Hyperresolution for Propositional Product Logic
Du
ˇ
san Guller
Department of Applied Informatics, Comenius University, Mlynsk
´
a dolina, 842 48 Bratislava, Slovakia
Keywords:
Hyperresolution, Product Logic, Automated Deduction, Fuzzy Logics, Many-valued Logics.
Abstract:
We provide the foundations of automated deduction in the propositional product logic. Particularly, we gen-
eralise the hyperresolution principle to the propositional product logic. We propose translation of a formula
to an equivalent satisfiable finite order clausal theory, which consists of order clauses - finite sets of order
literals of the augmented form: ε
1
 ε
2
where ε
i
is either the truth constant 0 or 1 or a conjunction of powers
of propositional atoms, and  is the connective P or ≺. P and ≺ are interpreted by the standard equality
and strict order on [0,1], respectively. We devise a hyperresolution calculus over order clausal theories, which
is refutation sound and complete for the finite case. By means of the translation and calculus, we solve the
deduction problem T |= φ for a finite theory T and a formula φ.
1 INTRODUCTION
Automated deduction in fuzzy (many-valued) log-
ics has gradually been receiving an attention from
logicians, informaticians, and engineers. The rea-
son is its growing application potential in many
fields, spanning from engineering to informatics,
such as fuzzy control and optimisation of both dis-
crete and continuous industrial processes, knowl-
edge representation and reasoning, ontology lan-
guages, the Semantic Web, the Web Ontology Lan-
guage (OWL), fuzzy description logics and ontolo-
gies, multi-step fuzzy (many-valued) inference, fuzzy
knowledge/expert systems. An important subclass
consists of t-norm fuzzy logics, with the special cases
of continuous and left-continuous t-norm (Klement
and Mesiar, 2005; Klement et al., 2013). The stan-
dard semantics of a t-norm fuzzy logic is formed by
the unit interval of real numbers [0, 1] equipped with
the standard order, supremum, infimum, the t-norm
and its residuum. The condition of left-continuity
ensures the existence of the unique residuum for a
given t-norm. The basic logics of continuous and left-
continuous t-norm are the BL (basic) (H
´
ajek, 2001)
and MTL (monodial t-norm) (Esteva and Godo, 2001)
ones, respectively. G
¨
odel logic is one of the simplest
t-norm fuzzy logics with the (idempotent) minimum
t-norm. By the Mostert-Shields theorem (Mostert and
Shields, 1957), a t-norm is continuous if and only if it
is isomorphic to an ordinal sum (countably many open
Partially supported by VEGA Grant 1/0592/14.
disjoint subintervals of the unit interval) of the prod-
uct and Łukasiewicz t-norms, completed by G
¨
odel
(minimum) t-norm. This is a useful mathematical
characterisation but infinitary, and hence, insufficient
for computational purposes. Our objective is to pro-
pose logic calculi suitable for automated deduction
and underlying procedures/algorithms for (in)finitely
summed t-norms and related fuzzy logics. However,
even the three fundamental continuous fuzzy logics
have not yet been investigated in a systematic way
from a computational logic perspective.
Descriptions of real-world problems may become
rather complex. So, efficient inference stipulates the
methods and techniques of automated deduction. The
early research in automated deduction had started in
the 1950s, basically focused on theorem proving. The
resolution method, devised by Robinson (Robinson,
1965b; Robinson, 1965a), is based on the following
inference rules:
(Binary resolution)
a ∨ B, ¬c ∨ D
(B ∨ D)θ
θ is a most general unifier of the atoms a and c;
(Hyperresolution)
a
1
∨ B
1
, . . . , a
n
∨ B
n
, ¬c
1
∨ ··· ∨ ¬c
n
∨ D
(B
1
∨ ··· ∨ B
n
∨ D)θ
θ is a most general unifier of the atoms a
i
and c
i
.
Both the rules/calculi are refutation complete and
sound: a clausal theory is unsatisfiable if and only
30
Guller, D.
Hyperresolution for Propositional Product Logic.
DOI: 10.5220/0006044300300041
In Proceedings of the 8th International Joint Conference on Computational Intelligence (IJCCI 2016) - Volume 2: FCTA, pages 30-41
ISBN: 978-989-758-201-1
Copyright
c
 2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

if the empty clause can be inferred. A large class of
refinements and strategies has been developed (Bach-
mair and Ganzinger, 1994; Bachmair and Ganzinger,
1998). Another direction in automated deduction con-
stitutes the Davis-Putnam-Logemann-Loveland pro-
cedure (DPLL) (Davis and Putnam, 1960; Davis
et al., 1962) and its refinements, e.g. chronologi-
cal backtracking is replaced with non-chronological
one using so-called conflict-driven clause learning
(CDCL) (Silva and Sakallah, 1996; Marques-Silva
and Sakallah, 1999). Most modern propositional
SAT solvers are based on the DPLL or CDCL proce-
dure, improved by various features (Biere et al., 2009;
Sch
¨
oning and Tor
´
an, 2013).
In recent years, we have investigated both the
propositional and first-order case of G
¨
odel logic. In
(Guller, 2010; Guller, 2012a), we have proposed an
extension of the DPLL procedure. In (Guller, 2012b;
Guller, 2016a; Guller, 2014; Guller, 2015a), we have
devised an extension of hyperresolution, augmented
by truth constants and the equality, P
P
P, strict order, ≺
≺
≺,
projection, ∆
∆
∆, operators. As a side result, we have
shown that unsatisfiable formulae are recursively enu-
merable (Guller, 2016b; Guller, 2015b).
Our exploration also concerns the propositional
product logic with the multiplication t-norm. We
have introduced an extension of the DPLL procedure
(Guller, 2013; Guller, 2016a). In this paper, we exam-
ine the resolution counterpart. Particularly, we gener-
alise the hyperresolution principle to the propositional
product logic. We propose translation of a formula
to an equivalent satisfiable finite order clausal theory,
which consists of order clauses - finite sets of order
literals of the augmented form: ε
1
 ε
2
where ε
i
is ei-
ther the truth constant 0 or 1 or a conjunction of pow-
ers of propositional atoms, and  is the connective P
or ≺. P and ≺ are interpreted by the standard equal-
ity and strict order on [0, 1], respectively. We devise
a hyperresolution calculus over order clausal theories,
which is refutation sound and complete for the finite
case. By means of the translation and calculus, we
solve the deduction problem T |= φ for a finite theory
T and a formula φ.
The paper is arranged as follows. Section 2 recalls
the propositional product logic. Section 3 presents
translation to clausal form. Section 4 proposes a hy-
perresolution calculus. Section 5 brings conclusions.
2 PROPOSITIONAL PRODUCT
LOGIC
Throughout the paper, we shall use the common no-
tions and notation of propositional logic. N, Z, R
designates the set of natural, integer, real numbers,
and =, ≤, < denotes the standard equality, order,
strict order on N, Z, R. We denote R
+
0
= {c |0 ≤
c ∈ R}, R
+
= {c |0 < c ∈ R}, [0, 1] = {c |c ∈ R, 0 ≤
c ≤ 1}; [0, 1] is the unit interval. The set of propo-
sitional atoms of the product logic will be denoted
as PropAtom. We assume truth constants - propo-
sitional atoms 0, 1 ∈ PropAtom; 0 denotes the false
and 1 the true in the product logic. By PropForm
we designate the set of all propositional formulae of
the product logic built up from PropAtom using the
connectives: ¬, negation, ∧, conjunction, &, strong
conjunction, ∨, disjunction, →, implication, and ↔,
equivalence. We introduce a new unary connective ∆,
Delta, and binary connectives P, equality, ≺, strict
order. By OrdPropForm we designate the set of all
so-called order propositional formulae of the prod-
uct logic built up from PropAtom using the connec-
tives: ¬, ∆, ∧, &, ∨, →, ↔, and P, ≺.
1
Note that
PropForm ⊆ OrdPropForm. In the paper, we shall
assume that PropAtom is countably infinite; hence,
both the sets of formulae are countably infinite. Let
ε
i
, 1 ≤ i ≤ n, be either an order formula or a set of
order formulae or a set of sets of order formulae, in
general. By atoms(ε
1
, . . . , ε
n
) ⊆ PropAtom we denote
the set of all atoms occurring in ε
1
, . . . , ε
n
. We define
the size of order formula |φ| : OrdPropForm −→ N by
recursion on the structure of φ:
|φ| =





1 if φ ∈ PropAtom,
1 + |φ
1
| if φ = φ
1
,
1 + |φ
1
| + |φ
2
| if φ = φ
1
 φ
2
.
Let T ⊆ OrdPropForm be finite. We define the size of
T as |T | =
∑
φ∈T
|φ|.
Let X, Y , Z be sets and f : X −→ Y a mapping.
By kXk we denote the set-theoretic cardinality of
X. The relationship of X being a finite subset of Y
is denoted as X ⊆
F
Y . Let Z ⊆ X. We designate
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 bijec-
tion δ : γ −→ X. Recall that X is countable if and
only if there exists a sequence of X . Let I be an in-
dex set and S
i
6=
/
0, i ∈ I, be sets. A selector S over
{S
i
|i ∈ I} is a mapping S : I −→
S
{S
i
|i ∈ I} such that
for all i ∈ I, S (i) ∈ S
i
. We denote Sel({S
i
|i ∈ I}) =
{S |S is a selector over {S
i
|i ∈ I}}. Let c ∈ R
+
. log c
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
and c
∗
∈ R
+
0
such that for all n ≥ n
0
,
f (n) ≤ c
∗
· g(n).
1
We assume a decreasing connective precedence: ¬, ∆,
&, P, ≺, ∧, ∨, →, ↔.
Hyperresolution for Propositional Product Logic
31

The product logic is interpreted by the standard
Π-algebra augmented by the operators P
P
P, ≺
≺
≺, ∆
∆
∆ for
the connectives P, ≺, ∆, respectively.
Π = ([0, 1], ≤,∨
∨
∨,∧
∧
∧, ·,⇒
⇒
⇒, ,P
P
P,≺
≺
≺,∆
∆
∆, 0, 1)
where ∨
∨
∨, ∧
∧
∧ denotes the supremum, infimum operator
on [0, 1];
a⇒
⇒
⇒b =
(
1 if a ≤ b,
b
a
else;
a =

1 if a = 0,
0 else;
aP
P
P b =

1 if a = b,
0 else;
a≺
≺
≺b =

1 if a < b,
0 else;
∆
∆
∆a =

1 if a = 1,
0 else.
Recall that Π is a complete linearly ordered lattice al-
gebra; ∨
∨
∨, ∧
∧
∧ is commutative, associative, idempotent,
monotone; 0, 1 is its neutral element; · is commuta-
tive, associative, monotone; 1 is its neutral element;
the residuum operator ⇒
⇒
⇒ of · satisfies the condition
of residuation:
for all a, b, c ∈ Π, a · b ≤ c ⇐⇒ a ≤ b⇒
⇒
⇒c; (1)
G
¨
odel negation satisfies the condition:
for all a ∈ Π, a = a⇒
⇒
⇒0; (2)
∆
∆
∆ satisfies the condition:
2
for all a ∈ Π, ∆
∆
∆a = aP
P
P 1. (3)
A valuation V of propositional atoms is a map-
ping V : PropAtom −→ [0, 1] such that V (0) = 0 and
V (1) = 1. Let φ ∈ OrdPropForm and V be a valua-
tion. We define the truth value kφk
V
∈ [0, 1] of φ in
V by recursion on the structure of φ as follows:
φ ∈ PropAtom, kφk
V
= V (φ);
φ = ¬φ
1
, kφk
V
= kφ
1
k
V
;
φ = ∆ φ
1
, kφk
V
= ∆
∆
∆kφ
1
k
V
;
φ = φ
1
 φ
2
, kφk
V
= kφ
1
k
V


kφ
2
k
V
,
 ∈ {∧, &, ∨, →, P, ≺};
φ = φ
1
↔ φ
2
, kφk
V
= (kφ
1
k
V
⇒
⇒
⇒kφ
2
k
V
)·
(kφ
2
k
V
⇒
⇒
⇒kφ
1
k
V
).
An order theory is a set of order formulae. Let φ, φ
0
∈
OrdPropForm and T ⊆ OrdPropForm. φ is true in V ,
written as V |= φ, iff kφk
V
= 1. V is a model of
T , in symbols V |= T , iff, for all φ ∈ T , V |= φ. φ
is a tautology iff, for every valuation V , V |= φ. φ
is equivalent to φ
0
, in symbols φ ≡ φ
0
, iff, for every
valuation V , kφk
V
= kφ
0
k
V
.
2
We assume a decreasing operator precedence: , ∆
∆
∆, ·,
P
P
P, ≺
≺
≺, ∧
∧
∧, ∨
∨
∨, ⇒
⇒
⇒.
3 TRANSLATION TO CLAUSAL
FORM
We firstly introduce a notion of power of proposi-
tional atom and a notion of conjunction of powers of
propositional atoms. Let a ∈ PropAtom − {0, 1} and
n ≥ 1. The n-th power of the propositional atom a, a
raised to the power of n, is the pair (a, n), written as
a
n
. A power a
1
is denoted as a; if it does not cause the
ambiguity with the denotation of the single atom a in
a given context. The set of all powers is designated as
PropPow. Let a
n
∈ PropPow. We define the size of a
n
as |a
n
| = n ≥ 1. A conjunction Cn of powers of propo-
sitional atoms is a non-empty finite set of powers such
that for all a
m
6= b
n
∈ Cn, a 6= b. A conjunction
{a
m
0
0
, . . . , a
m
n
n
} is written in the form a
m
0
0
&···&a
m
n
n
.
A conjunction {p} is called unit and denoted as p;
if it does not cause the ambiguity with the denota-
tion of the single power p in a given context. The
set of all conjunctions is designated as PropConj.
Let p ∈ PropPow, Cn, Cn
1
, Cn
2
∈ PropConj, V be a
valuation. The truth value kCnk
V
∈ [0, 1] of Cn =
a
m
0
0
&···&a
m
n
n
in V is defined by
kCnk
V
= ka
0
k
V
·····ka
0
k
V
| {z }
m
0
·····ka
n
k
V
·····ka
n
k
V
| {z }
m
n
.
We define the size of Cn as |Cn| =
∑
p∈Cn
|p| ≥ 1. By
p&Cn we denote {p} ∪ Cn where p 6∈ Cn. Cn
1
is a
subconjunction of Cn
2
, in symbols Cn
1
v Cn
2
, iff, for
all a
m
∈ Cn
1
, there exists a
n
∈ Cn
2
such that m ≤ n.
Cn
1
is a proper subconjunction of Cn
2
, in symbols
Cn
1
@ Cn
2
, iff Cn
1
v Cn
2
and Cn
1
6= Cn
2
.
We finally introduce order clauses in the prod-
uct logic. l is an order literal iff l = ε
1
 ε
2
, ε
i
∈
{0, 1} ∪ PropConj,  ∈ {P, ≺}. The set of all or-
der literals is designated as OrdPropLit. Let l =
ε
1
 ε
2
∈ OrdPropLit and V be a valuation. The truth
value klk
V
∈ [0, 1] of l in V is defined by klk
V
=
kε
1
k
V


kε
2
k
V
. Note that V |= l if and only if ei-
ther l = ε
1
P ε
2
, kε
1
P ε
2
k
V
= 1, kε
1
k
V
= kε
2
k
V
;
or l = ε
1
≺ ε
2
, kε
1
≺ ε
2
k
V
= 1, kε
1
k
V
< kε
2
k
V
.
We define the size of l as |l| = 1 + |ε
1
| + |ε
2
|. An
order clause is a finite set of order literals. Since
= is symmetric, P is commutative; hence, for all
ε
1
P ε
2
∈ OrdPropLit, we identify ε
1
P ε
2
and ε
2
P
ε
1
∈ OrdPropLit with respect to order clauses. An
order clause {l
0
, . . . , l
n
} 6=
/
0 is written in the form
l
0
∨ ··· ∨ l
n
. The empty order clause
/
0 is denoted as
. An order clause {l} is called unit and denoted
as l; if it does not cause the ambiguity with the de-
notation of the single order literal l in a given con-
text. We designate the set of all order clauses as
OrdPropCl. Let l, l
0
, . . . , l
n
∈ OrdPropLit and C,C
0
∈
FCTA 2016 - 8th International Conference on Fuzzy Computation Theory and Applications
32

OrdPropCl. We define the size of C as |C| =
∑
l∈C
|l|.
By l
0
∨ ··· ∨ l
n
∨C we denote {l
0
, . . . , l
n
} ∪C where,
for all i, i
0
≤ n and i 6= i
0
, l
i
6∈ C, l
i
6= l
i
0
. By C ∨ C
0
we denote C ∪ C
0
. C is a subclause of C
0
, in sym-
bols C v C
0
, iff C ⊆ C
0
. An order clausal theory is
a set of order clauses. A unit order clausal theory is
a set of unit order clauses. Let φ, φ
0
∈ OrdPropForm,
T, T
0
⊆ OrdPropForm, S, S
0
⊆ OrdPropCl, V be a val-
uation. C is true in V , written as V |= C, iff there
exists l
∗
∈ C such that V |= l
∗
. V is a model of S,
in symbols V |= S, iff, for all C ∈ S, V |= C. Let
ε
1
∈ {φ, T,C, S} and ε
2
∈ {φ
0
, T
0
,C
0
, S
0
}. ε
2
is a propo-
sitional consequence of ε
1
, in symbols ε
1
|= ε
2
, iff, for
every valuation V , if V |= ε
1
, then V |= ε
2
. ε
1
is satis-
fiable iff there exists a valuation V such that V |= ε
1
.
ε
1
is equisatisfiable to ε
2
iff ε
1
is satisfiable if and only
if ε
2
is satisfiable. Let S ⊆
F
OrdPropCl. We define
the size of S as |S| =
∑
C∈S
|C|. Let I = N ×N; a count-
ably infinite index set. Since PropAtom is countably
infinite, there exist O,
˜
A ⊆ PropAtom such that O ⊇
{0, 1}, O ∪
˜
A = PropAtom, O∩
˜
A =
/
0, both are count-
ably infinite,
˜
A = { ˜a | ∈ I}. Let A ⊆
˜
A. We denote
OrdPropForm
A
= {φ |φ ∈ OrdPropForm, atoms(φ) ⊆
O ∪ A} ⊆ OrdPropForm and OrdPropCl
A
= {C |C ∈
OrdPropCl,atoms(C) ⊆ O ∪ A} ⊆ OrdPropCl.
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.
Translation of an order formula or theory to
clausal form, is based on the following lemma:
Lemma 1. Let n
φ
, n
0
∈ N, φ ∈ OrdPropForm
/
0
, T ⊆
OrdPropForm
/
0
.
(I) There exist an index set J
φ
⊆ {(n
φ
, j) | j ∈ N} ⊆
I and S
φ
⊆
F
OrdPropCl
{ ˜a | ∈J
φ
}
such that ei-
ther J
φ
=
/
0 or J
φ
= {(n
φ
, j) | j ≤ n
J
φ
} for some
n
J
φ
(J
φ
is a non-empty interval of indices);
(a) kJ
φ
k ≤ 2 · |φ|;
(b) either J
φ
=
/
0, S
φ
= {} or J
φ
= S
φ
=
/
0 or
J
φ
6=
/
0,  6∈ S
φ
6=
/
0;
(c) there exists a valuation A and A |= φ if and
only if there exists a valuation A
0
and A
0
|=
S
φ
, satisfying A|
O
= A
0
|
O
;
(d) |S
φ
| ∈ O(|φ|); the number of all elementary
operations of the translation of φ to S
φ
, is
in O(|φ|); the time and space complexity
of the translation of φ to S
φ
, is in O(|φ| ·
(log(1 + n
φ
) + log |φ|));
(e) if S
φ
6=
/
0, {}, then J
φ
6=
/
0, for all C ∈ S
φ
,
/
0 6= atoms(C) ∩
˜
A ⊆ { ˜a | ∈ J
φ
}.
(II) There exist an index set J
T
⊆ {(i, j)|i ≥ n
0
} ⊆ I
and S
T
⊆ OrdPropCl
{ ˜a | ∈J
T
}
such that
(a) either J
T
=
/
0, S
T
= {} or J
T
= S
T
=
/
0 or
J
T
6=
/
0,  6∈ S
T
6=
/
0;
(b) there exists a valuation A and A |= T if and
only if there exists a valuation A
0
and A
0
|=
S
T
, satisfying A|
O
= A
0
|
O
;
(c) if T ⊆
F
OrdPropForm
/
0
, then J
T
⊆
F
{(i, j) | i ≥ n
0
}, kJ
T
k ≤ 2 · |T |, S
T
⊆
F
OrdPropCl
{ ˜a | ∈J
T
}
, |S
T
| ∈ O(|T |); the
number of all elementary operations of the
translation of T to S
T
, is in O(|T |); the time
and space complexity of the translation of
T to S
T
, is in O(|T | · log(1 + n
0
+ |T |));
(d) if S
T
6=
/
0, {}, then J
T
6=
/
0, for all C ∈ S
T
,
/
0 6= atoms(C) ∩
˜
A ⊆ { ˜a | ∈ J
T
}.
Proof. It is straightforward to prove the following
statements:
Let n
θ
∈ N and θ ∈ OrdPropForm
/
0
. There ex-
ists θ
0
∈ OrdPropForm
/
0
such that
(a) θ
0
≡ θ;
(b) |θ
0
| ≤ 2 · |θ|; θ
0
can be built up from θ via
a postorder traversal of θ with #O(θ) ∈
O(|θ|) and the time, space complexity in
O(|θ| · (log(1 + n
θ
) + log |θ|));
(c) θ
0
does not contain ¬ and ∆;
(d) θ
0
∈ {0, 1}; or for every subformula of
θ
0
of the form ε
1
 ε
2
,  ∈ {∧, &, ∨, ↔},
ε
i
6= 0, 1; for every subformula of θ
0
of the
form ε
1
→ ε
2
, ε
1
6= 0, 1, ε
2
6= 1; for ev-
ery subformula of θ
0
of the form ε
1
P ε
2
,
{ε
1
, ε
2
} 6⊆ {0, 1}; for every subformula of
θ
0
of the form ε
1
≺ ε
2
, ε
1
6= 1, ε
2
6= 0,
{ε
1
, ε
2
} 6⊆ {0, 1}.
(4)
The proof is by induction on the structure of θ.
Let n
θ
∈ N, θ ∈ OrdPropForm
/
0
−{0, 1}, (4c,d)
hold for θ; = (n
θ
, j ) ∈ {(n
θ
, j) | j ∈ N} ⊆ I
be an index, ˜a ∈
˜
A. There exist an index set
J = {(n
θ
, j) | j + 1 ≤ j ≤ n
J
} ⊆ {(n
θ
, j) | j ∈
N} ⊆ I for some n
J
, j ≤ n
J
, 6∈ J, and S ⊆
F
OrdPropCl
{ ˜a }∪{ ˜a | ∈J}
such that
(a) kJk ≤ |θ| − 1;
(b) there exists a valuation A and A |= ˜a ↔
θ ∈ OrdPropForm
{ ˜a }
if and only if there
exists a valuation A
0
and A
0
|= S, satisfying
A|
O∪{ ˜a }
= A
0
|
O∪{ ˜a }
;
(c) |S| ≤ 31 · |θ|, S can be built up from θ
via a preorder traversal of θ with #O(θ) ∈
O(|θ|);
(d) for all C ∈ S,
/
0 6= atoms(C) ∩
˜
A ⊆ { ˜a } ∪
{ ˜a | ∈ J}, ˜a P 1, ˜a ≺ 1 6∈ S.
(5)
Hyperresolution for Propositional Product Logic
33

Table 1: Binary interpolation rules for ∧, &, ∨, →, ↔, P, ≺.
Case
θ = θ
1
∧ θ
2
θ = θ
1
∧ θ
2
θ = θ
1
∧ θ
2
˜a ↔ (θ
1
∧ θ
2
)

˜a
1
≺ ˜a
2
∨ ˜a
1
P ˜a
2
∨ ˜a P ˜a
2
, ˜a
2
≺ ˜a
1
∨ ˜a P ˜a
1
, ˜a
1
↔ θ
1
, ˜a
2
↔ θ
2

(6)
|Consequent| = 15 + | ˜a
1
↔ θ
1
| + | ˜a
2
↔ θ
2
| ≤ 31 + | ˜a
1
↔ θ
1
| + | ˜a
2
↔ θ
2
|
θ = θ
1
&θ
2
θ = θ
1
&θ
2
θ = θ
1
&θ
2
˜a ↔ (θ
1
&θ
2
)

˜a P ˜a
1
& ˜a
2
, ˜a
1
↔ θ
1
, ˜a
2
↔ θ
2

(7)
|Consequent| = 5 + | ˜a
1
↔ θ
1
| + | ˜a
2
↔ θ
2
| ≤ 31 + | ˜a
1
↔ θ
1
| + | ˜a
2
↔ θ
2
|
θ = θ
1
∨ θ
2
θ = θ
1
∨ θ
2
θ = θ
1
∨ θ
2
˜a ↔ (θ
1
∨ θ
2
)

˜a
1
≺ ˜a
2
∨ ˜a
1
P ˜a
2
∨ ˜a P ˜a
1
, ˜a
2
≺ ˜a
1
∨ ˜a P ˜a
2
, ˜a
1
↔ θ
1
, ˜a
2
↔ θ
2

(8)
|Consequent| = 15 + | ˜a
1
↔ θ
1
| + | ˜a
2
↔ θ
2
| ≤ 31 + | ˜a
1
↔ θ
1
| + | ˜a
2
↔ θ
2
|
θ = θ
1
→ θ
2
, θ
2
6= 0
θ = θ
1
→ θ
2
, θ
2
6= 0
θ = θ
1
→ θ
2
, θ
2
6= 0
˜a ↔ (θ
1
→ θ
2
)

˜a
1
≺ ˜a
2
∨ ˜a
1
P ˜a
2
∨ ˜a
1
& ˜a P ˜a
2
, ˜a
2
≺ ˜a
1
∨ ˜a P 1, ˜a
1
↔ θ
1
, ˜a
2
↔ θ
2

(9)
|Consequent| = 17 + | ˜a
1
↔ θ
1
| + | ˜a
2
↔ θ
2
| ≤ 31 + | ˜a
1
↔ θ
1
| + | ˜a
2
↔ θ
2
|
θ = θ
1
↔ θ
2
θ = θ
1
↔ θ
2
θ = θ
1
↔ θ
2
˜a ↔ (θ
1
↔ θ
2
)

˜a
1
≺ ˜a
2
∨ ˜a
1
P ˜a
2
∨ ˜a
1
& ˜a P ˜a
2
, ˜a
2
≺ ˜a
1
∨ ˜a
2
P ˜a
1
∨ ˜a
2
& ˜a P ˜a
1
,
˜a
1
≺ ˜a
2
∨ ˜a
2
≺ ˜a
1
∨ ˜a P 1, ˜a
1
↔ θ
1
, ˜a
2
↔ θ
2

(10)
|Consequent| = 31 + | ˜a
1
↔ θ
1
| + | ˜a
2
↔ θ
2
| ≤ 31 + | ˜a
1
↔ θ
1
| + | ˜a
2
↔ θ
2
|
θ = θ
1
P θ
2
, θ
i
6= 0, 1
θ = θ
1
P θ
2
, θ
i
6= 0, 1
θ = θ
1
P θ
2
, θ
i
6= 0, 1
˜a ↔ (θ
1
P θ
2
)

˜a
1
P ˜a
2
∨ ˜a P 0, ˜a
1
≺ ˜a
2
∨ ˜a
2
≺ ˜a
1
∨ ˜a P 1, ˜a
1
↔ θ
1
, ˜a
2
↔ θ
2

(11)
|Consequent| = 15 + | ˜a
1
↔ θ
1
| + | ˜a
2
↔ θ
2
| ≤ 31 + | ˜a
1
↔ θ
1
| + | ˜a
2
↔ θ
2
|
θ = θ
1
≺ θ
2
, θ
1
6= 0, θ
2
6= 1
θ = θ
1
≺ θ
2
, θ
1
6= 0, θ
2
6= 1
θ = θ
1
≺ θ
2
, θ
1
6= 0, θ
2
6= 1
˜a ↔ (θ
1
≺ θ
2
)

˜a
1
≺ ˜a
2
∨ ˜a P 0, ˜a
2
≺ ˜a
1
∨ ˜a
2
P ˜a
1
∨ ˜a P 1, ˜a
1
↔ θ
1
, ˜a
2
↔ θ
2

(12)
|Consequent| = 15 + | ˜a
1
↔ θ
1
| + | ˜a
2
↔ θ
2
| ≤ 31 + | ˜a
1
↔ θ
1
| + | ˜a
2
↔ θ
2
|
The proof is by induction on the structure of θ using
the interpolation rules in Tables 1 and 2.
(I) By (4) for n
φ
, φ, there exists φ
0
∈
OrdPropForm
/
0
such that (4a–d) hold for n
φ
, φ, φ
0
. We
get three cases for φ
0
.
Case 1: φ
0
= 0. We put J
φ
=
/
0 ⊆ {(n
φ
, j) | j ∈ N} ⊆
I and S
φ
= {} ⊆
F
OrdPropCl
/
0
.
Case 2: φ
0
= 1. We put J
φ
=
/
0 ⊆ {(n
φ
, j) | j ∈ N} ⊆
I and S
φ
=
/
0 ⊆
F
OrdPropCl
/
0
.
Case 3: φ
0
6= 0, 1. We put j = 0 and =
(n
φ
, j ) ∈ {(n
φ
, j) | j ∈ N} ⊆ I. We get by (5) for
n
φ
, φ
0
, , ˜a that there exist J = {(n
φ
, j) | 1 ≤ j ≤
n
J
} ⊆ {(n
φ
, j) | j ∈ N} ⊆ I for some n
J
, j ≤ n
J
, 6∈ J,
S ⊆
F
OrdPropCl
{ ˜a }∪{ ˜a | ∈J}
, and (5a–d) hold for
φ
0
, ˜a , J, S. We put n
J
φ
= n
J
, J
φ
= {(n
φ
, j) | j ≤
n
J
φ
} ⊆ {(n
φ
, j) | j ∈ N} ⊆ I, S
φ
= { ˜a P 1} ∪ S ⊆
F
OrdPropCl
{ ˜a | ∈J
φ
}
. (II) straightforwardly follows
from (I). The lemma is proved.
Theorem 2. Let n
0
∈ N, φ ∈ OrdPropForm
/
0
, T ⊆
OrdPropForm
/
0
. There exist an index set J
φ
T
⊆
{(i, j) | i ≥ n
0
} ⊆ I and S
φ
T
⊆ OrdPropCl
{ ˜a | ∈J
φ
T
}
such
that
(i) there exists a valuation A and A |= T , A 6|= φ if
and only if there exists a valuation A
0
and A
0
|=
S
φ
T
, satisfying A|
O
= A
0
|
O
;
(ii) T |= φ if and only if S
φ
T
is unsatisfiable;
FCTA 2016 - 8th International Conference on Fuzzy Computation Theory and Applications
34

Table 2: Unary interpolation rules for →, P, ≺.
Case
θ = θ
1
→ 0
θ = θ
1
→ 0
θ = θ
1
→ 0
˜a ↔ (θ
1
→ 0)
{ ˜a
1
P 0 ∨ ˜a P 0, 0 ≺ ˜a
1
∨ ˜a P 1, ˜a
1
↔ θ
1
}
(13)
|Consequent| = 12 + | ˜a
1
↔ θ
1
| ≤ 31 + | ˜a
1
↔ θ
1
|
θ = θ
1
P 0
θ = θ
1
P 0
θ = θ
1
P 0
˜a ↔ (θ
1
P 0)
{ ˜a
1
P 0 ∨ ˜a P 0, 0 ≺ ˜a
1
∨ ˜a P 1, ˜a
1
↔ θ
1
}
(14)
|Consequent| = 12 + | ˜a
1
↔ θ
1
| ≤ 31 + | ˜a
1
↔ θ
1
|
θ = θ
1
P 1
θ = θ
1
P 1
θ = θ
1
P 1
˜a ↔ (θ
1
P 1)
{ ˜a
1
P 1 ∨ ˜a P 0, ˜a
1
≺ 1 ∨ ˜a P 1, ˜a
1
↔ θ
1
}
(15)
|Consequent| = 12 + | ˜a
1
↔ θ
1
| ≤ 31 + | ˜a
1
↔ θ
1
|
θ = 0 ≺ θ
1
θ = 0 ≺ θ
1
θ = 0 ≺ θ
1
˜a ↔ (0 ≺ θ
1
)
{0 ≺ ˜a
1
∨ ˜a P 0, ˜a
1
P 0 ∨ ˜a P 1, ˜a
1
↔ θ
1
}
(16)
|Consequent| = 12 + | ˜a
1
↔ θ
1
| ≤ 31 + | ˜a
1
↔ θ
1
|
θ = θ
1
≺ 1
θ = θ
1
≺ 1
θ = θ
1
≺ 1
˜a ↔ (θ
1
≺ 1)
{ ˜a
1
≺ 1 ∨ ˜a P 0, ˜a
1
P 1 ∨ ˜a P 1, ˜a
1
↔ θ
1
}
(17)
|Consequent| = 12 + | ˜a
1
↔ θ
1
| ≤ 31 + | ˜a
1
↔ θ
1
|
(iii) if T ⊆
F
OrdPropForm
/
0
, then J
φ
T
⊆
F
{(i, j) | i ≥ n
0
}, kJ
φ
T
k ∈ O(|T | + |φ|),
S
φ
T
⊆
F
OrdPropCl
{ ˜a | ∈J
φ
T
}
, |S
φ
T
| ∈ O(|T |+|φ|);
the number of all elementary operations of the
translation of T and φ to S
φ
T
, is in O(|T | + |φ|);
the time and space complexity of the translation
of T and φ to S
φ
T
, is in O(|T | · log(1 + n
0
+
|T |) + |φ| · (log(1 + n
0
) + log |φ|)).
Proof. We get by Lemma 1(II) for n
0
+ 1, T that
there exist J
T
⊆ {(i, j)|i ≥ n
0
+ 1} ⊆ I, S
T
⊆
OrdPropCl
{ ˜a | ∈J
T
}
, and Lemma 1(II a–d) hold for
n
0
+ 1, T , J
T
, S
T
. By (4) for n
0
, φ, there exists
φ
0
∈ OrdPropForm
/
0
such that (4a–d) hold for n
0
, φ,
φ
0
. We get three cases for φ
0
.
Case 1: φ
0
= 0. We put J
φ
T
= J
T
⊆ {(i, j)|i ≥
n
0
+ 1} ⊆ {(i, j)|i ≥ n
0
} ⊆ I and S
φ
T
= S
T
⊆
OrdPropCl
{ ˜a | ∈J
φ
T
}
.
Case 2: φ
0
= 1. We put J
φ
T
=
/
0 ⊆ {(i, j)|i ≥ n
0
} ⊆
I and S
φ
T
= {} ⊆ OrdPropCl
/
0
.
Case 3: φ
0
6= 0, 1. We put j = 0 and
= (n
0
, j ) ∈ {(n
0
, j) | j ∈ N} ⊆ I. We get by (5) for
n
0
, φ
0
, , ˜a that there exist J = {(n
0
, j) | 1 ≤ j ≤
n
J
} ⊆ {(n
0
, j) | j ∈ N} ⊆ I for some n
J
, j ≤ n
J
, 6∈ J,
S ⊆
F
OrdPropCl
{ ˜a }∪{ ˜a | ∈J}
, and (5a–d) hold for φ
0
,
˜a , J, S. We put J
φ
T
= J
T
∪{ }∪J ⊆ {(i, j)|i ≥ n
0
} ⊆ I
and S
φ
T
= S
T
∪ { ˜a ≺ 1} ∪ S ⊆ OrdPropCl
{ ˜a | ∈J
φ
T
}
.
The theorem is proved.
4 HYPERRESOLUTION OVER
ORDER CLAUSES
In this section, we propose an order hyperresolu-
tion calculus operating over order clausal theories,
and prove its refutational soundness, completeness.
At first, we introduce some basic notions and nota-
tion. Let l ∈ OrdPropLit. l is a contradiction iff
l = 0 P 1 or l = ε ≺ 0 or l = 1 ≺ ε or l = ε ≺ ε.
Let Cn ∈ PropConj and C ∈ OrdPropCl. We define
an auxiliary function simplify : ({0, 1} ∪ PropConj ∪
OrdPropLit ∪ OrdPropCl) × PropAtom × {0, 1} −→
{0, 1} ∪ PropConj ∪ OrdPropLit ∪ OrdPropCl as fol-
lows:
Hyperresolution for Propositional Product Logic
35

simplify(0, a, υ) = 0;
simplify(1, a, υ) = 1;
simplify(Cn, a, 0) =

0 if a ∈ atoms(Cn),
Cn else;
simplify(Cn, a, 1) =





1 if ∃n
∗
Cn = a
n
∗
,
Cn − a
n
∗
if ∃n
∗
a
n
∗
∈ Cn 6= a
n
∗
,
Cn else;
simplify(l, a, υ) = simplify(ε
1
, a, υ)  simplify(ε
2
, a, υ)
if l = ε
1
 ε
2
;
simplify(C, a, υ) = {simplify(l, a, υ)| l ∈ C}.
For an input expression, atom, truth constant, simplify
replaces every occurrence of the atom by the truth
constant in the expression, and returns a simplified
expression according to laws holding in Π. Let
Cn
1
, Cn
2
∈ PropConj and l
1
, l
2
∈ OrdPropLit. An-
other auxiliary function  : ({0, 1} ∪ PropConj) ×
({0, 1} ∪ PropConj) −→ {0, 1} ∪ PropConj is defined
as follows:
0  ε = ε  0 = 0;
1  ε = ε  1 = ε;
Cn
1
 Cn
2
= {a
m+n
|a
m
∈ Cn
1
, a
n
∈ Cn
2
} ∪
{a
n
|a
n
∈ Cn
1
, a 6∈ atoms(Cn
2
)} ∪
{a
n
|a
n
∈ Cn
2
, a 6∈ atoms(Cn
1
)}.
For two input expressions,  returns the product of
them. It can be extended to {0, 1} ∪ OrdPropLit
component-wisely.  : ({0, 1} ∪ OrdPropLit) ×
({0, 1} ∪ OrdPropLit) −→ {0, 1} ∪ OrdPropLit is de-
fined as
0  ε = ε  0 = 0;
1  ε = ε  1 = ε;
l
1
 l
2
= (ε
1
 ε
2
)  (υ
1
 υ
2
) if l
i
= ε
i

i
υ
i
,
 =

P if 
1
= 
2
=P,
≺ else.
Note that  is a binary commutative, associative op-
erator. We denote l
n
= l  ···  l
| {z }
n
, n ≥ 1, and say that
l
n
is the n-th power of l. Let I ⊆
F
N, l
i
∈ OrdPropLit,
α
i
≥ 1, i ∈ I. We define by recursion on I:

i∈I
l
α
i
i
=



1 if I =
/
0,
l
α
i
∗
i
∗



i∈I−{i
∗
}
l
α
i
i

if ∃i
∗
∈ I.
Let S ⊆ OrdPropCl. The basic order hyperresolu-
tion calculus is defined as follows. The first rule is the
central order hyperresolution one.
(Order hyperresolution rule) (18)
0 ≺ a
0
, . . . , 0 ≺ a
m
, a
0
≺ 1, . . . , a
m
≺ 1,
l
0
∨C
0
, . . . , l
n
∨C
n
∈ S
κ−1
n
_
i=0
C
i
∈ S
κ
;
atoms(l
0
, . . . , l
n
) = {a
0
, . . . , a
m
}⊆ PropAtom − {0, 1},
l
i
= Cn
i
1

i
Cn
i
2
, Cn
i
j
∈ PropConj,
there exist α
∗
i
≥ 1, i = 0, . . . , n, J
∗
⊆ { j | j ≤ m},
β
∗
j
≥ 1, j ∈ J
∗
, such that


n
i=0
l
α
∗
i
i




j∈J
∗
(a
j
≺ 1)
β
∗
j

is a contradiction.
If there exists a product of powers of the input order
literals l
0
, . . . , l
n
and of some so-called literals-guards
a
j
≺ 1, j ∈ J
∗
, which is a contradiction of the form ε ≺
ε, then we can derive the output order clause
W
n
i=0
C
i
consisting of the remainder order clauses C
i
, i ≤ n.
We say that
W
n
i=0
C
i
is an order hyperresolvent of 0 ≺
a
1
, . . . , 0 ≺ a
m
, a
1
≺ 1, . . . , a
m
≺ 1, l
0
∨C
0
, . . . , l
n
∨C
n
.
(Order contradiction rule) (19)
l ∨C ∈ S
κ−1
C ∈ S
κ
;
l is a contradiction.
If the order literal l is a contradiction, then it can be
removed from the input order clause l ∨ C. C is an
order contradiction resolvent of l ∨C.
(Order 0-simplification rule) (20)
a P 0,C ∈ S
κ−1
simplify(C, a, 0) ∈ S
κ
;
a ∈ atoms(C).
If a so-called literal-guard a P 0 is in the antecedent
order clausal theory and the input order clause C con-
tains the atom a, then C can be simplified using the
auxiliary function simplify. simplify(C, a, 0) is an or-
der 0-simplification resolvent of a P 0 and C. Anal-
ogously, C can be simplified with respect to a literal-
guard a P 1.
(Order 1-simplification rule) (21)
a P 1,C ∈ S
κ−1
simplify(C, a, 1) ∈ S
κ
;
a ∈ atoms(C).
simplify(C, a, 1) is an order 1-simplification resolvent
of a P 1 and C.
(Order 0-contradiction rule) (22)
a
α
0
0
&···&a
α
n
n
P 0 ∨C, 0 ≺ a
0
, . . . , 0 ≺ a
n
∈ S
κ−1
C ∈ S
κ
.
FCTA 2016 - 8th International Conference on Fuzzy Computation Theory and Applications
36

C is an order 0-contradiction resolvent of
a
α
0
0
&···&a
α
n
n
P 0 ∨C, 0 ≺ a
0
, . . . , 0 ≺ a
n
.
(Order 1-contradiction rule) (23)
a
α
0
0
&···&a
α
n
n
P 1 ∨C, a
i
≺ 1 ∈ S
κ−1
C ∈ S
κ
;
i ≤ n.
C is an order 1-contradiction resolvent of
a
α
0
0
&···&a
α
n
n
P 1 ∨ C and a
i
≺ 1. The last two
rules detect a contradictory set of order literals of the
form either {a
α
0
0
&···&a
α
n
n
P 0, 0 ≺ a
0
, . . . , 0 ≺ a
n
}
or {a
α
0
0
&···&a
α
n
n
P 1, a
i
≺ 1}, i ≤ n. In either
case, the remainder order clause C can be derived.
Note that all the rules are sound; for every rule, the
consequent order clausal theory is a propositional
consequence of the antecedent one.
Let S
0
=
/
0 ⊆ OrdPropCl. Let D = C
1
, . . . ,C
n
,
C
κ
∈ OrdPropCl, n ≥ 1. D is a deduction of C
n
from S
by order hyperresolution iff, for all 1 ≤ κ ≤ n, C
κ
∈ S,
or there exist 1 ≤ j
∗
k
≤ κ − 1, k = 0, . . . , m, such that
C
κ
is an order resolvent of C
j
∗
0
, . . . ,C
j
∗
m
∈ S
κ−1
using
Rule (18)–(23) with respect to S
κ−1
; S
κ
is defined by
recursion on 1 ≤ κ ≤ n as follows:
S
κ
= S
κ−1
∪ {C
κ
} ⊆ OrdPropCl.
D is a refutation of S iff C
n
= . We denote
clo
H
(S) = {C | there exists a deduction of C from S
by order hyperresolution}
⊆ OrdPropCl.
Lemma 3. Let S ⊆
F
OrdPropCl. clo
H
(S) ⊆
F
OrdPropCl.
Proof. Straightforward.
Lemma 4. Let A = {a
i
|1 ≤ i ≤ m} ⊆ PropAtom −
{0, 1}, S
1
= {0 ≺ a
i
|1 ≤ i ≤ m} ∪ {a
i
≺ 1|1 ≤
i ≤ m} ⊆ OrdPropCl, S
2
= {Cn
i
1

i
Cn
i
2
|Cn
i
j
∈
PropConj, 1 ≤ i ≤ n} ⊆ OrdPropCl, atoms(S
2
) = A,
S = S
1
∪ S
2
⊆ OrdPropCl, there not exist an applica-
tion of Rule (18) with respect to S. S is satisfiable.
Proof. S is unit. Note that an application of
Rule (18) with respect to S would derive . We de-
note PropConj
A
= {Cn |Cn ∈ PropConj, atoms(Cn) ⊆
A} ⊆ PropConj. Let Cn
1
, Cn
2
∈ PropConj
A
and
Cn
2
@ Cn
1
. We define
cancel(Cn
1
, Cn
2
) =
{a
r−s
|a
r
∈ Cn
1
, a
s
∈ Cn
2
, r > s} ∪
{a
r
|a
r
∈ Cn
1
, a 6∈ atoms(Cn
2
)} ∈ PropConj
A
.
We further denote
gen =

Cn
1
P Cn
2
|Cn
i
∈ PropConj
A
, there exist
/
0 6= I
∗
⊆ {i |1 ≤ i ≤ n}, α
∗
i
≥ 1, i ∈ I
∗
,
Cn
1
P Cn
2
= 
i∈I
∗
(Cn
i
1

i
Cn
i
2
)
α
∗
i

∪

Cn
1
≺ Cn
2
|Cn
i
∈ PropConj
A
, there exist
/
0 6= I
∗
⊆ {i |1 ≤ i ≤ n}, α
∗
i
≥ 1, i ∈ I
∗
,
J
∗
⊆ { j | 1 ≤ j ≤ m}, β
∗
j
≥ 1, j ∈ J
∗
,
Cn
1
≺ Cn
2
=


i∈I
∗
(Cn
i
1

i
Cn
i
2
)
α
∗
i




j∈J
∗
(a
j
≺ 1)
β
∗
j

⊆ OrdPropLit,
cnl =

Cn
1
 Cn
2
|Cn
i
∈ PropConj
A
, there exist
Cn
∗
1
 Cn
∗
2
∈ gen, Cn
∗
∈ PropConj
A
,
Cn
∗
@ Cn
∗
i
, Cn
i
= cancel(Cn
∗
i
, Cn
∗
)

⊆ OrdPropLit,
clo = gen ∪ cnl ⊆ OrdPropLit.
Then S
2
⊆ gen ⊆ clo.
For all Cn ∈ PropConj
A
, Cn ≺ Cn 6∈ gen, clo. (24)
The proof is straightforward; we have that there does
not exist an application of Rule (18) with respect to S.
A ∩ {0, 1} ⊆ (PropAtom − {0, 1}) ∩ {0, 1} =
/
0.
Let {0, 1} ⊆ X ⊆ {0, 1} ∪ A. A partial valuation V
is a mapping V : X −→ [0, 1] such that V (0) = 0
and V (1) = 1. We denote dom(V ) = X , {0, 1} ⊆
dom(V ) ⊆ {0, 1} ∪ A. We define a partial valuation
V
ι
by recursion on ι ≤ m in Table 3.
For all ι ≤ ι
0
≤ m, V
ι
is a partial valuation,
dom(V
ι
) = {0, 1} ∪ {a
1
, . . . , a
ι
}, V
ι
⊆ V
ι
0
.
(25)
The proof is by induction on ι ≤ m.
For all ι ≤ m, for all a ∈ dom(V
ι
) − {0, 1},
Cn
1
, Cn
2
∈ PropConj
A
and
atoms(Cn
i
) ⊆ dom(V
ι
) − {0, 1},
0 < V
ι
(a) < 1;
if Cn
1
P Cn
2
∈ clo, then kCn
1
k
V
ι
= kCn
2
k
V
ι
;
if Cn
1
≺ Cn
2
∈ clo, then kCn
1
k
V
ι
< kCn
2
k
V
ι
.
(26)
The proof is by induction on ι ≤ m.
atoms(S
1
) = {0, 1} ∪ A and atoms(S) =
atoms(S
1
)∪atoms(S
2
) = {0, 1}∪A. We put V = V
m
,
dom(V )
(25)
== {0, 1} ∪ {a
1
, . . . , a
m
} = {0, 1} ∪ A =
atoms(S).
For all a ∈ A, Cn
1
, Cn
2
∈ PropConj
A
,
0 < V (a) < 1;
if Cn
1
P Cn
2
∈ clo, then kCn
1
k
V
= kCn
2
k
V
;
if Cn
1
≺ Cn
2
∈ clo, then kCn
1
k
V
< kCn
2
k
V
.
(27)
Hyperresolution for Propositional Product Logic
37

Table 3: A partial valuation V
ι
.
V
0
= {(0, 0), (1, 1)};
V
ι
= V
ι−1
∪ {(a
ι
, λ
ι
)} (1 ≤ ι ≤ m),
E
ι−1
=







kCn
2
k
V
ι−1
kCn
1
k
V
ι−1
!
1
α





Cn
1
&a
α
ι
P Cn
2
∈ clo,
atoms(Cn
i
) ⊆ dom(V
ι−1
)







∪




kCn
2
k
V
ι−1

1
α




a
α
ι
P Cn
2
∈ clo,
atoms(Cn
2
) ⊆ dom(V
ι−1
)



,
D
ι−1
=







kCn
2
k
V
ι−1
kCn
1
k
V
ι−1
!
1
α





Cn
2
≺ Cn
1
&a
α
ι
∈ clo,
atoms(Cn
i
) ⊆ dom(V
ι−1
)







∪




kCn
2
k
V
ι−1

1
α




Cn
2
≺ a
α
ι
∈ clo,
atoms(Cn
2
) ⊆ dom(V
ι−1
)



,
U
ι−1
=







kCn
2
k
V
ι−1
kCn
1
k
V
ι−1
!
1
α





Cn
1
&a
α
ι
≺ Cn
2
∈ clo,
atoms(Cn
i
) ⊆ dom(V
ι−1
)







∪




kCn
2
k
V
ι−1

1
α




a
α
ι
≺ Cn
2
∈ clo,
atoms(Cn
2
) ⊆ dom(V
ι−1
)



,
λ
ι
=
(
W
W
W
D
ι−1
+
V
V
V
U
ι−1
2
if E
ι−1
=
/
0,
W
W
W
E
ι−1
else.
The proof is by (26) for m.
We put A = V ∪ {(a, 0) | a ∈ PropAtom −
dom(V )}; A is a valuation. Let l ∈ S. Then l ∈
OrdPropLit and atoms(l) ⊆ atoms(S) = dom(V ). We
get two cases for l.
Case 1: l ∈ S
1
, either l = 0 ≺ a or l = a ≺ 1.
Hence, a ∈ A, by (27) for a, either A(0) = V (0) = 0 <
V (a) = A(a) or A(a) = V (a) < 1 = V (1) = A(1),
A |= l.
Case 2: l ∈ S
2
, either l = Cn
1
P Cn
2
or l = Cn
1
≺
Cn
2
. Hence, l ∈ S
2
⊆ clo, either Cn
1
P Cn
2
∈ clo
or Cn
1
≺ Cn
2
∈ clo, Cn
1
, Cn
2
∈ PropConj
A
, by (27)
for Cn
1
, Cn
2
, either kCn
1
k
A
= kCn
1
k
V
= kCn
2
k
V
=
kCn
2
k
A
or kCn
1
k
A
= kCn
1
k
V
< kCn
2
k
V
= kCn
2
k
A
,
A |= l.
So, in both Cases 1 and 2, A |= l; A |= S; S is
satisfiable.
Lemma 5 (Reduction Lemma). Let A = {a
i
|i ≤ m} ⊆
PropAtom − {0, 1}, S
1
= {0 ≺ a
i
|i ≤ m} ∪ {a
i
≺
1|i ≤ m} ⊆ OrdPropCl, S
2
= {(
W
k
i
j=0
Cn
1
i
j

i
j
Cn
2
i
j
) ∨
C
i
|Cn
1
i
j
, Cn
2
i
j
∈ PropConj, i ≤ n} ⊆ OrdPropCl,
atoms(S
2
) = A, S = S
1
∪ S
2
⊆ OrdPropCl such that
for all S ∈ S el({{ j | j ≤ k
i
}
i
|i ≤ n}), there exists
an application of Rule (18) with respect to S
1
∪
{Cn
1
i
S(i)

i
S(i)
Cn
2
i
S(i)
|i ≤ n} ⊆ OrdPropCl. There ex-
ists
/
0 6= I
∗
⊆ {i|i ≤ n} such that
W
i∈I
∗
C
i
∈ clo
H
(S).
Proof. Analogous to the one of Proposition 2,
(Guller, 2009).
Let S ⊆ OrdPropCl. S is a guarded order clausal
theory iff, for all a ∈ atoms(S)−{0, 1}, either a P 0 ∈
S or 0 ≺ a, a ≺ 1 ∈ S or a P 1 ∈ S. Let l ∈ OrdPropLit
and a ∈ PropAtom − {0, 1}. l is a guard iff either l =
a P 0 or l = 0 ≺ a or l = a ≺ 1 or l = a P 1. We
denote guards(S) = {l |l ∈ S is a guard} ⊆ S.
Lemma 6 (Normalisation Lemma). Let S ⊆
F
OrdPropCl be guarded. There exists S
∗
⊆
F
clo
H
(S)
such that there exist A = {a
i
|1 ≤ i ≤ m} ⊆
PropAtom − {0, 1} for some m, S
1
= {0 ≺ a
i
|1 ≤
i ≤ m} ∪ {a
i
≺ 1|1 ≤ i ≤ m} ⊆ OrdPropCl, S
2
=
{
W
k
i
j=1
Cn
1
i
j

i
j
Cn
2
i
j
|Cn
1
i
j
, Cn
2
i
j
∈ PropConj, 1 ≤ i ≤
n} ⊆ OrdPropCl for some n; and atoms(S
2
) = A,
S
∗
= S
1
∪ S
2
, guards(S
∗
) = S
1
, S
∗
is guarded; S
∗
is
equisatisfiable to S.
Proof. Let B
0
= {b |b P 0 ∈ guards(S)} ⊆
atoms(S)−{0, 1} and B
1
= {b |b P 1 ∈ guards(S)} ⊆
atoms(S) − {0, 1}. Then, for all b ∈ B
0
, clo
H
(S) is
closed with respect to applications of Rule (20); for all
b ∈ B
1
, clo
H
(S) is closed with respect to applications
of Rule (21); clo
H
(S) is closed with respect to appli-
cations of Rule (19); clo
H
(S) is closed with respect
to applications of Rule (22); clo
H
(S) is closed with
respect to applications of Rule (23); the order clausal
theory in the antecedent is equisatisfiable to the one
in the consequent of every Rule (19), (20)–(23). By
Lemma 3 for S, clo
H
(S) ⊆
F
OrdPropCl. We put S
2
=
{C |C =
W
k
j=1
Cn
1
j

j
Cn
2
j
∈ clo
H
(S), Cn
1
j
, Cn
2
j
∈
PropConj, atoms(C) ∩ (B
0
∪ B
1
) =
/
0} ⊆
F
clo
H
(S), A = atoms(S
2
) ⊆ PropAtom − {0, 1},
S
1
= {0 ≺ a |a ∈ A, 0 ≺ a ∈ guards(S)} ∪ {a ≺
1|a ∈ A, a ≺ 1 ∈ guards(S)} ⊆ S ⊆
F
clo
H
(S),
S
∗
= S
1
∪ S
2
⊆
F
clo
H
(S). Hence, guards(S
∗
) = S
1
,
S
∗
is guarded; S
∗
is equisatisfiable to S.
Theorem 7 (Refutational Soundness and Complete-
ness). Let S ⊆
F
OrdPropCl be guarded.  ∈ clo
H
(S)
if and only if S is unsatisfiable.
Proof. (=⇒) Let A be a model of S and C ∈ clo
H
(S).
Then A |= C. The proof is by complete induction on
FCTA 2016 - 8th International Conference on Fuzzy Computation Theory and Applications
38

the length of a deduction of C from S by order hyper-
resolution. Let  ∈ clo
H
(S) and A be a model of S.
Hence, A |= , which is a contradiction; S is unsatis-
fiable.
(⇐=) Let  6∈ clo
H
(S). Then, by Lemma 6 for
S, there exists S
∗
⊆
F
clo
H
(S) such that there exist
A = {a
i
|1 ≤ i ≤ m} ⊆ PropAtom − {0, 1} for some
m, S
1
= {0 ≺ a
i
|1 ≤ i ≤ m} ∪ {a
i
≺ 1 |1 ≤ i ≤ m} ⊆
OrdPropCl, S
2
= {
W
k
i
j=1
Cn
1
i
j

i
j
Cn
2
i
j
|Cn
1
i
j
, Cn
2
i
j
∈
PropConj, 1 ≤ i ≤ n} ⊆ OrdPropCl for some n; and
atoms(S
2
) = A, S
∗
= S
1
∪ S
2
, S
∗
is equisatisfiable to
S;  6∈ clo
H
(S
∗
). We get two cases for S
∗
.
Case 1: S
∗
=
/
0. Then S
∗
is satisfiable, and S is
satisfiable.
Case 2: S
∗
6=
/
0. Then m, n ≥ 1, for all 1 ≤ i ≤
n, k
i
≥ 1, by Lemma 5 for S
∗
, there exists S
∗
∈
Sel({{ j |1 ≤ j ≤ k
i
}
i
|1 ≤ i ≤ n}) such that there
does not exist an application of Rule (18) with re-
spect to S
1
∪ {Cn
1
i
S
∗
(i)

i
S
∗
(i)
Cn
2
i
S
∗
(i)
|1 ≤ i ≤ n} ⊆
OrdPropCl. We put S
0
2
= {Cn
1
i
S
∗
(i)

i
S
∗
(i)
Cn
2
i
S
∗
(i)
|1 ≤
i ≤ n} ⊆ OrdPropCl, A
0
= atoms(S
0
2
) ⊆
F
PropAtom−
{0, 1}, S
0
1
= {0 ≺ a|0 ≺ a ∈ S
1
, a ∈ A
0
} ∪ {a ≺
1|a ≺ 1 ∈ S
1
, a ∈ A
0
} ⊆
F
OrdPropCl, S
0
= S
0
1
∪ S
0
2
⊆
OrdPropCl. Hence, atoms(S
0
2
) = A
0
, S
0
1
⊆ S
1
, S
0
=
S
0
1
∪ S
0
2
⊆ S
1
∪ S
0
2
, there does not exist an application
of Rule (18) with respect to S
0
; by Lemma 4 for S
0
, S
0
is satisfiable; S
1
∪ S
0
2
is satisfiable; S
∗
is satisfiable; S
is satisfiable.
So, in both Cases 1 and 2, S is satisfiable. The
theorem is proved.
Let S ⊆ S
0
⊆ OrdPropCl. S
0
is a guarded extension
of S iff S
0
is guarded and minimal with respect to ⊆.
Theorem 8 (Satisfiability Problem). Let S ⊆
F
OrdPropCl. S is satisfiable if and only if there ex-
ists a guarded extension S
0
⊆
F
OrdPropCl of S which
is satisfiable.
Proof. (=⇒) Let S be satisfiable and A be a model
of S. Then atoms(S) ⊆
F
PropAtom. We put
S
1
= {a P 0 | a ∈ atoms(S) − {0, 1}, A(a) = 0} ∪
{0 ≺ a | a ∈ atoms(S) − {0, 1}, 0 < A(a) < 1} ∪ {a ≺
1|a ∈ atoms(S) −{0, 1}, 0 < A(a) < 1}∪{a P 1|a ∈
atoms(S)−{0, 1}, A(a) = 1} ⊆
F
OrdPropCl and S
0
=
S
1
∪ S ⊆
F
OrdPropCl. Hence, S
0
is a guarded exten-
sion of S, for all l ∈ S
1
, A |= l; A |= S
1
; A |= S
0
; S
0
is
satisfiable.
(⇐=) Let there exist a guarded extension S
0
⊆
F
OrdPropCl of S which is satisfiable. Then S ⊆ S
0
is
satisfiable. The theorem is proved.
Corollary 9. Let n
0
∈ N, φ ∈ OrdPropForm
/
0
, T ⊆
F
OrdPropForm
/
0
. There exist J
φ
T
⊆
F
{(i, j) | i ≥ n
0
} and
S
φ
T
⊆
F
OrdPropCl
{ ˜a | ∈J
φ
T
}
such that T |= φ if and
only if, for every guarded extension S
0
⊆
F
OrdPropCl
of S
φ
T
,  ∈ clo
H
(S
0
).
Proof. An immediate consequence of Theorems 2, 7,
and 8.
We illustrate the solution to the deduction problem
with an example. We show that φ = (0 ≺ c)&(a & c ≺
b&c) → a ≺ b ∈ OrdPropForm is a tautology us-
ing the proposed translation to clausal form and the
order hyperresolution calculus. Let V be a valu-
ation. Let there exist p
∗
∈ {a, b, c} and V (p
∗
) ∈
{0, 1}. Then V |= φ. Hence, it suffices to exam-
ine the case that for all p ∈ {a, b, c}, 0 < V (p) < 1.
We put S
0
= {0 ≺ a, a ≺ 1, 0 ≺ b, b ≺ 1, 0 ≺ c, c ≺
1}. Let there exist p
∗
∈ { ˜a
5
, . . . , ˜a
7
, ˜a
10
, . . . , ˜a
13
}
and V (p
∗
) ∈ {0, 1}. Then V 6|= S
0
∪ S
φ
. Hence,
it suffices to examine the case that for all p ∈
{ ˜a
5
, . . . , ˜a
7
, ˜a
10
, . . . , ˜a
13
}, 0 < V (p) < 1. We put
S
1
= S
0
∪ {0 ≺ ˜a
i
|i ∈ {5, . . . , 7, 10, . . . , 13}} ∪ { ˜a
i
≺
1|i ∈ {5, . . . , 7, 10, . . . , 13}}. Let V ( ˜a
0
) = 1. Then
V 6|= {[1]} ⊆ S
1
∪S
φ
. Let V ( ˜a
0
) < 1. Then, from [16]
and [17], V ( ˜a
2
) ∈ {0, 1}, from [6], V ( ˜a
3
) = 1, from
[4], V ( ˜a
1
) = V ( ˜a
4
), from [8] and [9], V ( ˜a
4
) ∈ {0, 1},
V ( ˜a
1
) ∈ {0, 1}, from [3], V ( ˜a
2
) < V ( ˜a
1
), V ( ˜a
2
) =
0, V ( ˜a
4
) = V ( ˜a
1
) = 1, from [2], V ( ˜a
1
)·V ( ˜a
0
) =
V ( ˜a
2
), V ( ˜a
0
) = V ( ˜a
2
) = 0. We put S
2
= S
1
∪ { ˜a
0
P
0, ˜a
1
P 1, ˜a
2
P 0, ˜a
3
P 1, ˜a
4
P 1}. In Table 4, we de-
rive [21], [23] from S
2
∪ S
φ
. Let V ( ˜a
8
) = 0. Then
V 6|= {0 ≺ ˜a
10
, 0 ≺ ˜a
11
, [10]} ⊆ S
2
∪ S
φ
. Let V ( ˜a
8
) =
1. Then V 6|= { ˜a
10
≺ 1, ˜a
11
≺ 1, [10]} ⊆ S
2
∪ S
φ
. Let
V ( ˜a
9
) = 0. Then V 6|= {0 ≺ ˜a
12
, 0 ≺ ˜a
13
, [13]} ⊆
S
2
∪ S
φ
. Let V ( ˜a
9
) = 1. Then V 6|= { ˜a
12
≺ 1, ˜a
13
≺
1, [13]} ⊆ S
2
∪ S
φ
. We put S
3
= S
2
∪ {0 ≺ ˜a
8
, 0 ≺
˜a
9
, ˜a
8
≺ 1, ˜a
9
≺ 1}. In Table 4, we get a refutation
of S
3
∪ S
φ
. We conclude that there exists a refutation
of every guarded extension of S
φ
; by Corollary 9 for
φ, S
φ
, φ is a tautology.
5 CONCLUSIONS
We have generalised the hyperresolution principle to
the propositional product logic. We have proposed
translation of a formula to an equivalent satisfiable fi-
nite order clausal theory. Order clauses are finite sets
of order literals of the augmented form: ε
1
 ε
2
where
ε
i
is either the truth constant 0 or 1 or a conjunction
of powers of propositional atoms, and  is the con-
nective P or ≺. P and ≺ are interpreted by the stan-
dard equality and strict order on [0, 1], respectively.
We have devised a hyperresolution calculus over or-
der clausal theories. The calculus is refutation sound
Hyperresolution for Propositional Product Logic
39

Table 4: φ = (0 ≺ c)&(a &c ≺ b & c) → a ≺ b.
φ = (0 ≺ c) &(a& c ≺ b &c) → a ≺ b
n
˜a
0
≺ 1, ˜a
0
↔

(0 ≺ c)&(a & c ≺ b & c)
| {z }
˜a
1
→ a ≺ b
|{z}
˜a
2

o
(9)
n
˜a
0
≺ 1, ˜a
1
≺ ˜a
2
∨ ˜a
1
P ˜a
2
∨ ˜a
1
& ˜a
0
P ˜a
2
, ˜a
2
≺ ˜a
1
∨ ˜a
0
P 1,
˜a
1
↔ (0 ≺ c
|{z}
˜a
3
)&(a & c ≺ b &c
| {z }
˜a
4
), ˜a
2
↔ a
|{z}
˜a
5
≺ b
|{z}
˜a
6
o
(7), (12)
n
˜a
0
≺ 1, ˜a
1
≺ ˜a
2
∨ ˜a
1
P ˜a
2
∨ ˜a
1
& ˜a
0
P ˜a
2
, ˜a
2
≺ ˜a
1
∨ ˜a
0
P 1,
˜a
1
P ˜a
3
& ˜a
4
, ˜a
3
↔ 0 ≺ c
|{z}
˜a
7
, ˜a
4
↔ a & c
|{z}
˜a
8
≺ b & c
|{z}
˜a
9
,
˜a
5
≺ ˜a
6
∨ ˜a
2
P 0, ˜a
6
≺ ˜a
5
∨ ˜a
6
P ˜a
5
∨ ˜a
2
P 1, ˜a
5
P a, ˜a
6
P b
o
(16), (12)
n
˜a
0
≺ 1, ˜a
1
≺ ˜a
2
∨ ˜a
1
P ˜a
2
∨ ˜a
1
& ˜a
0
P ˜a
2
, ˜a
2
≺ ˜a
1
∨ ˜a
0
P 1,
˜a
1
P ˜a
3
& ˜a
4
, 0 ≺ ˜a
7
∨ ˜a
3
P 0, ˜a
7
P 0 ∨ ˜a
3
P 1, ˜a
7
P c,
˜a
8
≺ ˜a
9
∨ ˜a
4
P 0, ˜a
9
≺ ˜a
8
∨ ˜a
9
P ˜a
8
∨ ˜a
4
P 1, ˜a
8
↔ a
|{z}
˜a
10
& c
|{z}
˜a
11
, ˜a
9
↔ b
|{z}
˜a
12
& c
|{z}
˜a
13
,
˜a
5
≺ ˜a
6
∨ ˜a
2
P 0, ˜a
6
≺ ˜a
5
∨ ˜a
6
P ˜a
5
∨ ˜a
2
P 1, ˜a
5
P a, ˜a
6
P b
o
(7)
S
φ
=
(
˜a
0
≺ 1 [1]
˜a
1
≺ ˜a
2
∨ ˜a
1
P ˜a
2
∨ ˜a
1
& ˜a
0
P ˜a
2
[2]
˜a
2
≺ ˜a
1
∨ ˜a
0
P 1 [3]
˜a
1
P ˜a
3
& ˜a
4
[4]
0 ≺ ˜a
7
∨ ˜a
3
P 0 [5]
˜a
7
P 0 ∨ ˜a
3
P 1 [6]
˜a
7
P c [7]
˜a
8
≺ ˜a
9
∨ ˜a
4
P 0 [8]
˜a
9
≺ ˜a
8
∨ ˜a
9
P ˜a
8
∨ ˜a
4
P 1 [9]
˜a
8
P ˜a
10
& ˜a
11
[10]
˜a
10
P a [11]
˜a
11
P c [12]
˜a
9
P ˜a
12
& ˜a
13
[13]
˜a
12
P b [14]
˜a
13
P c [15]
˜a
5
≺ ˜a
6
∨ ˜a
2
P 0 [16]
˜a
6
≺ ˜a
5
∨ ˜a
6
P ˜a
5
∨ ˜a
2
P 1 [17]
˜a
5
P a [18]
˜a
6
P b
)
[19]
Rule (21) : [8][ ˜a
4
P 1] :
˜a
8
≺ ˜a
9
∨ 1 P 0 [20]
Rule (19) : [20] :
˜a
8
≺ ˜a
9
[21]
Rule (20) : [17][ ˜a
2
P 0] :
˜a
6
≺ ˜a
5
∨ ˜a
6
P ˜a
5
∨ 0 P 1 [22]
Rule (19) : [22] :
˜a
6
≺ ˜a
5
∨ ˜a
6
P ˜a
5
[23]
repeatedly Rule (18) : {0 ≺ p | p ∈ {a, b, c, ˜a
5
, ˜a
6
, ˜a
8
, . . . , ˜a
13
}},
{p ≺ 1 | p ∈ {a, b, c, ˜a
5
, ˜a
6
, ˜a
8
, . . . , ˜a
13
}};
[10][11][12][13][14][15][21];[18][19][23] :
 [24]
FCTA 2016 - 8th International Conference on Fuzzy Computation Theory and Applications
40

and complete for finite guarded order clausal theories.
A clausal theory is satisfiable if and only if there ex-
ists a satisfiable guarded extension of it. So, the SAT
problem of a finite order clausal theory can be reduced
to the SAT problem of a finite guarded order clausal
theory. By means of the translation and calculus, we
have solved the deduction problem T |= φ for a finite
theory T and a formula φ.
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