AN ORDER HYPERRESOLUTION CALCULUS FOR G
¨
ODEL LOGIC
General First-order Case
Du
ˇ
san Guller
Department of Applied Informatics, Comenius University, Mlynsk
´
a dolina, 842 48 Bratislava, Slovakia
Keywords:
G
¨
odel Logic : Resolution : Many-valued Logics : Automated Deduction
Abstract:
This paper addresses the deduction problem of a formula from a countable theory in the first-order G
¨
odel
logic from a perspective of automated deduction. Our approach is based on the translation of a formula
to an equivalent satisfiable CNF one, which contains literals of the augmented form: either a or a → b or
(a → b) → b or Qx c → a or a → Qx c where a, c are atoms different from 0 (the false), 1 (the true); b is an
atom different from 1; Q ∈ {∀,∃}; x is a variable occurring in c. A CNF formula is further translated to an
equivalent satisfiable finite order clausal theory, which consists of order clauses - finite sets of order literals
of the form: either a P b or Qx c P a or a P Qx c or a ≺ b or Qx c ≺ a or a ≺ Qx c where a, b, c are atoms;
Q ∈ {∀,∃}; x is a variable occurring in c. P and ≺ are interpreted by the equality and strict linear order on
[0,1], respectively. For an input theory, the proposed translation produces a so-called semantically admissible
order clausal theory. An order hyperresolution calculus, operating on semantically admissible order clausal
theories, is devised. The calculus is proved to be refutation sound and complete for the countable case.
1 INTRODUCTION
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 Prod-
uct logic, and Σ
1
-complete for G
¨
odel logic, as with
classical first-order logic. Among these fuzzy logics,
only G
¨
odel logic is recursively axiomatisable. Hence,
it is all important 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 (Baaz
et al., 2001; Baaz and Ferm
¨
uller, 2010), the prenex
fragment of G
¨
odel logic in presence of the projection
operator ∆ : [0,1] −→ [0,1],
∆a =

1 if a = 1,
0 else,
is investigated, denoted as the prenex G
∆
∞
. (Baaz et al.,
2001) solves the validity problem (VAL). A variant
of Herbrand’s Theorem for the prenex G
∆
∞
is proved,
which reduces the VAL problem of a formula in the
prenex G
∆
∞
to the VAL problem of an open formula in
G
∆
∞
. Further, a meta-level logic of order clauses is de-
fined, which is a fragment of classical one. An order
Partially supported by VEGA Grant 1/0979/12.
clause is a finite set of inequalities of the form either
a < b or a ≤ b where <, ≤ are meta-level binary pred-
icate symbols and a, b are atoms of G
∆
∞
considered
as meta-level terms. The semantics of the meta-level
logic of order clauses is given by classical interpre-
tations on [0,1], varying on assigned (truth) values to
atoms of G
∆
∞
(meta-level terms), which are the strict
dense linear order with endpoints on [0,1]; < is inter-
preted as the strict dense linear order with endpoints
and ≤ as its reflexive closure on [0,1]. A formula
in the prenex G
∆
∞
is valid if and only if a translation
of it to the order clause form is unsatisfiable with re-
spect to the semantics of the meta-level logic. In the
prenex G
∆
∞
, the problem of the unsatisfiability of a for-
mula cannot straightforwardly be reduced to the VAL
problem. Although the standard Skolemisation can be
used for the reduction of the VAL problem to the open
case, it does not preserve satisfiability. (Baaz and
Ferm
¨
uller, 2010) have shown that any conjunction of
formulae can be translated to an equivalent satisfiable
universal form via an alternative version of Skolemi-
sation. The ordered chaining calculi (Bachmair and
Ganzinger, 1998) may be used for resolution-style de-
duction over order clauses.
In the paper, we solve the deduction problem of a
formula from a countable theory in G
¨
odel logic. Our
approach is based on the translation of a formula to
Guller D..
An Order Hyperresolution Calculus for Gödel Logic - General First-order Case.
DOI: 10.5220/0004104103290342
In Proceedings of the 4th International Joint Conference on Computational Intelligence (FCTA-2012), pages 329-342
ISBN: 978-989-8565-33-4
Copyright
c
 2012 SCITEPRESS (Science and Technology Publications, Lda.)

an equivalent satisfiable CNF one, which contains lit-
erals of the augmented form: either a or a → b or
(a → b) → b or Qx c → a or a → Qx c where a, c
are atoms different from 0, 1; b is an atom differ-
ent from 1; Q ∈ {∀,∃}; x is a variable occurring in
c; Lemma 3.1, Section 3. A CNF formula is fur-
ther translated to an equivalent satisfiable finite order
clausal theory, which consists of order clauses - fi-
nite sets of order literals of the form: either a P b
or Qx c P a or a P Qx c or a ≺ b or Qx c ≺ a or
a ≺ Qx c where a, b, c are atoms; Q ∈ {∀, ∃}; x is
a variable occurring in c; Lemma 3.1, Section 3. P
and ≺ are interpreted by the equality and strict lin-
ear order on [0,1], respectively. They are added to
G
¨
odel logic as new binary connectives. The trans-
lation is based on so-called interpolation rules given
in Tables 2–4, Section 3. For an input theory, the
translation produces a so-called semantically admis-
sible order clausal theory, Section 4, Subsection 4.1.
Corollary 4.1 states that for an input countable the-
ory T and formula φ, there exists a countable se-
mantically admissible order clausal theory S
φ
T
such
that T |= φ if and only if S
φ
T
is unsatisfiable. In
case of a finite T , |S
φ
T
| ∈ O(|T |
2
+ |φ|
2
) and the time
as well as space complexity of the translation is in
O((|T |
2
+ |φ|
2
) · log(|T | + |φ|)). An order hyperres-
olution calculus, operating on semantically admissi-
ble order clausal theories, uses order hyperresolution
rules introduced in Tables 6 and 7, Section 4, Subsec-
tion 4.3. Most of the resolution rules of ordered chain-
ing calculi (Bachmair and Ganzinger, 1998) (e.g. the
factorised chaining rule) have non-empty residua in
their consequences; i.e. they infer new (in)equalities.
Many of them are only transitive consequences, un-
necessary for refutational argument. We avoid this
inefficiency using the hyperresolution principle; our
rules do not infer new (in)equalities being transitive
consequences, which confines search space consider-
ably. The calculus is proved to be refutation sound
and complete for the countable case, Theorem 4.4,
Section 4, Subsection 4.3.
The paper is organised as follows. Section 2 con-
cerns G
¨
odel logic. Section 3 deals with the transla-
tion to order clausal form. Section 4 proposes the or-
der hyperresolution calculus. Section 5 brings con-
clusions.
2 G
¨
ODEL LOGIC
Throughout the paper, we shall use the common
notions and notation of first-order logic. By L we
denote a 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. 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. 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 quantifiers: ∀, the universal quantifier, ∃,
the existential one. In addition, we introduce new
binary connectives P, equality, and ≺, strict order.
By OrdForm
L
we designate the set of all so-called
order formulae of L built up from Atom
L
and Var
L
using the connectives: ¬, ∧, ∨, →, P, ≺, and the
quantifiers: ∀, ∃.
1
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, in general. By
vars(ε
1
,...,ε
m
) ⊆ Var
L
| freevars(ε
1
,...,ε
m
) ⊆
Var
L
| boundvars(ε
1
,...,ε
m
) ⊆ Var
L
|
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 ` we denote the empty
sequence. By |ε
1
,...,ε
m
| = m we denote the length
of the sequence ε
1
,...,ε
m
. We define the concate-
nation of the 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
.
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 f [Z] = { f (z)|z ∈ Z}; f [Z] is
the image of Z under f ; and f |
Z
= {(z, f (z)) | z ∈ Z};
f |
Z
is the restriction of f onto Z. Let γ ≤ ω. A se-
quence δ of X is a bijection δ : γ −→ X. X is countable
if and only if there exists a sequence of X. Let X be a
set of non-empty sets. A selector S over X is a map-
ping S : X −→
S
X such that for all x ∈ X, S (x) ∈ x.
We denote S el(X ) = {S |S is a selector over X}. Let
f , g : N −→ R
+
0
. f is of the order of g, in sym-
bols f ∈ O(g), iff there exist n
0
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| ∈ N | |φ| ∈ N, is defined as the
number of nodes of its standard tree representation.
We define the size of T as |T | =
∑
φ∈T
|φ| ∈ N. By
1
We assume a decreasing connective and quantifier
precedence: ∀, ∃, ¬, ∧, →, P, ≺, ∨.

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(∀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
1
,. . . ,x
n
be a sequence
of variables 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 P
P
P and ≺
≺
≺ for
P and ≺, respectively.
G = ([0,1],≤, ∨, ∧,⇒, ,P
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;
a P
P
P b =

1 if a =
[0,1]
b,
0 else;
a ≺
≺
≺ b =

1 if a <
[0,1]
b,
0 else
where =
[0,1]
| <
[0,1]
is the equality | strict order on
[0,1]. We recall that G is a complete linearly ordered
lattice algebra; 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 exploited later,
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)
2
We assume a decreasing operator precedence: , ∧,
⇒, P
P
P, ≺
≺
≺, ∨.
An interpretation I for L is a triple

U
I
,{ f
I
| f ∈ Func
L
},{p
I
| p ∈ Pred
L
}

defined as usual. A variable assignment in I is a map-
ping Var
L
−→ U
I
. We denote the set of all variable
assignments in I as S
I
. Let t ∈ Term
L
; ¯v be a se-
quence of variables of L; φ ∈ OrdForm
L
; e ∈ S
I
. 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 ¯vk
I
e
∈ U
| ¯v|
I
of ¯v, the truth value kφk
I
e
∈ [0,1] of φ by
recursion on the structure of φ, as usual. A theory of
L is a set of formulae of L. An order theory of L is a
set of order formulae of L. Let φ,φ
0
∈ OrdForm
L
and
T,T
0
⊆ OrdForm
L
. φ is equivalent to φ
0
, in symbols
φ ≡ φ
0
, iff, for every interpretation I for L and e ∈ S
I
,
kφk
I
e
= kφ
0
k
I
e
. φ | T is equisatisfiable to φ
0
| T
0
iff φ |
T is satisfiable if and only if φ
0
| T
0
is satisfiable.
3 TRANSLATION TO ORDER
CLAUSAL FORM
At first, we introduce conjunctive normal form (CNF)
in G
¨
odel logic. In contrast to two-valued logic, we
have to consider an augmented set of literals appear-
ing in CNF formulae. Let l, φ ∈ Form
L
. l is a literal
of L iff either l = a or l = a → b or l = (a → b) → b
or l = Qx c → a or l = a → Qx c where a,c ∈ Atom
L
−
{0,1}, b ∈ Atom
L
− {1}, x ∈ vars(c). φ is a conjunc-
tive | disjunctive normal form of L, in symbols CNF |
DNF, iff either φ = 0 or φ = 1 or φ =
V
i≤n
W
j≤m
i
l
i
j
|
φ =
W
i≤n
V
j≤m
i
l
i
j
where l
i
j
is a literal of L. Let
D = l
0
∨ ··· ∨ l
n
∈ Form
L
, l
i
is a literal of L. We de-
note lits(D) = {l
0
,. . . ,l
n
} ⊆ Form
L
. D is a factor iff,
for all i < i
0
≤ n, l
i
6= l
i
0
. We now describe some gen-
eralisation of the translation in (Guller, 2010; Guller,
2012) to the first-order case. A similar approach ex-
ploiting the renaming subformulae technique can be
found in (Plaisted and Greenbaum, 1986; de la Tour,
1992; Nonnengart et al., 1998; Sheridan, 2004). Let
l ∈ OrdForm
L
. l is an order literal of L iff either
l = a P b or l = Qx c P a or l = a P Qx c or l = Qx c P
Q
0
yd or l = a ≺ b or l = Qx c ≺ a or l = a ≺ Qx c or
l = Qx c ≺ Q
0
yd where a,b,c, d ∈ Atom
L
, x ∈ vars(c),
y ∈ vars(d). An order clause of L is a finite set of or-
der literals of L; since =
[0,1]
is commutative, we iden-
tify the order literals ε
1
P ε
2
and ε
2
P ε
1
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 literal l in given context. We desig-
nate the set of all order clauses of L as OrdCl
L
. Let

l,l
0
,. . . ,l
n
be order literals of L and C,C
0
∈ OrdCl
L
.
We define the size of C as |C| =
∑
l∈C
|l| ∈ N. By
l
0
∨ · · · ∨ l
n
∨C we denote {l
0
} ∪ · · · ∪ {l
n
} ∪C where,
for all i,i
0
≤ n, i 6= i
0
, l
i
6∈ C and l
i
6= l
i
0
. By C ∨ C
0
we denote C ∪C
0
. C is a subclause of C
0
, in symbols
C v C
0
, iff C ⊆ C
0
. 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 I be an interpre-
tation for L and e ∈ S
I
. 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. Let S, S
0
⊆ OrdCl
L
. I is
a model of S, in symbols I |= S, iff, for all C ∈ S,
I |= C. C is a logical consequence of S, in symbols
S |= C, iff, for every model I of S for L, I |= C. S
0
is a logical consequence of S, in symbols S |= S
0
, iff,
for every model I of S for L, I |= S
0
. C | S is sat-
isfiable iff there exists a model of C | S for L. Let
φ,φ
0
∈ OrdForm
L
and T,T
0
⊆ OrdForm
L
. φ | T | C |
S is equisatisfiable to φ
0
| T
0
| C
0
| S
0
iff φ | T | C |
S is satisfiable if and only if φ
0
| T
0
| C
0
| S
0
is sat-
isfiable. Let S ⊆
F
OrdCl
L
. We define the size of S
as |S| =
∑
C∈S
|C| ∈ N. Let I = N × N; I is an infi-
nite countable set of indices. Let
˜
P = { ˜p | ∈ I} such
that
˜
P ∩ Pred
L
=
/
0;
˜
P is an infinite countable set of
new predicate symbols. From a computational point
of view, the worst case time and space complexity will
be estimated using the logarithmic cost measurement.
Let A be an algorithm. #O ∈ N denotes the number of
all basic operations executed by A. The translation to
order clausal form is based on the following lemma.
Lemma 3.1. Let φ ∈ Form
L
; T ⊆ Form
L
be count-
able; F ⊆ I such that there exists n
0
and F ∩
{(i, j)|i ≥ n
0
} =
/
0; n
φ
≥ n
0
.
(i) There exist either J
φ
=
/
0 or J
φ
= {(n
φ
, j)| j ≤
n
J
φ
}, J
φ
⊆
F
I, J
φ
∩ F =
/
0; a CNF ψ ∈
Form
L∪{ ˜p | ∈J
φ
}
; S
φ
⊆
F
OrdCl
L∪{ ˜p | ∈J
φ
}
such
that
(a) kJ
φ
k ≤ 2 · |φ|;
(b) either J
φ
= S
φ
=
/
0, or J
φ
=
/
0, S
φ
= {}, or
J
φ
6=
/
0,  6∈ S
φ
6=
/
0;
(c) there exists an interpretation A for L and
A |= φ ∈ Form
L
if and only if there exists
an interpretation A
0
for L ∪ { ˜p | ∈ J
φ
} and
A
0
|= ψ ∈ Form
L∪{ ˜p | ∈J
φ
}
, satisfying A =
A
0
|
L
;
(d) there exists an interpretation A for L and
A |= φ ∈ Form
L
if and only if there exists
an interpretation A
0
for L ∪ { ˜p | ∈ J
φ
} and
A
0
|= S
φ
⊆ OrdCl
L∪{ ˜p | ∈J
φ
}
, satisfying A =
A
0
|
L
;
(e) |ψ| ∈ O(|φ|
2
); the number of all basic opera-
tions of the translation of φ to ψ is in O(|φ|
2
);
the time and space complexity of the transla-
tion of φ to ψ is in O(|φ|
2
· log |φ|);
(f) |S
φ
| ∈ O(|φ|
2
); the number of all basic op-
erations 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 |φ|);
(g) if ψ 6= 0 and ψ 6= 1, then ψ =
V
i≤n
ψ
D
i
,
D
i
is a factor; J
φ
6=
/
0; for all i ≤ n
ψ
,
/
0 6=
preds(D
i
) ∩
˜
P ⊆ { ˜p | ∈ J
φ
}; for all i < i
0
≤
n
ψ
, lits(D
i
) 6= lits(D
i
0
);
(h) if S
φ
6=
/
0 and S
φ
6= {}, then J
φ
6=
/
0; for all
C ∈ S
φ
,
/
0 6= preds(C) ∩
˜
P ⊆ { ˜p | ∈ J
φ
}.
(ii) There exist J
T
⊆ I, J
T
∩ F =
/
0, and S
T
⊆
OrdCl
L∪{ ˜p | ∈J
T
}
being countable such that
(a) either J
T
= S
T
=
/
0, or J
T
=
/
0, S
T
= {}, or
J
T
6=
/
0,  6∈ S
T
6=
/
0;
(b) there exists an interpretation A for L and
A |= T ⊆ Form
L
if and only if there exists
an interpretation A
0
for L ∪ { ˜p | ∈ J
T
} and
A
0
|= S
T
⊆ OrdCl
L∪{ ˜p | ∈J
T
}
, satisfying A =
A
0
|
L
;
(c) if T ⊆
F
Form
L
, then J
T
⊆
F
I, kJ
T
k ≤ 2 · |T |;
S
T
⊆
F
OrdCl
L∪{ ˜p | ∈J
T
}
, |S
T
| ∈ O(|T |
2
); the
number of all basic operations of the trans-
lation of T to S
T
is in O(|T |
2
); the time and
space complexity of the translation of T to S
T
is in O(|T |
2
· log(1 + |T |));
(d) if S
T
6=
/
0 and S
T
6= {}, then J
T
6=
/
0; for all
C ∈ S
T
,
/
0 6= preds(C) ∩
˜
P ⊆ { ˜p | ∈ J
T
}.
Proof. Technical using interpolation. Let l ∈ C ∈
S
φ
|S
T
. Then either l = a P b or l = c P a or l =
a P c or l = a ≺ b or l = c ≺ a or l = a ≺ c, a,b ∈
atoms(S
φ
)| atoms(S
T
), c ∈ qatoms(S
φ
)| qatoms(S
T
).
Let θ ∈ Form
L
. There exists θ
0
∈ Form
L
such
that
(a) θ
0
≡ θ;
(b) |θ
0
| ≤ 2 · |θ|; θ
0
can be built up via a pos-
torder traversal of θ with #O ∈ O(|θ|),
the time and space complexity in O(|θ| ·
log|θ|);
(c) θ
0
does not contain ¬;
(d) either θ
0
= 0, or 0 is a subformula of θ
0
if
and only if 0 is a subformula of a subfor-
mula of θ
0
of the form ϑ → 0, ϑ 6= 0;
(e) either θ
0
= 1 or 1 is not a subformula of θ
0
.
(13)
The proof is by induction on the structure of θ.
In Table 1, for every form of literal, an order
clause 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.

Table 1: Translation of l to C.
Case: l C
1 a a P 1 |C| ≤ 3 · |l|
2 a → 0 a P 0 |C| ≤ 3 · |l|
3 a → b a ≺ b ∨ a P b |C| ≤ 3 · |l|
4 (a → 0) → 0 0 ≺ a |C| ≤ 3 · |l|
5 (a → b) → b b ≺ a ∨ b P 1 |C| ≤ 3 · |l|
6 Qx c → a Qx c ≺ a ∨ Qx c P a |C| ≤ 3 · |l|
7 a → Qx c a ≺ Qx c ∨ a P Qx c |C| ≤ 3 · |l|
a,b, c ∈ Atom
L
− {0,1}, x ∈ vars(c).
Let θ ∈ Form
L
− {0, 1}; (13c–e) hold for θ;
¯x be a sequence of variables of L, vars( ¯x) ⊇
vars(θ); G ⊆ I such that there exists n
1
and G ∩ {(i, j)| i ≥ n
1
} =
/
0; n
θ
≥ n
1
; =
(n
θ
, j ) ∈ I, ˜p ∈
˜
P, ar( ˜p ) = | ¯x|, { } ∩ G ⊆
{(i, j)|i ≥ n
1
} ∩ G =
/
0. There exist n
J
≥ j ,
J = {(n
θ
, j)| j + 1 ≤ j ≤ n
J
} ⊆
F
I, J ∩ (G ∪
{ }) =
/
0; a CNF ψ
s
∈ Form
L∪{ ˜p }∪{ ˜p | ∈J}
,
S
s
⊆
F
OrdCl
L∪{ ˜p }∪{ ˜p | ∈J}
, s = +,−, such
that for both s,
(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
0
for L ∪ { ˜p } ∪ { ˜p | ∈ J} and A
0
|=
ψ
+
∈ Form
L∪{ ˜p }∪{ ˜p | ∈J}
, satisfying A =
A
0
|
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 interpreta-
tion A
0
for L ∪ { ˜p } ∪ { ˜p | ∈ J} and
A
0
|= ψ
−
∈ Form
L∪{ ˜p }∪{ ˜p | ∈J}
, satisfy-
ing A = A
0
|
L∪{ ˜p }
;
(d) for every interpretation A for
L ∪ { ˜p } ∪ { ˜p | ∈ J}, A |= ψ
s
∈
Form
L∪{ ˜p }∪{ ˜p | ∈J}
if and only if
A |= S
s
⊆ OrdCl
L∪{ ˜p }∪{ ˜p | ∈J}
;
(e) there exists an interpretation A for L ∪
{ ˜p } and A |= ˜p (¯x) → θ ∈ Form
L∪{ ˜p }
if and only if there exists an interpreta-
tion A
0
for L ∪ { ˜p } ∪ { ˜p | ∈ J} and
A
0
|= S
+
⊆ OrdCl
L∪{ ˜p }∪{ ˜p | ∈J}
, satisfy-
ing A = A
0
|
L∪{ ˜p }
;
(f) there exists an interpretation A for L ∪
{ ˜p } and A |= θ → ˜p ( ¯x) ∈ Form
L∪{ ˜p }
if and only if there exists an interpreta-
tion A
0
for L ∪ { ˜p } ∪ { ˜p | ∈ J} and
A
0
|= S
−
⊆ OrdCl
L∪{ ˜p }∪{ ˜p | ∈J}
, satisfy-
ing A = A
0
|
L∪{ ˜p }
;
(14)
Table 3: 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) (27)
|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) P 0 ∨ ˜p
1
( ¯x) P 0, θ
1
→ ˜p
1
( ¯x)}
(28)
|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) (29)
|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) P 1, ˜p
1
( ¯x) → θ
1
}
(30)
|Consequent| = 6 + 2 · | ¯x| + | ˜p
1
( ¯x) → θ
1
| ≤ 15 · (1 + | ¯x|) + | ˜p
1
( ¯x) → θ
1
|
(g) |ψ
s
| ≤ 13 · |θ| · (1 + | ¯x|), ψ
s
can be built up
from θ and ¯x via a preorder traversal of θ
with #O ∈ O(|θ| · (1 + | ¯x|));
(h) |S
s
| ≤ 15 · |θ| · (1 + | ¯x|), S
s
can be built up
from θ and ¯x via a preorder traversal of θ
with #O ∈ O(|θ| · (1 + | ¯x|));
(i) ψ
s
=
V
i≤n
ψ
s
D
s
i
, D
s
i
6= ˜p ( ¯x) is a factor; for
all i ≤ n
ψ
s
,
/
0 6= preds(D
s
i
) ∩
˜
P ⊆ { ˜p } ∪
{ ˜p | ∈ J}; for all i < i
0
≤ n
ψ
s
, lits(D
s
i
) 6=
lits(D
s
i
0
);
(j) for all C ∈ S
s
,
/
0 6= preds(C) ∩
˜
P ⊆ { ˜p } ∪
{ ˜p | ∈ J}; ˜p ( ¯x) P 1, ˜p ( ¯x) ≺ 1 6∈ S
s
.
The proof is by induction on the structure of θ using
the interpolation rules in Tables 2–4.
(i) By (13) for φ ∈ Form
L
, there exists φ
0
∈ Form
L
such that (13a–e) hold for φ
0
. We then distinguish
three cases for φ
0
. Case 1: φ
0
= 0. We put J
φ
=
/
0 ⊆
F
I, J
φ
∩ F =
/
0; ψ = 0 ∈ Form
L
; S
φ
= {} ⊆
F
OrdCl
L
. Case 2: φ
0
= 1. We put J
φ
=
/
0 ⊆
F
I,
J
φ
∩ F =
/
0; ψ = 1 ∈ Form
L
; S
φ
=
/
0 ⊆
F
OrdCl
L
.
Case 3: φ
0
6= 0 and φ
0
6= 1. Let ¯x = varseq(φ
0
).
Let = (n
φ
,0) ∈ I, ˜p ∈
˜
P, ar( ˜p ) = | ¯x|. We get
by (14) for φ
0
, ¯x, F, n
0
, n
φ
, , ˜p that there ex-
ist n
J
+
, J
+
= {(n
φ
, j)| 1 ≤ j ≤ n
J
+
} ⊆
F
I, J
+
∩
(F ∪ { }) =
/
0; a CNF ψ
+
∈ Form
L∪{ ˜p }∪{ ˜p | ∈J
+
}
;
S
+
⊆
F
OrdCl
L∪{ ˜p }∪{ ˜p | ∈J
+
}
; and (14a,b,e,g–j) hold
for φ
0
, ¯x, ˜p , J
+
, ψ
+
, S
+
. We put n
J
φ
= n
J
+
,
J
φ
= { } ∪ J
+
⊆
F
I, J
φ
∩ F =
/
0; ψ = ˜p ( ¯x) ∧
ψ
+
∈ Form
L∪{ ˜p | ∈J
φ
}
; S
φ
= { ˜p ( ¯x) P 1} ∪ S
+
⊆
F
OrdCl
L∪{ ˜p | ∈J
φ
}
. (ii) straightforwardly follows from
(i).

Table 2: Binary interpolation rules for ∧, ∨, →.
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) (15)
|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 ˜p
1
( ¯x), ˜p ( ¯x) ≺ ˜p
2
( ¯x) ∨ ˜p ( ¯x) P ˜p
2
( ¯x),
˜p
1
( ¯x) → θ
1
, ˜p
2
( ¯x) → θ
2
)
(16)
|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) (17)
|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 ˜p ( ¯x), ˜p
2
( ¯x) ≺ ˜p ( ¯x) ∨ ˜p
2
( ¯x) P ˜p ( ¯x),
θ
1
→ ˜p
1
( ¯x), θ
2
→ ˜p
2
( ¯x)
)
(18)
|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) (19)
|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 ˜p
1
( ¯x) ∨ ˜p ( ¯x) ≺ ˜p
2
( ¯x) ∨ ˜p ( ¯x) P ˜p
2
( ¯x), ˜p
1
( ¯x) → θ
1
, ˜p
2
( ¯x) → θ
2
}
(20)
|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) (21)
|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 ˜p ( ¯x), ˜p
2
( ¯x) ≺ ˜p ( ¯x) ∨ ˜p
2
( ¯x) P ˜p ( ¯x), θ
1
→ ˜p
1
( ¯x), θ
2
→ ˜p
2
( ¯x)}
(22)
|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
,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) (23)
|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 ˜p
2
( ¯x) ∨ ˜p
1
( ¯x) ≺ ˜p
2
( ¯x) ∨ ˜p
1
( ¯x) P ˜p
2
( ¯x),
θ
1
→ ˜p
1
( ¯x), ˜p
2
( ¯x) → θ
2
)
(24)
|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) (25)
|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) P 1 ∨ ˜p ( ¯x) P 1, ˜p
2
( ¯x) ≺ ˜p ( ¯x) ∨ ˜p
2
( ¯x) P ˜p ( ¯x),
˜p
1
( ¯x) → θ
1
,θ
2
→ ˜p
2
( ¯x)
)
(26)
|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 ∀ and ∃.
Case:
∀x θ
1
Positive
interpolation
˜p ( ¯x) → ∀x θ
1
( ˜p ( ¯x) → ∀x ˜p
1
( ¯x)) ∧ ( ˜p
1
( ¯x) → θ
1
)
(31)
|Consequent| = 6 + 2 · | ¯x| + | ˜p
1
( ¯x) → θ
1
| ≤ 13 · (1 + | ¯x|) + | ˜p
1
( ¯x) → θ
1
|
Positive
interpolation
˜p ( ¯x) → ∀x θ
1
{ ˜p ( ¯x) ≺ ∀x ˜p
1
( ¯x) ∨ ˜p ( ¯x) P ∀x ˜p
1
( ¯x), ˜p
1
( ¯x) → θ
1
}
(32)
|Consequent| = 10 + 4 · | ¯x| + | ˜p
1
( ¯x) → θ
1
| ≤ 15 · (1 + | ¯x|) + | ˜p
1
( ¯x) → θ
1
|
Negative
interpolation
∀x θ
1
→ ˜p ( ¯x)
(∀x ˜p
1
( ¯x) → ˜p ( ¯x)) ∧ (θ
1
→ ˜p
1
( ¯x))
(33)
|Consequent| = 6 + 2 · | ¯x| + |θ
1
→ ˜p
1
( ¯x)| ≤ 13 · (1 + | ¯x|) + |θ
1
→ ˜p
1
( ¯x)|
Negative
interpolation
∀x θ
1
→ ˜p ( ¯x)
{∀x ˜p
1
( ¯x) ≺ ˜p ( ¯x) ∨ ∀x ˜p
1
( ¯x) P ˜p ( ¯x), θ
1
→ ˜p
1
( ¯x)}
(34)
|Consequent| = 10 + 4 · | ¯x| + |θ
1
→ ˜p
1
( ¯x)| ≤ 15 · (1 + | ¯x|) + |θ
1
→ ˜p
1
( ¯x)|
∃x θ
1
Positive
interpolation
˜p ( ¯x) → ∃x θ
1
( ˜p ( ¯x) → ∃x ˜p
1
( ¯x)) ∧ ( ˜p
1
( ¯x) → θ
1
)
(35)
|Consequent| = 6 + 2 · | ¯x| + | ˜p
1
( ¯x) → θ
1
| ≤ 13 · (1 + | ¯x|) + | ˜p
1
( ¯x) → θ
1
|
Positive
interpolation
˜p ( ¯x) → ∃x θ
1
{ ˜p ( ¯x) ≺ ∃x ˜p
1
( ¯x) ∨ ˜p ( ¯x) P ∃x ˜p
1
( ¯x), ˜p
1
( ¯x) → θ
1
}
(36)
|Consequent| = 10 + 4 · | ¯x| + | ˜p
1
( ¯x) → θ
1
| ≤ 15 · (1 + | ¯x|) + | ˜p
1
( ¯x) → θ
1
|
Negative
interpolation
∃x θ
1
→ ˜p ( ¯x)
(∃x ˜p
1
( ¯x) → ˜p ( ¯x)) ∧ (θ
1
→ ˜p
1
( ¯x))
(37)
|Consequent| = 6 + 2 · | ¯x| + |θ
1
→ ˜p
1
( ¯x)| ≤ 13 · (1 + | ¯x|) + |θ
1
→ ˜p
1
( ¯x)|
Negative
interpolation
∃x θ
1
→ ˜p ( ¯x)
{∃x ˜p
1
( ¯x) ≺ ˜p ( ¯x) ∨ ∃x ˜p
1
( ¯x) P ˜p ( ¯x), θ
1
→ ˜p
1
( ¯x)}
(38)
|Consequent| = 10 + 4 · | ¯x| + |θ
1
→ ˜p
1
( ¯x)| ≤ 15 · (1 + | ¯x|) + |θ
1
→ ˜p
1
( ¯x)|
4 HYPERRESOLUTION OVER
ORDER CLAUSES
4.1 Restrictions on Order Clauses
The described translation produces order clausal
theories in some restrictive form, which will be
utilised in devising an order hyperresolution calcu-
lus. Let π ∈ Form
L
. π is a quantified atom of L
iff π = 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
Qx a. Let ε
i
, 1 ≤ i ≤ n, be either an expression
or a set of expressions or a set of sets of expres-
sions, in general. By qatoms(ε
1
,. . . ,ε
n
) ⊆ QAtom
L
|
qatoms
Q
(ε
1
,. . . ,ε
n
) ⊆ QAtom
Q
L
we denote the set of
all quantified atoms | quantified atoms of the form
Qx a of L occurring in ε
1
,. . . ,ε
n
. Let Qx p(t
0
,. . . ,t
τ
) ∈
QAtom
L
, x ∈ vars(p(t
0
,. . . ,t
τ
)), and p(t
0
0
,. . . ,t
0
τ
) ∈
Atom
L
. We denote
p(t
0
,. . . ,t
τ
)[i] = t
i
,i ≤ τ,
boundindset(Qx p(t
0
,. . . ,t
τ
)) = {i |i ≤ τ,t
i
= x} 6=
/
0.
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
0
≤ k, r
i
< r
i
0
, 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
,t
r
i
∈ Term
L
,
freetermseq(p(t
0
0
,. . . ,t
0
τ
)) = t
0
0
,. . . ,t
0
τ
,t
0
i
∈ Term
L
,
freetermseq(p(t
0
0
,. . . ,t
0
τ
)/Qx p(t
0
,. . . ,t
τ
)) =
t
0
r
1
,. . . ,t
0
r
k
,t
0
r
i
∈ Term
L
.
Let l be an order literal of L. l is admissible iff l =
a  b, a,b ∈ Atom
L
∪ QAtom
L
. Let C ∈ OrdCl
L
. C
is admissible iff, for all l ∈ C, l is admissible. Let
S ⊆ OrdCl
L
. S is admissible iff, for all C ∈ S, C is
admissible. Let J ⊆ I and S ⊆ OrdCl
L∪{ ˜p | ∈J}
. S is
semantically admissible iff
(a) S is admissible;
(b) for all a ∈ qatoms(S), there exists
∗
∈ J such that
preds(a) = { ˜p
∗
};
(c) for all Qxa, Q
0
x
0
a
0
∈ qatoms(S), if
preds(a) = preds(a
0
), then Q = Q
0
, x = x
0
,
boundindset(Qx a) = boundindset(Q
0
x
0
a
0
).
Corollary 4.1. Let T ⊆ Form
L
be countable; φ ∈
Form
L
; F ⊆ I such that there exists n
0
and F ∩
{(i, j)|i ≥ n
0
} =
/
0. There exist J
φ
T
⊆ I, J
φ
T
∩ F =
/
0,
and S
φ
T
⊆ OrdCl
L∪{ ˜p | ∈J
φ
T
}
being countable 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, kJ
φ
T
k ∈ O(|T | +
|φ|); S
φ
T
⊆
F
OrdCl
L∪{ ˜p | ∈J
φ
T
}
, |S
φ
T
| ∈ O(|T |
2
+
|φ|
2
); the number of all basic 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
+ |φ|
2
) · log(|T | +
|φ|));
(iii) S
φ
T
is semantically admissible.
Proof. (i) We put J
n
0
= {(n
0
, j)| j ∈ N} ⊆ I and
G = F ∪ J
n
0
⊆ I. We get by Lemma 3.1(ii) for T ,
G, n
0
+ 1 that there exist J
T
⊆ I, J
T
∩ G =
/
0; S
T
⊆
OrdCl
L∪{ ˜p | ∈J
T
}
being countable; and 3.1(ii a–d)

hold for T , J
T
, S
T
. By (13) for φ ∈ Form
L
, there ex-
ists φ
0
∈ Form
L
such that (13a–e) hold for φ
0
. We
then distinguish three cases for φ
0
. Case 1: φ
0
=
0. We put J
φ
T
= J
T
⊆ I, J
φ
T
∩ F =
/
0, and S
φ
T
=
S
T
⊆ OrdCl
L∪{ ˜p | ∈J
φ
T
}
being countable. Case 2:
φ
0
= 1. We put J
φ
T
=
/
0 ⊆ I, J
φ
T
∩ F =
/
0, and S
φ
T
=
{} ⊆ OrdCl
L
being countable. Case 3: φ
0
6= 0
and φ
0
6= 1. Let ¯x = varseq(φ
0
). Let = (n
0
,0) ∈
I, ˜p ∈
˜
P, ar( ˜p ) = | ¯x|. We get by (14) for ∀ ¯x φ
0
,
¯x, F, n
0
, n
0
, , ˜p that there exist n
J
−
, J
−
=
{(n
0
, j)| 1 ≤ j ≤ n
J
−
} ⊆
F
I, J
−
∩ (F ∪ { }) =
/
0;
S
−
⊆
F
OrdCl
L∪{ ˜p }∪{ ˜p | ∈J
−
}
; and (14a,f,h,j) hold
for ∀ ¯x φ
0
, ¯x, ˜p , J
−
, S
−
. We put J
φ
T
= J
T
∪ { } ∪
J
−
⊆ I, J
φ
T
∩ F =
/
0, and S
φ
T
= S
T
∪ { ˜p ( ¯x) ≺ 1} ∪
S
−
⊆ OrdCl
L∪{ ˜p | ∈J
φ
T
}
being countable. (ii) and (iii)
straightforwardly follow from the translation via in-
terpolation. Let l ∈ C ∈ S
φ
T
. Then either l = a P b or
l = c P a or l = a P c or l = a ≺ b or l = c ≺ a or
l = a ≺ c, a,b ∈ atoms(S
φ
T
), c ∈ qatoms(S
φ
T
).
4.2 Substitutions
We assume the reader to be familiar with the standard
notions of substitutions. Let X = {x
1
,. . . ,x
n
} ⊆
F
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
.
We define id
L
: Var
L
−→ Var
L
, id
L
(x) = x. Let
ϑ ∈ Subst
L
. Let Qx a ∈ QAtom
L
, x ∈ vars(a). ϑ is
applicable to Qx a iff dom(ϑ) ⊇ freevars(Qx a) and
x 6∈ range(ϑ|
freevars(Qx a)
). We define the application
of ϑ to Qx a as (Qx a)ϑ = Qx a(ϑ|
freevars(Qx a)
∪x/x) ∈
QAtom
L
. Let ε
1
 ε
2
be an admissible order literal
of L. We define the application of ϑ to ε
1
 ε
2
as
(ε
1
 ε
2
)ϑ = ε
1
ϑ  ε
2
ϑ being an admissible order lit-
eral of L. Let ε, ε
0
be either expressions or sets of
expressions of L, in this context. ε
0
is an instance of
ε of L iff there exists ϑ ∈ Subst
L
such that ε
0
= εϑ. ε
0
is a variant of ε of L iff there exists a variable renam-
ing ρ ∈ Subst
L
such that ε
0
= ερ. Let C ∈ OrdCl
L
be
admissible and S ⊆ OrdCl
L
be admissible. C is an in-
stance | 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
.
Let E = E
0
,. . . ,E
n
, E
i
is a set of expressions of L,
in this context. We define the application of ϑ to E
as Eϑ = E
0
ϑ,. . . ,E
n
ϑ. ϑ is a unifier of L for E iff,
for all i ≤ n, ϑ is a unifier of L for E
i
. 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
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 gen-
eral unifiers of L for E.
Theorem 4.2 (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
∈ OrdCl
L
is admissible.
If there exists a unifier of L for E, then there ex-
ists a most general unifier θ of L for E such that
range(θ|
freevars(E)
) ∩ boundvars(E) =
/
0.
Proof. Standard.
4.3 Order Hyperresolution Rules
At first, we introduce some basic notions and notation
concerning chains of admissible order literals. Let
ε
1
,ε
2
∈ Atom
L
∪ QAtom
L
. ε
1
E ε
2
iff either ε
1
= ε
2
or ε
1
= 0 or ε
2
= 1; or ε
1
= ∀x a, x ∈ vars(a), there
exists t ∈ Term
L
and ε
2
= a(x/t ∪ id
L
|
vars(a)−{x}
); or
ε
2
= ∃x a, x ∈ vars(a), there exists t ∈ Term
L
and
ε
1
= a(x/t ∪ id
L
|
vars(a)−{x}
).
Let J ⊆ I and S ⊆ OrdCl
L∪{ ˜p | ∈J}
be se-
mantically admissible. For all ε
1
,ε
2
,ε
3
∈
Atom
L∪{ ˜p | ∈J}
∪ qatoms(S), if ε
1
E ε
2
E ε
3
,
then ε
1
E ε
3
.
(39)
The proof. A straightforward consequence of the se-
mantical admissibility of S.
Let R
i
= t
i
1
,. . . ,t
i
m
, t
i
j
∈ Term
L
, i = 1,2. We define
the union of R
1
and R
2
as
R
1
∪ R
2
= {t
1
1
,t
2
1
},. . . ,{t
1
m
,t
2
m
},
{t
1
j
,t
2
j
} ⊆
F
Term
L
.
Note that if m = 0, then R
1
∪ R
2
= `. Let J ⊆ I and
S ⊆ OrdCl
L∪{ ˜p | ∈J}
be semantically admissible. Let
ε
i
∈ Atom
L∪{ ˜p | ∈J}
∪ qatoms(S), i = 1, 2. We define
the sequence ε
1
E
E
E ε
2
of the form either
/
0 6= A ⊆
F
Atom
L∪{ ˜p | ∈J}
or
/
0 6= A ⊆
F
qatoms(S) or R
1
,. . . ,R
n
,
R
i
= {t
i
1
,t
i
2
} ⊆
F
Term
L
, in Table 5.
A chain Ξ of L is a sequence Ξ = ε
0

0
υ
0
,. . . ,ε
n

n
υ
n
; ε
i

i
υ
i
is an admissible order literal of L. ε
0
∈
Atom
L
∪ QAtom
L
is the beginning element of Ξ and
υ
n
∈ Atom
L
∪ QAtom
L
the ending element of Ξ.
ε
0
Ξυ
n
denotes Ξ together with its respective begin-
ning 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
=P, and for all i < n, υ
i
= ε
i+1
. Ξ is
an increasing chain of L iff, for all i < n, υ
i
E ε
i+1
.
Let Ξ = ε
0

0
υ
0
,. . . ,ε
n

n
υ
n
be an increasing chain of
L. Ξ is a strictly increasing chain of L iff there exists
i
∗
≤ n such that 
i
∗
=≺. Ξ is an unstrictly increasing
chain of L iff, for all i ≤ n, 
i
=P. Let Ξ be a chain
of L. Ξ is a contradiction of L iff εΞυ is a strictly in-
creasing chain of L and υ E ε. Let S ⊆ OrdCl
L
be ad-
missible, unit and Ξ = ε
0

0
υ
0
,. . . ,ε
n

n
υ
n
be a chain |

Table 5: ε
1
E
E
E ε
2
.
ε
1
E
E
E ε
2
=



















` if ε
1
= 0;
` if ε
2
= 1;
{ε
1
,ε
2
} if either ε
1
6= 0, ε
2
6= 1, ε
1
,ε
2
∈ Atom
L∪{ ˜p | ∈J}
, or ε
1
,ε
2
∈ qatoms(S);
freetermseq(ε
1
) ∪ freetermseq(ε
2
/ε
1
) if ε
1
∈ qatoms(S)
∀
,ε
2
∈ Atom
L∪{ ˜p | ∈J}
,preds(ε
1
) = preds(ε
2
);
freetermseq(ε
1
/ε
2
) ∪ freetermseq(ε
2
) if ε
1
∈ Atom
L∪{ ˜p | ∈J}
,ε
2
∈ qatoms(S)
∃
,preds(ε
1
) = preds(ε
2
).
an equality chain | an increasing chain | a strictly in-
creasing chain | an unstrictly increasing chain | a con-
tradiction of L. Ξ is a chain | an equality chain | an
increasing chain | a strictly increasing chain | an un-
strictly increasing chain | a contradiction of S iff, for
all i ≤ n, ε
i

i
υ
i
∈ S.
Let
˜
W = { ˜w
α
|ar( ˜w
α
) = 0,α < ω} such that
˜
W ∩
Func
L
=
/
0;
˜
W is an infinite countable set of new con-
stant symbols. Let J ⊆ I and S ⊆ OrdCl
L∪{ ˜p | ∈J}
be semantically admissible. A basic order hyper-
resolution calculus is defined in Table 6. The basic
order hyperresolution calculus can be generalised to
an order hyperresolution one in Table 7. Let L
0
=
L ∪ { ˜p | ∈ J} and S
0
= {0 ≺ 1} ∪ S ⊆ OrdCl
L
0
. Let
D = C
0
,. . . ,C
n
, C
κ
∈ OrdCl
L∪
˜
W ∪{ ˜p | ∈J}
. D is a de-
duction of C
n
from S by basic order | basic witnessing
order | order hyperresolution iff, for all κ ≤ n, C
κ
∈ S
0
,
or there exist j
∗
k
< κ, k ≤ m, such that C
κ
is an or-
der resolvent of C
0
j
∗
0
,. . . ,C
0
j
∗
m
using Rule (40)–(43) |
Rule (40)–(45) | Rule (46)–(48) where C
0
j
∗
k
is an in-
stance | a variant of C
j
∗
k
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
α
∗
} in case of Rule (44), (45),
L
κ−1
else;
S
κ
= S
κ−1
∪ {C
κ
} ⊆ OrdCl
L
κ
.
D is a refutation of S iff C
n
= . By
clo
BH
(S) ⊆ OrdCl
L∪
˜
W ∪{ ˜p | ∈J}
| clo
BW H
(S) ⊆
OrdCl
L∪
˜
W ∪{ ˜p | ∈J}
| clo
H
(S) ⊆ OrdCl
L∪
˜
W ∪{ ˜p | ∈J}
we denote the closure of S under basic order | basic
witnessing order | order hyperresolution.
Lemma 4.3 (Lifting Lemma). Let J ⊆ I and S ⊆
OrdCl
L∪{ ˜p | ∈J}
be semantically admissible. If C ∈
clo
BH
(S), then there exists C
∗
∈ clo
H
(S) such that C
is an instance of C
∗
of L ∪
˜
W ∪ { ˜p | ∈ J}.
Proof. By complete induction on the length of a de-
duction of C from S by basic order hyperresolu-
tion.
We are in position to prove the refutational sound-
ness and completeness of the order hyperresolution
calculus.
Theorem 4.4 (Refutational Soundness and Complete-
ness). Let J ⊆ I and S ⊆ OrdCl
L∪{ ˜p | ∈J}
be count-
able, semantically admissible.  ∈ clo
H
(S) if and
only if S is unsatisfiable.
Proof. (=⇒) Let A be a model of S for L ∪
˜
W ∪
{ ˜p | ∈ J} and C ∈ clo
H
(S). Then A |= C. The proof
is by complete induction on the length of a deduc-
tion of C from S by order hyperresolution. Let  ∈
clo
H
(S). Let A be a model of S for L ∪
˜
W ∪ { ˜p | ∈
J}. We get A |= , which is a contradiction. We con-
clude that S is unsatisfiable.
(⇐=) Let L contain a constant symbol, S 6=
/
0,
 6∈ clo
H
(S). We get by Lemma 4.3 for J, S,  that
 6∈ clo
BH
(S). It is straightforward to prove that there
exist L
∗
being an expansion of L ∪ { ˜p | ∈ J}, a re-
duction of L ∪
˜
W ∪ { ˜p | ∈ J}; and S
clo
⊆ OrdCl
L
∗
being countable, semantically admissible, S
clo
⊇ S,
 6∈ S
clo
, S
clo
= Inst
L
∗
(S
clo
), S
clo
= clo
BW H
(S
clo
);
the condition of completeness (49) (formulated be-
low) holds. Then S
clo
|= S and 0 ≺ 1 ∈ S
clo
.
We put S = {C |C ∈ S
clo
is unit, freevars(C) =
/
0} ⊆
OrdCl
L
∗
, U
A
= GTerm
L
∗
6=
/
0, B = GAtom
L
∗
∪
qatoms(S). Hence, 0,1 ∈ B; B is countable; there
exist 2 ≤ γ
B
≤ ω and a sequence δ : γ
B
−→ B of B
such that δ(0) = 0, δ(1) = 1. Let ε
1
,ε
2
∈ B. ε
1
PP ε
2
iff ε
1
= ε
2
or there exists an equality chain ε
1
Ξε
2
of
S. ε
1
≺≺ ε
2
iff there exists a strictly increasing chain
υ
1
Ξυ
2
of S and ε
1
E υ
1
Ξυ
2
E ε
2
. We can formulate
the condition of completeness as follows:
for all ε
1
,ε
2
∈ B,
either ε
1
≺≺ ε
2
or ε
1
PP ε
2
or ε
2
≺≺ ε
1
.
(49)
Note that 0 ≺≺ 1.
0 6PP 1; for all ε
1
∈ B, ε
1
6≺≺ 0, 1 6≺≺ ε
1
, ε
1
6≺≺ ε
1
.
(50)
The proof is straightforward.
Let {0,1} ⊆ X ⊆ B. A partial valuation V is a
mapping
V : X −→ [0, 1] such that V (0) = 0, V (1) = 1.

Table 6: Basic order hyperresolution calculus.
(Basic order hyperresolution rule) (40)
l
0
∨C
0
,.. . ,l
n
∨C
n
∈ S
I
κ
n
_
i=0
C
i
∈ S
κ+1
;
l
0
,.. . ,l
n
is a contradiction of L
κ
.
(Basic order hyperresolution rule of rank r) (41)
m
0
_
j=0
l
0
j
∨C
0
,.. . ,
m
n
_
j=0
l
n
j
∨C
n
∈ S
I
κ
n
_
i=0
C
i
∈ S
κ+1
;
for all i ≤ n, m
i
≤ r;
for all S ∈ S el({m
i
+ 1 |i ≤ n}), there exists a contradiction of {l
i
S(i)
|i ≤ n};
there does not exist
/
0 6= I ⊂ n + 1 such that for all S ∈ S el({m
i
+ 1 |i ∈ I}), there exists a contradiction of {l
i
S(i)
|i ∈ I}.
(Basic order ∀-saturation rule) (42)
ε
0

0
υ
0
∨C
0
,.. . ,ε
n

n
υ
n
∨C
n
∈ S
I
κ
χ ≺ µ ∨ χ P µ ∨
n
_
i=0
C
i
∈ S
κ+1
;
υ
n
∈ atoms(S
I
κ
),µ ∈ qatoms(S
I
κ
)
∀
,µ E υ
n
,υ
n
[min(boundindset(µ))] ∈ Var
L
,υ
n
[min(boundindset(µ))] 6∈ vars(freetermseq(υ
n
/µ)) ∪
S
n
i=0
freevars(C
i
);
ε
0

0
υ
0
,.. . ,ε
n

n
υ
n
is an increasing chain, υ
n
[min(boundindset(µ))] ∈
T
n
i=1
freevars(ε
i
) ∩
T
n−1
i=0
freevars(υ
i
);
ε
0
∈ atoms(S
I
κ
) − {0} ∪ qatoms(S
I
κ
),
if υ
n
[min(boundindset(µ))] 6∈ freevars(ε
0
), then χ = ε
0
, else qatoms(S
I
κ
)
∀
3 χ E ε
0
, υ
n
[min(boundindset(µ))] 6∈ freevars(χ).
(Basic order ∃-saturation rule) (43)
ε
0

0
υ
0
∨C
0
,.. . ,ε
n

n
υ
n
∨C
n
∈ S
I
κ
µ ≺ χ ∨ µ P χ ∨
n
_
i=0
C
i
∈ S
κ+1
;
ε
0
∈ atoms(S
I
κ
),µ ∈ qatoms(S
I
κ
)
∃
,ε
0
E µ, ε
0
[min(boundindset(µ))] ∈ Var
L
,ε
0
[min(boundindset(µ))] 6∈ vars(freetermseq(ε
0
/µ)) ∪
S
n
i=0
freevars(C
i
);
ε
0

0
υ
0
,.. . ,ε
n

n
υ
n
is an increasing chain, ε
0
[min(boundindset(µ))] ∈
T
n
i=1
freevars(ε
i
) ∩
T
n−1
i=0
freevars(υ
i
);
υ
n
∈ atoms(S
I
κ
) − {1} ∪ qatoms(S
I
κ
),
if ε
0
[min(boundindset(µ))] 6∈ freevars(υ
n
), then χ = υ
n
, else υ
n
E χ ∈ qatoms(S
I
κ
)
∃
, ε
0
[min(boundindset(µ))] 6∈ freevars(χ).
(Basic order ∀-witnessing rule) (44)
ε
0

0
υ
0
,.. . ,ε
n

n
υ
n
∈ S
I
κ
aγ ≺ υ
n
∈ S
κ+1
;
qatoms(S
I
κ
)
∀
3 ∀x a E ε
0

0
υ
0
,.. . ,ε
n

n
υ
n
is a strictly increasing chain such that for all i < n, 
i
=P, 
n
=≺, freevars(∀x a) ∪
S
n
i=0
freevars(ε
i

i
υ
i
) =
/
0;
˜w
α
∗
∈
˜
W , ˜w
α
∗
6∈ Func
L
κ
;
γ = x/ ˜w
α
∗
∈ Subst
L
κ+1
,dom(γ) = vars(a).
(Basic order ∃-witnessing rule) (45)
ε
0

0
υ
0
,.. . ,ε
n

n
υ
n
∈ S
I
κ
ε
0
≺ aγ ∈ S
κ+1
;
ε
0

0
υ
0
,.. . ,ε
n

n
υ
n
E ∃x a ∈ qatoms(S
I
κ
)
∃
is a strictly increasing chain such that 
0
=≺, for all 1 ≤ i ≤ n, 
i
=P,
S
n
i=0
freevars(ε
i

i
υ
i
) ∪ freevars(∃xa) =
/
0;
˜w
α
∗
∈
˜
W , ˜w
α
∗
6∈ Func
L
κ
;
γ = x/ ˜w
α
∗
∈ Subst
L
κ+1
,dom(γ) = vars(a).
S
I
κ
= Inst
L
κ
(S
κ
) ⊆ OrdCl
L
κ
.

Table 7: Order hyperresolution calculus.
(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
I
κ

n
_
i=0
m
i
_
j=1
l
i
j

θ ∈ S
κ+1
;
θ ∈ mgu
L
κ

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
E
E
E ε
1
0
,.. . ,υ
n−1
0
E
E
E ε
n
0
,υ
n
0
E
E
E ε
0
0

,
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}

;
there exists i
∗
≤ n such that 
i
∗
0
=≺.
(Order ∀-saturation rule) (47)
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
I
κ
χ ≺ µ ∨ χ P µ ∨

n
_
i=0
m
i
_
j=1
l
i
j

θ ∈ S
κ+1
;
θ ∈ mgu
L
κ

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
E
E
E ε
1
0
,.. . ,υ
n−1
0
E
E
E ε
n
0

,
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}

;
υ
n
0
θ ∈ atoms(S
I
κ
),µ ∈ qatoms(S
I
κ
)
∀
,µ E υ
n
0
θ,
υ
n
0
θ[min(boundindset(µ))] ∈ Var
L
,υ
n
0
θ[min(boundindset(µ))] 6∈ vars(freetermseq(υ
n
0
θ/µ)) ∪
S
n
i=0
freevars({l
i
j
θ|1 ≤ j ≤ m
i
});
υ
n
0
θ[min(boundindset(µ))] ∈
T
n
i=1
freevars(ε
i
0
θ) ∩
T
n−1
i=0
freevars(υ
i
0
θ);
ε
0
0
θ ∈ atoms(S
I
κ
) − {0} ∪ qatoms(S
I
κ
),
if υ
n
0
θ[min(boundindset(µ))] 6∈ freevars(ε
0
0
θ), then χ = ε
0
0
θ, else qatoms(S
I
κ
)
∀
3 χ E ε
0
0
θ, υ
n
0
θ[min(boundindset(µ))] 6∈ freevars(χ).
(Order ∃-saturation rule) (48)
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
I
κ
µ ≺ χ ∨ µ P χ ∨

n
_
i=0
m
i
_
j=1
l
i
j

θ ∈ S
κ+1
;
θ ∈ mgu
L
κ

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
E
E
E ε
1
0
,.. . ,υ
n−1
0
E
E
E ε
n
0

,
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}

;
ε
0
0
θ ∈ atoms(S
I
κ
),µ ∈ qatoms(S
I
κ
)
∃
,ε
0
0
θ E µ,
ε
0
0
θ[min(boundindset(µ))] ∈ Var
L
,ε
0
0
θ[min(boundindset(µ))] 6∈ vars(freetermseq(ε
0
0
θ/µ)) ∪
S
n
i=0
freevars({l
i
j
θ|1 ≤ j ≤ m
i
});
ε
0
0
θ[min(boundindset(µ))] ∈
T
n
i=1
freevars(ε
i
0
θ) ∩
T
n−1
i=0
freevars(υ
i
0
θ);
υ
n
0
θ ∈ atoms(S
I
κ
) − {1} ∪ qatoms(S
I
κ
),
if ε
0
0
θ[min(boundindset(µ))] 6∈ freevars(υ
n
0
θ), then χ = υ
n
0
θ, else υ
n
0
θ E χ ∈ qatoms(S
I
κ
)
∃
, ε
0
0
θ[min(boundindset(µ))] 6∈ freevars(χ).
For all i < i
0
≤ 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
0
j=0
ε
i
0
j

i
0
j
υ
i
0
j
∨
W
m
i
0
j=1
l
i
0
j
) =
/
0; S
I
κ
= Inst
L
κ
(S
κ
) ⊆ OrdCl
L
κ
.

We denote dom(V ) = X, {0,1} ⊆ dom(V ) ⊆ B. We
define a partial valuation V
α
by recursion on 2 ≤ α ≤
γ
B
as follows:
V
2
= {(0,0),(1, 1)};
V
α
= V
α−1
∪ {(δ(α − 1), λ
α−1
)}
(3 ≤ α ≤ γ
B
is a successor ordinal),
E
PP
α−1
= {V
α−1
(a)| a PP δ(α − 1),a ∈ dom(V
α−1
)},
D
≺≺
α−1
= {V
α−1
(a)| a ≺≺ δ(α − 1),a ∈ dom(V
α−1
)},
U
≺≺
α−1
= {V
α−1
(a)| δ(α − 1) ≺≺ a, a ∈ dom(V
α−1
)},
λ
α−1
=



W
D
≺≺
α−1
+
V
U
≺≺
α−1
2
if E
PP
α−1
=
/
0,
W
E
PP
α−1
else;
V
γ
B
=
[
α<γ
B
V
α
(γ
B
is a limit ordinal).
For all 2 ≤ α ≤ γ
B
, V
α
is a partial valuation,
dom(V
α
) = δ[α]; and for all 2 ≤ α ≤ α
0
≤ γ
B
,
V
α
⊆ V
α
0
.
(51)
The proof is by induction on 2 ≤ α ≤ γ
B
.
For all 2 ≤ α ≤ γ
B
, for all a,a
0
∈ dom(V
α
),
if a PP a
0
, then V
α
(a) = V
α
(a
0
);
if a ≺≺ a
0
, then V
α
(a) < V
α
(a
0
);
if V
α
(a) = 0, then a PP 0;
if V
α
(a) = 1, then a PP 1.
(52)
The proof is by induction on 2 ≤ α ≤ γ
B
.
We put V = V
γ
B
, dom(V )
(51)
== δ[γ
B
] = B;
f
A
(u
1
,. . . ,u
τ
) = f (u
1
,. . . ,u
τ
),
f ∈ Func
L
∗
,u
i
∈ U
A
;
p
A
(u
1
,. . . ,u
τ
) = V (p(u
1
,. . . ,u
τ
)),
p ∈ Pred
L
∗
,u
i
∈ U
A
;
A =

U
A
,{ f
A
| f ∈ Func
L
∗
},{p
A
| p ∈ Pred
L
∗
}

.
We get A |= S
clo
|= S ⊆ OrdCl
L∪{ ˜p | ∈J}
. We con-
clude that A|
L∪{ ˜p | ∈J}
is a model of S for L ∪
{ ˜p | ∈ J} and S is satisfiable. The theorem is
proved.
In Table 8, we show that φ = ∀x (q
1
(x) → q
2
) →
(∃x q
1
(x) → q
2
) ∈ Form
L
is logically valid using the
proposed translation to order clausal form and the ba-
sic order hyperresolution calculus.
5 CONCLUSIONS
The order hyperresolution calculus is amenable to
adding the projection operator ∆
3
to G
¨
odel logic, as
a unary connective of L. Henceforward, we sup-
pose that Form
L
designates the set of all formulae
of L built up from Atom
L
and Var
L
using the con-
nectives: ¬, ∆, ∧, ∨, →, P, ≺, and the quantifiers:
∀, ∃; OrdForm
L
designates the set of all order for-
mulae of L built up from Atom
L
and Var
L
using the
connectives: ¬, ∆, ∧, ∨, →, P, ≺, and the quan-
tifiers: ∀, ∃. We slightly modify the definition of
literal. Let l ∈ Form
L
. l is a literal of L iff either
l = a or l = a → b or l = (a → b) → b or l = ∆ d → a
or l = a → ∆ d or l = Qx c → a or l = a → Qx c
where a,c,d ∈ Atom
L
− {0,1}, b ∈ Atom
L
− {1},
x ∈ vars(c). The definition of order literal remains
unchanged. We add two rows to Table 1, given in
Table 9. We add unary interpolation rules for ∆,
Table 10. This way modified Lemma 3.1 will still
hold. Thanks to having the definition of order lit-
eral unchanged, the rest of the formal treatment re-
mains intact. So, in the countable case, we have pro-
posed a refutation sound and complete hyperresolu-
tion proof method over semantically admissible order
clausal theories together with an efficient translation
of theories in general G
¨
odel logic (with ∆) to such
clausal theories, and hence, we have solved the deduc-
tion problem of a formula from a theory in the context
of automated deduction.
REFERENCES
Baaz, M., Ciabattoni, A., and Ferm
¨
uller, C. G. (2001). Her-
brand’s theorem for prenex G
¨
odel logic and its conse-
quences for theorem proving. In Nieuwenhuis, R. and
Voronkov, A., editors, LPAR, volume 2250 of Lecture
Notes in Computer Science, pages 201–215. Springer.
Baaz, M., Ciabattoni, A., and Ferm
¨
uller, C. G. (2003). Hy-
persequent calculi for G
¨
odel logics - a survey. J. Log.
Comput., 13(6):835–861.
Baaz, M. and Ferm
¨
uller, C. G. (2010). A resolution mecha-
nism for prenex G
¨
odel logic. In Dawar, A. and Veith,
H., editors, CSL, volume 6247 of Lecture Notes in
Computer Science, pages 67–79. Springer.
Bachmair, L. and Ganzinger, H. (1994). Rewrite-based
equational theorem proving with selection and simpli-
fication. J. Log. Comput., 4(3):217–247.
Bachmair, L. and Ganzinger, H. (1998). Ordered chaining
calculi for first-order theories of transitive relations. J.
ACM, 45(6):1007–1049.
de la Tour, T. B. (1992). An optimality result for clause
form translation. J. Symb. Comput., 14(4):283–302.
3
Cf. INTRODUCTION.

Table 8: An example: φ = ∀x (q
1
(x) → q
2
) → (∃x q
1
(x) → q
2
).
φ = ∀x (q
1
(x) → q
2
) → (∃x q
1
(x) → q
2
)
{ ˜p
0
(x) ≺ 1,

∀x (q
1
(x) → q
2
)
| {z }
˜p
1
(x)
→ (∃x q
1
(x) → q
2
| {z }
˜p
2
(x)
)

→ ˜p
0
(x)} (26)
{ ˜p
0
(x) ≺ 1, ˜p
2
(x) ≺ ˜p
1
(x) ∨ ˜p
2
(x) P 1 ∨ ˜p
0
(x) P 1, ˜p
2
(x) ≺ ˜p
0
(x) ∨ ˜p
2
(x) P ˜p
0
(x), ˜p
1
(x) → ∀x (q
1
(x) → q
2
| {z }
˜p
3
(x)
),(∃xq
1
(x)
| {z }
˜p
4
(x)
→ q
2
|{z}
˜p
5
(x)
) → ˜p
2
(x)} (32),(26)
{ ˜p
0
(x) ≺ 1, ˜p
2
(x) ≺ ˜p
1
(x) ∨ ˜p
2
(x) P 1 ∨ ˜p
0
(x) P 1, ˜p
2
(x) ≺ ˜p
0
(x) ∨ ˜p
2
(x) P ˜p
0
(x), ˜p
1
(x) ≺ ∀x ˜p
3
(x) ∨ ˜p
1
(x) P ∀x ˜p
3
(x), ˜p
3
(x) → (q
1
(x)
|{z}
˜p
6
(x)
→ q
2
|{z}
˜p
7
(x)
),
˜p
5
(x) ≺ ˜p
4
(x) ∨ ˜p
5
(x) P 1 ∨ ˜p
2
(x) P 1, ˜p
5
(x) ≺ ˜p
2
(x) ∨ ˜p
5
(x) P ˜p
2
(x), ˜p
4
(x) → ∃x q
1
(x)
|{z}
˜p
8
(x)
,q
2
≺ ˜p
5
(x) ∨ q
2
P ˜p
5
(x)} (24),(36)
S
φ
=

˜p
0
(x) ≺ 1 [1]
˜p
2
(x) ≺ ˜p
1
(x) ∨ ˜p
2
(x) P 1 ∨ ˜p
0
(x) P 1 [2]
˜p
2
(x) ≺ ˜p
0
(x) ∨ ˜p
2
(x) P ˜p
0
(x) [3]
˜p
1
(x) ≺ ∀x ˜p
3
(x) ∨ ˜p
1
(x) P ∀x ˜p
3
(x) [4]
˜p
3
(x) ≺ ˜p
7
(x) ∨ ˜p
3
(x) P ˜p
7
(x) ∨ ˜p
6
(x) ≺ ˜p
7
(x) ∨
˜p
6
(x) P ˜p
7
(x) [5]
q
1
(x) ≺ ˜p
6
(x) ∨ q
1
(x) P ˜p
6
(x) [6]
˜p
7
(x) ≺ q
2
∨ ˜p
7
(x) P q
2
[7]
˜p
5
(x) ≺ ˜p
4
(x) ∨ ˜p
5
(x) P 1 ∨ ˜p
2
(x) P 1 [8]
˜p
5
(x) ≺ ˜p
2
(x) ∨ ˜p
5
(x) P ˜p
2
(x) [9]
˜p
4
(x) ≺ ∃x ˜p
8
(x) ∨ ˜p
4
(x) P ∃x ˜p
8
(x) [10]
˜p
8
(x) ≺ q
1
(x) ∨ ˜p
8
(x) P q
1
(x) [11]
q
2
≺ ˜p
5
(x) ∨ q
2
P ˜p
5
(x)

[12]
Rule (40) : [1][2] :
˜p
2
(x) ≺ ˜p
1
(x) ∨ ˜p
2
(x) P 1 [13]
Rule (40) : [3][13] :
˜p
2
(x) P ˜p
0
(x) ∨ ˜p
2
(x) ≺ ˜p
1
(x) [14]
Rule (40) : [1][13][14] :
˜p
2
(x) ≺ ˜p
1
(x) [15]
Rule (40) : [8][15] :
˜p
5
(x) ≺ ˜p
4
(x) ∨ ˜p
5
(x) P 1 [16]
Rule (40) : [9][16] :
˜p
5
(x) P ˜p
2
(x) ∨ ˜p
5
(x) ≺ ˜p
4
(x) [17]
Rule (40) : [15][16][17] :
˜p
5
(x) ≺ ˜p
4
(x) [18]
Rule (41) : [4][5][7][9][12][15] :
˜p
6
(x) ≺ ˜p
7
(x) ∨ ˜p
6
(x) P ˜p
7
(x) [19]
repeatedly Rule (43) : [6][7][11][19] :
.
.
.
.
.
.
.
.
.
∃x ˜p
8
(x) ≺ q
2
∨ ∃x ˜p
8
(x) P q
2
[20]
Rule (41) : [10][12][18][20] :
 [21]

Table 9: Translation of l to C.
Case: l C
8 ∆ d → a d ≺ 1 ∨ a P 1 |C| ≤ 3 · |l|
9 a → ∆ d a P 0 ∨ d P 1 |C| ≤ 3 · |l|
a,d ∈ Atom
L
− {0,1}.
Table 10: Unary interpolation rules for ∆.
Case:
∆θ
1
Positive
interpolation
˜p ( ¯x) → ∆ θ
1
( ˜p ( ¯x) → ∆ ˜p
1
( ¯x)) ∧ ( ˜p
1
( ¯x) → θ
1
)
(53)
|Consequent| = 5 + 2 · | ¯x| + | ˜p
1
( ¯x) → θ
1
| ≤ 13 · (1 + | ¯x|) + | ˜p
1
( ¯x) → θ
1
|
Positive
interpolation
˜p ( ¯x) → ∆ θ
1
{ ˜p ( ¯x) P 0 ∨ ˜p
1
( ¯x) P 1, ˜p
1
( ¯x) → θ
1
}
(54)
|Consequent| = 6 + 2 · | ¯x| + | ˜p
1
( ¯x) → θ
1
| ≤ 15 · (1 + | ¯x|) + | ˜p
1
( ¯x) → θ
1
|
Negative
interpolation
∆θ
1
→ ˜p ( ¯x)
(∆ ˜p
1
( ¯x) → ˜p ( ¯x)) ∧ (θ
1
→ ˜p
1
( ¯x))
(55)
|Consequent| = 5 + 2 · | ¯x| + |θ
1
→ ˜p
1
( ¯x)| ≤ 13 · (1 + | ¯x|) + |θ
1
→ ˜p
1
( ¯x)|
Negative
interpolation
∆θ
1
→ ˜p ( ¯x)
{ ˜p
1
( ¯x) ≺ 1 ∨ ˜p ( ¯x) P 1, θ
1
→ ˜p
1
( ¯x)}
(56)
|Consequent| = 6 + 2 · | ¯x| + |θ
1
→ ˜p
1
( ¯x)| ≤ 15 · (1 + | ¯x|) + |θ
1
→ ˜p
1
( ¯x)|
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odel logic. In Filipe, J. and Kacprzyk, J., editors,
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