An Order Hyperresolution Calculus for G

odel Logic with Truth

Constants and Equality, Strict Order, Delta

san Guller

Department of Applied Informatics, Comenius University, Mlynsk

a dolina, 842 48, Bratislava, Slovakia

Keywords:

odel Logic, Resolution, Many-valued Logics, Automated Deduction.

Abstract:

In (Guller, 2014), we have generalised the well-known hyperresolution principle to the ﬁrst-order G

odel logic

with truth constants. This paper is a continuation of our work. We propose a hyperresolution calculus suitable

for automated deduction in a useful expansion of G

odel logic by intermediate truth constants and the equality,

P, strict order, ≺

≺

≺, projection, ∆

∆

∆, operators. We solve the deduction problem of a formula from a countable

theory in this expansion. We expand G

odel logic by a countable set of intermediate truth constants ¯c, c ∈

(0,1). Our approach is based on translation of a formula to an equivalent satisﬁable ﬁnite order clausal theory,

consisting of order clauses. An order clause is a ﬁnite set of order literals of the form ε

 ε

where ε

an atom or a quantiﬁed atom, and  is the connective P or ≺. P and ≺ are interpreted by the equality

and standard strict linear order on [0,1], respectively. We shall investigate the so-called canonical standard

completeness, where the semantics of G

odel logic is given by the standard G-algebra and truth constants are

interpreted by ’themselves’. The hyperresolution calculus is refutation sound and complete for a countable

order clausal theory under a certain condition for the set of truth constants occurring in the theory. As an

interesting consequence, we get an afﬁrmative solution to the open problem of recursive enumerability of

unsatisﬁable formulae in G

odel logic with truth constants and the equality, P

P, strict order, ≺

≺

≺, projection, ∆

∆

∆,

operators.

1 INTRODUCTION

Current research in many-valued logics is mainly

concerned with left-continuous t-norm based log-

ics including the fundamental fuzzy logics: G

odel,

Łukasiewicz, and Product ones. Most explorations

of t-norm based logics are focused on tautologies and

deduction calculi with the only distinguished truth de-

gree 1, (H

ajek, 2001). However, in many real-world

applications, one may be interested in representation

and inference with explicit partial truth; besides the

truth constants 0, 1, intermediate truth constants are

involved in. In the literature, two main approaches to

expansions with truth constants, are described. His-

torically, the ﬁrst one has been introduced in (Pavelka,

1979), where the propositional Łukasiewicz logic is

augmented by truth constants ¯r, r ∈ [0,1], Pavelka’s

logic (PL). A formula of the form ¯r → φ evaluated to

1 expresses that the truth value of φ is greater than

or equal to r. In (Nov

ak et al., 1999), further develop-

ment of evaluated formulae, and in (H

ajek, 2001), Ra-

tional Pavelka’s logic (RPL) - a simpliﬁcation of PL,

Partially supported by VEGA Grant 1/0592/14.

are described. Another approach relies on traditional

algebraic semantics. Various completeness results

for expansions of t-norm based logics with countably

many truth constants are investigated, among others,

in (Esteva et al., 2001; Savick

y et al., 2006; Esteva

et al., 2007b; Esteva et al., 2007a; Esteva et al., 2009;

Esteva et al., 2010a; Esteva et al., 2010b).

In (Guller, 2012; Guller, 2015), we have gener-

alised the well-known hyperresolution principle to the

ﬁrst-order G

odel logic for the general case. Our ap-

proach is based on translation of a formula of G

odel

logic to an equivalent satisﬁable ﬁnite order clausal

theory, consisting of order clauses. We have intro-

duced a notion of quantiﬁed atom: a formula a is

a quantiﬁed atom if a = Qx p(t

,.. .,t

) where Q is

a quantiﬁer (∀, ∃); p(t

,.. .,t

) is an atom; x is a

variable occurring in p(t

,.. .,t

); for all i ≤ τ, ei-

ther t

= x or x does not occur in t

is a free term

in the quantiﬁed atom). The notion of quantiﬁed

atom is all important. It permits us to extend clas-

sical uniﬁcation to quantiﬁed atoms without any ad-

ditional computational cost. Two quantiﬁed atoms

Qx p(t

,.. .,t

) and Q

,.. .,t

) are uniﬁable if

Guller, D..

An Order Hyperresolution Calculus for Gödel Logic with Truth Constants and Equality, Strict Order, Delta.

In Proceedings of the 7th International Joint Conference on Computational Intelligence (IJCCI 2015) - Volume 2: FCTA, pages 31-46

ISBN: 978-989-758-157-1

 2015 by SCITEPRESS – Science and Technology Publications, Lda. All rights reser ved

Q = Q

, x = x

, p = p

, and the left-right sequence

of free terms of Qx p(t

,.. .,t

) is uniﬁable with the

left-right sequence of free terms of Q

,.. .,t

)

in the standard manner. An order clause is a ﬁnite set

of order literals of the form ε

ε

where ε

is an atom

or a quantiﬁed atom, and  is the connective P or ≺.

P and ≺ are interpreted by the equality and standard

strict linear order on [0,1], respectively. On the ba-

sis of the hyperresolution principle, a calculus operat-

ing over order clausal theories, has been devised. The

calculus is proved to be refutation sound and com-

plete for the countable case with respect to the stan-

dard G-algebra G = ([0,1], ≤,∨

∨

∨,∧

∧

∧,⇒

⇒

⇒, ,P

P,≺

≺

≺,0,1)

augmented by binary operators P

P and ≺

≺

≺ for P and

≺, respectively. As another step, one may incorpo-

rate a countable set of intermediate truth constants ¯c,

c ∈ (0,1), to get a modiﬁcation of the hyperresolution

calculus suitable for automated deduction with ex-

plicit partial truth (Guller, 2014). We shall investigate

the so-called canonical standard completeness, where

the semantics of G

odel logic is given by the stan-

dard G-algebra G and truth constants are interpreted

by ’themselves’. We say that a set {0,1} ⊆ X of

truth constants is admissible with respect to suprema

and inﬁma if, for all

0 6= Y

⊆ X and

∈ Y

. Then the hyperresolution calcu-

lus is refutation sound and complete for a countable

order clausal theory if the set of all truth constants

occurring in the theory, is admissible with respect to

suprema and inﬁma. This condition obviously covers

the case of ﬁnite order clausal theories. As an inter-

esting consequence, we get an afﬁrmative solution to

the open problem of recursive enumerability of unsat-

isﬁable formulae in G

odel logic with truth constants

and the operators P

P, ≺

≺

≺, ∆

∆

∆ on [0,1]:

Pb =



1 if a = b,

0 else;

a≺

≺

≺b =



1 if a < b,

0 else;

∆

∆a =



1 if a = 1,

0 else;

which strengthens a similar result for prenex formulae

of G

∆

∞

stated in Conclusion of (Baaz et al., 2012).

Some applications of our hyperresolution calculus

may lead to computational linguistics, to design and

analysis of scientiﬁc (natural) language processing

systems (Mandel

ıkov

a, 2012; Mandel

ıkov

a, 2014).

The paper is organised as follows. Section 2 gives

the basic notions and notation concerning the ﬁrst-

order G

odel logic. Section 3 deals with clause form

translation. In Section 4, we propose a hyperresolu-

tion calculus with truth constants and prove its refu-

tational soundness, completeness. Section 5 brings

conclusions.

2 FIRST-ORDER G

ODEL LOGIC

Throughout the paper, we shall use the common

notions and notation of ﬁrst-order logic. N | Z

designates the set of natural | integer numbers and ≤ |

< the standard order | strict order on N | Z. By L we

denote a ﬁrst-order language. Var

| Func

| Pred

Term

| GTerm

| Atom

| GAtom

denotes the set of

all variables | function symbols | predicate symbols |

terms | ground terms | atoms | ground atoms of L.

: Func

∪ Pred

−→ N denotes the mapping

assigning an arity to every function and predicate

symbol of L. We assume truth constants - nullary

predicate symbols 0,1 ∈ Pred

, ar

(0) = ar

(1) = 0;

0 denotes the false and 1 the true in L. Let

⊆ (0,1) be countable. In addition, we as-

sume a countable set of nullary predicate symbols

= { ¯c| ¯c ∈ Pred

,ar

( ¯c) = 0, c ∈ C

} ⊆ Pred

;

{0}, {1}, C

are pairwise disjoint. 0, 1,

¯c ∈ C

are called truth constants. We denote

Tcons

= {0,1} ∪ C

⊆ Pred

. Let X ⊆ Tcons

We denote X = {0|0 ∈ X } ∪ {1 | 1 ∈ X} ∪ {c | ¯c ∈

X ∩ C

} ⊆ [0,1]. We introduce a new unary

connective ∆, Delta, and binary connectives P,

equality, ≺, strict order. By OrdForm

we designate

the set of all so-called order formulae of L built

up from Atom

and Var

using the connectives:

¬, negation, ∆, ∧, conjunction, ∨, disjunction,

→, implication, ↔, equivalence, P, ≺, and the

quantiﬁers: ∀, the universal one, ∃, the existential

one.

In the paper, we shall assume that L is a

countable ﬁrst-order language; hence, all the above

mentioned sets of symbols and expressions are

countable. Let ε | ε

, 1 ≤ i ≤ m | υ

, 1 ≤ i ≤ n,

be either an expression or a set of expressions or

a set of sets of expressions of L, in general. By

vars(ε

,.. .,ε

) ⊆ Var

| freevars(ε

,.. .,ε

) ⊆

Var

| boundvars(ε

,.. .,ε

) ⊆ Var

funcs(ε

,.. .,ε

) ⊆ Func

| preds(ε

,.. .,ε

) ⊆

Pred

| atoms(ε

,.. .,ε

) ⊆ Atom

we denote

the set of all variables | free variables | bound

variables | function symbols | predicate symbols |

atoms of L occurring in ε

,.. .,ε

. ε is closed

iff freevars(ε) =

0. By ` we denote the empty

sequence. By |ε

,.. .,ε

| = m we denote the length

of the sequence ε

,.. .,ε

. We deﬁne the concate-

nation of the sequences ε

,.. .,ε

and υ

,.. .,υ

as (ε

,.. .,ε

),(υ

,.. .,υ

) = ε

,.. .,ε

,υ

,.. .,υ

Note that concatenation of sequences is associative.

Let X, Y , Z be sets, Z ⊆ X; f : X −→ Y be a map-

ping. By kX k we denote the set-theoretic cardinal-

ity of X. X being a ﬁnite subset of Y is denoted as

We assume a decreasing connective and quantiﬁer prece-

dence: ∀, ∃, ¬, ∆, P, ≺, ∧, ∨, →, ↔.

FCTA 2015 - 7th International Conference on Fuzzy Computation Theory and Applications

X ⊆

Y . We designate P (X) = {x |x ⊆ X}; P (X) is

the power set of X; P

(X) = {x | x ⊆

X}; P

(X) is

the set of all ﬁnite subsets of X; f [Z] = { f (z) |z ∈ Z};

f [Z] is the image of Z under f ; f |

= {(z, f (z))| z ∈

Z}; f |

is the restriction of f onto Z. Let γ ≤ ω. A

sequence δ of X is a bijection δ : γ −→ X . Recall that

X is countable if and only if there exists a sequence of

X. Let I be a set and S

0, i ∈ I, be sets. A selector

S over {S

|i ∈ I} is a mapping S : I −→

|i ∈ I}

such that for all i ∈ I, S (i) ∈ S

. We denote S el({S

|i ∈

I}) = {S |S is a selector over {S

|i ∈ I}}. R desig-

nates the set of real numbers and ≤ | < the standard

order | strict order on R. We denote R

= {c | 0 ≤ c ∈

R}, R

= {c |0 < c ∈ R}; [0,1] = {c |0 ≤ c ≤ 1, c ∈

R}; [0,1] is the unit interval. Let c ∈ R

. log c de-

notes the binary logarithm of c. Let f , g : N −→ R

f is of the order of g, in symbols f ∈ O(g), iff there

exist n

∈ N and c

∗

∈ R

such that for all n ≥ n

f (n) ≤ c

∗

· g(n).

Let t ∈ Term

, φ ∈ OrdForm

, T ⊆

OrdForm

the number of nodes of its standard tree representa-

tion. We deﬁne the size of T as |T | =

∑

φ∈T

|φ|. By

varseq(φ), vars(varseq(φ)) ⊆ Var

, we denote the se-

quence of all variables of L occurring in φ which is

built up via the left-right preorder traversal of φ. For

example, varseq(∃w(∀x p(x, x,z) ∨ ∃y q(x, y, z))) =

w,x, x,x,z,y,x, y, z and |w, x,x, x,z,y, x,y,z| = 9. A se-

quence of variables will often be denoted as ¯x, ¯y, ¯z,

etc. Let Q ∈ {∀, ∃} and ¯x = x

,.. .,x

be a sequence

of variables of L. By Q ¯x φ we denote Qx

... Qx

φ.

odel logic is interpreted by the standard G-

algebra augmented by the operators P

P, ≺

≺

≺, ∆

∆

∆ for the

connectives P, ≺, ∆, respectively.

G = ([0, 1],≤,∨

∨

∨,∧

∧

∧,⇒

⇒

⇒, ,P

P,≺

≺

≺,∆

∆

∆,0,1)

where ∨

∨

∨ | ∧

∧

∧ denotes the supremum | inﬁmum operator

on [0,1];

a⇒

⇒

⇒b =



1 if a ≤ b,

b else;

a =



1 if a = 0,

0 else;

Pb =



1 if a = b,

0 else;

a≺

≺

≺b =



1 if a < b,

0 else;

∆

∆a =



1 if a = 1,

0 else.

Recall that G is a complete linearly ordered lattice al-

gebra; ∨

∨

∨ | ∧

∧

∧ is commutative, associative, idempotent,

monotone; 0 | 1 is its neutral element; the residuum

operator ⇒

⇒

⇒ of ∧

∧

∧ satisﬁes the condition of residuation:

for all a,b,c ∈ G, a∧

∧

∧b ≤ c ⇐⇒ a ≤ b⇒

⇒

⇒c; (1)

odel negation satisﬁes the condition:

for all a ∈ G, a = a⇒

⇒

⇒0; (2)

the following properties, which will be exploited later,

hold:

for all a,b,c ∈ G,

a∨

∨

∨b∧

∧

∧c = (a∨

∨

∨b)∧

∧

∧(a∨

∨

∨c),

(distributivity of ∨

∨

∨ over ∧

∧

∧) (3)

a∧

∧

∧(b∨

∨

∨c) = a∧

∧

∧b∨

∨

∨a∧

∧

∧c,

(distributivity of ∧

∧

∧ over ∨

∨

∨) (4)

a⇒

⇒

⇒b∨

∨

∨c = (a⇒

⇒

⇒b)∨

∨

∨(a⇒

⇒

⇒c), (5)

a⇒

⇒

⇒b∧

∧

∧c = (a⇒

⇒

⇒b)∧

∧

∧(a⇒

⇒

⇒c), (6)

a∨

∨

∨b⇒

⇒

⇒c = (a⇒

⇒

⇒c)∧

∧

∧(b⇒

⇒

⇒c), (7)

a∧

∧

∧b⇒

⇒

⇒c = (a⇒

⇒

⇒c)∨

∨

∨(b⇒

⇒

⇒c), (8)

a⇒

⇒

⇒(b⇒

⇒

⇒c) = a∧

∧

∧b⇒

⇒

⇒c, (9)

((a⇒

⇒

⇒b)⇒

⇒

⇒b)⇒

⇒

⇒b = a⇒

⇒

⇒b, (10)

(a⇒

⇒

⇒b)⇒

⇒

⇒c = ((a⇒

⇒

⇒b)⇒

⇒

⇒b)∧

∧

∧(b⇒

⇒

⇒c)∨

∨

∨c, (11)

(a⇒

⇒

⇒b)⇒

⇒

⇒0 = ((a⇒

⇒

⇒0)⇒

⇒

⇒0)∧

∧

∧(b⇒

⇒

⇒0), (12)

∆

∆a = aP

P1. (13)

An interpretation I for L is a triple



,{ f

| f ∈

Func

},{p

| p ∈ Pred

}



deﬁned as follows: U

0 is the universum of I ; every f ∈ Func

is in-

terpreted as a function f

: U

( f )

−→ U

; ev-

ery p ∈ Pred

is interpreted as a [0,1]-relation p

(p)

−→ [0,1]. A variable assignment in I is a

mapping Var

−→ U

. We denote the set of all vari-

able assignments in I as S

. Let e ∈ S

and u ∈ U

A variant e[x/u] ∈ S

of e with respect to x and u is

deﬁned as

e[x/u](z) =



u if z = x,

e(z) else.

Let t ∈ Term

, ¯x be a sequence of variables of L, φ ∈

OrdForm

. In I with respect to e, we deﬁne the value

ktk

∈ U

of t by recursion on the structure of t, the

value k ¯xk

∈ U

| ¯x|

of ¯x, the truth value kφk

∈ [0,1] of

φ by recursion on the structure of φ, as usual. Notice

that 0

= 0, 1

= 1, for all ¯c ∈

, ¯c

= c, kφ

↔

= (kφ

⇒

⇒kφ

)∧

∧

∧(kφ

⇒

⇒kφ

). Let φ be

closed. Then, for all e,e

∈ S

, kφk

= kφk

. Let

e ∈ S

0. We denote kφk

= kφk

Let L | L

be a ﬁrst-order language and I | I

an interpretation for L | L

. L

is an expansion of L

iff Func

⊇ Func

and Pred

⊇ Pred

; on the other

side, we say L is a reduct of L

. I

is an expansion

of I to L

iff L

is an expansion of L, U

= U

, for

all f ∈ Func

, f

= f

, for all p ∈ Pred

, p

= p

;

We assume a decreasing operator precedence: , ∆

∆

∆, P

P, ≺

≺

≺,

∧

∧, ∨

∨

∨, ⇒

⇒

⇒.

An Order Hyperresolution Calculus for Gödel Logic with Truth Constants and Equality, Strict Order, Delta

on the other side, we say I is a reduct of I

to L, in

symbols I = I

An order theory of L is a set of order formulae of

L. Let φ,φ

∈ OrdForm

, T ⊆ OrdForm

, e ∈ S

. φ

is true in I with respect to e, written as I |=

φ, iff

kφk

= 1. I is a model of φ, in symbols I |= φ, iff,

for all e ∈ S

, I |=

φ. I is a model of T , in symbols

I |= T , iff, for all φ ∈ T , I |= φ. φ is a logically valid

formula iff, for every interpretation I for L, I |= φ.

φ is equivalent to φ

, in symbols φ ≡ φ

, iff, for every

interpretation I for L and e ∈ S

, kφk

= kφ

. We

denote tcons(φ) = {0, 1}∪ (preds(φ) ∩C

) ⊆ Tcons

and tcons(T ) = {0,1} ∪ (preds(T ) ∩C

) ⊆ Tcons

3 TRANSLATION TO CLAUSAL

FORM

In the propositional case (Guller, 2010), we have pro-

posed some translation of a formula to an equivalent

CNF containing literals of the form either a or a → b

or (a → b) → b where a is a propositional atom and b

is either a propositional atom or the propositional con-

stant 0. An output equivalent CNF may be of expo-

nential size with respect to the input formula; we had

laid no restrictions on use of the distributivity law (3)

during translation to conjunctive normal form. To

avoid this disadvantage, we have devised translation

to CNF via interpolation using new atoms, which pro-

duces an output CNF of linear size at the cost of being

only equisatisﬁable to the input formula. A similar

approach exploiting the renaming subformulae tech-

nique can be found in (Plaisted and Greenbaum, 1986;

de la Tour, 1992; H

ahnle, 1994; Nonnengart et al.,

1998; Sheridan, 2004). A CNF is further translated to

a ﬁnite set of order clauses. An order clause is a ﬁnite

set of order literals of the form ε

 ε

where ε

is ei-

ther a propositional atom or a propositional constant,

0, 1, and  ∈ {P, ≺}.

We now describe some generalisation of the men-

tioned translation to the ﬁrst-order case. At ﬁrst, we

introduce a notion of quantiﬁed atom. Let a ∈ Form

a is a quantiﬁed atom of L iff a = Qx p(t

,.. .,t

)

where p(t

,.. .,t

) ∈ Atom

, x ∈ vars(p(t

,.. .,t

)),

either t

= x or x 6∈ vars(t

). QAtom

⊆ Form

de-

notes the set of all quantiﬁed atoms of L. QAtom

⊆

QAtom

, Q ∈ {∀,∃}, denotes the set of all quantiﬁed

atoms of L of the form Qx a. Let ε | ε

, 1 ≤ i ≤ m | υ

1 ≤ i ≤ n, be either an expression or a set of expres-

sions or a set of sets of expressions of L, in general.

By qatoms(ε

,.. .,ε

) ⊆ QAtom

we denote the set of

all quantiﬁed atoms of L occurring in ε

,.. .,ε

. We

denote qatoms

(ε

,.. .,ε

) = qatoms(ε

,.. .,ε

) ∩

QAtom

, Q ∈ {∀, ∃}. Let Qx p(t

,.. .,t

) ∈ QAtom

and p(t

,.. .,t

) ∈ Atom

. We denote

boundindset(Qx p(t

,.. .,t

)) = {i |i ≤ τ,t

= x} 6=

Let I = {i|i ≤ τ,x 6∈ vars(t

)} and r

,.. .,r

, r

≤ τ,

k ≤ τ, for all 1 ≤ i < i

≤ k, r

< r

, be a sequence

such that {r

|1 ≤ i ≤ k} = I. We denote

freetermseq(Qx p(t

,.. .,t

)) = t

,.. .,t

freetermseq(p(t

,.. .,t

)) = t

,.. .,t

We further introduce order clauses in G

odel logic.

Let l ∈ OrdForm

. l is an order literal of L iff

l = ε

ε

, ε

∈ Atom

∪QAtom

,  ∈ {P,≺}. The set

of all order literals of L is designated as OrdLit

⊆

OrdForm

. An order clause of L is a ﬁnite set of

order literals of L; since = is commutative, for all

P ε

∈ OrdLit

, we identify ε

P ε

and ε

∈ OrdLit

with respect to order clauses. An order

clause {l

,.. .,l

} is written in the form l

∨ ·· · ∨ l

The order clause

0 is called the empty order clause

and denoted as . An order clause {l} is called a unit

order clause and denoted as l; if it does not cause the

ambiguity with the denotation of the single order lit-

eral l in given context. We designate the set of all or-

der clauses of L as OrdCl

. Let l,l

,.. .,l

∈ OrdLit

and C,C

∈ OrdCl

. We deﬁne the size of C as

|C| =

∑

l∈C

|l|. By l ∨ C we denote {l} ∪ C where

l 6∈ C. Analogously, by l

∨ ··· ∨ l

∨ C we denote

}∪·· ·∪{l

}∪C where, for all i, i

≤ n, i 6= i

, l

6∈ C

and l

6= l

. By C ∨C

we denote C ∪ C

. C is a sub-

clause of C

, in symbols C v C

, iff C ⊆ C

. An order

clausal theory of L is a set of order clauses of L. A

unit order clausal theory is a set of unit order clauses.

Let φ, φ

∈ OrdForm

, T, T

⊆ OrdForm

, S,S

⊆

OrdCl

, I be an interpretation for L, e ∈ S

. Note that

I |=

l if and only if either l = ε

P ε

, kε

P ε

1, kε

= kε

; or l = ε

≺ ε

, kε

≺ ε

= 1,

kε

< kε

. C is true in I with respect to e,

written as I |=

C, iff there exists l

∗

∈ C such that

I |=

∗

. I is a model of C, in symbols I |= C, iff,

for all e ∈ S

, I |=

C. I is a model of S, in sym-

bols I |= S, iff, for all C ∈ S, I |= C. φ

| T

| C

| S

is a logical consequence of φ | T | C | S, in symbols

φ|T |C |S |= φ

, iff, for every model I of φ |

T | C | S for L, I |= φ

. φ | T | C | S is

satisﬁable iff there exists a model of φ | T | C | S for

L. Note that both  and  ∈ S are unsatisﬁable. φ |

T | C | S is equisatisﬁable to φ

| T

| C

| S

iff φ | T |

C | S is satisﬁable if and only if φ

| T

| C

| S

is sat-

isﬁable. We denote tcons(S) = {0,1} ∪ (preds(S) ∩

) ⊆ Tcons

. Let S ⊆

OrdCl

. We deﬁne the

size of S as |S| =

∑

C∈S

|C|. l is a simpliﬁed order lit-

eral of L iff l = ε

 ε

, {ε

,ε

} 6⊆ Tcons

, {ε

,ε

} 6⊆

QAtom

. The set of all simpliﬁed order literals of L

FCTA 2015 - 7th International Conference on Fuzzy Computation Theory and Applications

is designated as SimOrdLit

⊆ OrdLit

. We denote

SimOrdCl

= {C |C ∈ OrdCl

,C ⊆ SimOrdLit

} ⊆

OrdCl

. Let

6∈ Func

;

is a new function sym-

bol. Let I = N × N; I is an inﬁnite countable set of

indices. Let

P = { ˜p | ∈ I} such that

P ∩ Pred

P is an inﬁnite countable set of new predicate sym-

bols.

From a computational point of view, the worst

case time and space complexity will be estimated us-

ing the logarithmic cost measurement. Let A be an

algorithm. #O

(In) ≥ 1 denotes the number of all el-

ementary operations executed by A on an input In.

3.1 Substitutions

We assume the reader to be familiar with the standard

notions and notation of substitutions. We introduce

a few deﬁnitions and denotations; some of them are

slightly different from the standard ones, but found to

be more convenient. Let X = {x

|1 ≤ i ≤ n} ⊆ Var

A substitution ϑ of L is a mapping ϑ : X −→ Term

ϑ may be written in the form x

/ϑ(x

),.. .,x

/ϑ(x

We denote dom(ϑ) = X ⊆

Var

and range(ϑ) =

x∈X

vars(ϑ(x)) ⊆

Var

. The set of all substitutions

of L is designated as Subst

. Let ϑ,ϑ

∈ Subst

. ϑ

is a variable renaming of L iff ϑ : dom(ϑ) −→ Var

for all x,x

∈ dom(ϑ), x 6= x

, ϑ(x) 6= ϑ(x

). We deﬁne

: Var

−→ Var

, id

(x) = x. Let t ∈ Term

. ϑ is

applicable to t iff dom(ϑ) ⊇ vars(t) = freevars(t). Let

ϑ be applicable to t. We deﬁne the application tϑ ∈

Term

of ϑ to t by recursion on the structure of t in the

standard manner. Let range(ϑ) ⊆ dom(ϑ

). We deﬁne

the composition of ϑ and ϑ

as ϑ ◦ ϑ

: dom(ϑ) −→

Term

, ϑ ◦ ϑ

(x) = ϑ(x)ϑ

, ϑ ◦ ϑ

∈ Subst

, dom(ϑ ◦

) = dom(ϑ), range(ϑ ◦ ϑ

) = range(ϑ

range(ϑ)

Note that composition of substitutions is associative.

is a regular extension of ϑ iff dom(ϑ

) ⊇ dom(ϑ),

dom(ϑ)

= ϑ, ϑ

dom(ϑ

)−dom(ϑ)

is a variable renam-

ing such that range(ϑ

dom(ϑ

)−dom(ϑ)

)∩range(ϑ) =

Let a ∈ Atom

. ϑ is applicable to a iff dom(ϑ) ⊇

vars(a) = freevars(a). Let ϑ be applicable to a and

a = p(t

,.. .,t

). We deﬁne the application of ϑ to a as

aϑ = p(t

ϑ,.. .,t

ϑ) ∈ Atom

. Let Qx a ∈ QAtom

ϑ is applicable to Qx a iff dom(ϑ) ⊇ freevars(Qx a)

and x 6∈ range(ϑ|

freevars(Qxa)

). Let ϑ be applicable

to Qx a. We deﬁne the application of ϑ to Qxa as

(Qx a)ϑ = Qx a(ϑ|

freevars(Qxa)

∪ x/x) ∈ QAtom

. Let

ε

∈ OrdLit

. ϑ is applicable to ε

ε

iff, for both

i, ϑ is applicable to ε

. Let ϑ be applicable to ε

 ε

Then, for both i, ϑ is applicable to ε

, dom(ϑ) ⊇

freevars(ε

), dom(ϑ) ⊇ freevars(ε

) ∪ freevars(ε

) =

freevars(ε

 ε

). We deﬁne the application of ϑ to

 ε

as (ε

 ε

)ϑ = ε

ϑ  ε

ϑ ∈ OrdLit

. Let

E ⊆ A, A = Term

| A = Atom

| A = QAtom

A = OrdLit

. ϑ is applicable to E iff, for all ε ∈ E, ϑ

is applicable to ε. Let ϑ be applicable to E. Then, for

all ε ∈ E, ϑ is applicable to ε, dom(ϑ) ⊇ freevars(ε),

dom(ϑ) ⊇

ε∈E

freevars(ε) = freevars(E). We deﬁne

the application of ϑ to E as Eϑ = {εϑ |ε ∈ E} ⊆ A.

Let ε, ε

∈ A | ε, ε

∈ OrdCl

. ε

is an instance of

ε of L iff there exists ϑ

∗

∈ Subst

such that ε

εϑ

∗

. ε

is a variant of ε of L iff there exists a vari-

able renaming ρ

∗

∈ Subst

such that ε

= ερ

∗

. Let

C ∈ OrdCl

and S ⊆ OrdCl

. C is an instance | a

variant of S of L iff there exists C

∗

∈ S such that

C is an instance | a variant of C

∗

of L. We denote

Inst

(S) = {C |C is an instance of S of L} ⊆ OrdCl

and Vrnt

(S) = {C |C is a variant of S of L} ⊆

OrdCl

ϑ is a uniﬁer of L for E iff Eϑ is a singleton

set. Note that there does not exist a uniﬁer for

Let θ ∈ Subst

. θ is a most general uniﬁer of L for

E iff θ is a uniﬁer of L for E, and for every uni-

ﬁer ϑ of L for E, there exists γ

∗

∈ Subst

such that

ϑ|

freevars(E)

= θ|

freevars(E)

◦ γ

∗

. By mgu

(E) ⊆ Subst

we denote the set of all most general uniﬁers of L for

E. Let

E = E

,.. .,E

, E

⊆ A

, either A

= Term

or A

= Atom

or A

= QAtom

or A

= OrdLit

ϑ is applicable to E iff, for all i ≤ n, ϑ is applica-

ble to E

. Let ϑ be applicable to E. Then, for all

i ≤ n, ϑ is applicable to E

, dom(ϑ) ⊇ freevars(E

dom(ϑ) ⊇

i≤n

freevars(E

) = freevars(E). We de-

ﬁne the application of ϑ to E as Eϑ = E

ϑ,.. .,E

ϑ,

ϑ ⊆ A

. ϑ is a uniﬁer of L for E iff, for all i ≤ n, ϑ

is a uniﬁer of L for E

. Note that if there exists i

∗

≤ n

and E

∗

0, then there does not exist a uniﬁer for E. θ

is a most general uniﬁer of L for E iff θ is a uniﬁer of

L for E, and for every uniﬁer ϑ of L for E, there exists

∗

∈ Subst

such that ϑ|

freevars(E)

= θ|

freevars(E)

◦ γ

∗

By mgu

(E) ⊆ Subst

we denote the set of all most

general uniﬁers of L for E.

Theorem 3.1 (Uniﬁcation Theorem). Let E =

,.. .,E

, either E

⊆

Term

or E

⊆

Atom

. If

there exists a uniﬁer of L for E, then there exists

∗

∈ mgu

(E) such that range(θ

∗

vars(E)

) ⊆ vars(E).

Proof. By induction on kvars(E)k; a modiﬁcation of

the proof of Theorem 2.3 (Uniﬁcation Theorem) in

(Apt, 1988), Section 2.4, pp. 5–6.

Theorem 3.2 (Extended Uniﬁcation Theorem). Let

E = E

,.. .,E

, either E

⊆

Term

or E

⊆

Atom

or E

⊆

QAtom

or E

⊆

OrdLit

, and

boundvars(E) ⊆ V ⊆

Var

. If there exists a uniﬁer

of L for E, then there exists θ

∗

∈ mgu

(E) such that

range(θ

∗

freevars(E)

) ∩V =

Proof. A straightforward consequence of Theo-

rem 3.1.

An Order Hyperresolution Calculus for Gödel Logic with Truth Constants and Equality, Strict Order, Delta

3.2 A Formal Treatment

Translation of an order formula or a theory to clausal

form, is based on the following lemma:

Lemma 3.3. Let n

∈ N, φ ∈ OrdForm

, T ⊆

OrdForm

(I) There exist either J

0 or J

= {(n

, j)| j ≤

}, J

⊆ {(n

, j)| j ∈ N}, and S

⊆

SimOrdCl

L∪{ ˜p | ∈J

}

such that

(a) kJ

k ≤ 2 · |φ|;

(b) either J

0, S

= {} or J

= S

0 or

0,  6∈ S

A |= φ if and only if there exists an interpre-

tation A

for L ∪ { ˜p | ∈ J

} and A

|= S

satisfying A = A

;

(d) |S

| ∈ O(|φ|

); the number of all elemen-

tary operations of the translation of φ to

, is in O(|φ|

); the time and space com-

plexity of the translation of φ to S

, is in

O(|φ|

· (log(1 + n

) + log |φ|));

(e) if S

0,{}, then J

0, for all C ∈ S

0 6= preds(C) ∩

P ⊆ { ˜p | ∈ J

};

(f) for all a ∈ qatoms(S

), there exists

∗

∈ J

and preds(a) = { ˜p

∗

};

(g) for all ∈ J

, there exists a sequence ¯x

of variables of L and ˜p ( ¯x) ∈ atoms(S

)

satisfying, for all a ∈ atoms(S

) and

preds(a) = { ˜p }, a = ˜p ( ¯x); if there ex-

ists a

∗

∈ qatoms(S

) and preds(a

∗

) = { ˜p },

then there exists Qx ˜p ( ¯x) ∈ qatoms(S

)

satisfying, for all a ∈ qatoms(S

) and

preds(a) = { ˜p }, a = Qx ˜p ( ¯x);

(h) tcons(S

) ⊆ tcons(φ).

(II) There exist J

⊆ {(i, j)| i ≥ n

} and S

⊆

SimOrdCl

L∪{ ˜p | ∈J

}

such that

(a) either J

0, S

= {} or J

= S

0 or

0,  6∈ S

(b) there exists an interpretation A for L and

A |= T if and only if there exists an in-

terpretation A

for L ∪ { ˜p | ∈ J

} and

|= S

, satisfying A = A

;

OrdForm

, then J

⊆

{(i, j)|i ≥ n

}, kJ

k ≤ 2 · |T |, S

⊆

SimOrdCl

L∪{ ˜p | ∈J

}

, |S

| ∈ O(|T |

);

the number of all elementary opera-

tions of the translation of T to S

, is in

O(|T |

); the time and space complex-

ity of the translation of T to S

, is in

O(|T |

· log(1 + n

+ |T |));

(d) if S

0,{}, then J

0, for all C ∈ S

0 6= preds(C) ∩

P ⊆ { ˜p | ∈ J

};

(e) for all a ∈ qatoms(S

), there exists

∗

∈ J

and preds(a) = { ˜p

∗

};

(f) for all ∈ J

, there exists a sequence ¯x

of variables of L and ˜p ( ¯x) ∈ atoms(S

)

satisfying, for all a ∈ atoms(S

) and

preds(a) = { ˜p }, a = ˜p ( ¯x); if there exists

∗

∈ qatoms(S

) and preds(a

∗

) = { ˜p },

then there exists Qx ˜p ( ¯x) ∈ qatoms(S

)

satisfying, for all a ∈ qatoms(S

) and

preds(a) = { ˜p }, a = Qx ˜p ( ¯x);

(g) tcons(S

) ⊆ tcons(T ).

Proof. Technical, using interpolation. It is straight-

forward to prove the following statements:

Let n

∈ N and θ ∈ OrdForm

. There exists

∈ OrdForm

such that

(a) θ

≡ θ;

(b) |θ

| ≤ 2 · |θ|; θ

can be built up from θ via

a postorder traversal of θ with #O(θ) ∈

O(|θ|) and the time, space complexity in

O(|θ| · (log(1 + n

) + log |θ|));

does not contain ¬ and ∆;

(d) θ

∈ Tcons

; or for every subformula of

of the form ε

 ε

,  ∈ {∧,∨,↔}, ε

0,1, {ε

,ε

} 6⊆ Tcons

; for every subfor-

mula of θ

of the form ε

→ ε

, ε

0,1, ε

6= 1, {ε

,ε

} 6⊆ Tcons

; for ev-

ery subformula of θ

of the form ε

P ε

{ε

,ε

} 6⊆ Tcons

; for every subformula

of θ

of the form ε

≺ ε

, ε

6= 1, ε

6= 0,

{ε

,ε

} 6⊆ Tcons

; for every subformula

of θ

of the form Qx ε

, Q ∈ {∀,∃}, ε

6∈

Tcons

;

(e) tcons(θ

) ⊆ tcons(θ).

(14)

The proof is by induction on the structure of θ.

Let n

∈ N, θ ∈ OrdForm

− {0,1}, (14c,d)

hold for θ; ¯x be a sequence of variables,

vars(θ) ⊆ vars( ¯x) ⊆ Var

; = (n

, j ) ∈

{(n

, j)| j ∈ N}, ˜p ∈

P, ar( ˜p ) = | ¯x|.

There exist J = {(n

, j)| j + 1 ≤ j ≤ n

} ⊆

{(n

, j)| j ∈ N}, j ≤ n

, 6∈ J, and S ⊆

SimOrdCl

L∪{ ˜p }∪{ ˜p | ∈J}

such that

(a) kJk ≤ |θ| − 1;

(b) there exists an interpretation A

for L ∪ { ˜p } and A |= ˜p ( ¯x) ↔

θ ∈ OrdForm

L∪{ ˜p }

if and only if

there exists an interpretation A

for

L ∪ { ˜p } ∪ { ˜p | ∈ J} and A

|= S,

satisfying A = A

L∪{ ˜p }

;

from θ and

( ¯x) via a preorder traversal

of θ with #O(θ,

( ¯x)) ∈ O(|θ| · (1 + | ¯x|));

(15)

FCTA 2015 - 7th International Conference on Fuzzy Computation Theory and Applications

(d) for all C ∈ S,

0 6= preds(C) ∩

P ⊆ { ˜p } ∪

{ ˜p | ∈ J}, ˜p ( ¯x) P 1, ˜p ( ¯x) ≺ 1 6∈ S;

(e) for all a ∈ qatoms(S), there exists

∗

∈ J

and preds(a) = { ˜p

∗

};

(f) for all ∈ { } ∪ J, ˜p ( ¯x) ∈ atoms(S) satis-

fying, for all a ∈ atoms(S) and preds(a) =

{ ˜p }, a = ˜p ( ¯x); ˜p 6∈ preds(qatoms(S)),

for all ∈ J, if there exists a

∗

∈ qatoms(S)

and preds(a

∗

) = { ˜p }, then there exists

Qx ˜p ( ¯x) ∈ qatoms(S) satisfying, for all

a ∈ qatoms(S) and preds(a) = { ˜p }, a =

Qx ˜p ( ¯x);

(g) tcons(S) = tcons(θ).

The proof is by induction on the structure of θ using

the interpolation rules in Table 1.

(I) By (14) for n

, φ, there exists φ

∈ OrdForm

such that (14a–e) hold for n

, φ, φ

. We distin-

guish three cases for φ

. Case 1: φ

∈ Tcons

−

{1}. We put J

0 ⊆ {(n

, j)| j ∈ N} and S

{} ⊆

SimOrdCl

. Case 2: φ

= 1. We put

0 ⊆ {(n

, j)| j ∈ N} and S

0 ⊆

SimOrdCl

Case 3: φ

6∈ Tcons

. We put ¯x = varseq(φ

), j =

0, = (n

, j ), ar( ˜p ) = | ¯x|. We get by (15) for

, φ

, ¯x, , ˜p that there exist J = {(n

, j)|1 ≤

j ≤ n

} ⊆ {(n

, j)| j ∈ N}, j ≤ n

, 6∈ J, S ⊆

SimOrdCl

L∪{ ˜p }∪{ ˜p | ∈J}

, and (15a–g) hold for φ

¯x, ˜p , J, S. We put n

= n

, J

= {(n

, j)| j ≤

} ⊆ {(n

, j)| j ∈ N}, S

= { ˜p ( ¯x) P 1} ∪ S ⊆

SimOrdCl

L∪{ ˜p | ∈J

}

. (II) straightforwardly follows

from (I). The lemma is proved.

The described translation produces order clausal

theories in some restrictive form, which will be

utilised in inference using our order hyperresolution

calculus to get shorter deductions in average case, cf.

Section 4. Let P ⊆

P and S ⊆ OrdCl

L∪P

. S is admis-

sible iff

(a) for all a ∈ qatoms(S), preds(a) ⊆ P;

(b) for all ˜p ∈ P, there exists a sequence ¯x of vari-

ables of L and ˜p( ¯x) ∈ atoms(S) satisfying, for all

a ∈ atoms(S) and preds(a) = { ˜p}, a is an instance

of ˜p( ¯x) of L ∪ P; if there exists a

∗

∈ qatoms(S)

and preds(a

∗

) = { ˜p}, then there exists Qx ˜p( ¯x) ∈

qatoms(S) satisfying, for all a ∈ qatoms(S) and

preds(a) = { ˜p}, a is an instance of Qx ˜p( ¯x) of

L ∪ P.

(a) and (b) imply that for all Qx a, Q

∈ qatoms(S),

if preds(a) = preds(a

), then Q = Q

, x = x

boundindset(Qx a) = boundindset(Q

Theorem 3.4. Let n

∈ N, φ ∈ OrdForm

, T ⊆

OrdForm

. There exist J

⊆ {(i, j)|i ≥ n

} and S

⊆

SimOrdCl

L∪{ ˜p | ∈J

}

such that

(i) there exists an interpretation A for L and A |=

T , A 6|= φ if and only if there exists an interpre-

tation A

for L ∪ { ˜p | ∈ J

} and A

|= S

, sat-

isfying A = A

;

(ii) if T ⊆

OrdForm

, then J

⊆

{(i, j)|i ≥

}, kJ

k ∈ O(|T | + |φ|), S

⊆

SimOrdCl

L∪{ ˜p | ∈J

}

, |S

| ∈ O(|T |

+ |φ|

);

the number of all elementary operations of the

translation of T and φ to S

, is in O(|T |

+|φ|

);

the time and space complexity of the translation

of T and φ to S

, is in O(|T |

· log(1 + n

|T |) + |φ|

· (log(1 + n

) + log |φ|));

(iii) S

is admissible;

(iv) tcons(S

) ⊆ tcons(φ) ∪ tcons(T ).

Proof. Similar to that of Lemma 3.3(I). We get by

Lemma 3.3(II) for n

+ 1, T that there exist J

⊆

{(i, j)|i ≥ n

+ 1}, S

⊆ SimOrdCl

L∪{ ˜p | ∈J

}

, and

Lemma 3.3(II a–g) hold for n

+ 1, T , J

, S

. By

(14) for n

, φ, there exists φ

∈ OrdForm

such that

(14a–e) hold for n

, φ, φ

. We distinguish three

cases for φ

. Case 1: φ

∈ Tcons

− {1}. We put

= J

⊆ {(i, j) |i ≥ n

+ 1} ⊆ {(i, j) |i ≥ n

} and

= S

⊆ SimOrdCl

L∪{ ˜p | ∈J

}

. Case 2: φ

= 1.

We put J

0 ⊆ {(i, j)|i ≥ n

} and S

= {} ⊆

SimOrdCl

. Case 3: φ

6∈ Tcons

. We put ¯x =

varseq(φ

), j = 0, = (n

, j ), ar( ˜p ) = | ¯x|. We

get by (15) for n

, ∀ ¯x φ

, ¯x, , ˜p that there exist

J = {(n

, j)|1 ≤ j ≤ n

} ⊆ {(n

, j)| j ∈ N}, j ≤ n

6∈ J, S ⊆

SimOrdCl

L∪{ ˜p }∪{ ˜p | ∈J}

, and (15a–g)

hold for ∀ ¯x φ

, ¯x, ˜p , J, S. We put J

= J

∪ { } ∪

J ⊆ {(i, j)|i ≥ n

} and S

= S

∪ { ˜p ( ¯x) ≺ 1} ∪ S ⊆

SimOrdCl

L∪{ ˜p | ∈J

}

. The theorem is proved.

Corollary 3.5. Let n

∈ N, φ ∈ OrdForm

, T ⊆

OrdForm

. There exist J

⊆ {(i, j)|i ≥ n

} and S

⊆

SimOrdCl

L∪{ ˜p | ∈J

}

such that

(i) T |= φ if and only if S

is unsatisﬁable;

(ii) if T ⊆

OrdForm

, then J

⊆

{(i, j)|i ≥

}, kJ

k ∈ O(|T | + |φ|), S

⊆

SimOrdCl

L∪{ ˜p | ∈J

}

, |S

| ∈ O(|T |

+ |φ|

);

the number of all elementary operations of the

translation of T and φ to S

, is in O(|T |

+|φ|

);

the time and space complexity of the translation

of T and φ to S

, is in O(|T |

· log(1 + n

|T |) + |φ|

· (log(1 + n

) + log |φ|));

(iii) S

is admissible;

(iv) tcons(S

) ⊆ tcons(φ) ∪ tcons(T ).

An Order Hyperresolution Calculus for Gödel Logic with Truth Constants and Equality, Strict Order, Delta

Table 1: Interpolation rules.

Case Laws

θ = θ

∧ θ

˜p ( ¯x) ↔ θ

∧ θ

(

˜p ( ¯x) ≺ ˜p

( ¯x) ∨ ˜p (¯x) P ˜p

( ¯x), ˜p ( ¯x) ≺ ˜p

( ¯x) ∨ ˜p (¯x) P ˜p

( ¯x),

˜p

( ¯x) ≺ ˜p ( ¯x) ∨ ˜p

( ¯x) P ˜p ( ¯x) ∨ ˜p

( ¯x) ≺ ˜p ( ¯x) ∨ ˜p

( ¯x) P ˜p ( ¯x), ˜p

( ¯x) ↔ θ

, ˜p

( ¯x) ↔ θ

)

(6), (8) (16)

|Consequent| = 24 + 16 · | ¯x| + | ˜p

( ¯x) ↔ θ

| + | ˜p

( ¯x) ↔ θ

| ≤ 27 · (1 + | ¯x|) + | ˜p

( ¯x) ↔ θ

| + | ˜p

( ¯x) ↔ θ

θ = θ

∨ θ

˜p ( ¯x) ↔ (θ

∨ θ

)

(

˜p ( ¯x) ≺ ˜p

( ¯x) ∨ ˜p (¯x) P ˜p

( ¯x) ∨ ˜p (¯x) ≺ ˜p

( ¯x) ∨ ˜p (¯x) P ˜p

( ¯x), ˜p

( ¯x) ≺ ˜p ( ¯x) ∨ ˜p

( ¯x) P ˜p ( ¯x),

˜p

( ¯x) ≺ ˜p ( ¯x) ∨ ˜p

( ¯x) P ˜p ( ¯x), ˜p

( ¯x) ↔ θ

, ˜p

( ¯x) ↔ θ

)

(5), (7) (17)

|Consequent| = 24 + 16 · | ¯x| + | ˜p

( ¯x) ↔ θ

| + | ˜p

( ¯x) ↔ θ

| ≤ 27 · (1 + | ¯x|) + | ˜p

( ¯x) ↔ θ

| + | ˜p

( ¯x) ↔ θ

θ = θ

→ θ

,θ

6= 0

˜p ( ¯x) ↔ (θ

→ θ

)

{ ˜p

( ¯x) ≺ ˜p

( ¯x) ∨ ˜p

( ¯x) P ˜p

( ¯x) ∨ ˜p (¯x) P ˜p

( ¯x), ˜p

( ¯x) ≺ ˜p

( ¯x) ∨ ˜p (¯x) P 1, ˜p

( ¯x) ↔ θ

, ˜p

( ¯x) ↔ θ

}

(18)

|Consequent| = 15 + 9 · | ¯x| + | ˜p

( ¯x) ↔ θ

| + | ˜p

( ¯x) ↔ θ

| ≤ 27 · (1 + | ¯x|) + | ˜p

( ¯x) ↔ θ

| + | ˜p

( ¯x) ↔ θ

θ = θ

→ 0

˜p ( ¯x) ↔ (θ

→ 0)

{ ˜p

( ¯x) P 0 ∨ ˜p ( ¯x) P 0, 0 ≺ ˜p

( ¯x) ∨ ˜p (¯x) P 1, ˜p

( ¯x) ↔ θ

}

(19)

|Consequent| = 12 + 4 · | ¯x| + | ˜p

( ¯x) ↔ θ

| ≤ 27 · (1 + | ¯x|) + | ˜p

( ¯x) ↔ θ

θ = θ

↔ θ

˜p ( ¯x) ↔ (θ

↔ θ

)

(

˜p

( ¯x) ≺ ˜p

( ¯x) ∨ ˜p

( ¯x) P ˜p

( ¯x) ∨ ˜p (¯x) P ˜p

( ¯x), ˜p

( ¯x) ≺ ˜p

( ¯x) ∨ ˜p

( ¯x) P ˜p

( ¯x) ∨ ˜p (¯x) P ˜p

( ¯x),

˜p

( ¯x) ≺ ˜p

( ¯x) ∨ ˜p

( ¯x) ≺ ˜p

( ¯x) ∨ ˜p (¯x) P 1, ˜p

( ¯x) ↔ θ

, ˜p

( ¯x) ↔ θ

)

(20)

|Consequent| = 27 + 17 · | ¯x| + | ˜p

( ¯x) ↔ θ

| + | ˜p

( ¯x) ↔ θ

| ≤ 27 · (1 + | ¯x|) + | ˜p

( ¯x) ↔ θ

| + | ˜p

( ¯x) ↔ θ

θ = θ

P θ

˜p ( ¯x) ↔ (θ

P θ

)

{ ˜p

( ¯x) P ˜p

( ¯x) ∨ ˜p (¯x) P 0, ˜p

( ¯x) ≺ ˜p

( ¯x) ∨ ˜p

( ¯x) ≺ ˜p

( ¯x) ∨ ˜p (¯x) P 1, ˜p

( ¯x) ↔ θ

, ˜p

( ¯x) ↔ θ

}

(21)

|Consequent| = 15 + 8 · | ¯x| + | ˜p

( ¯x) ↔ θ

| + | ˜p

( ¯x) ↔ θ

| ≤ 27 · (1 + | ¯x|) + | ˜p

( ¯x) ↔ θ

| + | ˜p

( ¯x) ↔ θ

θ = θ

≺ θ

˜p ( ¯x) ↔ (θ

≺ θ

)

{ ˜p

( ¯x) ≺ ˜p

( ¯x) ∨ ˜p (¯x) P 0, ˜p

( ¯x) ≺ ˜p

( ¯x) ∨ ˜p

( ¯x) P ˜p

( ¯x) ∨ ˜p (¯x) P 1, ˜p

( ¯x) ↔ θ

, ˜p

( ¯x) ↔ θ

}

(22)

|Consequent| = 15 + 8 · | ¯x| + | ˜p

( ¯x) ↔ θ

| + | ˜p

( ¯x) ↔ θ

| ≤ 27 · (1 + | ¯x|) + | ˜p

( ¯x) ↔ θ

| + | ˜p

( ¯x) ↔ θ

θ = ∀x θ

˜p ( ¯x) ↔ ∀x θ

{ ˜p ( ¯x) P ∀x ˜p

( ¯x), ˜p

( ¯x) ↔ θ

}

(23)

|Consequent| = 5 + 2 · | ¯x| + | ˜p

( ¯x) ↔ θ

| ≤ 27 · (1 + | ¯x|) + | ˜p

( ¯x) ↔ θ

θ = ∃x θ

˜p ( ¯x) ↔ ∃x θ

{ ˜p ( ¯x) P ∃x ˜p

( ¯x), ˜p

( ¯x) ↔ θ

}

(24)

|Consequent| = 5 + 2 · | ¯x| + | ˜p

( ¯x) ↔ θ

| ≤ 27 · (1 + | ¯x|) + | ˜p

( ¯x) ↔ θ

FCTA 2015 - 7th International Conference on Fuzzy Computation Theory and Applications

Proof. Let T |= φ. Then, for every interpretation A

for L, A 6|= T or A |= φ; by Theorem 3.4(i), there does

not exist an interpretation A

for L ∪ { ˜p | ∈ J

} and

|= S

; S

is unsatisﬁable.

Let S

is unsatisﬁable. Then, for every interpre-

tation A

for L ∪ { ˜p | ∈ J

}, A

6|= S

; by Theo-

rem 3.4(i), there does not exist an interpretation A for

L and A |= T , A 6|= φ; for every interpretation A for

L, A 6|= T or A |= φ; T |= φ; (i) holds.

(ii–iv) are the same as Theorem 3.4(ii–iv); (ii–iv)

hold. The corollary is proved.

4 HYPERRESOLUTION OVER

ORDER CLAUSES

In this section, we propose an order hyperresolu-

tion calculus with truth constants operating over order

clausal theories, and prove its refutational soundness,

completeness.

4.1 Order Hyperresolution Rules

At ﬁrst, we introduce some basic notions and notation

concerning chains of order literals. A chain Ξ of L is a

sequence Ξ = ε



,.. .,ε



, ε



∈ OrdLit

such that for all i < n, υ

= ε

i+1

. ε

is the beginning

element of Ξ and υ

the ending element of Ξ. ε

Ξυ

denotes Ξ together with its respective beginning and

ending element. Let Ξ = ε



,.. .,ε



be a

chain of L. Ξ is an equality chain of L iff, for all

i ≤ n, 

=P. Ξ is an increasing chain of L iff there

exists i

∗

≤ n such that 

∗

=≺. Ξ is a contradiction

of L iff Ξ is an increasing chain of L of the form

Ξ0 or 1 Ξ υ

or ε

Ξε

. Let S ⊆ OrdCl

be unit

and Ξ = ε



,.. .,ε



be a chain | an equality

chain | an increasing chain | a contradiction of L. Ξ

is a chain | an equality chain | an increasing chain | a

contradiction of S iff, for all i ≤ n, ε



∈ S.

Let

W = { ˜w | ∈ I} such that

W ∩ (Func

∪

{

}) =

W is an inﬁnite countable set of

new function symbols. Let L contain a con-

stant (nullary function) symbol. Let P ⊆

P and

S ⊆ OrdCl

L∪P

. We denote GOrdCl

= {C |C ∈

OrdCl

is closed} ⊆ OrdCl

, GInst

(S) = {C |C ∈

GOrdCl

is an instance of S of L} ⊆ GOrdCl

ordtcons(S) = {0 ≺ 1} ∪ {0 ≺ ¯c | ¯c ∈ tcons(S) ∩

} ∪ { ¯c ≺ 1 | ¯c ∈ tcons(S) ∩C

} ∪ { ¯c

≺ ¯c

| ¯c

, ¯c

∈

tcons(S) ∩C

< c

} ⊆ GOrdCl

. A basic order

hyperresolution calculus is deﬁned in Table 2.

The basic order hyperresolution calculus can be

generalised to an order hyperresolution one in Ta-

ble 3. Let L

= L ∪ P, a reduct of L ∪

W ∪ P, and

0 ⊆ GOrdCl

| OrdCl

. Let D = C

,.. .,C

∈ GOrdCl

L∪

W∪P

| OrdCl

L∪

W∪P

, n ≥ 1. D is a de-

duction of C

from S by basic order hyperresolution

iff, for all 1 ≤ κ ≤ n, C

∈ ordtcons(S)∪GInst

κ−1

(S),

or there exist 1 ≤ j

∗

≤ κ − 1, k = 1,. ..,m, such that

is a basic order resolvent of C

∗

,.. .,C

∗

∈ S

κ−1

using Rule (25)–(31) with respect to L

κ−1

and S

κ−1

;

D is a deduction of C

from S by order hyperresolu-

tion iff, for all 1 ≤ κ ≤ n, C

∈ ordtcons(S) ∪ S, or

there exist 1 ≤ j

∗

≤ κ − 1, k = 1,.. .,m, such that C

is an order resolvent of C

∗

,.. .,C

∗

∈ S

κ−1

using Rule

(32)–(38) with respect to L

κ−1

and S

κ−1

where C

∗

a variant of C

∗

∈ S

κ−1

of L

κ−1

; L

and S

are deﬁned

by recursion on 1 ≤ κ ≤ n as follows:







κ−1

∪ { ˜w} in case of Rule (30),(31) |

(37),(38),

κ−1

else;

= S

κ−1

∪ {C

} ⊆ GOrdCl

| OrdCl

= Vrnt

) ⊆ OrdCl

D is a refutation of S iff C

= . We denote

clo

(S) = {C |there exists a deduction of C from S

by basic order hyperresolution}

⊆ GOrdCl

L∪

W∪P

clo

(S) = {C |there exists a deduction of C from S

by order hyperresolution}

⊆ OrdCl

L∪

W∪P

4.2 Refutational Soundness and

Completeness

We are in position to prove the refutational soundness

and completeness of the order hyperresolution calcu-

lus. At ﬁrst, we list some auxiliary lemmata.

Lemma 4.1 (Lifting Lemma). Let L contain a con-

stant symbol. Let P ⊆

P and S ⊆ OrdCl

L∪P

. Let

C ∈ clo

(S). There exists C

∗

∈ clo

(S) such that

C is an instance of C

∗

of L ∪

W ∪ P.

Proof. Technical, analogous to the standard one.

Lemma 4.2 (Reduction Lemma). Let L contain a

constant symbol. Let P ⊆

P and S ⊆ OrdCl

L∪P

Let {

j=0



∨C

|i ≤ n} ⊆ clo

(S) such that

for all S ∈ S el({{ j | j ≤ k

}

|i ≤ n}), there ex-

ists a contradiction of {ε

S(i)



S(i)

|i ≤ n} ⊆

GOrdCl

L∪

W∪P

. There exists

0 6= I

∗

⊆ {i | i ≤ n} such

that

i∈I

∗

∈ clo

(S).

An Order Hyperresolution Calculus for Gödel Logic with Truth Constants and Equality, Strict Order, Delta

Table 2: Basic order hyperresolution rules.

(Basic order hyperresolution rule) (25)

∨C

,. .. ,l

∨C

∈ S

κ−1

i=0

∈ S

;

,. .. ,l

is a contradiction of L

κ−1

i=0

is a basic order hyperresolvent of l

∨C

,. .. ,l

∨C

(Basic order trichotomy rule) (26)

a,b ∈ atoms(S

κ−1

),a ∈ C

,b 6∈ Tcons

,qatoms(S) =

a ≺ b ∨ a P b ∨ b ≺ a ∈ S

(Basic order trichotomy rule) (27)

a,b ∈ atoms(S

κ−1

) − {0,1}, {a,b} 6⊆ Tcons

,qatoms(S) 6=

a ≺ b ∨ a P b ∨ b ≺ a ∈ S

a ≺ b ∨ a P b ∨ b ≺ a is a basic order trichotomy resolvent of a and b.

(Basic order ∀-quantiﬁcation rule) (28)

∀x a ∈ qatoms

∀

κ−1

)

∀x a ≺ aγ ∨ ∀x a P aγ ∈ S

;

t ∈ GTerm

κ−1

,γ = x/t ∈ Subst

κ−1

,dom(γ) = {x} = vars(a).

∀x a ≺ aγ ∨ ∀x a P aγ is a basic order ∀-quantiﬁcation resolvent of ∀x a.

(Basic order ∃-quantiﬁcation rule) (29)

∃x a ∈ qatoms

∃

κ−1

)

aγ ≺ ∃x a ∨ aγ P ∃x a ∈ S

;

t ∈ GTerm

κ−1

,γ = x/t ∈ Subst

κ−1

,dom(γ) = {x} = vars(a).

aγ ≺ ∃x a ∨ aγ P ∃x a is a basic order ∃-quantiﬁcation resolvent of ∃xa.

(Basic order ∀-witnessing rule) (30)

∀x a ∈ qatoms

∀

κ−1

),b ∈ atoms(S

κ−1

) ∪ qatoms(S

κ−1

)

aγ ≺ b ∨ b P ∀x a ∨ b ≺ ∀x a ∈ S

;

˜w ∈

W − Func

κ−1

,ar( ˜w) = |freetermseq(∀xa),freetermseq(b)|,

γ = x/ ˜w(freetermseq(∀xa),freetermseq(b)) ∈ Subst

,dom(γ) = {x} = vars(a).

aγ ≺ b ∨ b P ∀x a ∨ b ≺ ∀x a is a basic order ∀-witnessing resolvent of ∀xa and b.

(Basic order ∃-witnessing rule) (31)

∃x a ∈ qatoms

∃

κ−1

),b ∈ atoms(S

κ−1

) ∪ qatoms(S

κ−1

)

b ≺ aγ ∨ ∃xa P b ∨ ∃x a ≺ b ∈ S

;

˜w ∈

W − Func

κ−1

,ar( ˜w) = |freetermseq(∃xa),freetermseq(b)|,

γ = x/ ˜w(freetermseq(∃xa),freetermseq(b)) ∈ Subst

,dom(γ) = {x} = vars(a).

b ≺ aγ ∨ ∃xa P b ∨ ∃x a ≺ b is a basic order ∃-witnessing resolvent of ∃xa and b.

Proof. Technical, analogous to the one of Proposi-

tion 2, (Guller, 2009).

Lemma 4.3 (Unit Lemma). Let L contain a constant

symbol. Let P ⊆

P and S ⊆ OrdCl

L∪P

. Let  6∈

clo

(S) = {

j=0



|ι < γ}, γ ≤ ω. There exists

∗

∈ S el({{ j | j ≤ k

}

|ι < γ}) such that there does

not exist a contradiction of {ε

∗

(ι)



∗

(ι)

∗

(ι)

|ι <

γ} ⊆ GOrdCl

L∪

W∪P

FCTA 2015 - 7th International Conference on Fuzzy Computation Theory and Applications

Table 3: Order hyperresolution rules.

(Order hyperresolution rule) (32)

j=0



∨

j=1

,. .. ,

j=0



∨

j=1

∈ S

κ−1



i=0

j=1



θ ∈ S

;

for all i < i

≤ n,

freevars(

j=0



∨

j=1

) ∩ freevars(

j=0



∨

j=1

) =

θ ∈ mgu

κ−1



j=0



,. .. ,l

,. .. ,

j=0



,. .. ,l

,{υ

,ε

},. .. ,{υ

n−1

,ε

},{a, b}



dom(θ) = freevars



{ε



| j ≤ k

,i ≤ n}, {l

|1 ≤ j ≤ m

,i ≤ n}



a = ε

,b = 1 or a = υ

,b = 0 or a = ε

,b = υ

, there exists i

∗

≤ n such that 

∗

=≺ .



i=0

j=1



θ is an order hyperresolvent of

j=0



∨

j=1

,. .. ,

j=0



∨

j=1

(Order trichotomy rule) (33)

a,b ∈ atoms(S

κ−1

),a ∈

,b 6∈ Tcons

,qatoms(S) =

a ≺ b ∨ a P b ∨ b ≺ a ∈ S

(Order trichotomy rule) (34)

a,b ∈ atoms(S

κ−1

) − {0,1}, {a,b} 6⊆ Tcons

,qatoms(S) 6=

a ≺ b ∨ a P b ∨ b ≺ a ∈ S

;

vars(a) ∩vars(b) =

a ≺ b ∨ a P b ∨ b ≺ a is an order trichotomy resolvent of a and b.

(Order ∀-quantiﬁcation rule) (35)

∀x a ∈ qatoms

∀

κ−1

)

∀x a ≺ a ∨ ∀x a P a ∈ S

∀x a ≺ a ∨ ∀x a P a is an order ∀-quantiﬁcation resolvent of ∀xa.

(Order ∃-quantiﬁcation rule) (36)

∃x a ∈ qatoms

∃

κ−1

)

a ≺ ∃x a ∨ a P ∃x a ∈ S

a ≺ ∃x a ∨ a P ∃x a is an order ∃-quantiﬁcation resolvent of ∃xa.

(Order ∀-witnessing rule) (37)

∀x a ∈ qatoms

∀

κ−1

),b ∈ atoms(S

κ−1

) ∪ qatoms(S

κ−1

)

aγ ≺ b ∨ b P ∀x a ∨ b ≺ ∀x a ∈ S

;

freevars(∀xa) ∩ freevars(b) =

˜w ∈

W − Func

κ−1

,ar( ˜w) = |freetermseq(∀xa),freetermseq(b)|,

γ = x/ ˜w(freetermseq(∀xa),freetermseq(b))∪ id|

vars(a)−{x}

∈ Subst

,dom(γ) = {x} ∪ (vars(a) − {x}) = vars(a).

aγ ≺ b ∨ b P ∀x a ∨ b ≺ ∀x a is an order ∀-witnessing resolvent of ∀xa and b.

(Order ∃-witnessing rule) (38)

∃x a ∈ qatoms

∃

κ−1

),b ∈ atoms(S

κ−1

) ∪ qatoms(S

κ−1

)

b ≺ aγ ∨ ∃xa P b ∨ ∃x a ≺ b ∈ S

;

freevars(∃xa) ∩ freevars(b) =

˜w ∈

W − Func

κ−1

,ar( ˜w) = |freetermseq(∃xa),freetermseq(b)|,

γ = x/ ˜w(freetermseq(∃xa),freetermseq(b))∪ id|

vars(a)−{x}

∈ Subst

,dom(γ) = {x} ∪ (vars(a) − {x}) = vars(a).

b ≺ aγ ∨ ∃xa P b ∨ ∃x a ≺ b is an order ∃-witnessing resolvent of ∃xa and b.

An Order Hyperresolution Calculus for Gödel Logic with Truth Constants and Equality, Strict Order, Delta

Proof. Technical, a straightforward consequence of

onig’s Lemma and Lemma 4.2.

Let {0,1} ⊆ X ⊆ [0, 1]. X is admissible with re-

spect to suprema and inﬁma iff, for all

0 6= Y

⊆ X

and

∈ Y

. Let {0,1} ⊆

Tc ⊆ Tcons

. Tc is admissible with respect to

suprema and inﬁma iff {0, 1} ⊆ Tc ⊆ [0,1] is admis-

sible with respect to suprema and inﬁma.

Theorem 4.4 (Refutational Soundness and Complete-

ness). Let L contain a constant symbol. Let P ⊆

S ⊆ OrdCl

L∪P

, tcons(S) be admissible with respect to

suprema and inﬁma.  ∈ clo

(S) if and only if S is

unsatisﬁable.

Proof. (=⇒) Let A be a model of S for L ∪ P and

C ∈ clo

(S) ⊆ OrdCl

L∪

W∪P

. Then there exists an

expansion A

of A to L ∪

W ∪ P such that A

|= C.

The proof is by complete induction on the length of a

deduction of C from S by order hyperresolution. Let

 ∈ clo

(S) and A be a model of S for L ∪P. Hence,

there exists an expansion A

of A to L ∪

W ∪ P such

that A

|= , which is a contradiction; S is unsatisﬁ-

able.

(⇐=) Let  6∈ clo

(S). Then, by Lemma 4.1 for

S, ,  6∈ clo

(S); we have L,

W are countable,

P ⊆

P, S ⊆ OrdCl

L∪P

, clo

(S) ⊆ GOrdCl

L∪

W∪P

;

P, L ∪ P, OrdCl

L∪P

, S, L ∪

W ∪ P, GOrdCl

L∪

W∪P

clo

(S) are countable; there exists γ

≤ ω and  6∈

clo

(S) = {

j=0



|ι < γ

}; by Lemma 4.3 for

S, there exists S

∗

∈ Sel({{ j | j ≤ k

}

|ι < γ

}) and

there does not exist a contradiction of {ε

∗

(ι)



∗

(ι)

∗

(ι)

|ι < γ

} ⊆ GOrdCl

L∪

W∪P

. We put S =

{ε

∗

(ι)



∗

(ι)

∗

(ι)

|ι < γ

} ⊆ GOrdCl

L∪

W∪P

. Then

ordtcons(S) ⊆ clo

(S), S ⊇ ordtcons(S) is count-

able, unit, (q)atoms(S) ⊆ (q)atoms(clo

(S)); there

does not exist a contradiction of S. We have L con-

tains a constant symbol. Hence, there exists cn

∗

∈

Func

, ar

(cn

∗

) = 0. We put

∗

= funcs(S) ∩

W ⊆

∗

∩ (Func

∪ {

}) ⊆

W ∩(Func

∪ {

}) =

= GTerm

L∪

∗

∪P

,cn

∗

∈ U

B = atoms(S) ∪ qatoms(S) ⊆

GAtom

L∪

∗

∪P

∪ QAtom

L∪

∗

∪P

We have S is countable. Then tcons(S) =

atoms(ordtcons(S)) ⊆ atoms(S) ⊆ B, B = tcons(S) ∪

(B − tcons(S)), tcons(S) ∩ (B − tcons(S)) =

atoms(S), qatoms(S), B, tcons(S), B − tcons(S) are

countable; there exist γ

≤ ω and a sequence δ

−→ B − tcons(S) of B − tcons(S). Let ε

,ε

∈ B.

,ε

iff there exists an equality chain ε

Ξε

S. Note that , is a binary symmetric transitive rela-

tion on B. ε

Cε

iff there exists an increasing chain

Ξε

of S. Note that C is a binary transitive relation

on B.

06, 1,1 6, 0,0 C 1,1 6 0, (39)

for all ε ∈ B, ε6 0,1 6 ε,ε6 ε.

The proof is straightforward; we have that there does

not exist a contradiction of S. Note that C is also

irreﬂexive and a partial strict order on B.

Let tcons(S) ⊆ X ⊆ B. A partial valuation V

is a mapping V : X −→ [0,1] such that V (0) = 0,

V (1) = 1, for all ¯c ∈ tcons(S) ∩ C

, V ( ¯c) = c. We

denote dom(V ) = X, tcons(S) ⊆ dom(V ) ⊆ B. We

deﬁne a partial valuation V

by recursion on α ≤ γ

as follows:

= {(0, 0),(1,1)} ∪ {( ¯c, c)| ¯c ∈ tcons(S) ∩C

};

= V

α−1

∪ {(δ

(α − 1),λ

α−1

)}

(1 ≤ α ≤ γ

is a successor ordinal),

α−1

= {V

α−1

(a)|a ,δ

(α − 1),a ∈ dom(V

α−1

)},

α−1

= {V

α−1

(a)|a Cδ

(α − 1),a ∈ dom(V

α−1

)},

α−1

= {V

α−1

(a)|δ

(α − 1)C a,a ∈ dom(V

α−1

)},

α−1

(

α−1

if E

α−1

else;

[

α<γ

(γ

is a limit ordinal).

For all α ≤ α

≤ γ

, V

is a partial valuation,

dom(V

) = tcons(S) ∪ δ

[α], V

⊆ V

(40)

The proof is by induction on α ≤ γ

We list some auxiliary statements without proofs:

If qatoms(S) =

0, then qatoms(clo

(S)) =

0. (41)

tcons(S) = tcons(clo

(S)). (42)

For all a,b ∈ atoms(clo

(S)) ∪

qatoms(clo

(S)), there exist a deduction

,.. .,C

, n ≥ 1, from S by basic order hyper-

resolution, associated L

, S

⊆ GOrdCl

such that a,b ∈ atoms(S

) ∪ qatoms(S

(43)

For all

0 6= A ⊆

atoms(clo

(S)) ∪

qatoms(clo

(S)), there exist a deduction

,.. .,C

, n ≥ 1, from S by basic order hyper-

resolution, associated L

, S

⊆ GOrdCl

such that A ⊆ atoms(S

) ∪ qatoms(S

(44)

FCTA 2015 - 7th International Conference on Fuzzy Computation Theory and Applications

For all a ∈ tcons(S) ∩ C

, b ∈ B − tcons(S),

either aC b or a, b or b Ca.

(45)

Let qatoms(S) 6=

0. For all a,b ∈ B − {0,1},

either aC b or (a = b or a , b) or bC a.

(46)

For all α ≤ γ

, for all a,b ∈ dom(V

if a, b, then V

(a) = V

(b);

if aC b, then V

(a) < V

(b);

if V

(a) = 0, then a = 0 or a ,0;

if V

(a) = 1, then a = 1 or a ,1;

for all α < γ

[dom(V

)] is admissible with respect to

suprema and inﬁma.

(47)

The proof is by induction on α ≤ γ

using the as-

sumption that tcons(S) is admissible with respect to

suprema and inﬁma.

We put V = V

, dom(V )

(40)

== tcons(S) ∪ δ[γ

] =

tcons(S) ∪ (B − tcons(S)) = B. We further list some

other auxiliary statements without proofs:

For all a,b ∈ B,

if a, b, then V (a) = V (b);

if aC b, then V (a) < V (b).

(48)

For all Qxa ∈ qatoms(clo

(S)) and u ∈ U

a(x/u) ∈ atoms(clo

(S)).

(49)

For all a ∈ B,

if a = ∀xb, then V (a) =

u∈U

V (b(x/u));

if a = ∃xb, then V (a) =

u∈U

V (b(x/u)).

(50)

We put

,.. .,u

) =







f (u

,.. .,u

)

if f ∈ Func

L∪

∗

∪P

∗

else,

f ∈ Func

L∪

W∪P

∈ U

;

,.. .,u

) =







V (p(u

,.. .,u

))

if p(u

,.. .,u

) ∈ B,

0 else,

p ∈ Pred

L∪

W∪P

∈ U

;

A =



,{ f

| f ∈ Func

L∪

W∪P

| p ∈ Pred

L∪

W∪P

}



an interpretation for L ∪

W ∪ P.

For all C ∈ S and e ∈ S

, C(e|

freevars(C)

) ∈

clo

(S).

(51)

It is straightforward to prove that for all a ∈ B

and e ∈ S

, kak

= V (a). Let l = ε

P ε

∈

S and e ∈ S

. Then ε

,ε

∈ B, ε

,ε

, by (48)

for ε

, ε

, V (ε

) = V (ε

), klk

= kε

P ε

kε

Pkε

= V (ε

PV (ε

) = 1. Let l = ε

≺

∈ S and e ∈ S

. Then ε

,ε

∈ B, ε

Cε

, by (48)

for ε

, ε

, V (ε

) < V (ε

), klk

= kε

≺ ε

kε

≺

≺kε

= V (ε

)≺

≺

≺V (ε

) = 1. So, for all

l ∈ S and e ∈ S

, for both the cases l = ε

P ε

∈ S

and l = ε

≺ ε

∈ S, klk

= 1; klk

= 1. Let C ∈

S ⊆ OrdCl

L∪P

and e ∈ S

. Then e : Var

−→ U

freevars(C) ⊆

Var

, e|

freevars(C)

∈ Subst

L∪

∗

∪P

dom(e|

freevars(C)

) = freevars(C), range(e|

freevars(C)

) =

0; e|

freevars(C)

is applicable to C; by (51) for

C, e, C(e|

freevars(C)

) ∈ clo

(S), there exists l

∗

∈

C(e|

freevars(C)

) and l

∗

∈ S, kl

∗

= 1; there exists l

∗∗

∈

C ∈ OrdCl

L∪P

and l

∗∗

∈ OrdLit

L∪P

⊆ OrdLit

L∪

∗

∪P

freevars(l

∗∗

) ⊆ freevars(C); e|

freevars(l

∗∗

)

is appli-

cable to l

∗∗

, l

∗∗

(e|

freevars(l

∗∗

)

) = l

∗

; for all t ∈

Term

L∪

∗

∪P

, a ∈ Atom

L∪

∗

∪P

∪ QAtom

L∪

∗

∪P

, l ∈

OrdLit

L∪

∗

∪P

, ktk

= t(e|

vars(t)

) = kt(e|

vars(t)

kak

= ka(e|

freevars(a)

, klk

= kl(e|

freevars(l)

;

the proof is by induction on t and by deﬁnition;

∗∗

= kl

∗∗

(e|

freevars(l

∗∗

)

= kl

∗

= 1; A |=

A |= S, A|

L∪P

|= S; S is satisﬁable. The theorem is

proved.

Consider S = {0 ≺ a}∪{a ≺

|n ≥ 2} ⊆ OrdCl

a ∈ Pred

− Tcons

, ar

(a) = 0. tcons(S) is not ad-

missible with respect to suprema and inﬁma; for {0}

and {

|n ≥ 2},

{0} =

{

|n ≥ 2} = 0, 0 6∈ {

|n ≥

2}. S is unsatisﬁable; both the cases kak

= 0 and

kak

> 0 lead to A 6|= S for every interpretation A

for L. However,  6∈ clo

(S) = S ∪ {0 ≺ 1} ∪ {0 ≺

|n ≥ 2} ∪ {

≺ 1 |n ≥ 2} ∪ {

≺

> n

≥

2} ∪ {

≺ a ∨

P a ∨ a ≺

|n ≥ 2} ∪ {

P a ∨ a ≺

|n ≥ 2} ∪ {

≺ a ∨ a ≺

|n ≥ 2}, using Rules (33)

and (32); clo

(S) contains the order clauses from S,

from ordtcons(S), and some superclauses of them.

So, the condition on tcons(S) being admissible with

respect to suprema and inﬁma, is necessary.

The deduction problem of a formula from a theory

can be solved as follows:

Corollary 4.5. Let L contain a constant symbol. Let

∈ N, φ ∈ OrdForm

, T ⊆ OrdForm

, tcons(T )

be admissible with respect to suprema and in-

ﬁma. There exist J

⊆ {(i, j) |i ≥ n

} and S

⊆

SimOrdCl

L∪{ ˜p | ∈J

}

such that tcons(S

) is admissi-

ble with respect to suprema and inﬁma; T |= φ if and

only if  ∈ clo

Proof. By Corollary 3.5 for n

, φ, T , there exist

⊆ {(i, j) |i ≥ n

},S

⊆ SimOrdCl

L∪{ ˜p | ∈J

}

An Order Hyperresolution Calculus for Gödel Logic with Truth Constants and Equality, Strict Order, Delta

Table 4: An example: φ = ∀x (q

(x) → 0.3) → (∃x q

(x) → 0.5).

φ = ∀x (q

(x) → 0.3) → (∃x q

(x) → 0.5)

˜p

(x) ≺ 1, ˜p

(x) ↔



∀x (q

(x) → 0.3)

| {z }

˜p

(x)

→ (∃x q

(x) → 0.5

| {z }

˜p

(x)

)



(18)

˜p

(x) ≺ 1, ˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P ˜p

(x) ∨ ˜p

(x) P ˜p

(x), ˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P 1, ˜p

(x) ↔ ∀x (q

(x) → 0.3

| {z }

˜p

(x)

), ˜p

(x) ↔ (∃x q

(x)

| {z }

˜p

(x)

→ 0.5

|{z}

˜p

(x)

)

(23), (18)

˜p

(x) ≺ 1, ˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P ˜p

(x) ∨ ˜p

(x) P ˜p

(x), ˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P 1, ˜p

(x) P ∀x ˜p

(x), ˜p

(x) ↔ (q

(x)

|{z}

˜p

(x)

→ 0.3

|{z}

˜p

(x)

˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P ˜p

(x) ∨ ˜p

(x) P ˜p

(x), ˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P 1, ˜p

(x) ↔ ∃x q

(x)

|{z}

˜p

(x)

, ˜p

(x) P 0.5

(18), (24)

˜p

(x) ≺ 1, ˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P ˜p

(x) ∨ ˜p

(x) P ˜p

(x), ˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P 1, ˜p

(x) P ∀x ˜p

(x),

˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P ˜p

(x) ∨ ˜p

(x) P ˜p

(x), ˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P 1, ˜p

(x) P q

(x), ˜p

(x) P

0.3,

˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P ˜p

(x) ∨ ˜p

(x) P ˜p

(x), ˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P 1, ˜p

(x) P ∃x ˜p

(x), ˜p

(x) P q

(x), ˜p

(x) P 0.5

(

˜p

(x) ≺ 1 [1]

˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P ˜p

(x) ∨ ˜p

(x) P ˜p

(x) [2]

˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P 1 [3]

˜p

(x) P ∀x ˜p

(x) [4]

˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P ˜p

(x) ∨ ˜p

(x) P ˜p

(x) [5]

˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P 1 [6]

˜p

(x) P q

(x) [7]

˜p

(x) P 0.3 [8]

˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P ˜p

(x) ∨ ˜p

(x) P ˜p

(x) [9]

˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P 1 [10]

˜p

(x) P ∃x ˜p

(x) [11]

˜p

(x) P q

(x) [12]

˜p

(x) P 0.5

)

[13]

Rule (32) : [1][3] :

˜p

(x) ≺ ˜p

(x) [14]

Rule (32) : [10][14] :

˜p

(x) ≺ ˜p

(x) [15]

repeatedly Rule (32) : [9][15] :

˜p

(x) P ˜p

(x) [16]

Rule (35) : ∀x ˜p

(x) :

∀x ˜p

(x) ≺ ˜p

(x) ∨ ∀x ˜p

(x) P ˜p

(x) [17]

0.3 ≺ 0.5 ∈ ordtcons(S

)

0.3 ≺ 0.5 [18]

repeatedly Rule (32) : [4][5][8][13][14][16][17][18] :

˜p

(x) ≺ ˜p

(x) ∨ ˜p

(x) P ˜p

(x) [19]

Rule (38) : ∃x ˜p

(x),0.5 :

0.5 ≺ ˜p

( ˜w

(0,0)

) ∨ ∃x ˜p

(x) ≺ 0.5 ∨ ∃x ˜p

(x) P 0.5 [20]

repeatedly Rule (32) : [11][13][15][20] :

0.5 ≺ ˜p

( ˜w

(0,0)

) [21]

repeatedly Rule (32) : [7][8][12][19]; ˜w

(0,0)

: [18][21] :

 [22]

and Corollary 3.5(i,iv) hold for φ, T , S

; we have

tcons(T ) is admissible with respect to suprema

and inﬁma, tcons(S

) ⊆ tcons(φ) ∪ tcons(T );

tcons(φ) ⊆

Tcons

, tcons(S

) is admissible with

respect to suprema and inﬁma; we have T |= φ if

and only if S

is unsatisﬁable; by Theorem 4.4 for

{ ˜p | ∈ J

}, S

, S

is unsatisﬁable if and only if

 ∈ clo

); T |= φ if and only if  ∈ clo

The corollary is proved.

In Table 4, we show that φ = ∀x(q

(x) → 0.3) →

(∃x q

(x) → 0.5) ∈ OrdForm

is logically valid using

the translation to order clausal form and the order hy-

perresolution calculus.

5 CONCLUSIONS

In the paper, we have proposed a modiﬁcation of the

hyperresolution calculus from (Guller, 2012; Guller,

FCTA 2015 - 7th International Conference on Fuzzy Computation Theory and Applications

2015) which is suitable for automated deduction in

the ﬁrst-order G

odel logic with explicit partial truth.

odel logic is expanded by a countable set of inter-

mediate truth constants ¯c, c ∈ (0,1). We have mod-

iﬁed translation of a formula to an equivalent satis-

ﬁable ﬁnite order clausal theory, consisting of order

clauses. An order clause is a ﬁnite set of order literals

of the form ε

 ε

where ε

is an atom or a quanti-

ﬁed atom, and  is the connective P or ≺. P and

≺ are interpreted by the equality and standard strict

linear order on [0,1], respectively. We have investi-

gated the so-called canonical standard completeness,

where the semantics of G

odel logic is given by the

standard G-algebra and truth constants are interpreted

by ’themselves’. The modiﬁed hyperresolution cal-

culus is refutation sound and complete for a count-

able order clausal theory if the set of truth constants

occurring in the theory, is admissible with respect to

suprema and inﬁma. This condition covers the case of

ﬁnite order clausal theories. We have solved the de-

duction problem of a formula from a countable theory.

As an interesting consequence, we get an afﬁrmative

solution to the open problem of recursive enumerabil-

ity of unsatisﬁable formulae in G

odel logic with truth

constants and the equality, P

P, strict order, ≺

≺

≺, projec-

tion, ∆

∆

∆, operators.

Corollary 5.1. The set of unsatisﬁable formulae of L

is recursively enumerable.

Proof. Let φ ∈ OrdForm

. Then φ contains a ﬁnite

number of truth constants and tcons({φ}) is admissi-

ble with respect to suprema and inﬁma. φ is unsatisﬁ-

able if and only if {φ} |= 0. Hence, the problem that φ

is unsatisﬁable can be reduced to the deduction prob-

lem {φ} |= 0 after a constant number of steps. Let

∈ N. By Corollary 4.5 for n

, 0, {φ}, there exist

{φ}

⊆ {(i, j) |i ≥ n

}, S

{φ}

⊆ SimOrdCl

L∪{ ˜p | ∈J

{φ}

}

and tcons(S

{φ}

) is admissible with respect to suprema

and inﬁma, {φ} |= 0 if and only if  ∈ clo

{φ}

); if

{φ} |= 0, then  ∈ clo

{φ}

) and we can decide it

after a ﬁnite number of steps. This straightforwardly

implies that the set of unsatisﬁable formulae of L is

recursively enumerable. The corollary is proved.

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