[ieee 2006 international conference on machine learning and cybernetics - dalian, china...

Proceedings of the Fifth International Conference on Machine Learning and Cybernetics, Dalian, 13-16 August 2006

1-4244-0060-0/06/$20.00 ©2006 IEEE 2548

A BOOLEAN PRUNING METHOD FOR IMPROVING TABLEAU REASONING EFFICIENCY IN FIRST-ORDER MULTIPLE-VALUED LOGICS

QUAN LIU1, 2, ZHI-MING CUI1, JI-GUI SUN2, WAN-JUN YU2

1 College of Computer Science and Technology, Soochow University, Soochow 215006,China 2 Ministry of Education Laboratory of Symbol Calculation and Knowledge Engineering, Jilin University, Changchun

130012,China E-MAIL: [email protected], [email protected],

Abstract: Tableau method with quantifiers in first-order

multiple-valued logic has uniform rules of extension, and sound and completeness have been proved by Zabel and so on. The number of branches increases in exponent with the increasing of truth-value, which will affect the performance efficiency of machine. A Boolean pruning method is proposed in this paper, which simplified the extended rules of logic formula with quantifiers in first-order multiple-valued greatly by linking the signed formula and upper/lower bound of set. In addition, through the analyzing of Boolean pruning method, simplified tableau reasoning method for a kind of special regular logic formulae in first-order multiple-valued was founded, which made logic tableau reasoning method with quantifiers in first-order multiple-valued is similar to classical logic tableau method.

Keywords: Multiple-valued logics; Quantifier; Tableau method;

Upper/lower set; Regular formula

1. Introduction

Semantic tableau method was brought forward by Beth in 1959 and Hintikka in 1955, later AI researchers imported it to automated theorem proving [1]. For the closeness of its logical semantic definition, it can be expanded to non-classical logics after some expansion of formulae structure sets for different logic system. With the in-deep study of multiple-valued logic, proving theory, algebra, axiomatization and application etc., tableau method was introduced into multiple-valued logic [2, 3] and it has very good application in the field of formal verification [4], nature language processing[5], fuzzy control [6], expert system [7], database, knowledge expression[8] etc. in the late of 1980’s.

In multiple-valued logics, tableau method with quantifiers was introduced by Carnielli in 1987, Zabel found satisfiable expansion rules in theory in 1993 and gave

a proving of soundness and completeness [9, 10]. But it is very difficult for automated theorem proving, it is necessary to find a reasoning method with less expansion branches. This paper that provides Boolean pruning method, which linked the signed formula and upper/lower, set and simplified the expansion rules of formula with quantifiers in first order. After further discussion of Boolean pruning method, a simplified tableau reasoning method of special regular formula in first-order multiple-valued was founded, which made logic tableau reasoning method with quantifiers in first-order multiple-valued be similar to tableau method of classical logic. To save space, please see reference [9, 10, 11] for the unexplained signs and concept.

2. The outline of multiple-valued logics

Multiple-valued logics have similar language structure with classical logic. A first-order language L is a triple <Θ，

Λ，α> , where Θ is a non-empty family of predicate connective symbols, Λ is a non-empty family of quantifier and α defines the arity of each connective. For example, the language of first-order three-valued Kleene logic is defined by LKle=<﹁/1, ∨/2, ∧/2, ∀, ∃>.

Multiple-valued logics are different from other non-classical logic. It starts from the reconstructing of two-value semantic, expands truth values set, redefines the meaning of language component of truth value connective, quantifier, predicate and formula etc [12]. So multiple-valued logics have big difference with two-value logic on semantic and its reasoning method is reconstructed around the redefined semantic. The set of truth values N is either the unit interval on the rational numbers, denoted with [0,1], or it is a finite set of rational numbers of the

form ｛０, 11−n ,…, 1

2−−

nn

,1｝, where n= N , N denotes the cardinality of N. IfＬ＝<Θ, Λ, α> is a first-order language then we call a triple Α=<N, A, Q> a first-order matrix for L,


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where N is a set of truth values and A assigns to each θ∈Θ a function Ａ(θ): Ｎα(θ)→N, Q assigns to each λ∈Λ a function Q(λ): Ρ＋(N)→N (we abbreviate 2N\∅ by Ρ＋(N)). Q(λ) is called the distribution function of the quantifier λ. For example, matrix of first-order n-valued Kleene logic is defined by: ﹁ i=1-I, i ∨ j=max{i,j}, i ∧ j=min{i,j}, Q(∀)=min, Q(∃)=max.

Let ℒ be a first-order logic. A first-order structure M over Σ is a pair <D,I>, where D is a non-empty set, called the domain and an interpretation I that designed to constant, function, predicate in a non-empty domain D as follow:

1. For every constant symbol, an element in D is assigned.

2. For every n-arity function symbol, a function is assigned. In other word, for function I(f), if f∈FΣ then Dα(f)→D.

3. For every n-arity predicate symbol, a predicate is assigned. In other word, for function I(p), if p ∈PΣ thenＤα(p)→N．

A variable assignment is a function β: Var→D. For d∈D the d-variant at x of β is defined as

d x=y dxβ (y)= β(y) x≠y

Given a structure M and a variable assignment β we define a valuation function νM, β: LΣ→N:

1. If φ＝p (t1,…,tn) then νM, β(φ)=I(p)( νM, β(t1),…,νM,

β(tn)). 2. If φ ＝ θ(φ1,…, φn) then νM, β(φ)=A(θ)(νM,

β(φ1),…,νM,β( φn)). 3. The distribution of φ at x is d M,

β,x(φ)={ dxM βν

,(φ )d∈D}. If φ ＝ (λx)ϕ then νM,

β(φ)=Q(λ)(d M, β,x(ϕ)) .

3. Tableau with quantifiers in first-order multiple-valued logics

For the change of truth-value, the corresponding signs of signed tableau system will be reinterpreted. The signs will not be T and F (1 and 0) under two-value logic but truth-value set. Let SN⊆P+(N) be a family of truth-value sets. Let φ be a formula and S∈SN. Then we call the expression Sφ signed formula. If p is an atomic formula, then Sp is a signed literal. A signed formula Sφ is satiable if there is a structure M and a variable assignment β such that νM,β (φ)∈S.

In multiple-valued logic in first order, signed formula

with quantifier S (λx)φ(x) can get the tableau rules by quantifier distribution function )φ(ZS ijijJjIi i∈∈ ∧∨ , where Sij⊆N, Zij belongs to the following two ground items: 1. Skolem constant c, which doesn’t show in the process of proving; 2, an arbitrary ground item t.

Theorem 1. Let (Q(λ))-1(S)={I∅≠I⊆N，Q(λ) (I)∈S}. Then a sign quantified formula S (λx)φ(x) is satisfied if there is an I∈(Q(λ))-1(S) such that

1. for each i∈I, there is a constant term ci not occurring in φ(x) such that {i}φ (ci) and

2. for any ground term t, Iφ(t) are simultaneously satisfiable.

Theorem 1 can conveniently be expressed as rules of Skolem item and signs, these rules have been proved to be sound and complete. If φ be a formula in a multiple-valued logics and ∅ ≠ S⊆N, then φ is S-valid if there exists a closed tableau for (N\S) φ. The form of rules can be expressed as follow:

S (λx)φ(x) (1)

{ 11i }φ( 1c ) . .

{11ki }φ(

1kc )

I1φ(t1)

… … …

{ 1mi }φ( 1c ) . .

{mki1 }φ(

mkc )

Imφ(tm) Where (Q (λ))-1(S)={I,…,Im}, Ij={ 1ji ,…,

jjki },

c1,c2,… is new Skolem constant, and t1,…,tm are random ground items. For a certain j, the corresponding expansion only has Ijφ(t) is enough when Ij={ij}, otherwise highlighted formula like Nφ(t) can be deleted.

For example, in three-value Kleene logic in first-order, {0}(∀x)φ(x) can be expanded according to formulation (1), where (Q(∀))-1({0})={{0},{0, 2

1 },{0,1},{0, 21 ,1}}. The

expansion result is as follow: {0}(∀x)φ(x) (2)

{0}φ(c) {0}φ(t1)

{0}φ(c) { 2

1 }φ(d)

{0, 21 }φ(t2)

{0}φ(c) { 2

1 }φ(d) {1}φ(e)

{0, 21 ,1}φ(t3)

{0}φ(c)

{1}φ(e) {0,1}φ(t4)

The number of branches can be close to 2n-2 for every

truth value set in (Q(λ))-1(S) must be expanded. The simple formula (2) shows that the expansion of {0}(∀x)φ(x) becomes a reasonable branch that will make tableau very


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large. If one more i∈N is added to S, the maximum number of tableau expansion branches will be 2N-1. Tableau will be very difficult to be realized when N is relatively large. So it is very necessary to search after simplified tableau expansion rules of multiple-valued logics.

4. The improvement on tableau method by Boolean set

When the above-mentioned discussion is pieced together with classical logic, it can be found that {0}(∀x)φ(x) (which can be expressed as F(∀x)φ(x) in classical logic) is obviously not the simplest expansion rules, it is also reliable and complete if it is expanded as

)(}0{)()}(0{

cxx

φφ∀ . In theorem 1, every truth value set in

(Q(λ))-1(S) expands under Skeleon qualifications and expresses the distribution of truth value in the form of Fφ(c) and Iφ(t), where F, I⊆N, c is ‘newly added’ and t is a random item. It is worth mentioning that these distributions sets occupy particular roles in 2N Boolean set.

For example, let S be a random set, power set of S be P (S), semi-order relation be set inclusion relation, then (P (S), ⊆) is the lattice, the reason is that A⊆A ∪ B, B⊆A ∪ B for any A, B∈P(S). So A ∪ B is the upper bound of {A, B}, if C is the upper bound of {A, B}, namely A⊆C, B⊆C, then A ∪ B⊆C, A ∪ B is the minimum upper bound of {A, B}, {A, B}=A ∩ B=A ∪ B; A ∩ B⊆A, A ∩ B⊆B, so A ∩ B is the lower bound of {A, B}, namely the maximum lower bound. Here we linked the elements of P (S) with distribution of truth-value.

Definition 1. Let ∅ ≠ F⊆N, upset generated by Boolean set lattice 2N is U(F)={X | X⊆N, X ∩ F≠∅}. Let ∅ ≠ I⊆N, down set generated by Boolean set lattice 2N is D (I)={X |X⊆N, ∅≠X⊆I}.

Theorem 2. Let N is finite truth value set and S⊆N such that:

1. If (Q(λ))-1(S)= U(F), then S (λx)φ(x) is satisfiable if there is a new Skolem constant c such that Fφ(c) is satisfiable.

2. If (Q(λ))-1(S)=D(I), then S (λx)φ(x) is satisfiable if Iφ(t) is satisfiable for all ground terms t.

Proof. 1. (only if:) Assume M ,β be such that νM,β((λx)φ(x))∈S. Thus

the distribution I=dM,β,x(φ(x)) is in (Q(λ))-1(S) and in U(F). By definition of U(F): I ∩ F≠∅, thus consider i∈I ∩ F. By i∈I and the definition of a distribution there must be a ground term t such that νM,β(φ(t))=i. We define M by

setting ∑＝Σ ∪ {c}, I M (p(t))=IM(p(t)) if c occurs in p. Then we have ν M ,β(φ(c))= ν M ,β(φ(t))= νM,β(φ(t))=i∈F, in other word, Fφ(c) is satisfiable.

(if:) Assume i=νM,β(φ(t))= ν M ,β(φ(c))∈F. By definition of

D(I), i∈dM,β,x(φ(x))=I. Hence, by definition of U(F): I∈U(F), we have νM,β((λx)φ(x))= Q(λ)(dM,β,x(φ(x)))= Q(λ)(I)∈S, in other word, S (λx)φ(x) is satisfiable.

2. (only if:) Assume M ,β be such that νM,β((λx)φ(x))∈S. Thus

the distribution J=dM,β,x(φ(x)) is in (Q(λ))-1(S) and in D(I). By definition of D(I): I⊇J≠∅, for an arbitrary term t one has νM,β(φ(t))∈J⊆I, in other word, Iφ(t) is satisfiable.

(if:) Assume νM, β (φ(t))∈ I for all ground terms t. By

definition, this entails ∅≠ dM,β,x(φ(x)) ⊆I, hence dM,β,x(φ(x)) ∈ D(I) ＝ (Q(λ))-1(S), From this one immediately concludes νM,β((λx)φ(x))∈S. In other word, S (λx)φ(x) is satisfiable. ▍

For example, in formulation (2) of three-value Kleene logic in first-order, because of (Q(∀))-1({0})=U({0})= {{0}, {0, 2

1 }, {0,1}, {0, 21 ,1}}, the rule shown in

formulation (2) can be simplified to:

{0}(∀x)φ(x) {0}φ(c)

In addition, Because of (Q(∃))-1({0})=D({0})=

{{0}}, rule

{0}(∃x)φ(x) {0}φ(t)

Holds. The conclusion is similar to classical logic. The combination of tableau method with Boolean set

simplifies tableau expanding branches, we name this method as Boolean Pruning in this paper.

Formulae with symbolic form, such as Fφ(c), are used to depict quantifiers of upper bound of 2N set corresponding to distribution. Signed formulae with symbolic form, such as Iφ(t) are used to depict quantifiers of lower bound of 2N set corresponding to distribution.

In reality, not all (Q(λ))-1(S) can be expressed as upper/lower bound of set or combined DNF expression, so the above result has certain particularity, (Q(λ))-1(S) can be expressed as two parts of M R, where M can be regarded ∪as the upper/lower set or combined DNF expression. In this


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case, at least the part corresponding to M can be simplified and it will simplify tableau rules greatly. It is obvious that time efficiency and space efficiency are O (nk) and O (n*k) respectively in Zabel method, while O (n*k) and O(n) respectively in tableau method. Boolean pruning tableau method has great improvement on efficiency and realization than Zabel method. In the following we will deal with a group of formulae with special symbolic forms and offer much more simplified multiple-valued logics Boolean pruning tableau rules.

5. Tableau with regular formula

In classical first-order logic, all quantified formulas and their negations are grouped into two categories, those that act universally, which are called γ-formulas and those that act existentially, which are called δ-formulas [12,13]. The groups and the notions of instance are given as follows:

Universal Existential

γ δ (∀x)φ ¬(∃x)φ

(∃x)φ ¬(∀x)φ

In tableau with free variable, expansion rules of

formulas with quantifier are as follows:

γ δ γ(x) δ(f(x1,…,xn))

x is a free variable f is a new Skolem function and x1,…,xn is variable on the branch

Definite 2. Let φ∈L and i∈N, ≥iφ and ≤iφ is

called a regular formula, where ≥i={j| j∈N, j≥i}, ≤i ={j| j∈N, j≤i}.

Theorem 3. Let ℒ be any n-valued regular first-order logic.

1. ≥ i(∀x)φ(x)( ≤ i(∃x)φ(x)) is satisfiable iff ≥iφ(t)(≤iφ(t))is satisfiable for any term t.

2. ≤ i(∀x)φ(x)( ≥ i(∃x)φ(x)) is satisfiable iff ≤iφ(c)(≥iφ(c)) is satisfiable for a new Skolem constant c.

Proof. 1. (only if) Let ≥i(∀x)φ(x) be satisfiable. Then in some model M

for some assignment β νM, β((∀x)φ(x)) {∈ i, i+ 1

1−n ,…,1}

iff

()(min

,xd

xMDdφν β∈ ) {i∈ ，i+ 1

1−n ,…, 1}

By definition of min, this value is unique, say ( )(min

,xd

xMDdφν β∈

)＝i0 且 i≤i0≤1

Thus, for any d D∈ )(

,xd

xMφν β ∈{i0,…, 1} (3)

Thus, for all ground terms φν ββ )(, t

xM = νM,β(φ{t/x}) (4) Thus, for any term t, by ⑶ and ⑷

)(, tM φν β ∈{i0,…, 1}⊆{i, i+ 11−n ,…,1}.

(If:) Let ≥iφ(t) be satisfiable for every ground term t.

Hence, for all t, )(, tM φν β ∈{i, i+ 1

1−n ,…,1}.

And by ⑷: )()(,

xtxM

φν ββ ∈{i, i+ 11−n ,…, 1} (5)

By assumption, for each d∈D there is a td∈T such that β(td)=d. By this fact and ⑸:

)(min,

xdxMDdφν β∈ =

)(min )(,xdt

xMDdφν ββ∈

= )(min )(,xt

xMDdφν ββ∈

∈{I, i+ 11−n ,…, 1}

≤i(∃x)φ(x) is similar to be proved. 2. (Only if:) Let ≤i(∀x)φ(x) be satisfiable. Then in some model

M for some assignment β νM, β((∀x)φ(x))∈{０,…, i－ 1

1−n ，i}

this holds iff ( )(min

,xd

xMDdφν β∈

)∈{０,…, i－ 11−n , i}

By definition of min, this value is unique, say, ( )(min

,xd

xMDdφν β∈

)＝i0 and 0≤i0≤i－ 11−n

Choose a d0∈D such that )(0,xd

xMφν β

=i0. Now, just

as in the classical case, extend M to M´ by a constant c that does not occur elsewhere and set I(c)=d0 in M´. By this and ⑷ we have:

νM, βφ(c)= )()(, xcxM φν ββ = )(0,

xdxM

φν β=i0

∈{０,…, i－ 11−n , i}

(If:) Let ≤ iφ(c) be satisfiable for a constant c. By

definition,


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νM, βφ(c)∈{０,…,i－ 11−n , i}.

By ⑷ )()(, xc

xM φν ββ ∈{０, …, i－ 11−n ，i}

By definition of min, )(min

,xd

xMDdφν β∈

∈{０, …, i－ 11−n ，i}

≥i(∃x)φ(x) is similar to be proved. ▍ By theorem 3 and Boolean Pruning Method, it is very

easy to be expanded to regular formulas. Let ℒ be an arbitrary first-order logic, then rules of tableau with quantifiers can be expanded as follows:

≥i(∀x)φ(x) ≤i(∀x)φ(x)

≥iφ(t) ≤iφ(f(x1,…,xn))

≤i(∃x)φ(x) ≥i(∃x)φ(x)

≤iφ(t) ≥iφ(f(x1,…,xn))

Where f is a new function symbol and x1,…,xn are variables occurring on branch.

≥ i(∀x)φ(x) and ≤ i(∃x)φ(x) can be regarded as γ-formulas in classical logics. ≤i(∀x)φ(x) and ≥i(∃x)φ(x) can be regarded as δ-formulas in classical logics. Hence regular formulas similar to be deduced to classical logics.

By definition 2, ≤i = {0，…，i}＝[0，1]∩N and ≥i = {i，…，1}＝[0，1]∩N. Hence for arbitrary i∈N there is {i}= ≥ i ∩≤ i. The special regular formulas have universality and have reasoning rules as follows:

Contraction rule: Splitting rule:

S1φ S1∩S2φ S2φ S1φ

S1∩S2φ S2φ

These rules are obviously sound for choice of S1 and S2. Then tableau rule with singleton signs can be described forward:

{i}(∀x)φ(x) {i}(∃x)φ(x)

≥iφ(t) ≤iφ(t) {i}φ(c) {i}φ(c)

We obtain a derivation of {i}(∀x)φ(x) as follows: (1) {i}(∀x)φ(x)

(2) ≥i∩≤i(∀x)φ(x) (3) ≥i(∀x)φ(x) (4) ≤i(∀x)φ(x) (5) ≥iφ(t) (6) ≤iφ(c) (7) ≥iφ(c) (8) {i}φ(c)

In the procedure, we use one time ≤i(∀x)φ(x) and two times ≥i(∀x)φ(x). (2) is from {i} = ≥i ∩ ≤i , (3), (4) come from spitting rules, (5), (6) come from theorem 2, (7) come from (3) by substitution {c/t}, (8) come from contraction rule.

For example, truth value is {0, 21 ,1} in three –valued

logics {０}(∀x)φ(x)

≥０φ(t) （{0, 21 ,1}φ(t)）

0φ(c) The conclusions similar to classical logics.

6. Conclusions

This paper introduced multiple-valued logic tableau reasoning method with quantifier, and linked Boolean set to get simplified method, Boolean pruning method, which simplified the tableau branch greatly and improved the situation that it is hard for machine to realize multiple-valued logic tableau method. Through the analysis of multiple-valued regular logic formula, the relationship between multiple-valued logic formula with quantifier and classical formula with quantifier was found, reasoning method of multiple-valued regular formula was constructed and reasoning efficiency was improved greatly. It shows that some non-classical logic problem can be solved by classical logic technology in some extent and it is helpful for the realization with reasoning machine.

Acknowledgements

This paper is supported by National Natural Science Foundation of China (60073039, 60273080) and supported by computer information process project of 211 important subject in soochow university.

References

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[2] Carnielli W A, “Systematization of finite many-valued logics through the method of tableaux”, Journal of Symbolic Logic, vol 52, No. 2, pp. 473-493, 1987.

[3] Bessonet C G, “A many-valued approach to deduction and reasoning for artifical intelligence”,Boston: Kluwer Academic Publishers, 1991.

[4] Hähnle R and Kernig W. “Verification of switch level designs with many-valued logic”. In: Voronkov A, Proc. LPAR’93, St.Petersburg, Russia, pp.158-169, 1993.

[5] Kerber M and Kohlhase M, “A tableau calculus for partial functions”, In: Collegium Logicum. Annals of the Kurt-Gödel-Society, New York: Springer-Verlag. pp. 21-49, 1996.

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[7] Pearl J., “Probabilistic reasoning in intelligent system：networks of plausible inference”, Revise second edition, Morgan Kaufmann,1994.

[8] Messing B., “Combining knowledge with many-valued Logics”, Data Knowledge Engineering, Boston: Kluwer Academic Publishers, 1997.

[9] Zabel R, “Proof theory of finity-valued logics”, [Ph D dissertation], Institut für Algebra und Diskrete Mathematic, TU Wien, 1993.

[10] Hähnle R, “Uniform notation of tableaux rules for multiple-valuedd logics”, In: Proc. International Symposium on Multiple-valuedd Logic, Los Alamitos: IEEE Press, 1991, 238-245

[11] Quan Liu, Jigui Sun, Zhiming Cui, “A method of simplifying many-valued generalized quantifiers tableau rules based on boolean pruning”, Chinese journal of computer, Vol 28, No. 9, pp. 1514-1518, 2005.

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