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УДК 510.643
Unification and Inference Rules in the Multi-modal Logic of Knowledge and Linear Time LTK
Stepan I. Bashmakov*
Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041
Russia
Received 10.12.2015, received in revised form 10.01.2016, accepted 15.02.2016 We study unification of formulas in multi-modal LTK logic and give a syntactic description of all formulas which are non-unificable in this logic. Passive inference rules are considered, it is shown that in LTK logic there is a finite basis for passive rules.
Keywords: unification, modal temporal logic, passive inference rules. DOI: 10.17516/1997-1397-2016-9-2-149-157.
Introduction
The research of unification for various logic systems is one of the most rapidly developing areas of modern mathematical logic. Arisen in the field of Computer Science, primarily in the form of a question about the possibility to transform two different terms in syntactically equivalent by replacing the variables of certain other terms ( [1,2]), from the time the task has changed course on the study of semantic equivalence ( [3,4]).
For most of the non-classical logics (modal, pseudo-boolean, temporal, etc.), there are special dual equational theories of special algebraic systems, so their problems are reduced to the corresponding logical-unificational counterparts ( [5-7]). Basic unificational problem can be viewed as a complex issue: whether the formula is to be transformed into a theorem after replacing the variables (keeping the same values of the coefficients parameters). This issue was investigated and partly resolved (including V.V. Rybakov [8-10]), for intuitionistic and modal logics S4 and Grz.
Unification in intuitionistic logic and in propositional modal logic over the K4 investigated by S.Ghilardi [11-15] (with applications of projective algebra ideas and technology based on projective formulas). In these works the problem of constructing the finite complete sets of unifiers was solved for the considered logic, efficient algorithms were found. Such an approach proved to be a a useful and effective in dealing with the admissibility and the basis of admissible rules (Jerabek [16-18], Iemhoff, Metcalfe [19,20]). Indeed, the existence of computable finite sets of unifiers follows directly solution of the admissibility problem.
Temporal logic is also very dynamic area of mathematical logic and computer science (including Gabbay h Hodkinson [21-23]). In particular, LTL (linear temporal logic) has a significant application in the field of Computer Science (Manna, Pnueli [24,25], Vardi [26,27]). Solving the problem of admissibility of rules in the LTL was proposed by Rybakov [28], basis of admissible
* [email protected] © Siberian Federal University. All rights reserved
rules in LTL by Babenyshev and Rybakov in [29] (without the operator Until [30]). Solution of the unificational problem for LTL has also been found by Rybakov [31,32] and proposed for basic modal and intuitionistic logic in [33,34]. Particularly, in [31] It proved that not all unified in LTL formula are projective, and in [32] proved the projectivity of any unified formula in LTLu (it is a fragment of LTL, only with the operator Until). In the paper of Dzik and Wojtylak [35] they obtained the same result for ¿''O.
Research conducted in the present paper, primarily based on the approach proposed in [36]. The key focus here is on the description of non-unifiable formulas in a wide class of modal logics. Especially, it proposed the criteria of non-unifiable (with the proofs) for modal extensions of S4 (Theorem 1.4 below) and [K4 + = (Theorem 1.5). The aim of this article is to investigate the question of unification in linear temporal logic (LTK).
1. Fundamental definitions and notations
Before describing the main results, we introduce the most important definitions and notations. Proofs for the most of propositions, consequences and the theorems in this section are detailed in [36].
First, we define a unified formula in this logic. Let A is a logic with the formula $(p, q) which describes the equivalent formula. We say that a is equivalent to ¡3 in A,and we write a ¡3 if
^(a,ft). For convenience, ¡3) we denote a = ¡3.
Definition 1.1. Formula a(pi,... ,pn) is unifiable in an algebraic logic A iff there is a tuple of formulas 5i,... ,5n such that a(5i,..., 5n).
Definition 1.2. Formulas a(pi,... ,pn) and ¡3(pi,... ,pn) are said to be unifiable in algebraic logic A iff there is a tuple of formulas 5i,... ,5n such that a(5i,..., 5n) = ¡3 (¿i,..., 5n). In this case, the tuple ¿i,... ,5n is called an unifier for these two formulas.
Corollary 1.3 (2.7 from [36]). For all logics SIL, S4ext, K4 + m± = ± unifiers for formulas can be effectively found among sequences of formulas t end
For example, by setting t everywhere in the performance of the variable p and ± otherwise.
Theorem 1.4 (2.10 from [36]). For any modal logic A extending S4 and any modal formula a, a is not unifiable in A iff the formula Oa ^ \fpeVar(a) ^P a ^—P if provable in A.
Theorem 1.5 (2.11 from [36]). For any modal logic A extending K4, where = ± £ A and any modal formula a, a is not unifiable at A iff formula Oa a a ^ \/P£Var(a) ^P a ^—P is provable at A.
Definition 1.6. Rule r := A/B is a consequence of the rules ri := Ai/Bi ,...,rn := An/Bn in logic L ■ vA £ Var(L) = [A\A N (a = t), va £ L}: if
then
ViA N (ai = T) ^ (¡i =
A N (a = T) ^ (3 = T).
Let us recall the definition of algebra formulas, Lindenbaum algebra. Let For is the set of all formulas in the language of logic. We will use the following notation: A = B■ (A^ B)a(B ^ A). We write A =L B, if A = B £ L. Suppose that the logic L has theorem of replacing equivalent.
Namely, if © is any binary logic connective (for example a, V), and A1, A2, Bi h B2 are the formulas, then
(Ai =L Bi, A2 =L B2) ^ Ai © A2 =L Bi © B2, and if ® is any unary logical connective, then
Ai =L Bi ^ ®Ai =L ®Bi.
Lindenbaum algebra A/= has the basic set For=, where
For= := {[A]=\A € For, [A]= := {B\B =L A}}.
[A]= © [B]= := [A © B]^. We define an algebra A/= as follows: A/= = (For=, a, v, □}, where
A =l B ■ (A ^ B) a (B ^ A) € L.
Theorem 1.7 (Lindenbaum). For any modal formula a, modal logic L and variables xi,... ,xn € Var(L): a(xi,... ,xn) € L ■ a(xi,... ,xn) = t is a truth on A/=.
2. Semantics LTK
Alphabet of the language Lltk includes a countable set of propositional variables P := {pi,... ,pn,... }, braces (,) default Boolean operations and a variety of single modal operators □ ,,, ^ ..., ^n}. Every propositional variable p € P is well-formed formulae (wff), and if A is wff, then so are □^A, □eA, □iA(i € I). We abbreviate Fma(LLTK) as a set of all wff in language Lltk (hereinafter referred to the formula will be understood as formula from the set Fma(LLTK)). Logic operations O^, Oe, Oi determined through □i as follows:
O^ = -i^-i, Oe = -^e-, Oi = — □—. The values of described modal operators are defined as follows: □^A: A is a truth at the current time and in any future; □ A: A is a truth at a given moment of time; □iA means that A is a truth in all informational points which available to the agent i. Semantics for the language Lltk models linear and discrete stream of the computational process, in which each point in time is associated with a natural number n. Semantically, our logic is defined on the Kripke frames.
Definition 2.1. k-modal Kripke-frame is a tuple F = (WF,Ri,..., Rk}, where WF is a nonempty set of worlds and each Ri is some binary relation on WF.
Definition 2.2. Let F = (WF, Ri,..., Rk} is Kripke-frame, and vRi Ri-cluster is a subset CRi € WF such that vv, z € CRi : vRiz&zRiv and vz € WF, vv € CRi : ((vRiz&zRiv) ^ z € CRi). For any relation Ri, CRi(v) is the Ri-cluster s.t. v € CRi or cluster, generated by the element v. Ri-cluster called: degenerate, if it consists of a single Ri-irreflexive point; simple if it consists of a single Ri-reflexive point; proper, if it contains at least two Ri-reflexive points.
Definition 2.3. LTK-frame is a k+2-modal Kripke-frame F = (Wp, Ri,..., Rk, Re, Rwhere: 0) WF is the disjoint union of non-empty sets Ct, t € N: WF := |JteN Cf';
b) Ri, ...Rk are some equivalence relations within each cluster Cft;
c) Re is universal S5-relation of equivalence at any Cft € WF:
vw, z € WF(wRez ■ (w € Ct)&(z € Ct));
d) R^ is linear, reflexive, transitive binary temporal relation on WF, specifying linear order of clusters (simple chain):
vv,z € WF(vR^z & 3i,j € N((v € Ci)&(z € Cj)&(i < j)));
Also hold the following properties of matching these relations:
1) wRez & (wR^z)fo(zR^w);
2) wRiz ^ wRez.
We denote class of all such frames LTK.
Definition 2.4. For two R^ -clusters Cm and Cj notation CmRtCj indicates that vw € Cm, vz € Cj is performed (wR^z). Thus, Cm is R^-precursor of cluster Cj, and Cj is R^-follower of cluster Cm.
Frames of this class model a situation in which each agent has all the information in the current temporary state C*. Any temporary state C* (i.e R^-cluster) consists of a set of information points available at t. The relation R^ is a connection into a linear stream of information points, wherein for two points w and z term wR^z means that either w and z are available at the time t, or z will be available at subsequent times in relation to w. Relation Re connects all information points potentially available at the same moment of time, thus it represents knowledge that is potentially available at any given time. Each relation Ri, i = l,...,n, reflects the information available to a particular agent i.
Definition 2.5. Model MF on a LTK-frame F is a tuple MF = (F, V), where V is a valuation of a set of propositional letters p € P on the frame, i.e vp € P [V(p) ç WF]. Given a model MF = (F, V), where F is a LTK-frame WF. Then vw € WF :
a) (F, w) II v p & w € V(p);
b) (F, w) iiv O^A &vz € Wf(wR^z ^ (F, z) IIv A);
c) (F, w) IIV D^A &vz € WF(wRel ^ (F, z) IIV A);
d) vi € I, (F,w) IIV DiA &vz € WF(wRiz ^ (F,z) IIV A).
The relation IIV here means truth relation on the element w of model M. Namely, (F, w) IIV A means that A true on the element w in model (F, V). If the formula A true on any element of the frame F with any valuation V, we called A true on the frame F and write F I A.
Definition 2.6. The logic LTK is the set of all LTK-valid formulae on all frames: LTK := {A € Fma(LLTK )|VF € LTK (F II A)}. If А belongs to LTK, then we say that А is a theorem of LTK.
3. A criterion of non-unifiability
We immediately begin with the proof of the main statement of this section. Theorem 3.1. Any modal formula A is non-unifiable in LTK iff formula
\j o^p a o^-p
peVar(A)
is a theorem in LTK.
Proof. 1. Prove the theorem by contradiction. Assume that
□^A ^ y o^p a O<-p € LTK,
pEVar(A)
but at the same time, the formula A is unifiable in LTK.
Then by definition of unifier, there is a substitution (unifier) g s.t. g(A) € LTK. By the fact
that LTK is closed under substitution, we obtain g ^A ^
e LTK.
v O^p a O^-p
peVar(A)
Let us consider LTK-frame Fi with all single element clusters (i.e vt : Ct = a). Consider the valuation V for all variables q of formulas g(p), where p € Var(A), on the Fi : V(q) = 0. Then it is easy to check by the induction on the length of any formula B constructed on variables q that:
vb € Fi, vc € Fi : b llV B ■ c llV B.
Consequently,
vb € Fi : b Iv y O^g(p) a O^-g(p).
peVar(A)
At the same time,
vb € Fi : b llV □ ¿g(A).
Thereby,
vb € Fi : b lvg I □^A ^ which contradicts the hypothesis:
g I □¿A^
y o^p a o^-p
peVar(A)
y O^p a O^-p | € LTK.
peVar(A)
2. On the contrary, say that the formula A is non-unifiable in LTK, but at the same time
□ ¿A ->•
\/peVar(A) O^p a O^-p € LTK. Then, by finitary approximability of LTK, there is
a certain root frame F that disproves this formula: 3a € F : (F, a} jlV□¿A ^
y o^p a o<-p
peVar(A)
. Assume this element a as the root
That is (F, a}llv□¿A h (F, a} lv^lpeVar{A) O^p a O^-p of the frame Fi (Fi = a¿). By (F, a} lv \fpeVar(A) O^p a O^-p , vp € Var(A) : either
(1) vb € FiaR^b : b llV p,
or
(2) vb € FiaR<b : b jfVp.
Choose a substitution g for all of the variables p from the formula A as follows: vp € Var(A) : g(p) = t if (1) and g(p) = ± in the case of (2). Then g is a unifier of the formula A. Indeed, if we take any frame F2, any cluster a2 € F2 and any valuation V2:
a2 lV2 A ■ a llV A.
Therefore, the formula A is unifiable in LTK. □
4. Passive inference rules
Definition 4.1. Let r := A1,...,An/3 be an inference rule in the logic LTK. The rule r called passive for LTK if for any substitution g of formulas instead of variables in r never g(A1) £ LTK&... &g(An) £ LTK. In other words r is a passive rule if formulas from its premise have no common unifiers.
Proposition 4.2. For multi-modal logic LTK the rules rn := ^iSiSi—S--S- form a
basis for all passive rules for LTK.
Proof. It is true that □ s V— SPi A—s—Pi ^ \Zis.iS.n —SP A —S—P £ LTK, and hence by Theorem 3.1 formula An = \J —SPi A —S—Pi does not unifiable in modal logic LTK, i.e any rule rn is passive. Let us assume that a rule t1 := A1,..., An/3 is passive for LTK. Then the rule t2 := A1 A - ■■ A An/3 is also passive for LTK and formula A1 A ■ ■ ■ A An is not unifiable in LTK. Applying Theorem 4.1 we conclude
□ s(A1 A ■■■A An) ^
V —SP a ■
peVar(A1A---AAn)
>S—P
LTK.
Using the premise of rule t2 we conclude
\f —SP a —S—P
peVar(A1A---AAn)
and then applying the rule rn, where n is the number of variables in the conjunction of A1 a ■ ■ ■ a An, we can derive the formula From ± ^ ¡3 £ LTK, in its turn holds ¡3. Thus, all rn really represent all passive rules in LTK. □
Now we consider the possibility of reduction infinite (due to an unlimited number of variables) basis of passive rules in LTK that was obtained in the Proposition 4.1 to a finite and more simple form.
Let us remind that the rule r is a consequence of the rules ri £ X, i £ I in a logic L, if for any algebra A £ Var(L) and vi £ I: A N ri ^ A N r. Accordingly, a rule r is not a consequence of the rules ri £ X, i £ I otherwise. A rule r true in the algebra A if for any substitution of elements from algebra instead of the variables of a rule r if all formulas from the premise of a rule r is true, then a conclusion formula of r is also true.
—<P a —s—p
Theorem 4.3. In multi-modal logic LTK the rule r := —-— ^— is a basis for all passive inference rules
Proof. According to Proposition 4.2, it suffices to show that the rules rn (vn) are a consequence of r (r h rn, vn).
Suppose that it is not true:
V1SiSn —SPi a —S—Pi
rn := -^^-
n ±
is not a consequence of the rule r. Hence there is a finitely generated algebra A, in which the rule r is valid (A N r), but rn is not (A ^ rn), thus vi £ (1,... ,n) there is ai £ A : \/ —sai a —<—ai = T. Get a subalgebra A1 of algebra A generated by such elements ai, 1 < i < n,
(Ai = Ai(ai,... ,an) c A). Ai is a S4.3-algebra on □ <. By Lemma 4.3.18 from [10] Kripke-frame A+, associated to Ai has a single element reflexive maximal cluster C. By the definition A+, va € Ai, a c A+. By hypothesis of proof, \Ji<i<n O<ai a O<-ai € Ai, because Ai is a subalgebra A, on the construction. Then \Ji<i<n O<ao a O<-ai = t = A+, but it is impossible on a single element reflexive maximal cluster (C i<i<n O<ai a O<-ai), and hence vi<i<n O<ai a O<-ai € A+ that contradict with the proof conditions. □
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Унификация и правила вывода в многомодальной логике знания и линейного времени LTK
Степан И. Башмаков
Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041
Россия
В статье исследуется унификация формул в многомодальной логике LTK и предложено синтаксическое описание всех формул, которые не являются унифицируемыми в данной логике. Рассмотрен вопрос пассивных правил вывода, показано, что в логике LTK есть конечный базис для пассивных правил.
Ключевые слова: унификация, модальная темпоральная логика, пассивные правила вывода.
