Rybakov ( [17-20]), there was proposed another approach to modelling such reasoning, namely, by means formulas of a multi-valueted logic. Examples of formulas in multi-valueted logics and their interpretation from the point of view of agents can be found in his cited works.
Let us give two more examples.
1. Consider two agents: i h j. A situation when the agent i believes that starting from the event the event y will always be true in the future, while the agent j believes that starting from the event the event y is possible in the future, we can describe by the following formula:
((□i y) Si ^ A ((Oj —y) Sj ^.
Note that this formula is unsatisfiable in logics with the usual interpretation of indices by means of various reachability relations, while in a model MZ, with a certain y and ^ and not coinciding with each other Vi and Vj, the formula can be satisfiable.
2. Consider three agents: i, j h k. Let the agent i believes that, starting from the event the event y is possible in the future, and the agent j believes that, starting from the event the event —y is possible in the future. In this case, the agent k concludes that both events are possible. This situation can be described by the following formula:
(((Oi y) Si ^) A ((Oj —y) Sj ^ ^ (Ok y A Ok —y).
2. Satisfiability in the logic L (MZ)
In Section 2., to decide the problem of satisfiability in the logic L (MZ), we introduce a class of some special finite models K (Ma) and prove that the formula in the language of our logic is satisfiable if and only if it is satisfiable in some model Mc from the class K (Ma).
Class of models K (Ma)
Let there be given a segment A = [d-, d+] c Z, where d- ^ -4, d+ ^ 5, and some points c-, c+ € [d-, d+], c+ >4, c- < -3. Let Prop(y) be the set of propositional variables of a formula y. Then,
Ma = ([d-, d+] , 4, Next', Vi, ..., V^ ,
where the relation Next on [d-, d+) coincides with the relation Next, and at d+ it is defined in the following away: we consider that Next (d+) = c+ + 1 and Next (c- — 1) = d-, that is, we determine the relation Next on the segment [d-, d+].
The relation 4 on Ma we determine in the following away:
1. if xi ^ X2, then xi 4 X2,
2. on the segments [d-, c- — 1] and [c+ + 1, d+] the relation 4 is an equivalence relation.
If xi, x2 € [d-, d+], then the case, when a 4 b and a = b we denote as xi x2.
Valuations Vi, ..., Vk are the valuations of variables from the set Prop(y).
Rules for calculating the truth for any valuation Vi we determine in the same away as in the model MZ replacing in all items of Definition 1.5 the relation Next on Next , relation ^ on 4, and the symbol < on —.
Let K (Ma) be a class of all possible models Ma, that is, models obtained for all possible values of d-, d+, c- and c+, satisfied the conditions indicated above, and for all possible valuations of the variables from Prop(y).
Suppose, that there is some model MZ (3):
Mz = Z, <, Next, Vi, ..., Vk)
and a point a G MZ with a valuation Vj such that a lbVj ty. Obviously, we can take a =1, that
is,
(Mz, 1) llVj ty-
Let us show that by transforming the model MZ in a certain way, we can construct for this formula some model Me G K (Ma) such that
(Mc, 1) lbj. ty-
Model MC
First, we separately transform the positive part
M+ = (N, Next, V1, ...,Vk}
of the model MZ and obtain a model M+ that is a part of the model Me. We use in our proof the results from the paper by V. V.Rybakov [17] obtained for M + . Then we completely construct the model Me.
I. The positive part of M+ is constructed in the paper by V. V. Rybakov [17] and has the following construction.
Let there be given points n, c+ and d+ from N, and such that 4 ^ c+ < c+ + 1 < d+. Then, M+ = ^[1, d+] , Next', Vi, ..., Vk) , (4)
where the relation Next on [1, d+) coincides the relation Next, and at d+ it is defined in the following away: we consider d+ = c+ + 1, that is, Next (d+) = c+ + 1. Thus, we determine the relation Next on the finite segment [1, d+].
The relation ^ on M+ we determine in the following away:
1. if xi ^ x2, then xi ^ X2;
2. on the segment [c+ + 1, d+] the relation ^ is an equivalence relation.
If x1, x2 G [1, d+], then the case when a ^ b and a = b we denote as x1 — x2.
i+c
points of the model MZ.
etail the "t-main stages of constructing the model M
First, for every element b G MZ we define the following values. 1. For any valuation Vi, i G [1,&],
Valuations V1, ... ,Vk on each point of M+ coincide with the valuations of the corresponding Now we describe in detail the main stages of constructing the model M+
Subi(b) = {a € Sub(ty) | b II^ a}.
In fact, Subi(b) defines formula valuations on Mc. Obviously, there are exist such
possible different sets and this number is finite.
2. Compose from the valuations one more set Subi(b):
D(b) = {Subi(b) | i € [1, k]}.
It is convenient to represent D(b) as a column:
( Sub1(b) \ Sub2 (b)
\ Subk (b) )
Obviously, there are no more than
2\\Sub(^)\\ x ... x = 2k'^Sub(v)W (5)
such distinct columns. Denote this set as D. The power of the set D is also a finite number.
3. We put in the model M+:
F(b) = {D(c) | c > b}.
Since D is a finite set and N is infinite, there is an element c+ ^ 4 such that V d ^ c+ and Vg > c+ the following holds: F(d) = F(g) (the need to choose the number 4 is justified in Lemma 9 in [17] when estimating the power of the model Mc).
4. For any x € M+, a set of realizators is the minimum interval R(x) = [x, y], where for any two subformulas y-\Ujy2 and Nj yi from the set Sub(y) there is a valuation Vi such that the following holds:
((x Iby. fi Uj A (x ¥v. ^
^ | (B y G R(x)(y IIV. w)) A (y z (x < z<y)(z IIV. fi))
A
A
Ilvi Njfi ^ (x + 1) G R(x)
Denote by Rls(x) the largest point y in R(x). The minimum interval R(x) can be large, but it exists.
Consider the interval R(c+) = [c+, Rls(c+)].
By the definition of the point c+, there is a point d such that d ^ Rls(c+) + 2 and D(c+) = D(d). Take the least d = d+ with this property and delete from M+ all points that are strictly larger than d+. We put Next (d+) = c+ + 1 in the model M+.
Thus we obtain the positive part of the model Mc - the model M+.
Rules for calculating the truth for any valuation Vi in this part of the model we define in the same away as in the model MZ replacing in all items of Definition 1.5 the relation Next on Next , the relation ^ on 4, and the symbol < on —.
II. Construct now the model Mc.
Let there be given the points d+ h c+, which we define above when construct the model M + , and points d- and c- from Z \ N and such that d- + 1 < c- < —3 (the need to choose the number —3 is justified in Theorem 3.1 when estimating the power of the model Mc). Then,
Mr
^[d", d+] , Next', V1, ..., Vky.
where the relation Next on [d", d+) coincides with the relation Next, and at d+ is defined in the following away: we consider that Next (d+) = c+ + 1 and Next (c" — 1) = d", that is, we determine the relation Next on [d", d+].
The relation 4 on Mc we determine in the following away:
1. if xi ^ x2, then xi 4 x2,
2. on the segments [d", c" — 1] and [c+ + 1, d+] the relation 4 is an equivalence relation.
x
If x1, x2 € [d-, d+], then the case when a ^ b and a = b we denote as x1 — x2. Valuations V1, ... ,Vk on each point of the models Mc coincide with valuations of the corresponding points of the model MZ.
Now we describe in detail points d- and c- of the model Mc. Fist, we define the following values for every element b € MZ.
The first items 1 and 2 are the same as items 1 h 2 from I, where we define Subj (b) and D(b). 3. For each element b < 0 in Mc, we put:
Since D is a finite set and Z \ N is infinite, then there is an element c- ^ 0 such that V d ^ c-and V g < c- the equality G(d) = G(g) is true.
4. For any x € Mc, x ^ 0, a set of realizators is the minimum interval R(x) = [y, x], where for any two subformulas y1Sjy2 from the set Sub(y) there is a valuation Vi such that the following holds:
((x IIvi Sj y2) A (x¥Vi ^
Denote by Rls(x) the largest point y in R(x). The minimum interval R(x) can be large, but it exists.
Consider the interval R(c-) = [Rls(c-), c-].
By the definition of the point c-, there is a point d such that d < Rls(c-) and D(c-) = D(d). Take the largest d = d- with this property and delete from Mc all points that are strictly least than d- .
Thus we obtain the model Mc € K (Ma).
The rules for calculating the truth for any valuation Vi in this part of the model we define in the same away as in the model MZ replacing in all items the definitions 1.5 the relation Next on Next , the relation ^ on ^ and the symbol < on —.
Let there be given a formula y in the language of the logic L (MZ), a model MZ and constructed according to the given formula and the model - the model Mc.
Lemma 2.1. For any subformula ^ € Sub(y), for any element a € MZ and any valuation Vj the following holds:
(Mz, a) IIvi ^ ^^ (Mc, a) IIvi ^.
Proof. We prove by induction on the length of the subformula ^ € Sub(y).
I. Given the models M+ and M+, the statement of the lemma proved for formulas constructed from the propositional variables, Boolean logical operators, and temporal operators Nj and Uj by V. V.Rybakov [17] (Lemmas 7 and 8). As it is follows from the remark after Definition 1.5, we have that the lemma also holds for formulas with operators Oj and Oj.
Moreover, based on the construction of the point d+ and due to the fact that we did not change the model MZ from the point d- to the point c+, it is easy to see that for any element a € MZ the proof of this lemma for the case when formulas consist of propositional variables,
G(b) = {D(c) | c < b}.
(6)
Boolean logical operators, and temporary operators Nj and Uj, almost literally repeats the proofs from Lemmas 7 and 8 in [17].
Now we prove the case when ^ = yi Sj y2.
Suppose that the lemma holds for the formulas yi and y2.
Necessity.
II. Let the following holds:
(Mz,a) yi Sj y2. (7)
Show that the following also holds: (Mc,a) llv yi Sj y2. (8)
Formula (7) is equivalent to the truth of two statements at the same time:
3 b (b < a) (MZ, b) llVi y2 and (9)
Vc ((b<c < a) ^ (Mz,c) llv. yi). (10)
If b € [d-, d+], then by the inductive hypothesis and by the definition of the relation 4 on Mc we obtain that (9) yields
3 b (b 4 a) (Mc,b) llv y2, (11)
and (10) yields
Vc ((b — c 4 a) ^ (Mc, c) lb^ yi). (12)
But the truth of the statements (11) and (12) means the truth of the statement (8), therefore, for the case when b € [d-, d+], the necessity is proven.
Suppose now that b < d-. Hence, by the definition of the operator Sj, it is easy to obtain that if a ^ d-, then
(Mz, d-) llv, yi Sj y2.
By the construction of the points c- and d- we have D(c-) = D(d-). Hence, (Mz, c-) llv yi Sj y2-
But d- by its construction lies to the left of the interval of realizators R(c-), for which the condition (6) holds. Therefore, there is also a point bi € (d-, c-) such that
(Mz, bi) llvi y2■ (13)
Since bi > b, then by (10) we also have: Vc ((bi < c < a) ^ (Mz, c) lb^ yi) ■ (14)
But by the inductive hypothesis and by the definition of the relation 4 on Mc, (13) and (14) yield the truth of the following two statements:
3 b (b 4 a) (Mc, b) llvi y2, Vc ((b — c 4 a) ^ (Mc, c) llv, yi),
from where we finally conclude that the statement (8) holds in the model Mc. Thus, the necessity is proven. Sufficiency.
III. Let the following holds:
(Mc, a) IIvi y1 Sj y2. (15)
Show that the following is also holds:
By the inductive hypothesis and by the definition of the relation ^ on Mc, (17) and (18) yield
3b (b < a) (Mz, b) IIv y2, Vc ((b < c < a) ^ (Mz, c) IIv, y1),
which means that the condition (16) holds for the model MZ.
Therefore, the sufficiency is also proven. □
Suppose, that in some model MZ there is a point a and a valuation Vj such that (MZ, a) IIVi. Obviously, we can take a = 1, that is,
(Mz, 1) Ibi y.
Hence, Lemma 2.1 yields,
Lemma 2.2. Let there be given a formula y in the language of the logic L (MZ), some model MZ and a model Mc that constructed by the given formula and the model. Then
(Mz, 1) IIv, y ^^ (Mc, 1) IIv, y. Thus,
Theorem 2.3. If a formula y in the language of the logic L (MZ) satisfiable in a model of the class K (Mz), then it satisfiable in a model of the class K (Ma).
Let us prove the converse statement.
Theorem 2.4. If a formula y in the language of the logic L (MZ) satisfiable in a model of the class K (Ma), then it satisfiable in a model of the class K (MZ).
Proof. Let in some model Ma we have
where y is a formula in the language of the logic L (MZ).
Consider a model MZ with the following valuations V1, V2, ... ,Vk of its elements.
(Mz, a) II v y1 Sj y2. The statement (15) is equivalent to the truth of two statements:
(16)
3 b (b ^ a) (Mc, b) IIvi y2, Vc ((b — c ^ a) ^ (Mc, c) IIvi y 1).
(17)
(18)
(Ma, a) IIv, y,
1. Let a € MZ and a € [d-, d+ ].
Then the valuations Vi,V2, ... ,Vk of any element of the model MZ on [d-, d+] coincides with the valuations of the corresponded elements in the model Ma .
2. Let a € MZ and a € [d-, d+ ].
It is easy to see that if a > d+, then we can express a as a = c+ + 1 + j +1 ■ (d+ — c+), t > 1, where c+ + 1 + j ^ d+, j ^ 0.
Similarly, if a < d-, then we can express a as a = c- — 1 — j — t ■ (c- — d-), t ^ 1, where c- — 1 — j > d-, j > 0.
Thereat, the valuations of c+ + 1 + j +1 ■ (d+ — c+) € MZ, t > 1, coincide with the valuations of c+ + 1 + j ^ d+ from the model Ma, j ^ 0.
Similarly, the valuations of c- — 1 — j — t ■ (c- — d-) € MZ, t > 1, coincide with the valuations of c- — 1 — j ^ d- from the model Ma, j ^ 0.
Taking into account the given relations on the model Ma, it is easy to obtain by induction on the length of the formula that for any element s of the model Ma, any valuation Vi and any subformula ^ € Sub(y) the following holds:
(Ma, s) lhv, ^ ^ (Mz, s) lhv, (20)
Thereat, (20) yields
(Mz, a) lb* y■ (21)
Thus, (19) yields (21), that is, the theorem is proven. □
3. Decidability of the logic L (MZ)
In Section 3 we prove that the satisfiability problem in the logic L (MZ) is decidable, that is, there is an algorithm that for the final steps determines whether the formula y is satisfiable in the language of the given logic on some model from the class K (MZ) or not. To prove the decidability for any formula y in the language of the logic L (MZ), we construct a finite number of finite models, which sufficient for checking the satisfiability of the formula y. These models have the similar construction as the models Ma from Section 2.
V.V. Rybakov in Lemma 9 in [17] prove, that for any formula y in the language of the logic L (K) the following holds:
(M+, 1) llv* y ^ (Mc, 1) lv* y,
where Mc is a model, obtained in a special way from the model M+ and having the power at most f(y) = + 3 elements, where k is the number of different valuations in the
model MZ.
Let us prove a similar result for our models Mc. Theorem 3.1. Let there be given a formula y in the language of the logic L (MZ). Then,
(Mc, 1) lv* y ^ (Mc, 1) llv* y,
where Mc is a model obtained in a special way from the model Mc and that has a power no more than f (y) = 2 ■ (2fc'HSub(v)ll) + 5 elements, where k is a number of different valuations in the model MZ.
Proof. Let
(Mc, 1) Ibi y. (22)
I. First we transform the "positive" part of the model Mc by deleting from it certain intervals [x, y), where x > 1.
Recall that c- > 4. Consider the largest s1 € [2, c+ — 1] such that
D(2) = D(s1).
If s1 = 2, then delete the interval [2, s1) from the model Mc and denote the obtained model as M1. In this case, we assume that Next (1) = s1 and that for any x1, x2 € [2, s1) the following holds:
x1 ^ x2 in M1 ^^ x1 ^ x2 in Mc.
Valuations V1, ... ,Vk for all elements included in the model M1, we leave the same as they were for these elements in the model Mc.
If s1 = 2, then we do not change anything and assume that M1 = Mc. Denote in ascending order the elements of the model M1
.. ., —2, —1, 0, 1, S1, a2, as, .. . ,
where ai is the i + 1-th element of the model M1.
Let s1 = 2. Prove that (22) is holds if and only if the following holds:
(M1, 1) IIv y. (23)
The statement (23) is equivalent to the statement that for any element s € M1, any subformula ^ € Sub(y) and any valuation Vi the following holds:
(Mc, s) IIVi ^ ^ (M1, s) IIVi ^. (24)
We prove this statement by induction on the length of the formula
The truth of the statement (24) for Boolean operators is obvious. The proof for the operators Nj and Uj repeats the corresponding proof for these operators in Lemma 9 in [17]. Suppose that (24) is proved for y1 and y2. Let us prove that it is true for y1 Sj y2. By the definition of Sj, the statement
(Mc, s) Ibi y1 Sj y2 (25)
is equivalent to
3 b [(b ^ s) (Mc,b) IIVi y2 AV c ((b — c ^ s) ^ (Mc, c) IIVi y1)], (26)
and the definition
(M1, s) IIvi y1 Sj y2 (27)
is equivalent to
3 b [(b ^ s) (M1, b) IIvi y2 AV c ((b — c ^ s) ^ (M1, c) IIv y1)]. (28)
1. Let b € [2, s\).
Any element x € [2, s\) of Mc also belongs to M\, hence, by the inductive hypothesis and by the definition of the relation 4 in the model M\, (26) yields (28), and hence, yields (27).
2. Let b € [2, s\).
The s > s\. From the definition of the operator Sj it is easy to obtain that
(Mi, si) llv* yi Sj y2. Since D(s\) = D(2), we have (Mi, 2) llv* yi Sj y2.
If
(Mi, 2) llv* y2, then, since D(s\) = D(2), we have (Mi, si) llv* y2. Consequently, (25) is also equivalent to the statement 3 bi [(bi 4 s) (Mc, bi) llv* y2 A Vc ((bi ^ c 4 s) ^ (Mc, c) llv* yi)], (29)
where bi < 2 or bi = s\. But such elements belong M\. Then by the inductive hypothesis and by the definition of the relation 4 in the model M\, (29) yields
3 bi [(bi 4 s) (Mi, bi) llv* y2 a vc ((bi -< c 4 s) ^ (Mi, c) llv* yi)], (30)
which is equivalent to the statement (27).
3. Since all elements of the model M\ also belong to the model Mc, using the inductive assumption and taking into account the definition of the relation 4 in the model M\ , it is easy to carry out the proof in the opposite direction, that is, to obtain that (27) yields (25).
Thus, the statement (24), and, consequently, the statement (23), holds. Now we transform the model M\ in the same way. Consider the maximum s2 € [a2, c+ — 1] such that
D(a2) = D(s2).
If s2 = a2, then delete the interval [a2, s2) from the model Mc and denote the obtained model as M2. In this case, we assume that Next (s\) = s2 and that for any x\, x2 € [a2, s2) the following holds:
xi 4 X2 in M2 ^^ xi 4 X2 in Mi.
Valuations V\, ... ,Vk of all elements included in the model M2 we leave the same as they were for these elements in the model M\ .
If s2 = x2, then we do not change anything and assume that M2 = M\. We denote in ascending order the elements of the model M2:
—2, —1, 0, 1, s\, s2, b3, ... ,
where bi is the i + 1-th element of M2.
Further, in a similar way, we obtain from the model M2 the model Ms and so on. When we obtain at the k-th step that sk = c+ — 1, then we skip the point c+, and on the k + 1-th step we consider the interval [c+ + 1, d+), and then we continue the transformation in the same way.
A proof that for any element s € Mk, any subformula ^ € Sub(y), and any valuation Vi the following holds:
(Mk-1, s) Ibi ^ ^ (Mk, s) IK* ^ (31)
is similar to those given above by induction on the length of the formula Consequently, we obtain that (22) holds if and only if the following statement holds:
(Mk, 1) IIvi y.
The number t of steps than we can take is equal to the number of the different sets D(x), x € [2, d+], where x € {1, c+, d+}, that is, does not exceed the value 2k'\\Sub(v)\ (5).
Thereat, the number of positive elements in the model Mt, obtained at the last step, obviously does not exceed the value
2k-\\Sub(v)\\ +3. (32)
II. After we obtain the model Mt, we similarly continue transformation of the "negative part" of this model, deleting from it certain intervals (y, x], where x < 0. Recall, that c- < —3. Consider the least r1 € [c- + 1, 0] such that
D(0) = D(n).
If r1 = 0, then we delete the interval (r1, 0] from the model Mt and denote the obtained model M-1,t. In this case, we assume that Next (r1) = 1 and for any x1, x2 € (r1, 0] the following holds:
x1 ^ x2 in M-1,t ^^ x1 ^ x2 in Mt.
Valuations V1, ... ,Vk of all elements included in the model M-1,t we leave the same as they were in these elements in the model Mt.
If x = 0, then we do not change anything and assume that Mt = M-1,t. The elements of the model M-1tt we denote as:
.. .,ts, t2, r1, 1, s1, a2, as, ... .
Now we transform the model M-1tt in the same way. Consider the least r2 € [c- + 1, t2] such that
D(t2) = D(r2).
If r2 = t2, then we delete the interval (r2, t2] from the model M-1,t and denote the obtained model M-2,t. In this case, we assume that Next (r2) = r1 and for any x1, x2 € (r2, t2] the following holds:
x1 ^ x2 in M-2,t ^^ x1 ^ x2 in M-1,t.
Valuations V\, ... ,Vk of all elements included in the model M-2,t we leave the same as they were in these elements in the model M-i,t.
If r2 = t2, then we do not change anything and assume that 21 — M — i.t.
The elements of the model M-2,t we denote as:
. .., U4, U3, r2, ri, 1, si, s2, b3, ... .
Further, in a similar way, moving from larger negative numbers to smaller ones, we obtain from the model M-2,t the model M-3,t, from M-3,t the model M-4,t and so on. When on the l-th we obtain that ri = c- + 1, then we skip the point c- and on the l + 1-th step we consider the interval (d-, c- — 1], and then continue the transformations in the same way.
The numberm of steps than we can take is equal to the number of different sets D(x), x € [d-, 0], where x € {c-, d-}, that is, does not exceed the value 2k'llSub(v)l1 (5).
Then the number of positive elements in the model M-m,t, obtained at the last step, obviously does not exceed the value
2kllSub(v)l1 +2. (33)
From estimations (33) and (32) we have
\M-m,t\ < 2 ■ (V ^M11) +5. (34)
A proof that for any element s € M-m,t, any subformula tf € Sub(y), and any valuation Vi the following holds:
(Mc, s) llv* tf ^ (M-m,t, s) llv* tf (35)
is similar to the proof in Part I.
First, we prove by induction on the length of the subformula tf that
(Mt, s) llv* tf ^ (M-i,t, s) llv* tf, (36)
then we prove at each step in a similar way that
(M-i,t, s) llv* tf ^ (M-(i+i),t, s) llv* tf, (37)
and thus we arrive at the statement (35).
Let us prove the truth of the statement (36). We prove it by induction on the length of the formula tf. For Boolean operators, the proof is obvious. Then, we make the inductive assumption that the statement is proved for the formulas yi and y2.
Let tf = yi Uj y2 and
(Mt,s) llv* yi Uj y2, (38)
that is
3 b [(s 4 b) A (b llv* y2) AV c [(s 4 c ^ b) ^ (c llv* yi)]].
If b € (ri, 0], then, as we have in I.1 and I.3, (36) follows from the inductive assumption and from the definition of the relation 4 in the model M-itt.
Let b € (ri, 0]. Thereat, we obtain from the properties of the temporal operator Uj that
(Mt, ri) llv* yi Uj y2,
and hence, since D(r1) = D(0), (Mt, 0) IIvi y1 Uj y2.
If
(Mt, 0) IIvi y2, then, since D(r1) = D(0), (Mt, T'1) IIvi y2,
But then there is a point b1 > 0 or b1 = r1 such that
3 b1 [(s ^ b1) A ((Mt, b1) IIvi y2) A Vc [(s ^ c — b1) ^ ((Mt, c) IIv* y1)]] .
Hence, by the inductive hypothesis and by the definition of the relation ^ in the M-1tt, we have that now in the model M-1tt the following holds:
3 b1 [(s ^ b1) A ((M-1,t, b1) IK* y2) A Vc [(s ^ c — b1) ^ ((M-1,t, c) IIv* y1)]],
which is equivalent to the statement
(M-1,t, s) IIvi y1 Uj y2. (39)
Consequently, (38) yields (39).
Further, since all elements of the model M-1,t belong to the model Mt, then as in Part I, by the inductive hypothesis and by the definition of the relation ^ in the model M-1,t, it is easy to obtain that (39) yields (38).
Proofs for the operators Nj and Sj are similar. In the case when the corresponding point b € (r1, 0], the necessity and sufficiency follow from the inductive hypothesis and from the definition of the relation ^ in the model M-1tt. And in the case when b € (r1, 0], we use the condition D(r1) = D(0).
Thus, the statement (36) holds. In m — 1 steps, we get the statement (35).
Denote the model M-m,t = Mc. Obviously, the statement (35) is equalent to the fact that for any formula y in the language of the logic L (MZ) and any valuation Vi the following holds:
(Mc, 1) IIvi y ^^ (M , 1) IIv* y.
We estimate the power of the Mc in (34). Thus, the theorem is proved. □
Lemma 2.2 and Theorem 3.1 together give us the following result.
Lemma 3.2. Let there be given a formula y in the language of the logic L (MZ), some model MZ and a model Mc that constructed by the given formula and the model. Then
(Mz, 1) Iv* y ^^ (Mc, 1) Iv* y.
For a given formula y, we denote
f (y) =2 • (> \\Sub(v)\\j +5, f1(y) = 2k'WSub(v)\\ +2,
f2 (y) = 2k-\Sub(v)\ +3.
Model Ma,v
Let us describe one more class of models constructed for a formula y - class of models Mc,v, a special case of which are the models Mc.
Let there be given a formula y.
The class K (Mc,v) is the class of models Ma with
A=[ —f(y),f2(y)] .
Obviously, that K (Mc>v) is a finite set of finite models and Mc € K (Mc>v), where Mc is a model obtained in Theorem 3.1.
Now it is easy to get the main result.
Theorem 3.3. The problem of satisfiability in the logic L (MZ) is decidable.
Proof. Let there be given a formula y in the language of the logic L (MZ). In a finite number of steps, we can check its satisfiability at any point of any model from the class K (Mc,v), which contains the model Mc.
If the formula y is satisfiable in some model from the class K (Mc,v), then by Theorem 2.4, it is satisfiable in some model from the class K (MZ).
If the formula y is refutable in some model from the class K (Mc,v), then it is refutable in the model Mc, and hence, by Lemma 3.2, y is refutable in some model from the class K (MZ). □
Besides, if the formula y is valid in all models from the class K (Mc,v), then there are no models in the class K (MZ) where y is refutable. Indeed, suppose the converse, that is, there exists MZ such that y is refutable at a. As we wrote above, we can take a =1. But by Lemma 3.2, the formula y is refutable in the model Mc € K (Mc,v).
Finally,
Theorem 3.4. The logic L (MZ) is decidable.
The research was supported by Russian Foundation for Basic Research and by Krasnoyarsk Regional Science Foundation (Grant 18-41-240005) and by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of Regional Centers for Mathematics Research and Education (Agreement no. 075-02-2020-1534/1).
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Выполнимость во временной логике с мультиозначиванием, основанной на Z
Владимир Р. Кияткин Анна В. Кошелева
Сибирский федеральный университет Красноярск, Российская Федерация
Аннотация. Статья продолжает серию работ В. В. Рыбакова, посвященных свойствам логик с мультиозначиванием и в которых предложен новый подход для моделирования знаний и рассуждений агентов в мультиагентной среде. В нашей работе доказано, что проблема выполнимости во временной логике с мультиозначиванием, основанной на Z, разрешима.
Ключевые слова: временная логика, мультиозначивание, логика знаний, мультиагентная логика, проблема выполнимости в логике, разрешающие алгоритмы, представления знаний, мультиагент-ные системы.
