An Order Hyperresolution Calculus for G
¨
odel Logic with Truth
Constants
Duˇsan Guller
Department of Informatics, Alexander Dubˇcek University of Trenˇc´ın,
ˇ
Studentsk´a 2, 911 50 Trenˇc´ın, Slovakia
Keywords:
G¨odel Logic, Resolution, Many-valued Logics, Automated Deduction.
Abstract:
In (Guller, 2012), we have generalised the well-known hyperresolution principle to the first-order G¨odel logic
for the general case. This paper is a continuation of our work. We propose a modification of the hyperreso-
lution calculus suitable for automated deduction with explicit partial truth. We expand the first-order 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
1
⋄ ε
2
where ⋄ is a connective either ≖ or ≺. ≖ 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 the first-order G¨odel logic is given by the standard G-algebra
and truth constants are interpreted by themselves. The modified hyperresolution calculus is refutation sound
and complete for a countable order clausal theory under a certain condition for suprema and infima of sets of
the truth constants occurring in the theory.
1 INTRODUCTION
Current research in many-valued logics is mainly con-
cerned with left-continuous t-norm based logics in-
cluding the three fundamental fuzzy logics: G¨odel,
Łukasiewicz, and Product ones. From a syntactical
point of view, classical many-valued deduction cal-
culi are widely studied, especially Hilbert-style ones.
In addition, a perspective from automated deduction
has received attractivity during the last two decades.
A considerable effort has been made in development
of SAT solvers for the problem of Boolean satisfia-
bility. SAT solvers may exploit either complete so-
lution methods (called complete or systematic SAT
solvers) or incomplete or hybrid ones. Complete
SAT solvers are mostly based on the Davis-Putnam-
Logemann-Loveland procedure (DPLL) (Davis and
Putnam, 1960; Davis et al., 1962) or resolution proof
methods (Robinson, 1965b; Robinson, 1965a; Gal-
lier, 1985), improvedby various features, (Biere et al.,
2009). t-norm based logics are logics of compara-
tive truth: the residuum of a t-norm satisfies, for all
x,y ∈ [0,1], x → y = 1 if and only if x ≤ y. Since im-
plication is interpreted by a residuum, in the proposi-
tional case, a formula of the form φ → ψ is a conse-
quence of a theory if kφk
A
≤ kψk
A
for every model A
Partially supported by VEGA Grant 1/0592/14.
of the theory. Most explorations of t-norm based log-
ics are focused on tautologies and deduction calculi
with the only distinguished truth degree 1, (H´ajek,
2001). However, in many real-world applications, one
may be interested in representationand inferencewith
explicit partial truth; besides the truth constants 0, 1,
intermediate truth constants are involved in. In the lit-
erature, two main approaches to expansions with truth
constants, are described. Historically, first one has
been introduced in (Pavelka, 1979), where the propo-
sitional Łukasiewicz logic is augmented by truth con-
stants ¯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 development of evaluated for-
mulae, and in (H´ajek, 2001), Rational Pavelka’s logic
(RPL) - a simplification of PL, are described. An-
other approach relies on traditional algebraic seman-
tics. Various completeness results for expansions of
t-norm based logics with countably many truth con-
stants are investigated, among others, in (Esteva et al.,
2001; Savick´y et al., 2006; Esteva et al., 2007b; Es-
teva et al., 2007a; Esteva et al., 2009; Esteva et al.,
2010a; Esteva et al., 2010b).
Concerning the three fundamental first-order
fuzzy logics, the set of logically valid formulae is Π
2
-
complete for Łukasiewicz logic, Π
2
-hard for Product
37
Guller D..
An Order Hyperresolution Calculus for Gödel Logic with Truth Constants.
DOI: 10.5220/0005073700370052
In Proceedings of the International Conference on Fuzzy Computation Theory and Applications (FCTA-2014), pages 37-52
ISBN: 978-989-758-053-6
Copyright
c
 2014 SCITEPRESS (Science and Technology Publications, Lda.)

logic, and Σ
1
-complete for G¨odel logic, as with classi-
cal first-order logic. Among these fuzzy logics, only
G¨odel logic is recursively axiomatisable. Hence, it
was necessary to provide a proof method suitable for
automated deduction, as one has done for classical
logic. In contrast to classical logic, we cannot make
shifts of quantifiers arbitrarily and translate a formula
to an equivalent (satisfiable) prenex form. In (Guller,
2012), we have generalised the well-known hyperres-
olution principle to the first-order G¨odel logic for the
general case. Our approach is based on translation of
a formulaof G¨odel logic to an equivalentsatisfiable fi-
nite order clausal theory, consisting of order clauses.
We have introduced a notion of quantified atom: a
formula a is a quantified atom if a = Qx p(t
0
,... ,t
τ
)
where Q is a quantifier (∀, ∃); p(t
0
,... ,t
τ
) is an atom;
x is a variable occurring in p(t
0
,... ,t
τ
); for all i ≤ τ,
either t
i
= x or x does not occur in t
i
. An order clause
is a finite set of order literals of the form ε
1
⋄ ε
2
where
ε
i
is either an atom or a quantified atom; and ⋄ is a
connective either ≖ or ≺. ≖ and ≺ are interpreted by
the equality and standard strict linear order on [0,1],
respectively. For an input theory of G¨odel logic, the
proposed translation produces a so-called admissible
order clausal theory. On the basis of the hyperres-
olution principle, a calculus operating over admissi-
ble order clausal theories, has been devised. The cal-
culus is proved to be refutation sound and complete
for the countable case with respect to the standard
G-algebra G = ([0,1],≤,∨
∨
∨,∧
∧
∧,⇒
⇒
⇒,
,≖
≖
≖,≺
≺
≺,0,1) aug-
mented by binary operators ≖
≖
≖ and ≺
≺
≺ for ≖ and ≺,
respectively, cf. Section 2. As another step, one may
incorporate a countable set of intermediate truth con-
stants ¯c, c ∈ (0,1), to get a modification of our hy-
perresolution calculus suitable for automated deduc-
tion with explicit partial truth. We shall investigate
the so-called canonical standard completeness, where
the semantics of the first-order G¨odel logic is given
by the standard G-algebra G and truth constants are
interpreted by themselves. Note that the Hilbert-style
calculus for the first-order G¨odel logic introduced in
(H´ajek, 2001) is not suitable for expansion with truth
constants. We have φ ⊢ ψ if and only if φ |= ψ (wrt.
G). However, that cannot be preserved after adding
truth constants. Let c ∈ (0,1) and a be an atom dif-
ferent from a constant. Then ¯c |= a (¯c is unsatisfi-
able) but 6|= ¯c → a, 6⊢ ¯c → a, ¯c 6⊢ a (from the sound-
ness and the deduction-detachment theorem for this
calculus). So, we cannot achieve a strict canonical
standard completenessafter expansion with truth con-
stants. On the other side, such a completeness can be
feasible for our hyperresolutioncalculus under certain
condition. We say that a set X of truth constants is ad-
missible with respect to suprema and infima if, for all
Y
1
,Y
2
⊆ X ∪ {0,1} and
W
W
W
Y
1
=
V
V
V
Y
2
, either
W
W
W
Y
1
∈ Y
1
,
V
V
V
Y
2
∈ Y
2
, or
W
W
W
Y
1
6∈ Y
1
,
V
V
V
Y
2
6∈ Y
2
(constants are inter-
preted by themselves). Then the hyperresolution cal-
culus is refutation sound and completefor 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.
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
By L we denotea first-order language. Var
L
| Func
L
|
Pred
L
| Term
L
| GTerm
L
| Atom
L
| GAtom
L
denotes
the set of all variables | function symbols | predicate
symbols | terms | ground terms | atoms | ground
atoms of L. ar
L
: Func
L
∪ Pred
L
−→ N denotes
the mapping assigning an arity to every function and
predicate symbol of L. We assume nullary predicate
symbols 0,1 ∈ Pred
L
, ar
L
(0) = ar
L
(1) = 0; 0
denotes the false and 1 the true in L. In addition, we
assume a countable set of nullary predicate symbols
C
L
= { ¯c| ¯c ∈ Pred
L
,ar
L
( ¯c) = 0,c ∈ (0,1)} ⊆ Pred
L
.
0, 1, ¯c ∈ C
L
are called truth constants. We denote
Tcons
L
= {0,1} ∪
C
L
⊆ Pred
L
. Let X ⊆ Tcons
L
.
We denote X = {0|0 ∈ X} ∪ {1|1 ∈ X} ∪ {c| ¯c ∈
C
L
} ⊆ [0,1]. By Form
L
we designate the set of
all formulae of L built up from Atom
L
and Var
L
using the connectives: ¬, negation, ∧, conjunction,
∨, disjunction, →, implication, and the quanti-
fiers: ∀, the universal quantifier, ∃, the existential
one. In addition, we introduce new binary connec-
tives ≖, equality, and ≺, strict order. We denote
Con = {¬,∧,∨,→, ≖, ≺}. By OrdForm
L
we des-
ignate the set of all so-called order formulae of L
built up from Atom
L
and Var
L
using the connec-
tives in Con and the quantifiers: ∀, ∃.
1
Note that
OrdForm
L
⊇ Form
L
. 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
i
, 1 ≤ i ≤ m | υ
i
, 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
1
,... ,ε
m
) ⊆ Var
L
| freevars(ε
1
,... , ε
m
) ⊆
Var
L
| boundvars(ε
1
,... , ε
m
) ⊆ Var
L
|
1
We assume a decreasing connective and quantifier
precedence: ∀, ∃, ¬, ∧, →, ≖, ≺, ∨.
FCTA2014-InternationalConferenceonFuzzyComputationTheoryandApplications
38

preds(ε
1
,... , ε
m
) ⊆ Pred
L
| atoms(ε
1
,... , ε
m
) ⊆
Atom
L
we denote the set of all variables | free
variables | bound variables | predicate symbols |
atoms of L occurring in ε
1
,... , ε
m
. ε is closed
iff freevars(ε) =
/
0. By ℓ we denote the empty
sequence. By |ε
1
,... , ε
m
| = m we denote the length
of a sequence ε
1
,... , ε
m
. We define the concate-
nation of sequences ε
1
,... ,ε
m
and υ
1
,... , υ
n
as
(ε
1
,... , ε
m
),(υ
1
,... , υ
n
) = ε
1
,... , ε
m
,υ
1
,... , υ
n
.
Note that concatenation of sequences is associative.
Let X, Y, Z be sets, Z ⊆ X; f : X −→ Y be a map-
ping. By kXk we denote the set-theoretic cardinal-
ity of X. X being a finite subset of Y is denoted as
X ⊆
F
Y. We designate P (X) = {x|x ⊆ X}; P (X) is
the power set of X; P
F
(X) = {x|x ⊆
F
X}; P
F
(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
= {(z, f(z))|z ∈
Z}; f|
Z
is the restriction of f onto Z. Let γ ≤ ω. A
sequence δ of X is a bijection δ : γ −→ X. X is count-
able if and only if there exists a sequence of X. Let I
be a set and S
i
6=
/
0, i ∈ I, be sets. A selector S over
{S
i
|i∈ I} is a mappingS : I −→
S
{S
i
|i ∈ I} such that
for all i ∈ I, S (i) ∈ S
i
. We denote S el({S
i
|i ∈ I}) =
{S |S is a selector over {S
i
|i ∈ I}}. Let c ∈ R
+
. logc
denotes the binary logarithm of c. Let f, g : N −→ R
+
0
.
f is of the order of g, in symbols f ∈ O(g), iff there
exist n
0
∈ N and c
∗
∈ R
+
0
such that for all n ≥ n
0
,
f(n) ≤ c
∗
· g(n).
Let t ∈ Term
L
, φ ∈ OrdForm
L
, T ⊆
F
OrdForm
L
.
The size of t | φ, in symbols |t| | |φ|, is defined as
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
L
, 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(∀xp(x,x, z) ∨ ∃yq(x,y,z))) =
w,x, x,x,z,y,x,y, z and |w,x,x,x, z,y,x,y,z| = 9. Let
Q ∈ {∀, ∃} and ¯x = x
1
,... , x
n
be a sequence of vari-
ables of L. By Q¯xφ we denote Qx
1
...Qx
n
φ.
G¨odel logic is interpreted by the standard G-
algebra augmented by binary operators ≖
≖
≖ and ≺
≺
≺ for
≖ and ≺, respectively.
G = ([0, 1], ≤,∨
∨
∨,∧
∧
∧,⇒
⇒
⇒,
,≖
≖
≖,≺
≺
≺,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;
a≖
≖
≖ b =

1 if a = b,
0 else;
a≺
≺
≺b =

1 if a < b,
0 else.
We recall that G is a complete linearly ordered lat-
tice algebra; ∨
∨
∨ | ∧
∧
∧ is commutative, associative, idem-
potent, 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)
G¨odel negation
satisfies the condition:
for all a ∈ G,
a = a⇒
⇒
⇒0; (2)
the following properties, which will be exploitedlater,
hold:
2
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)
An interpretation I for L is a triple

U
I
,{ f
I
| f ∈
Func
L
},{p
I
| p ∈ Pred
L
}

defined as follows: U
I
6=
/
0 is the universum of I ; every f ∈ Func
L
is inter-
preted as a function f
I
: U
ar( f)
I
−→ U
I
; every p ∈
Pred
L
is interpretedas a [0, 1]-relation p
I
:U
ar(p)
I
−→
[0,1]; particularly, 0
I
= 0, 1
I
= 1, for all ¯c ∈
C
L
,
¯c
I
= c. A variable assignment in I is a mapping
Var
L
−→ U
I
. We denote the set of all variable as-
signments in I as S
I
. Let e ∈ S
I
and u ∈ U
I
. A vari-
ant e[x/u] ∈ S
I
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
L
, ¯x be a sequence of variables of L, φ ∈
OrdForm
L
. In I with respect to e, we define the value
ktk
I
e
∈ U
I
of t by recursion on the structure of t, the
value k ¯xk
I
e
∈ U
| ¯x|
I
of ¯x, the truth value kφk
I
e
∈ [0,1] of
φ by recursion on the structure of φ, as usual. Let φ
be closed. Then, for all e,e
′
∈ S
I
, kφk
I
e
= kφk
I
e
′
. Let
e ∈ S
I
6=
/
0. We denote kφk
I
= kφk
I
e
.
Let L | L
′
be a first-order language and I | I
′
be
an interpretation for L | L
′
. L
′
is an expansion of L
2
We assume a decreasing operator precedence:
, ∧
∧
∧,
⇒
⇒
⇒, ≖
≖
≖, ≺
≺
≺, ∨
∨
∨.
AnOrderHyperresolutionCalculusforGödelLogicwithTruthConstants
39

iff Func
L
′
⊇ Func
L
and Pred
L
′
⊇ Pred
L
; 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
I
′
= U
I
, for
all f ∈ Func
L
, f
I
′
= f
I
, for all p ∈ Pred
L
, p
I
′
= p
I
;
on the other side, we say I is a reduct of I
′
to L, in
symbols I = I
′
|
L
.
A theory of L is a set of formulae of L. An or-
der theory of L is a set of order formulae of L. Let
φ,φ
′
∈ OrdForm
L
, T ⊆ OrdForm
L
, e ∈ S
I
. φ is true in
I with respect to e, written as I |=
e
φ, iff kφk
I
e
= 1. I
is a model of φ, in symbols I |= φ, iff, for all e ∈ S
I
,
I |=
e
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 equiva-
lent to φ
′
, in symbols φ ≡ φ
′
, iff, for every interpre-
tation I for L and e ∈ S
I
, kφk
I
e
= kφ
′
k
I
e
. We de-
note tcons(φ) = {0,1} ∪ (preds(φ) ∩
C
L
) ⊆ Tcons
L
and tcons(T) = { 0,1} ∪ (preds(T) ∩C
L
) ⊆ Tcons
L
.
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
constant 0. An output equivalent CNF may be of ex-
ponential 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 transla-
tion to CNF via interpolation using new atoms, which
produces an output CNF of linear size at the cost of
being only equisatisfiable to the input formula. A
similar approach exploiting the renaming subformu-
lae technique can be found in (Plaisted and Green-
baum, 1986; de la Tour, 1992; H¨ahnle, 1994; Non-
nengart et al., 1998). 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
1
⋄ ε
2
where ε
i
is ei-
ther a propositional atom or a propositional constant,
0, 1, and ⋄ ∈ {≖, ≺}.
We have described some generalisation of the
mentioned translation to the first-order case in
(Guller, 2012). At first, we recall the notion of quan-
tified atom. Let a ∈ Form
L
. a is a quantified atom of
L iff a = Qx p(t
0
,... ,t
τ
) where p(t
0
,... ,t
τ
) ∈ Atom
L
,
x ∈ vars(p(t
0
,... ,t
τ
)), either t
i
= x or x 6∈ vars(t
i
).
QAtom
L
⊆ Form
L
denotes the set of all quantified
atoms of L. QAtom
Q
L
⊆ QAtom
L
, Q ∈ {∀,∃}, de-
notes the set of all quantified atoms of L of the
form Qxa. Let ε | ε
i
, 1 ≤ i ≤ m | υ
i
, 1 ≤ i ≤ n,
be either an expression or a set of expressions or
a set of sets of expressions of L, in general. By
qatoms(ε
1
,... , ε
m
) ⊆ QAtom
L
we denote the set of
all quantified atoms of L occurring in ε
1
,... , ε
m
. We
denote qatoms
Q
(ε
1
,... , ε
m
) = qatoms(ε
1
,... , ε
m
) ∩
QAtom
Q
L
, Q ∈ {∀,∃}. Let Qx p(t
0
,... ,t
τ
) ∈ QAtom
L
and p(t
′
0
,... ,t
′
τ
) ∈ Atom
L
. Let I = {i|i ≤ τ,x 6∈
vars(t
i
)} and r
1
,... , r
k
, r
i
≤ τ, k ≤ τ, for all 1 ≤ i <
i
′
≤ k, r
i
< r
i
′
, be a sequence such that {r
i
|1 ≤ i ≤
k} = I. We denote
freetermseq(Qx p(t
0
,... ,t
τ
)) = t
r
1
,... ,t
r
k
,
freetermseq(p(t
′
0
,... ,t
′
τ
)) = t
′
0
,... ,t
′
τ
.
We further introduce conjunctive normal form
(CNF) in G¨odel logic. In contrast to two-valued logic,
we have to consider an augmented set of literals ap-
pearing in CNF formulae. Let l,φ ∈ Form
L
. l is a lit-
eral of L iff either l = a or l = b → c or l = (a → d) →
d or l = a → e or l = e → a, a ∈ Atom
L
− Tcons
L
,
b ∈ Atom
L
− {0, 1}, c ∈ Atom
L
− {1}, d ∈ (Atom
L
−
Tcons
L
) ∪ {0}, e ∈ QAtom
L
, {b,c} 6⊆ Tcons
L
. The
set of all literals of L is designated as Lit
L
⊆ Form
L
.
φ is a conjunctive | disjunctive normal form of L,
in symbols CNF | DNF, iff either φ ∈ Tcons
L
or
φ =
V
i≤n
W
j≤m
i
l
i
j
| φ =
W
i≤n
V
j≤m
i
l
i
j
, l
i
j
∈ Lit
L
. Let
D = l
1
∨ ··· ∨ l
n
∈ Form
L
, l
i
∈ Lit
L
. We denote
lits(D) = {l
1
,... , l
n
} ⊆ Lit
L
. D is a factor iff, for all
1 ≤ i < i
′
≤ n, l
i
6= l
i
′
.
We finally introduce order clauses in G¨odel logic.
Let l ∈ OrdForm
L
. l is an order literal of L iff
l = ε
1
⋄ ε
2
, ε
i
∈ Atom
L
∪ QAtom
L
, ⋄ ∈ {≖, ≺}. The
set of all order literals of L is designatedas OrdLit
L
⊆
OrdForm
L
. An order clause of L is a finite set of or-
der literals of L; since = is commutative, we iden-
tify, for all ε
1
≖ ε
2
∈ OrdLit
L
, ε
1
≖ ε
2
and ε
2
≖
ε
1
∈ OrdLit
L
with respect to order clauses. An order
clause {l
1
,... , l
n
} is written in the form l
1
∨ · ·· ∨ l
n
.
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
L
. Let l,l
0
,... , l
n
∈ OrdLit
L
and C,C
′
∈ OrdCl
L
. 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
0
∨ · ·· ∨ l
n
∨ C we denote
{l
0
}∪···∪{l
n
}∪C where, for all i, i
′
≤ n, i 6= i
′
, l
i
6∈ C
and l
i
6= l
i
′
. By C ∨ C
′
we denote C ∪C
′
. C is a sub-
clause of C
′
, in symbols C ⊑ 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
L
, T, T
′
⊆ OrdForm
L
, S, S
′
⊆
OrdCl
L
, I be an interpretationfor L, e ∈ S
I
. Note that
FCTA2014-InternationalConferenceonFuzzyComputationTheoryandApplications
40

I |=
e
l if and only if either l = ε
1
≖ ε
2
, kε
1
≖ ε
2
k
I
e
=
1, kε
1
k
I
e
= k ε
2
k
I
e
; or l = ε
1
≺ ε
2
, kε
1
≺ ε
2
k
I
e
= 1,
kε
1
k
I
e
< kε
2
k
I
e
. C is true in I with respect to e,
written as I |=
e
C, iff there exists l
∗
∈ C such that
I |=
e
l
∗
. I is a model of C, in symbols I |= C, iff,
for all e ∈ S
I
, I |=
e
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
′
|T
′
|C
′
|S
′
, iff, for every model I of φ |
T | C | S for L, I |= φ
′
|T
′
|C
′
|S
′
. 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) ∩
C
L
) ⊆ Tcons
L
. Let S ⊆
F
OrdCl
L
. We define the
size of S as |S| =
∑
C∈S
|C|. l is a simplified order lit-
eral of L iff l = ε
1
⋄ ε
2
, {ε
1
,ε
2
} 6⊆ Tcons
L
, {ε
1
,ε
2
} 6⊆
QAtom
L
. The set of all simplified order literals of L
is designated as SimOrdLit
L
⊆ OrdLit
L
. We denote
SimOrdCl
L
= {C|C ∈ OrdCl
L
,C ⊆ SimOrdLit
L
} ⊆
OrdCl
L
. Let
˜
f
0
6∈ Func
L
;
˜
f
0
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
L
=
/
0;
˜
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
A
(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
i
|1 ≤ i ≤ n} ⊆ Var
L
.
A substitution ϑ of L is a mapping ϑ : X −→ Term
L
.
ϑ may be written in the form x
1
/ϑ(x
1
),... , x
n
/ϑ(x
n
).
We denote dom(ϑ) = X ⊆
F
Var
L
and range(ϑ) =
S
x∈X
vars(ϑ(x)) ⊆
F
Var
L
. The set of all substitutions
of L is designated as Subst
L
. Let ϑ,ϑ
′
∈ Subst
L
. ϑ is
a variable renamingof L iff ϑ : dom(ϑ) −→ Var
L
, and
for all x,x
′
∈ dom(ϑ), x 6= x
′
, ϑ(x) 6= ϑ(x
′
). We define
id
L
: Var
L
−→ Var
L
, id
L
(x) = x. Let t ∈ Term
L
. ϑ
is applicable to t iff dom(ϑ) ⊇ vars(t). Let ϑ be ap-
plicable to t. We define the application tϑ ∈ Term
L
of
ϑ to t by recursion on the structure of t in the stan-
dard manner. Let range(ϑ) ⊆ dom(ϑ
′
). We define
the composition of ϑ and ϑ
′
as ϑ ◦ ϑ
′
: dom(ϑ) −→
Term
L
, ϑ◦ ϑ
′
(x) = ϑ(x)ϑ
′
, ϑ ◦ ϑ
′
∈ Subst
L
, dom(ϑ◦
ϑ
′
) = dom(ϑ), range(ϑ ◦ ϑ
′
) = range(ϑ
′
|
range(ϑ)
).
Note that composition of substitutions is associa-
tive. ϑ
′
is a regular extension of ϑ iff dom(ϑ
′
) ⊇
dom(ϑ), ϑ
′
|
dom(ϑ)
= ϑ, ϑ
′
|
dom(ϑ
′
)−dom(ϑ)
is a vari-
able renaming such that range(ϑ
′
|
dom(ϑ
′
)−dom(ϑ)
) ∩
range(ϑ) =
/
0. Let a ∈ Atom
L
. ϑ is applicable to a
iff dom(ϑ) ⊇ vars(a). Let ϑ be applicable to a and
a = p(t
1
,... ,t
τ
). We define the application of ϑ to a as
aϑ = p(t
1
ϑ,.. . ,t
τ
ϑ) ∈ Atom
L
. Let Qxa ∈ QAtom
L
.
ϑ is applicable to Qxa iff dom(ϑ) ⊇ freevars(Qxa)
and x 6∈ range(ϑ|
freevars(Qxa)
). Let ϑ be applicable
to Qxa. We define the application of ϑ to Qxa as
(Qxa)ϑ = Qxa(ϑ|
freevars(Qxa)
∪ x/x) ∈ QAtom
L
. Let
ε
1
⋄ε
2
∈ OrdLit
L
. ϑ is applicable to ε
1
⋄ε
2
iff, for both
i, ϑ is applicable to ε
i
. Let ϑ be applicable to ε
1
⋄ ε
2
.
Then, for both i, ϑ is applicable to ε
i
, dom(ϑ) ⊇
freevars(ε
i
), dom(ϑ) ⊇ freevars(ε
1
) ∪ freevars(ε
2
) =
freevars(ε
1
⋄ ε
2
). We define the application of ϑ to
ε
1
⋄ ε
2
as (ε
1
⋄ ε
2
)ϑ = ε
1
ϑ ⋄ ε
2
ϑ ∈ OrdLit
L
. Let
E ⊆ A, A = Term
L
| A = Atom
L
| A = QAtom
L
|
A = OrdLit
L
. 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) ⊇
S
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
L
. ε
′
is an instance of
ε of L iff there exists ϑ
∗
∈ Subst
L
such that ε
′
=
εϑ
∗
. ε
′
is a variant of ε of L iff there exists a vari-
able renaming ρ
∗
∈ Subst
L
such that ε
′
= ερ
∗
. Let
C ∈ OrdCl
L
and S ⊆ OrdCl
L
. 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
L
(S) = {C|C is an instance of S of L} ⊆ OrdCl
L
and Vrnt
L
(S) = {C|C is a variant of S of L} ⊆
OrdCl
L
.
ϑ is a unifier of L for E iff Eϑ is a singleton
set. Note that there does not exist a unifier for
/
0.
Let θ ∈ Subst
L
. 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
L
such that
ϑ|
freevars(E)
= θ|
freevars(E)
◦ γ
∗
. By mgu
L
(E) ⊆ Subst
L
we denote the set of all most general unifiers of L for
E. Let
E = E
0
,... , E
n
, E
i
⊆ A
i
, either A
i
= Term
L
or A
i
= Atom
L
or A
i
= QAtom
L
or A
i
= OrdLit
L
.
ϑ is applicable to
E iff, for all i ≤ n, ϑ is applica-
ble to E
i
. Let ϑ be applicable to E. Then, for all
i ≤ n, ϑ is applicable to E
i
, dom(ϑ) ⊇ freevars(E
i
),
dom(ϑ) ⊇
S
i≤n
freevars(E
i
) = freevars(
E). We de-
fine the application of ϑ to
E as Eϑ = E
0
ϑ,.. . , E
n
ϑ,
E
i
ϑ ⊆ A
i
. ϑ is a unifier of L for E iff, for all i ≤ n, ϑ
is a unifier of L for E
i
. Note that if there exists i
∗
≤ n
and E
i
∗
=
/
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 forevery unifier ϑ of L for E, there exists
γ
∗
∈ Subst
L
such that ϑ|
freevars(
E)
= θ|
freevars(E)
◦ γ
∗
.
AnOrderHyperresolutionCalculusforGödelLogicwithTruthConstants
41

By mgu
L
(E) ⊆ Subst
L
we denote the set of all most
general unifiers of L for E.
Theorem 3.1 (Unification Theorem). Let
E =
E
0
,... , E
n
, either E
i
⊆
F
Term
L
or E
i
⊆
F
Atom
L
. If
there exists a unifier of L for
E, then there exists
θ
∗
∈ mgu
L
(
E) such that range(θ
∗
|
vars(
E)
) ⊆ vars(
E).
Proof. By induction on kvars(
E)k.
Theorem 3.2 (Extended Unification Theorem). Let
E = E
0
,... , E
n
, either E
i
⊆
F
Term
L
or E
i
⊆
F
Atom
L
or E
i
⊆
F
QAtom
L
or E
i
⊆
F
OrdLit
L
, and
boundvars(
E) ⊆ V ⊆
F
Var
L
. If there exists a unifier
of L for E, then there exists θ
∗
∈ mgu
L
(E) such that
range(θ
∗
|
freevars(
E)
) ∩V =
/
0.
Proof. A straightforward consequence of Theo-
rem 3.1.
3.2 A Formal Treatment
Translation of a formula or a theory to CNF and
clausal form, is based on the following lemma:
Lemma 3.3. Let n
φ
,n
0
∈ N, φ ∈ Form
L
, T ⊆ Form
L
.
(I) There exist either J
φ
=
/
0 or J
φ
= {(n
φ
, j)| j ≤
n
J
φ
}, J
φ
⊆ {(n
φ
, j)| j ∈ N}; a CNF ψ ∈
Form
L∪{ ˜p
| ∈J
φ
}
, S
φ
⊆
F
SimOrdCl
L∪{ ˜p | ∈J
φ
}
such that
(a) kJ
φ
k ≤ 2· |φ|;
(b) there exists an interpretation A for L and
A |= φ if and only if there exists an interpre-
tation A
′
for L ∪ { ˜p | ∈ J
φ
} and A
′
|= ψ,
satisfying A = A
′
|
L
;
(c) there exists an interpretation A for L and
A |= φ if and only if there exists an interpre-
tation A
′
for L ∪ { ˜p | ∈ J
φ
} and A
′
|= S
φ
,
satisfying A = A
′
|
L
;
(d) |ψ| ∈ O(|φ|
2
); the number of all elementary
operations of the translation of φ to ψ, is in
O(|φ|
2
); the time and space complexity of the
translation of φ to ψ, is in O(|φ|
2
· (log(1 +
n
φ
) + log|φ|));
(e) |S
φ
| ∈ O(|φ|
2
); the number of all elementary
operations of the translation of φ to S
φ
, is in
O(|φ|
2
); the time and space complexity of the
translation of φ to S
φ
, is in O(|φ|
2
· (log(1+
n
φ
) + log|φ|));
(f) for all a ∈ qatoms(ψ), there exists
∗
∈ J
φ
and preds(a) = { ˜p
∗
};
(g) for all
∈ J
φ
, there exist a sequence ¯x of vari-
ables of L and ˜p ( ¯x) ∈ atoms(ψ) satisfying,
for all a ∈ atoms(ψ) and preds(a) = { ˜p
},
a = ˜p ( ¯x); if there exists a
∗
∈ qatoms(ψ)
and preds(a
∗
) = { ˜p }, then there exists
Qx ˜p ( ¯x) ∈ qatoms(ψ) satisfying, for all
a ∈ qatoms(ψ) and preds(a) = { ˜p
}, a =
Qx ˜p ( ¯x);
(h) for all a ∈ qatoms(S
φ
), there exists
∗
∈ J
φ
and preds(a) = { ˜p
∗
};
(i) for all
∈ J
φ
, there exist a sequence ¯x of vari-
ables of L and ˜p
( ¯x) ∈ atoms(S
φ
) satisfying,
for all a ∈ atoms(S
φ
) and preds(a) = { ˜p },
a = ˜p
( ¯x); if there exists a
∗
∈ qatoms(S
φ
)
and preds(a
∗
) = { ˜p }, then there exists
Qx ˜p
( ¯x) ∈ qatoms(S
φ
) satisfying, for all
a ∈ qatoms(S
φ
) and preds(a) = { ˜p }, a =
Qx ˜p
( ¯x);
(j) tcons(S
φ
) ⊆ tcons(φ).
(II) There exist J
T
⊆ {(i, j) |i ≥ n
0
} and S
T
⊆
SimOrdCl
L∪{ ˜p
| ∈J
T
}
such that
(a) there exists an interpretation A for L and
A |= T if and only if there exists an interpre-
tation A
′
for L ∪ { ˜p
| ∈ J
T
} and A
′
|= S
T
,
satisfying A = A
′
|
L
;
(b) if T ⊆
F
Form
L
, then J
T
⊆
F
{(i, j)|i ≥ n
0
},
kJ
T
k ≤ 2· |T|; S
T
⊆
F
SimOrdCl
L∪{ ˜p
| ∈J
T
}
,
|S
T
| ∈ O(|T|
2
); the number of all elemen-
tary operations of the translation of T to
S
T
, is in O(|T|
2
); the time and space com-
plexity of the translation of T to S
T
, is in
O(|T|
2
· log(1+ n
0
+ |T|));
(c) for all a ∈ qatoms(S
T
), there exists
∗
∈ J
T
and preds(a) = { ˜p
∗
};
(d) for all
∈ J
T
, there exist a sequence ¯x of vari-
ables of L and ˜p ( ¯x) ∈ atoms(S
T
) satisfying,
for all a ∈ atoms(S
T
) and preds(a) = { ˜p
},
a = ˜p ( ¯x); if there exists a
∗
∈ qatoms(S
T
)
and preds(a
∗
) = { ˜p
}, then there exists
Qx ˜p ( ¯x) ∈ qatoms(S
T
) satisfying, for all
a ∈ qatoms(S
T
) and preds(a) = { ˜p
}, a =
Qx ˜p ( ¯x);
(e) tcons(S
T
) ⊆ tcons(T).
FCTA2014-InternationalConferenceonFuzzyComputationTheoryandApplications
42

Proof. Technical using interpolation.
Let n
θ
∈ N and θ ∈ Form
L
. There exists θ
′
∈
Form
L
such that
(a) θ
′
≡ θ;
(b) |θ
′
| ≤ 2· |θ|; θ
′
can be built up via a pos-
torder traversal of θ with #O(θ) ∈ O(|θ|)
and the time, space complexity in O(|θ| ·
(log(1+ n
θ
) + log|θ|));
(c) θ
′
does not contain ¬;
(d) θ
′
∈ Tcons
L
; or 1 is not a subformula of
θ
′
; for every subformula of θ
′
of the form
ε
1
⋄ ε
2
, ⋄ ∈ {∧,∨}, ε
i
6= 0,1, {ε
1
,ε
2
} 6⊆
Tcons
L
; for every subformula of θ
′
of the
form ε
1
→ ε
2
, ε
1
6= 0,1, ε
2
6= 1, {ε
1
,ε
2
} 6⊆
Tcons
L
;
(e) tcons(θ
′
) ⊆ tcons(θ).
(13)
The proof is by induction on the structure of θ.
Let l ∈ Lit
L
. There existsC ∈ SimOrdCl
L
such
that
(a) for every interpretation A for L, for all e ∈
S
A
, A |=
e
l if and only if A |=
e
C;
(b) |C| ≤ 3 · |l|, C can be built up from l with
#O(l) ∈ O(|l|).
(14)
In Table 1, for every form of l, C is assigned so that
for every interpretation A for L, for all e ∈ S
A
, A |=
e
l
if and only if A |=
e
C.
Let n
θ
∈ N, θ ∈ Form
L
− Tcons
L
, (13c–e)
hold for θ; ¯x be a sequence of variables,
vars(θ) ⊆ vars( ¯x) ⊆ Var
L
;
= (n
θ
, j ) ∈
{(n
θ
, j)| j ∈ N}, ˜p
∈
˜
P, ar( ˜p ) = | ¯x|.
There exist J = {(n
θ
, j)| j
+ 1 ≤ j ≤
n
J
} ⊆ {(n
θ
, j)| j ∈ N}, j
≤ n
J
, 6∈ J;
a CNF ψ
s
∈ Form
L∪{ ˜p
}∪{ ˜p | ∈J}
, S
s
⊆
F
SimOrdCl
L∪{ ˜p
}∪{ ˜p | ∈J}
, s = +, −, such that
for both s,
(15)
Table 1: Translation of l to C, a, b ∈ Atom
L
− Tcons
L
, ¯c ∈
C
L
, d ∈ QAtom
L
.
Case l C |l| |C|
1 a a ≖ 1 |a| |a| + 2 ≤ 3· |l|
2 a → 0 a ≖ 0 |a| + 2 |a| + 2 ≤ 3· |l|
3 ¯c → b ¯c ≺ b ∨ ¯c ≖ b |b| + 2 2· |b|+ 4 ≤ 3· |l|
4 a → ¯c a ≺ ¯c∨ a ≖ ¯c |a| + 2 2· |a| + 4 ≤ 3· |l|
5 a → b a ≺ b∨ a ≖ b |a| +|b| + 1 2· |a|+ 2·|b| + 2≤ 3· |l|
6 (a → 0) → 0 0 ≺ a |a| + 4 |a| + 2 ≤ 3· |l|
7 (a → b) → b b ≺ a ∨b ≖ 1 |a| + 2· |b| +2 |a| + 2· |b| + 3 ≤ 3· |l|
8 a → d a ≺ d ∨ a ≖ d |a| + |d| + 1 2· |a| + 2 · |d| + 2≤ 3 · |l|
9 d → a d ≺ a∨ d ≖ a |a| + |d| + 1 2· |a| + 2 ·|d| + 2≤ 3 ·|l|
(a) kJk ≤ |θ| − 1;
(b) there exists an interpretation A for L ∪
{ ˜p
} and A |= ˜p ( ¯x) → θ ∈ Form
L∪{ ˜p
}
if
and only if there exists an interpretation A
′
for L ∪ { ˜p
} ∪ { ˜p | ∈ J} and A
′
|= ψ
+
,
satisfying A = A
′
|
L∪{ ˜p
}
;
(c) there exists an interpretation A for L ∪
{ ˜p
} and A |= θ → ˜p ( ¯x) ∈ Form
L∪{ ˜p
}
if
and only if there exists an interpretation A
′
for L ∪ { ˜p
} ∪ { ˜p | ∈ J} and A
′
|= ψ
−
,
satisfying A = A
′
|
L∪{ ˜p
}
;
(d) there exists an interpretation A for L ∪
{ ˜p
} and A |= ˜p ( ¯x) → θ ∈ Form
L∪{ ˜p
}
if
and only if there exists an interpretation A
′
for L ∪ { ˜p
} ∪ { ˜p | ∈ J} and A
′
|= S
+
,
satisfying A = A
′
|
L∪{ ˜p
}
;
(e) there exists an interpretation A for L ∪
{ ˜p
} and A |= θ → ˜p ( ¯x) ∈ Form
L∪{ ˜p
}
if
and only if there exists an interpretation A
′
for L ∪ { ˜p
} ∪ { ˜p | ∈ J} and A
′
|= S
−
,
satisfying A = A
′
|
L∪{ ˜p
}
;
(f) |ψ
s
| ≤ 15· |θ| · (1+ | ¯x|), ψ
s
can be built up
from θ and
˜
f
0
( ¯x) via a preorder traversal
of θ with #O(θ,
˜
f
0
( ¯x)) ∈ O(|θ| · (1+ | ¯x|));
(g) |S
s
| ≤ 15· |θ| · (1+ | ¯x|), S
s
can be built up
from θ and
˜
f
0
( ¯x) via a preorder traversal
of θ with #O(θ,
˜
f
0
( ¯x)) ∈ O(|θ| · (1+ | ¯x|));
(h) for all a ∈ qatoms(ψ
s
), there exists
∗
∈ J
and preds(a) = { ˜p
∗
};
(i) for all
∈ { } ∪ J, ˜p ( ¯x) ∈ atoms(ψ
s
)
satisfying, for all a ∈ atoms(ψ
s
)
and preds(a) = { ˜p
}, a = ˜p ( ¯x);
˜p
6∈ preds(qatoms(ψ
s
)), for all ∈ J,
if there exists a
∗
∈ qatoms(ψ
s
) and
preds(a
∗
) = { ˜p
}, then there exists
Qx ˜p
( ¯x) ∈ qatoms(ψ
s
) satisfying, for all
a ∈ qatoms(ψ
s
) and preds(a) = { ˜p
},
a = Qx ˜p ( ¯x);
(j) for all a ∈ qatoms(S
s
), there exists
∗
∈ J
and preds(a) = { ˜p
∗
};
(k) for all
∈ { } ∪ J, ˜p ( ¯x) ∈ atoms(S
s
)
satisfying, for all a ∈ atoms(S
s
)
and preds(a) = { ˜p
}, a = ˜p ( ¯x);
˜p
6∈ preds(qatoms(S
s
)), for all ∈ J,
if there exists a
∗
∈ qatoms(S
s
) and
preds(a
∗
) = { ˜p
}, then there exists
Qx ˜p ( ¯x) ∈ qatoms(S
s
) satisfying, for all
a ∈ qatoms(S
s
) and preds(a) = { ˜p
},
a = Qx ˜p
( ¯x);
(l) tcons(θ) = tcons(ψ
s
) = tcons(S
s
).
The proof is by induction on the structure of θ using
the interpolation rules in Tables 2–5.
(I) By (13) for n
φ
, φ, there exists φ
′
∈ Form
L
AnOrderHyperresolutionCalculusforGödelLogicwithTruthConstants
43

Table 2: Binary interpolation rules for ∧ and ∨.
Case Laws
θ = θ
1
∧ θ
2
Positive interpolation
˜p
( ¯x) → θ
1
∧ θ
2
( ˜p ( ¯x) → ˜p
1
( ¯x))∧ ( ˜p
( ¯x) → ˜p
2
( ¯x)) ∧ ( ˜p
1
( ¯x) → θ
1
) ∧ ( ˜p
2
( ¯x) → θ
2
)
(6) (16)
|Consequent| = 9+ 4 · | ¯x| + | ˜p
1
( ¯x) → θ
1
| + | ˜p
2
( ¯x) → θ
2
| ≤ 13· (1+ | ¯x|)+ | ˜p
1
( ¯x) → θ
1
| + | ˜p
2
( ¯x) → θ
2
|
Positive interpolation
˜p
( ¯x) → θ
1
∧ θ
2
{ ˜p ( ¯x) ≺ ˜p
1
( ¯x) ∨ ˜p
( ¯x) ≖ ˜p
1
( ¯x), ˜p
( ¯x) ≺ ˜p
2
( ¯x) ∨ ˜p
( ¯x) ≖ ˜p
2
( ¯x), ˜p
1
( ¯x) → θ
1
, ˜p
2
( ¯x) → θ
2
}
(17)
|Consequent| = 12+ 8· | ¯x| + | ˜p
1
( ¯x) → θ
1
| + | ˜p
2
( ¯x) → θ
2
| ≤ 15· (1+ | ¯x|)+ | ˜p
1
( ¯x) → θ
1
| + | ˜p
2
( ¯x) → θ
2
|
Negative interpolation
θ
1
∧ θ
2
→ ˜p
( ¯x)
( ˜p
1
( ¯x) → ˜p
( ¯x) ∨ ˜p
2
( ¯x) → ˜p
( ¯x)) ∧ (θ
1
→ ˜p
1
( ¯x))∧ (θ
2
→ ˜p
2
( ¯x))
(8) (18)
|Consequent| = 9+ 4 · | ¯x| + |θ
1
→ ˜p
1
( ¯x)|+ |θ
2
→ ˜p
2
( ¯x)| ≤ 13· (1+ | ¯x|) + |θ
1
→ ˜p
1
( ¯x)|+ |θ
2
→ ˜p
2
( ¯x)|
Negative interpolation
θ
1
∧ θ
2
→ ˜p
( ¯x)
{ ˜p
1
( ¯x) ≺ ˜p
( ¯x)∨ ˜p
1
( ¯x) ≖ ˜p
( ¯x), ˜p
2
( ¯x) ≺ ˜p
( ¯x) ∨ ˜p
2
( ¯x) ≖ ˜p
( ¯x),θ
1
→ ˜p
1
( ¯x), θ
2
→ ˜p
2
( ¯x)}
(19)
|Consequent| = 12+ 8· | ¯x| + |θ
1
→ ˜p
1
( ¯x)|+ |θ
2
→ ˜p
2
( ¯x)| ≤ 15· (1+ | ¯x|) + |θ
1
→ ˜p
1
( ¯x)|+ |θ
2
→ ˜p
2
( ¯x)|
θ = θ
1
∨ θ
2
Positive interpolation
˜p
( ¯x) → (θ
1
∨ θ
2
)
( ˜p ( ¯x) → ˜p
1
( ¯x)∨ ˜p
( ¯x) → ˜p
2
( ¯x))∧ ( ˜p
1
( ¯x) → θ
1
) ∧ ( ˜p
2
( ¯x) → θ
2
)
(5) (20)
|Consequent| = 9+ 4 · | ¯x| + | ˜p
1
( ¯x) → θ
1
| + | ˜p
2
( ¯x) → θ
2
| ≤ 13· (1+ | ¯x|)+ | ˜p
1
( ¯x) → θ
1
| + | ˜p
2
( ¯x) → θ
2
|
Positive interpolation
˜p
( ¯x) → (θ
1
∨ θ
2
)
{ ˜p ( ¯x) ≺ ˜p
1
( ¯x) ∨ ˜p
( ¯x) ≖ ˜p
1
( ¯x)∨ ˜p
( ¯x) ≺ ˜p
2
( ¯x)∨ ˜p
( ¯x) ≖ ˜p
2
( ¯x), ˜p
1
( ¯x) → θ
1
, ˜p
2
( ¯x) → θ
2
}
(21)
|Consequent| = 12+ 8· | ¯x| + | ˜p
1
( ¯x) → θ
1
| + | ˜p
2
( ¯x) → θ
2
| ≤ 15· (1+ | ¯x|)+ | ˜p
1
( ¯x) → θ
1
| + | ˜p
2
( ¯x) → θ
2
|
Negative interpolation
(θ
1
∨ θ
2
) → ˜p
( ¯x)
( ˜p
1
( ¯x) → ˜p
( ¯x)) ∧ ( ˜p
2
( ¯x) → ˜p
( ¯x))∧ (θ
1
→ ˜p
1
( ¯x)) ∧ (θ
2
→ ˜p
2
( ¯x))
(7) (22)
|Consequent| = 9+ 4 · | ¯x| + |θ
1
→ ˜p
1
( ¯x)|+ |θ
2
→ ˜p
2
( ¯x)| ≤ 13· (1+ | ¯x|) + |θ
1
→ ˜p
1
( ¯x)|+ |θ
2
→ ˜p
2
( ¯x)|
Negative interpolation
(θ
1
∨ θ
2
) → ˜p
( ¯x)
{ ˜p
1
( ¯x) ≺ ˜p
( ¯x)∨ ˜p
1
( ¯x) ≖ ˜p
( ¯x), ˜p
2
( ¯x) ≺ ˜p
( ¯x) ∨ ˜p
2
( ¯x) ≖ ˜p
( ¯x),θ
1
→ ˜p
1
( ¯x), θ
2
→ ˜p
2
( ¯x)}
(23)
|Consequent| = 12+ 8· | ¯x| + |θ
1
→ ˜p
1
( ¯x)|+ |θ
2
→ ˜p
2
( ¯x)| ≤ 15· (1+ | ¯x|) + |θ
1
→ ˜p
1
( ¯x)|+ |θ
2
→ ˜p
2
( ¯x)|
such that (13a–e) hold for n
φ
, φ, φ
′
. We distin-
guish three cases for φ
′
. Case 1: φ
′
∈ Tcons
L
−
{1}. We put J
φ
=
/
0 ⊆ {(n
φ
, j)| j ∈ N}, ψ = φ
′
∈
Form
L
, S
φ
= {} ⊆
F
SimOrdCl
L
. Case 2: φ
′
= 1.
We put J
φ
=
/
0 ⊆ {(n
φ
, j)| j ∈ N}, ψ = 1 ∈ Form
L
,
S
φ
=
/
0 ⊆
F
SimOrdCl
L
. Case 3: φ
′
6∈ Tcons
L
. We
put ¯x = varseq(φ
′
), j
= 0, = (n
φ
, j ), ar( ˜p ) =
| ¯x|. We get by (15) for n
φ
, φ
′
, ¯x,
, ˜p that there
exist J = {(n
φ
, j)|1 ≤ j ≤ n
J
} ⊆ {(n
φ
, j)| j ∈ N},
j
≤ n
J
, 6∈ J, a CNF ψ
+
∈ Form
L∪{ ˜p
}∪{ ˜p | ∈J}
,
S
+
⊆
F
SimOrdCl
L∪{ ˜p
}∪{ ˜p | ∈J}
, and (15a,b,d,f–l)
hold for φ
′
, ¯x, ˜p
, J, ψ
+
, S
+
. We put n
J
φ
= n
J
,
J
φ
= {(n
φ
, j)| j ≤ n
J
φ
} ⊆ {(n
φ
, j)| j ∈ N}, ψ = ˜p
( ¯x)∧
ψ
+
∈ Form
L∪{ ˜p
| ∈J
φ
}
, S
φ
= { ˜p
( ¯x) ≖ 1} ∪ S
+
⊆
F
SimOrdCl
L∪{ ˜p
| ∈J
φ
}
. (II) straightforwardly follows
from (I).
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. Let
P ⊆
˜
P and S ⊆ OrdCl
L∪P
. S is admissible iff
(a) for all a ∈ qatoms(S), preds(a) ⊆ P;
(b) for all ˜p ∈ P, there exist a sequence ¯x of vari-
FCTA2014-InternationalConferenceonFuzzyComputationTheoryandApplications
44

Table 3: Binary interpolation rules for →.
Case Laws
θ = θ
1
→ θ
2
,θ
2
6= 0
Positive interpolation
˜p
( ¯x) → (θ
1
→ θ
2
)
( ˜p ( ¯x) → ˜p
2
( ¯x)∨ ˜p
1
( ¯x) → ˜p
2
( ¯x)) ∧ (θ
1
→ ˜p
1
( ¯x))∧ ( ˜p
2
( ¯x) → θ
2
)
(9), (8) (24)
|Consequent| = 9+ 4· | ¯x| + |θ
1
→ ˜p
1
( ¯x)| +| ˜p
2
( ¯x) → θ
2
| ≤ 13· (1 + | ¯x|) + |θ
1
→ ˜p
1
( ¯x)|+ | ˜p
2
( ¯x) → θ
2
|
Positive interpolation
˜p
( ¯x) → (θ
1
→ θ
2
)
{ ˜p ( ¯x) ≺ ˜p
2
( ¯x)∨ ˜p
( ¯x) ≖ ˜p
2
( ¯x)∨ ˜p
1
( ¯x) ≺ ˜p
2
( ¯x) ∨ ˜p
1
( ¯x) ≖ ˜p
2
( ¯x),θ
1
→ ˜p
1
( ¯x), ˜p
2
( ¯x) → θ
2
}
(25)
|Consequent| = 12+ 8· | ¯x| + |θ
1
→ ˜p
1
( ¯x)| +| ˜p
2
( ¯x) → θ
2
| ≤ 15· (1 + | ¯x|) + |θ
1
→ ˜p
1
( ¯x)| +| ˜p
2
( ¯x) → θ
2
|
Negative interpolation
(θ
1
→ θ
2
) → ˜p
( ¯x)
(( ˜p
1
( ¯x) → ˜p
2
( ¯x)) → ˜p
2
( ¯x) ∨ ˜p
( ¯x)) ∧ ( ˜p
2
( ¯x) → ˜p
( ¯x))∧ ( ˜p
1
( ¯x) → θ
1
) ∧ (θ
2
→ ˜p
2
( ¯x))
(11), (3), (1) (26)
|Consequent| = 13+ 6· | ¯x| + | ˜p
1
( ¯x) → θ
1
| + |θ
2
→ ˜p
2
( ¯x)| ≤ 13· (1+ | ¯x|) +| ˜p
1
( ¯x) → θ
1
| + |θ
2
→ ˜p
2
( ¯x)|
Negative interpolation
(θ
1
→ θ
2
) → ˜p
( ¯x)
{ ˜p
2
( ¯x) ≺ ˜p
1
( ¯x) ∨ ˜p
2
( ¯x) ≖ 1∨ ˜p
( ¯x) ≖ 1, ˜p
2
( ¯x) ≺ ˜p
( ¯x) ∨ ˜p
2
( ¯x) ≖ ˜p
( ¯x), ˜p
1
( ¯x) → θ
1
,θ
2
→ ˜p
2
( ¯x)}
(27)
|Consequent| = 15+ 8· | ¯x| + | ˜p
1
( ¯x) → θ
1
| + |θ
2
→ ˜p
2
( ¯x)| ≤ 15· (1+ | ¯x|) +| ˜p
1
( ¯x) → θ
1
| + |θ
2
→ ˜p
2
( ¯x)|
Table 4: Unary interpolation rules for →.
Case Laws
θ = θ
1
→ 0
Positive interpolation
˜p
( ¯x) → (θ
1
→ 0)
( ˜p ( ¯x) → 0∨ ˜p
1
( ¯x) → 0) ∧ (θ
1
→ ˜p
1
( ¯x))
(9), (8) (28)
|Consequent| = 8+ 2· | ¯x| + |θ
1
→ ˜p
1
( ¯x)| ≤ 13· (1+ | ¯x|) +|θ
1
→ ˜p
1
( ¯x)|
Positive interpolation
˜p
( ¯x) → (θ
1
→ 0)
{ ˜p ( ¯x) ≖ 0 ∨ ˜p
1
( ¯x) ≖ 0,θ
1
→ ˜p
1
( ¯x)}
(29)
|Consequent| = 6+ 2· | ¯x| + |θ
1
→ ˜p
1
( ¯x)| ≤ 15· (1+ | ¯x|) +|θ
1
→ ˜p
1
( ¯x)|
Negative interpolation
(θ
1
→ 0) → ˜p
( ¯x)
(( ˜p
1
( ¯x) → 0) → 0∨ ˜p
( ¯x)) ∧ ( ˜p
1
( ¯x) → θ
1
)
(11) (30)
|Consequent| = 8+ 2· | ¯x| + | ˜p
1
( ¯x) → θ
1
| ≤ 13· (1 + | ¯x|) + | ˜p
1
( ¯x) → θ
1
|
Negative interpolation
(θ
1
→ 0) → ˜p
( ¯x)
{0 ≺ ˜p
1
( ¯x) ∨ ˜p
( ¯x) ≖ 1, ˜p
1
( ¯x) → θ
1
}
(31)
|Consequent| = 6+ 2· | ¯x| + | ˜p
1
( ¯x) → θ
1
| ≤ 15· (1 + | ¯x|) + | ˜p
1
( ¯x) → θ
1
|
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 Qxa,Q
′
x
′
a
′
∈ qatoms(S),
if preds(a) = preds(a
′
), then Q = Q
′
, x = x
′
,
boundindset(Qxa) = boundindset(Q
′
x
′
a
′
).
Theorem 3.4. Let n
0
∈ N, φ ∈ Form
L
, T ⊆
Form
L
. There exist J
φ
T
⊆ {(i, j)|i ≥ n
0
} and S
φ
T
⊆
SimOrdCl
L∪{ ˜p
| ∈J
φ
T
}
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
φ
T
} and A
′
|= S
φ
T
, sat-
isfying A = A
′
|
L
;
(ii) if T ⊆
F
Form
L
, then J
φ
T
⊆
F
{(i, j)|i ≥
n
0
}, kJ
φ
T
k ∈ O(|T| + |φ|); S
φ
T
⊆
F
SimOrdCl
L∪{ ˜p
| ∈J
φ
T
}
, |S
φ
T
| ∈ O(|T|
2
+ |φ|
2
);
the number of all elementary operations of the
translation of T and φ to S
φ
T
, is in O(|T|
2
+|φ|
2
);
the time and space complexity of the translation
of T and φ to S
φ
T
, is in O(|T|
2
· log(1 + n
0
+
|T|) + |φ|
2
· (log(1+ n
0
) + log|φ|));
(iii) S
φ
T
is admissible;
(iv) tcons(S
φ
T
) ⊆ tcons(φ) ∪ tcons(T).
Proof. We get by Lemma 3.3(II) for n
0
+ 1, T
that there exist J
T
⊆ {(i, j) |i ≥ n
0
+ 1}, S
T
⊆
SimOrdCl
L∪{ ˜p
| ∈J
T
}
, and 3.3(II a–e) hold for n
0
+ 1,
T, J
T
, S
T
. By (13) for n
0
, φ, there exists φ
′
∈ Form
L
such that (13a–e) hold for n
0
, φ, φ
′
. We distinguish
three cases for φ
′
. Case 1: φ
′
∈ Tcons
L
− {1}. We
AnOrderHyperresolutionCalculusforGödelLogicwithTruthConstants
45

Table 5: Unary interpolation rules for ∀ and ∃.
Case
∀xθ
1
Positive interpolation
˜p
( ¯x) → ∀xθ
1
( ˜p ( ¯x) → ∀x ˜p
1
( ¯x))∧ ( ˜p
1
( ¯x) → θ
1
)
(32)
|Consequent| = 6+ 2· | ¯x| + | ˜p
1
( ¯x) → θ
1
| ≤ 13· (1 + | ¯x|) + | ˜p
1
( ¯x) → θ
1
|
Positive interpolation
˜p
( ¯x) → ∀xθ
1
{ ˜p ( ¯x) ≺ ∀x ˜p
1
( ¯x) ∨ ˜p
( ¯x) ≖ ∀x ˜p
1
( ¯x), ˜p
1
( ¯x) → θ
1
}
(33)
|Consequent| = 10+ 4· | ¯x| + | ˜p
1
( ¯x) → θ
1
| ≤ 15· (1 + | ¯x|) + | ˜p
1
( ¯x) → θ
1
|
Negative interpolation
∀xθ
1
→ ˜p
( ¯x)
(∀x ˜p
1
( ¯x) → ˜p
( ¯x)) ∧ (θ
1
→ ˜p
1
( ¯x))
(34)
|Consequent| = 6+ 2· | ¯x| + |θ
1
→ ˜p
1
( ¯x)| ≤ 13· (1+ | ¯x|) +|θ
1
→ ˜p
1
( ¯x)|
Negative interpolation
∀xθ
1
→ ˜p
( ¯x)
{∀x ˜p
1
( ¯x) ≺ ˜p
( ¯x)∨ ∀x ˜p
1
( ¯x) ≖ ˜p
( ¯x),θ
1
→ ˜p
1
( ¯x)}
(35)
|Consequent| = 10+ 4· | ¯x| + |θ
1
→ ˜p
1
( ¯x)| ≤ 15· (1+ | ¯x|) +|θ
1
→ ˜p
1
( ¯x)|
∃xθ
1
Positive interpolation
˜p
( ¯x) → ∃xθ
1
( ˜p ( ¯x) → ∃x ˜p
1
( ¯x))∧ ( ˜p
1
( ¯x) → θ
1
)
(36)
|Consequent| = 6+ 2· | ¯x| + | ˜p
1
( ¯x) → θ
1
| ≤ 13· (1 + | ¯x|) + | ˜p
1
( ¯x) → θ
1
|
Positive interpolation
˜p
( ¯x) → ∃xθ
1
{ ˜p ( ¯x) ≺ ∃x ˜p
1
( ¯x) ∨ ˜p
( ¯x) ≖ ∃x ˜p
1
( ¯x), ˜p
1
( ¯x) → θ
1
}
(37)
|Consequent| = 10+ 4· | ¯x| + | ˜p
1
( ¯x) → θ
1
| ≤ 15· (1 + | ¯x|) + | ˜p
1
( ¯x) → θ
1
|
Negative interpolation
∃xθ
1
→ ˜p
( ¯x)
(∃x ˜p
1
( ¯x) → ˜p
( ¯x)) ∧ (θ
1
→ ˜p
1
( ¯x))
(38)
|Consequent| = 6+ 2· | ¯x| + |θ
1
→ ˜p
1
( ¯x)| ≤ 13· (1+ | ¯x|) +|θ
1
→ ˜p
1
( ¯x)|
Negative interpolation
∃xθ
1
→ ˜p
( ¯x)
{∃x ˜p
1
( ¯x) ≺ ˜p
( ¯x)∨ ∃x ˜p
1
( ¯x) ≖ ˜p
( ¯x),θ
1
→ ˜p
1
( ¯x)}
(39)
|Consequent| = 10+ 4· | ¯x| + |θ
1
→ ˜p
1
( ¯x)| ≤ 15· (1+ | ¯x|) +|θ
1
→ ˜p
1
( ¯x)|
put J
φ
T
= J
T
⊆ { (i, j) |i ≥ n
0
+ 1} ⊆ {(i, j) |i ≥ n
0
}
and S
φ
T
= S
T
⊆ SimOrdCl
L∪{ ˜p
| ∈J
φ
T
}
. Case 2: φ
′
= 1.
We put J
φ
T
=
/
0 ⊆ {(i, j)|i ≥ n
0
} and S
φ
T
= {} ⊆
SimOrdCl
L
. Case 3: φ
′
6∈ Tcons
L
. We put ¯x =
varseq(φ
′
), j
= 0, = (n
0
, j ), ar( ˜p ) = | ¯x|. We
get by (15) for n
0
, ∀ ¯xφ
′
, ¯x, , ˜p that there exist J =
{(n
0
, j)|1 ≤ j ≤ n
J
} ⊆ {(n
0
, j)| j ∈ N}, j
≤ n
J
, 6∈ J,
S
−
⊆
F
SimOrdCl
L∪{ ˜p
}∪{ ˜p | ∈J}
, and (15e,g,j–l)hold
for ∀ ¯xφ
′
, ¯x, ˜p
, J, S
−
. We put J
φ
T
= J
T
∪ {
} ∪ J ⊆
{(i, j)|i ≥ n
0
} and S
φ
T
= S
T
∪ { ˜p
( ¯x) ≺ 1} ∪ S
−
⊆
SimOrdCl
L∪{ ˜p
| ∈J
φ
T
}
.
Corollary 3.5. Let n
0
∈ N, φ ∈ Form
L
, T ⊆
Form
L
. There exist J
φ
T
⊆ {(i, j)|i ≥ n
0
} and S
φ
T
⊆
SimOrdCl
L∪{ ˜p
| ∈J
φ
T
}
such that
(i) T |= φ if and only if S
φ
T
is unsatisfiable;
(ii) if T ⊆
F
Form
L
, then J
φ
T
⊆
F
{(i, j)|i ≥
n
0
}, kJ
φ
T
k ∈ O(|T| + |φ|); S
φ
T
⊆
F
SimOrdCl
L∪{ ˜p
| ∈J
φ
T
}
, |S
φ
T
| ∈ O(|T|
2
+ |φ|
2
);
the number of all elementary operations of the
translation of T and φ to S
φ
T
, is in O(|T|
2
+|φ|
2
);
the time and space complexity of the translation
of T and φ to S
φ
T
, is in O(|T|
2
· log(1 + n
0
+
|T|) + |φ|
2
· (log(1+ n
0
) + log|φ|));
(iii) S
φ
T
is admissible;
(iv) tcons(S
φ
T
) ⊆ tcons(φ) ∪ tcons(T).
Proof. A straightforward consequence of Theo-
rem 3.4.
4 ORDER HYPERRESOLUTION
RULES
At first, we introduce some basic notions and notation
concerningchains of order literals. A chain Ξ of L is a
sequence Ξ = ε
0
⋄
0
υ
0
,... , ε
n
⋄
n
υ
n
, ε
i
⋄
i
υ
i
∈ OrdLit
L
,
such that for all i < n, υ
i
= ε
i+1
. ε
0
is the beginning
element of Ξ and υ
n
the ending element of Ξ. ε
0
Ξυ
n
denotes Ξ together with its respective beginning and
ending element. Let Ξ = ε
0
⋄
0
υ
0
,... ,ε
n
⋄
n
υ
n
be a
chain of L. Ξ is an equality chain of L iff, for all
i ≤ n, ⋄
i
=≖. Ξ is an increasing chain of L iff there
exists i
∗
≤ n such that ⋄
i
∗
=≺. Ξ is a contradiction
of L iff Ξ is an increasing chain of L of the form
ε
0
Ξ0 or 1Ξυ
n
or ε
0
Ξε
0
. Let S ⊆ OrdCl
L
be unit
and Ξ = ε
0
⋄
0
υ
0
,... , ε
n
⋄
n
υ
n
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
i
⋄
i
υ
i
∈ S.
Let
˜
W = { ˜w
| ∈ I} such that
˜
W ∩ (Func
L
∪
{
˜
f
0
}) =
/
0;
˜
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
L
= {C|C ∈
OrdCl
L
is closed} ⊆ OrdCl
L
, GInst
L
(S) = {C|C ∈
GOrdCl
L
is an instance of S of L} ⊆ GOrdCl
L
,
ordtcons(S) = { ¯c
1
≺ ¯c
2
| ¯c
1
, ¯c
2
∈ tcons(S),c
1
< c
2
} ⊆
GOrdCl
L
. A basic order hyperresolution calculus is
defined in Table 6. The basic order hyperresolution
calculus can be generalised to an order hyperresolu-
tion one in Table 7. Let L
0
= L ∪ P, a reduct of
L ∪
˜
W ∪ P, and S
0
=
/
0 ⊆ GOrdCl
L
0
| OrdCl
L
0
. Let
FCTA2014-InternationalConferenceonFuzzyComputationTheoryandApplications
46

Table 6: Basic order hyperresolution rules.
(Basic order hyperresolution rule) (40)
l
0
∨C
0
,. . ., l
n
∨C
n
∈ S
κ−1
n
_
i=0
C
i
∈ S
κ
;
l
0
,. . ., l
n
is a contradiction of L
κ−1
.
W
n
i=0
C
i
is a basic order hyperresolvent of l
0
∨C
0
,. . ., l
n
∨C
n
.
(Basic order trichotomy rule) (41)
a, b ∈ atoms(S
κ−1
),{a,b} 6⊆ Tcons
L
,qatoms(S) 6=
/
0
a ≺ b∨ a ≖ b∨ b ≺ a ∈ S
κ
.
a ≺ b∨ a ≖ b∨ b ≺ a is a basic order trichotomy resolvent of a and b.
(Basic order ∀-quantification rule) (42)
∀xa ∈ qatoms
∀
(S
κ−1
)
∀xa ≺ aγ∨ ∀xa ≖ aγ ∈ S
κ
;
t ∈ GTerm
L
κ−1
,γ = x/t ∈ Subst
L
κ−1
,dom(γ) = {x} = vars(a).
∀xa ≺ aγ∨ ∀xa ≖ aγ is a basic order ∀-quantification resolvent of ∀xa.
(Basic order ∃-quantification rule) (43)
∃xa ∈ qatoms
∃
(S
κ−1
)
aγ ≺ ∃xa∨ aγ ≖ ∃xa ∈ S
κ
;
t ∈ GTerm
L
κ−1
,γ = x/t ∈ Subst
L
κ−1
,dom(γ) = {x} = vars(a).
aγ ≺ ∃xa∨ aγ ≖ ∃xa is a basic order ∃-quantification resolvent of ∃xa.
(Basic order ∀-witnessing rule) (44)
∀xa ∈ qatoms
∀
(S
κ−1
),b ∈ atoms(S
κ−1
) ∪ qatoms(S
κ−1
)
aγ ≺ b∨ b ≖ ∀xa∨ b ≺ ∀xa ∈ S
κ
;
˜w ∈
˜
W − Func
L
κ−1
,ar( ˜w) = |freetermseq(∀xa), freetermseq(b)|,γ = x/ ˜w(freetermseq(∀xa), freetermseq(b)) ∈ Subst
L
κ
,dom(γ) = {x} = vars(a).
aγ ≺ b∨ b ≖ ∀xa∨ b ≺ ∀xa is a basic order ∀-witnessing resolvent of ∀xa and b.
(Basic order ∃-witnessing rule) (45)
∃xa ∈ qatoms
∃
(S
κ−1
),b ∈ atoms(S
κ−1
) ∪ qatoms(S
κ−1
)
b ≺ aγ∨ ∃xa ≖ b ∨∃xa ≺ b ∈ S
κ
;
˜w ∈
˜
W − Func
L
κ−1
,ar( ˜w) = |freetermseq(∃xa), freetermseq(b)|,γ = x/ ˜w(freetermseq(∃xa), freetermseq(b)) ∈ Subst
L
κ
,dom(γ) = {x} = vars(a).
b ≺ aγ∨ ∃xa ≖ b ∨∃xa ≺ b is a basic order ∃-witnessing resolvent of ∃xa and b.
D = C
1
,... ,C
n
, C
κ
∈ GOrdCl
L∪
˜
W∪P
| OrdCl
L∪
˜
W∪P
,
n ≥ 1. D is a deduction of C
n
from S by basic
order hyperresolution iff, for all 1 ≤ κ ≤ n, C
κ
∈
ordtcons(S) ∪ GInst
L
κ−1
(S), or there exist 1 ≤ j
∗
k
≤
κ − 1, k = 1,...,m, such that C
κ
is a basic order re-
solvent of C
j
∗
1
,... ,C
j
∗
m
∈ S
κ−1
using Rule (40)–(45)
with respect to L
κ−1
and S
κ−1
; D is a deduction ofC
n
from S by order hyperresolution iff, for all 1 ≤ κ ≤ n,
C
κ
∈ ordtcons(S) ∪ S, or there exist 1 ≤ j
∗
k
≤ κ − 1,
k = 1,...,m, such that C
κ
is an order resolvent of
C
′
j
∗
1
,... ,C
′
j
∗
m
∈ S
Vr
κ−1
using Rule (46)–(51) with respect
to L
κ−1
and S
κ−1
where C
′
j
∗
k
is a variant of C
j
∗
k
∈ S
κ−1
of L
κ−1
; L
κ
and S
κ
are defined by recursion on 1 ≤
κ ≤ n as follows:
L
κ
=



L
κ−1
∪ { ˜w} in case of Rule (44),(45) |
(50),(51),
L
κ−1
else;
S
κ
= S
κ−1
∪ {C
κ
} ⊆ GOrdCl
L
κ
| OrdCl
L
κ
,
S
Vr
κ
= Vrnt
L
κ
(S
κ
) ⊆ OrdCl
L
κ
.
D is a refutation of S iff C
n
= . We denote
clo
BH
(S) = {C|there exists a deduction of C from S
by basic order hyperresolution}
⊆ GOrdCl
L∪
˜
W∪P
,
clo
H
(S) = {C|there exists a deduction of C from S
by order hyperresolution}
⊆ OrdCl
L∪
˜
W∪P
.
Lemma 4.1 (Lifting Lemma). Let L contain a con-
stant symbol. Let P ⊆
˜
P and S ⊆ OrdCl
L∪P
. Let
C ∈ clo
BH
(S). There exists C
∗
∈ clo
H
(S) such that
C is an instance of C
∗
of L ∪
˜
W ∪ P.
Proof. Straightforward.
Lemma 4.2 (Reduction Lemma). Let L contain a
constant symbol. Let P ⊆
˜
P and S ⊆ OrdCl
L∪P
.
Let {
W
k
i
j=0
ε
i
j
⋄
i
j
υ
i
j
∨C
i
|i ≤ n} ⊆ clo
BH
(S) such that
for all S ∈ S el({{ j| j ≤ k
i
}
i
|i ≤ n} ), there ex-
ists a contradiction of {ε
i
S (i)
⋄
i
S (i)
υ
i
S (i)
|i ≤ n} ⊆
AnOrderHyperresolutionCalculusforGödelLogicwithTruthConstants
47

Table 7: Order hyperresolution rules.
(Order hyperresolution rule) (46)
k
0
_
j=0
ε
0
j
⋄
0
j
υ
0
j
∨
m
0
_
j=1
l
0
j
,. . .,
k
n
_
j=0
ε
n
j
⋄
n
j
υ
n
j
∨
m
n
_
j=1
l
n
j
∈ S
Vr
κ−1

n
_
i=0
m
i
_
j=1
l
i
j

θ ∈ S
κ
;
for all i < i
′
≤ n, freevars(
W
k
i
j=0
ε
i
j
⋄
i
j
υ
i
j
∨
W
m
i
j=1
l
i
j
) ∩ freevars(
W
k
i
′
j=0
ε
i
′
j
⋄
i
′
j
υ
i
′
j
∨
W
m
i
′
j=1
l
i
′
j
) =
/
0,
θ ∈ mgu
L
κ−1

W
k
0
j=0
ε
0
j
⋄
0
j
υ
0
j
,l
0
1
,. . ., l
0
m
0
,. . .,
W
k
n
j=0
ε
n
j
⋄
n
j
υ
n
j
,l
n
1
,. . ., l
n
m
n
,{υ
0
0
,ε
1
0
}, . . ., {υ
n−1
0
,ε
n
0
}, {a, b}

,
dom(θ) = freevars

{ε
i
j
⋄
i
j
υ
i
j
| j ≤ k
i
,i ≤ n},{l
i
j
|1 ≤ j ≤ m
i
,i ≤ n}

,a = ε
0
0
,b = 1 or a = υ
n
0
,b = 0 or a = ε
0
0
,b = υ
n
0
, there exists i
∗
≤ n such that ⋄
i
∗
0
=≺ .

W
n
i=0
W
m
i
j=1
l
i
j

θ is an order hyperresolvent of
W
k
0
j=0
ε
0
j
⋄
0
j
υ
0
j
∨
W
m
0
j=1
l
0
j
,. . .,
W
k
n
j=0
ε
n
j
⋄
n
j
υ
n
j
∨
W
m
n
j=1
l
n
j
.
(Order trichotomy rule) (47)
a, b ∈ atoms(S
Vr
κ−1
),{a,b} 6⊆ Tcons
L
,qatoms(S) 6=
/
0
a ≺ b∨ a ≖ b∨ b ≺ a ∈ S
κ
;
vars(a)∩ vars(b) =
/
0.
a ≺ b∨ a ≖ b∨ b ≺ a is an order trichotomy resolvent of a and b.
(Order ∀-quantification rule) (48)
∀xa ∈ qatoms
∀
(S
κ−1
)
∀xa ≺ a∨ ∀xa ≖ a ∈ S
κ
.
∀xa ≺ a∨ ∀xa ≖ a is an order ∀-quantification resolvent of ∀xa.
(Order ∃-quantification rule) (49)
∃xa ∈ qatoms
∃
(S
κ−1
)
a ≺ ∃xa∨ a ≖ ∃xa ∈ S
κ
.
a ≺ ∃xa∨ a ≖ ∃xa is an order ∃-quantification resolvent of ∃xa.
(Order ∀-witnessing rule) (50)
∀xa ∈ qatoms
∀
(S
Vr
κ−1
),b ∈ atoms(S
Vr
κ−1
) ∪ qatoms(S
Vr
κ−1
)
aγ ≺ b∨ b ≖ ∀xa ∨b ≺ ∀xa ∈ S
κ
;
freevars(∀xa)∩ freevars(b) =
/
0, ˜w ∈
˜
W − Func
L
κ−1
,ar( ˜w) = |freetermseq(∀xa), freetermseq(b)|,
γ = x/ ˜w(freetermseq(∀xa), freetermseq(b))∪ id|
vars(a)−{x}
∈ Subst
L
κ
,dom(γ) = {x} ∪ (vars(a) − {x}) = vars(a).
aγ ≺ b∨ b ≖ ∀xa∨ b ≺ ∀xa is an order ∀-witnessing resolvent of ∀xa and b.
(Order ∃-witnessing rule) (51)
∃xa ∈ qatoms
∃
(S
Vr
κ−1
),b ∈ atoms(S
Vr
κ−1
) ∪ qatoms(S
Vr
κ−1
)
b ≺ aγ∨ ∃xa ≖ b∨ ∃ xa ≺ b ∈ S
κ
;
freevars(∃xa)∩ freevars(b) =
/
0, ˜w ∈
˜
W − Func
L
κ−1
,ar( ˜w) = |freetermseq(∃xa), freetermseq(b)|,
γ = x/ ˜w(freetermseq(∃xa), freetermseq(b))∪ id|
vars(a)−{x}
∈ Subst
L
κ
,dom(γ) = {x} ∪ (vars(a) − {x}) = vars(a).
b ≺ aγ∨ ∃xa ≖ b ∨∃xa ≺ b is an order ∃-witnessing resolvent of ∃xa and b.
GOrdCl
L∪
˜
W∪P
. There exists
/
0 6= I
∗
⊆ {i|i ≤ n} such
that
W
i∈I
∗
C
i
∈ clo
BH
(S).
Proof. Straightforward.
Lemma 4.3 (Unit Lemma). Let L contain a constant
symbol. Let P ⊆
˜
P and S ⊆ OrdCl
L∪P
. Let  6∈
clo
BH
(S) = {
W
k
ι
j=0
ε
ι
j
⋄
ι
j
υ
ι
j
|ι < γ}, γ ≤ ω. There exists
S
∗
∈ Sel({{ j| j ≤ k
ι
}
ι
|ι < γ}) such that there does
not exist a contradiction of {ε
ι
S
∗
(ι)
⋄
ι
S
∗
(ι)
υ
ι
S
∗
(ι)
|ι <
γ} ⊆ GOrdCl
L∪
˜
W∪P
.
Proof. An immediate consequence of K¨onig’s
Lemma and Lemma 4.2.
We are in position to prove the refutational sound-
ness and completeness of the order hyperresolution
calculus. Let {0,1} ⊆ X ⊆ Tcons
L
. X is admissi-
ble with respect to suprema and infima iff, for all
Y
1
,Y
2
⊆ X and
W
W
W
Y
1
=
V
V
V
Y
2
, either
W
W
W
Y
1
∈Y
1
,
V
V
V
Y
2
∈Y
2
,
or
W
W
W
Y
1
6∈ Y
1
,
V
V
V
Y
2
6∈ Y
2
.
Theorem 4.4 (Refutational Soundnessand Complete-
ness). Let L contain a constant symbol. Let P ⊆
˜
P,
S ⊆ OrdCl
L∪P
, tcons(S) be admissible with respect to
FCTA2014-InternationalConferenceonFuzzyComputationTheoryandApplications
48

suprema and infima.  ∈ clo
H
(S) if and only if S is
unsatisfiable.
Proof. (=⇒) Let A be a model of S for L ∪ P and
C ∈ clo
H
(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
H
(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
H
(S). Then, by Lemma 4.1 for
S, ,  6∈ clo
BH
(S); we have L,
˜
P,
˜
W are countable,
P ⊆
˜
P, S ⊆ OrdCl
L∪P
, clo
BH
(S) ⊆ GOrdCl
L∪
˜
W∪P
;
P, L ∪ P, OrdCl
L∪P
, S, L ∪
˜
W ∪ P, GOrdCl
L∪
˜
W∪P
,
clo
BH
(S) are countable; there exists γ
1
≤ ω and  6∈
clo
BH
(S) = {
W
k
ι
j=0
ε
ι
j
⋄
ι
j
υ
ι
j
|ι < γ
1
}; by Lemma 4.3 for
S, there exists S
∗
∈ S el({{ j| j ≤ k
ι
}
ι
|ι < γ
1
}) and
there does not exist a contradiction of {ε
ι
S
∗
(ι)
⋄
ι
S
∗
(ι)
υ
ι
S
∗
(ι)
|ι < γ
1
} ⊆ GOrdCl
L∪
˜
W∪P
. We put S =
{ε
ι
S
∗
(ι)
⋄
ι
S
∗
(ι)
υ
ι
S
∗
(ι)
|ι < γ
1
} ⊆ GOrdCl
L∪
˜
W∪P
. Then
S ⊇ ordtcons(S) is countable, unit, (q)atoms(S) ⊆
(q)atoms(clo
BH
(S)). We put
U
A
=

GTerm
L∪P
if qatoms(S) =
/
0,
GTerm
L∪
˜
W∪P
else,
U
A
6=
/
0,
and B = atoms(S) ∪ qatoms(S) ⊆ GAtom
L∪
˜
W∪P
∪
QAtom
L∪
˜
W∪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)) =
/
0, atoms(S), qatoms(S), B, tcons(S),
B − tcons(S) are countable; there exist γ
2
≤ ω and
a sequence δ
2
: γ
2
−→ B − tcons(S) of B − tcons(S).
Let ε
1
,ε
2
∈ B. ε
1
, ε
2
iff there exists an equality chain
ε
1
Ξε
2
of S. Note that , is a binary symmetric transi-
tive relation on B. ε
1
⊳ ε
2
iff there exists an increasing
chain ε
1
Ξε
2
of S. Note that ⊳ is a binary transitive
relation on B.
06,1,16, 0,0⊳ 1,1⋪0, (52)
for all ε ∈ B,ε⋪0,1⋪ε,ε⋪ ε.
The proof is straightforward; we have that there does
not exist a contradiction of S. Note that ⊳ 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
L
, V ( ¯c) = c. We denote
dom(V ) = X, tcons(S) ⊆ dom(V ) ⊆ B. We define a
partial valuation V
α
by recursion on α ≤ γ
2
in Table 8.
Table 8: V
α
.
V
0
= {(0,0),(1,1)}∪
{( ¯c, c)| ¯c ∈ tcons(S) ∩
C
L
};
V
α
= V
α−1
∪ {(δ
2
(α− 1),λ
α−1
)}
(1 ≤ α ≤ γ
2
is a successor ordinal),
E
α−1
= {V
α−1
(a)| a, δ
2
(α− 1),a ∈ dom(V
α−1
)},
D
α−1
= {V
α−1
(a)| a⊳ δ
2
(α− 1),a ∈ dom(V
α−1
)},
U
α−1
= {V
α−1
(a)| δ
2
(α− 1)⊳ a,a ∈ dom(V
α−1
)},
λ
α−1
=
(
W
W
W
D
α−1
+
V
V
V
U
α−1
2
if E
α−1
=
/
0,
W
W
W
E
α−1
else;
V
γ
2
=
[
α<γ
2
V
α
(γ
2
is a limit ordinal)
For all α ≤ α
′
≤ γ
2
, V
α
is a partial valuation,
dom(V
α
) = tcons(S) ∪ δ
2
[α], V
α
⊆ V
α
′
.
(53)
The proof is by induction on α ≤ γ
2
.
For all α ≤ γ
2
, for all a, b ∈ dom(V
α
),
if a,b, then V
α
(a) = V
α
(b);
if a⊳b, then V
α
(a) < V
α
(b).
(54)
The proof is by induction on α ≤ γ
2
using the as-
sumption that tcons(S) is admissible with respect to
suprema and infima.
We put V = V
γ
2
, dom(V )
(53)
== tcons(S) ∪ δ
2
[γ
2
] =
B.
For all a,b ∈ B = dom(V ),
if a,b, then V (a) = V (b);
if a⊳b, then V (a) < V (b);
if a = ∀xc, then V (a) =
V
V
V
u∈U
A
V (c(x/u));
if a = ∃xc, then V (a) =
W
W
W
u∈U
A
V (c(x/u)).
(55)
The proof. A straightforward consequence of (54).
AnOrderHyperresolutionCalculusforGödelLogicwithTruthConstants
49

We put
if qatoms(S) =
/
0,
f
A
(u
1
,... ,u
τ
) = f(u
1
,... , u
τ
),
f ∈ Func
L∪P
,u
i
∈ U
A
;
p
A
(u
1
,... , u
τ
) =

V (p(u
1
,... ,u
τ
)) if p(u
1
,... , u
τ
) ∈ B,
0 else,
p ∈ Pred
L∪P
,u
i
∈ U
A
;
A =

U
A
,{ f
A
| f ∈ Func
L∪P
},
{p
A
| p ∈ Pred
L∪P
}

,
an interpretation for L ∪ P;
if qatoms(S) 6=
/
0,
f
A
(u
1
,... ,u
τ
) = f(u
1
,... , u
τ
),
f ∈ Func
L∪
˜
W∪P
,u
i
∈ U
A
;
p
A
(u
1
,... , u
τ
) =

V (p(u
1
,... ,u
τ
)) if p(u
1
,... , u
τ
) ∈ B,
0 else,
p ∈ Pred
L∪
˜
W∪P
,u
i
∈ U
A
;
A =

U
A
,{ f
A
| f ∈ Func
L∪
˜
W∪P
},
{p
A
| p ∈ Pred
L∪
˜
W∪P
}

,
an interpretation for L ∪
˜
W ∪ P.
Hence, it is straightforward to prove that for all
a ∈ B and e ∈ S
A
, kak
A
e
= V (a); for all l ∈ S
and e ∈ S
A
, klk
A
e
= 1; for all C ∈ S and e ∈
S
A
, e|
freevars(C)
∈ Subst
L∪
˜
W∪P
, dom(e|
freevars(C)
) =
freevars(C), range(e|
freevars(C)
) =
/
0, C(e|
freevars(C)
) ∈
clo
BH
(S), there exists l
∗
∈ C(e|
freevars(C)
) and
l
∗
∈ S, kl
∗
k
A
e
= 1; there exists l
∗∗
∈ C and
l
∗
= l
∗∗
(e|
freevars(C)
), kl
∗∗
k
A
e
= kl
∗∗
(e|
freevars(C)
)k
A
e
=
kl
∗
k
A
e
= 1; A |=
e
C; A |= S, A|
L∪P
|= S; S is satisfi-
able.
Consider S = {0 ≺ a}∪{a ≺
1
n
|n ∈ N} ⊆ OrdCl
L
,
a ∈ Pred
L
− Tcons
L
, ar
L
(a) = 0. tcons(S) is not ad-
missible with respect to suprema and infima; for {0}
and {
1
n
|n ∈ N},
W
W
W
{0} =
V
V
V
{
1
n
|n ∈ N} = 0, 0 ∈ {0},
0 6∈ {
1
n
|n ∈ N}. S is unsatisfiable; both the cases
kak
A
= 0 and kak
A
> 0 lead to A 6|= S for every in-
terpretation A for L. However,  6∈ clo
H
(S) = S. So,
the condition on tcons(S) being admissible with re-
spect to suprema and infima, is necessary.
Corollary 4.5. Let L contain a constant symbol. Let
n
0
∈ N, φ ∈ Form
L
, T ⊆ Form
L
, tcons(T) be admis-
sible with respect to suprema and infima. There exist
J
φ
T
⊆ {(i, j)|i ≥ n
0
} and S
φ
T
⊆ SimOrdCl
L∪{ ˜p
| ∈J
φ
T
}
such that tcons(S
φ
T
) is admissible with respect to
suprema and infima; T |= φ if and only if  ∈
clo
H
(S
φ
T
).
Proof. A straightforward consequence of Corol-
lary 3.5 and Theorem 4.4.
In Table 9, we show that φ = ∀x(q
1
(x) → 0.3) →
(∃xq
1
(x) → 0.5) ∈ Form
L
is logically valid using the
proposed translation to order clausal form and the or-
der hyperresolution calculus.
5 CONCLUSIONS
In the paper, we have proposed a modification of the
hyperresolution calculus from (Guller, 2012) which
is suitable for automated deduction with explicit par-
tial truth. The first-order G¨odel logic is expanded
by a countable set of intermediate truth constants ¯c,
c ∈ (0,1). We have modified translation of a formula
to an equivalent satisfiable finite order clausal theory,
consisting of order clauses. An order clause is a fi-
nite set of order literals of the form ε
1
⋄ ε
2
where ⋄
is a connective either ≖ or ≺. ≖ and ≺ are inter-
preted by the equality and standard strict linear or-
der on [0,1], respectively. We have investigated the
so-called canonical standard completeness, where the
semantics of the first-orderG¨odel logic is given by the
standard G-algebra and truth constants are interpreted
by themselves. The modified 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 covers the case
of finite order clausal theories.
Let φ ∈ Form
L
; φ contains a finite number of truth
constants. Then the problem that φ is unsatisfiable
can be reduced to the deduction problem φ |= 0 (after
a constant number of steps). As an immediate conse-
quence of Corollary 3.5 and Theorem 4.4, if φ |= 0,
then we can decide it after a finite number of steps.
This straightforwardly implies that the set of unsatis-
fiable formulae of L (in the general first-order G¨odel
logic with intermediate truth constants) is recursively
enumerable.
FCTA2014-InternationalConferenceonFuzzyComputationTheoryandApplications
50

Table 9: An example: φ = ∀x(q
1
(x) → 0.3) → (∃xq
1
(x) → 0.5).
φ = ∀x(q
1
(x) → 0.3) → (∃xq
1
(x) → 0.5)
{ ˜p
0
(x) ≺ 1,

∀x(q
1
(x) →
0.3)
|
{z }
˜p
1
(x)
→ (∃xq
1
(x) →
0.5
|
{z }
˜p
2
(x)
)

→ ˜p
0
(x)} (27)
{ ˜p
0
(x) ≺ 1, ˜p
2
(x) ≺ ˜p
1
(x) ∨ ˜p
2
(x) ≖ 1∨ ˜p
0
(x) ≖ 1, ˜p
2
(x) ≺ ˜p
0
(x) ∨ ˜p
2
(x) ≖ ˜p
0
(x), ˜p
1
(x) → ∀x(q
1
(x) → 0.3
|
{z }
˜p
3
(x)
),(∃xq
1
(x)
|
{z }
˜p
4
(x)
→
0.5
|{z}
˜p
5
(x)
) → ˜p
2
(x)} (33),(27)
{ ˜p
0
(x) ≺ 1, ˜p
2
(x) ≺ ˜p
1
(x) ∨ ˜p
2
(x) ≖ 1∨ ˜p
0
(x) ≖ 1, ˜p
2
(x) ≺ ˜p
0
(x) ∨ ˜p
2
(x) ≖ ˜p
0
(x), ˜p
1
(x) ≺ ∀x ˜p
3
(x) ∨ ˜p
1
(x) ≖ ∀x ˜p
3
(x), ˜p
3
(x) → (q
1
(x)
|
{z}
˜p
6
(x)
→
0.3
|{z}
˜p
7
(x)
),
˜p
5
(x) ≺ ˜p
4
(x) ∨ ˜p
5
(x) ≖ 1∨ ˜p
2
(x) ≖ 1, ˜p
5
(x) ≺ ˜p
2
(x) ∨ ˜p
5
(x) ≖ ˜p
2
(x), ˜p
4
(x) → ∃x q
1
(x)
|{z}
˜p
8
(x)
,
0.5 ≺ ˜p
5
(x) ∨ 0.5 ≖ ˜p
5
(x)} (25),(37)
S
φ
=
(
˜p
0
(x) ≺ 1 [1]
˜p
2
(x) ≺ ˜p
1
(x) ∨ ˜p
2
(x) ≖ 1∨
˜p
0
(x) ≖ 1 [2]
˜p
2
(x) ≺ ˜p
0
(x) ∨ ˜p
2
(x) ≖ ˜p
0
(x) [3]
˜p
1
(x) ≺ ∀x ˜p
3
(x) ∨ ˜p
1
(x) ≖ ∀x ˜p
3
(x) [4]
˜p
3
(x) ≺ ˜p
7
(x) ∨ ˜p
3
(x) ≖ ˜p
7
(x) ∨ ˜p
6
(x) ≺ ˜p
7
(x) ∨
˜p
6
(x) ≖ ˜p
7
(x) [5]
q
1
(x) ≺ ˜p
6
(x) ∨ q
1
(x) ≖ ˜p
6
(x) [6]
˜p
7
(x) ≺ 0.3∨ ˜p
7
(x) ≖ 0.3 [7]
˜p
5
(x) ≺ ˜p
4
(x) ∨ ˜p
5
(x) ≖ 1∨
˜p
2
(x) ≖ 1 [8]
˜p
5
(x) ≺ ˜p
2
(x) ∨ ˜p
5
(x) ≖ ˜p
2
(x) [9]
˜p
4
(x) ≺ ∃x ˜p
8
(x) ∨ ˜p
4
(x) ≖ ∃x ˜p
8
(x) [10]
˜p
8
(x) ≺ q
1
(x) ∨ ˜p
8
(x) ≖ q
1
(x) [11]
0.5 ≺ ˜p
5
(x) ∨ 0.5 ≖ ˜p
5
(x)
)
[12]
Rule (46) : [1][2] :
˜p
2
(x) ≺ ˜p
1
(x) ∨
˜p
2
(x) ≖ 1 [13]
Rule (46) : [3][13] :
˜p
2
(x) ≖ ˜p
0
(x) ∨ ˜p
2
(x) ≺ ˜p
1
(x) [14]
Rule (46) : [1][13][14] :
˜p
2
(x) ≺ ˜p
1
(x) [15]
Rule (46) : [8][15] :
˜p
5
(x) ≺ ˜p
4
(x) ∨
˜p
5
(x) ≖ 1 [16]
Rule (46) : [9][16] :
˜p
5
(x) ≖ ˜p
2
(x) ∨ ˜p
5
(x) ≺ ˜p
4
(x) [17]
Rule (46) : [15][16][17] :
˜p
5
(x) ≺ ˜p
4
(x) [18]
Rule (48) : ∀x ˜p
3
(x) :
∀x ˜p
3
(x) ≺ ˜p
3
(x) ∨ ∀x ˜p
3
(x) ≖ ˜p
3
(x) [19]
repeatedly Rule (46) : [4][5][7][9][12][15][19] :
˜p
6
(x) ≺ ˜p
7
(x) ∨ ˜p
6
(x) ≖ ˜p
7
(x) [20]
Rule (51) : ∃x ˜p
8
(x),
0.5 :
0.5 ≺ ˜p
8
( ˜w
(0,0)
) ∨ ∃x ˜p
8
(x) ≺ 0.5∨ ∃x ˜p
8
(x) ≖ 0.5 [21]
repeatedly Rule (46) : [10][12][18][21] :
0.5 ≺ ˜p
8
( ˜w
(0,0)
) [22]
repeatedly Rule (46) : [6][7][11][20]; ˜w
(0,0)
: [22] :
 [23]
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