A DPLL Procedure for the Propositional Product Logic
Du
ˇ
san Guller
Department of Applied Informatics, Comenius University, Mlynsk
´
a dolina, 842 48 Bratislava, Slovakia
Keywords:
Product Logic, DPLL Procedure, Many-valued Logics, Automated Deduction.
Abstract:
In the paper, we investigate the deduction problem of a formula from a finite theory in the propositional
Product logic from a perspective of automated deduction. Our approach is based on translation of a formula to
an equivalent satisfiable finite order clausal theory, consisting of order clauses. An order clause is a finite set
of order literals of the form ε
1
 ε
2
where ε
i
is either a conjunction of propositional atoms or the propositional
constant 0 (false) or 1 (true), and  is a connective either P or ≺. P and ≺ are interpreted by the equality
and standard strict linear order on [0,1], respectively. A variant of the DPLL procedure, operating over order
clausal theories, is proposed. The DPLL procedure is proved to be refutation sound and complete for finite
order clausal theories.
1 INTRODUCTION
A considerable effort has been made in development
of SAT solvers for the problem of Boolean satisfi-
ability, especially in the last decade. SAT solvers
may exploit either complete solution methods (called
complete or systematic SAT solvers) or incomplete
or hybrid ones. Complete SAT solvers are mostly
based on the Davis-Putnam-Logemann-Loveland pro-
cedure (DPLL) (Davis and Putnam, 1960; Davis et al.,
1962) improved by various features. One of the lat-
est overviews of development of SAT solvers may
be found in (Biere et al., 2009). Research in many-
valued logics mainly concerns finitely-valued ones.
Thank to finiteness of truth value sets of these logics,
almost straightforward extensions of results achieved
in classical logic are feasible. The DPLL procedure
has been firstly generalised for regular clauses over
a linearly ordered truth value set (H
¨
ahnle, 1996). In
(Many
`
a et al., 1998), it is described an implemen-
tation of this regular DPLL procedure with the ex-
tended two-sided Jeroslow-Wang literal selection rule
defined in (H
¨
ahnle, 1996). A signed DPLL procedure
over a finite truth value set is introduced in (Beckert
et al., 2000). It is based on a branching rule form-
ing branches for every truth value. So, the branch-
ing factor equals the cardinality of the truth value set.
The branching factor can be decreased by a quotient
of the truth value set wrt. a suitable equivalence.
A slight modification of that equivalence enables a
Partially supported by VEGA Grant 1/0979/12.
generalisation to an infinite truth value set as well
(Guller, 2009). Another signed variant of the DPLL
procedure for a countable clausal theory over an ar-
bitrary truth value set is proposed in (Guller, 2009).
In some sense, the DPLL procedure may be viewed
like ”anti-resolution”. Thus, its branching rule, with
finite branching factor, may be considered as if a
”signed anti-hyperresolution rule”. The procedure is
refutation complete if the finitary disjunction condi-
tion for the set of signs occurring in the input count-
able clausal theory is satisfied. Infinitely-valued log-
ics have not yet been explored so widely as finitely-
valued ones. It is not known any general approach
as signed logic one in the finitely-valued case. A so-
lution of the SAT and VAL problems strongly varies
on a chosen infinitely-valued logic. The same holds
for translation of a formula to clause form, the ex-
istence of which is not guaranteed in general. Re-
sults in this area have been achieved in several ways,
since infinite truth value sets form distinct algebraic
structures. One approach may be based on reduction
from the infinitely-valued case to the finitely-valued
one, as it has been done e.g. for the VAL problem in
the propositional infinitely-valued Łukasiewicz logic
in (Mundici, 1987; Aguzzoli and Ciabattoni, 2000).
Another approach exploits reduction of the SAT prob-
lem to mixed integer programming (MIP) (H
¨
ahnle,
1994a; H
¨
ahnle, 1997). In (Guller, 2010), we have
devised a variant of the DPLL procedure with clause
form translation for finite theories in the propositional
G
¨
odel logic. The results have been generalised to the
213
Guller D..
A DPLL Procedure for the Propositional Product Logic.
DOI: 10.5220/0004557402130224
In Proceedings of the 5th International Joint Conference on Computational Intelligence (FCTA-2013), pages 213-224
ISBN: 978-989-8565-77-8
Copyright
c
 2013 SCITEPRESS (Science and Technology Publications, Lda.)

countable case in (Guller, 2012).
Product logic (H
´
ajek et al., 1996; Metcalfe et al.,
2004; Savick
´
y et al., 2006) is one of the fundamental
fuzzy logics, based on the product t-norm. It has been
discovered much later than G
¨
odel and Łukasiewicz
logics, known before the beginning of research on
fuzzy theory. In the paper, we investigate the de-
duction problem of a formula from a finite theory
in the propositional Product logic from a perspective
of automated deduction. Our approach is based on
translation of a formula to an equivalent satisfiable fi-
nite order clausal theory, consisting of order clauses,
Lemma 3.1, Theorem 3.2, Section 3. An order clause
is a finite set of order literals of the form ε
1
ε
2
where
ε
i
is either a conjunction of propositional atoms or
the propositional constant 0 (false) or 1 (true), and
 is a connective either P or ≺. P and ≺ are in-
terpreted by the equality and standard strict order on
[0, 1], respectively. The trichotomy over order literals:
either ε
1
≺ ε
2
or ε
1
P ε
2
or ε
2
≺ ε
1
, naturally invokes
proposing a variant of the DPLL procedure with a tri-
chotomy branching rule as an algorithm for deciding
the satisfiability of a finite order clausal theory. The
DPLL procedure with its basic rules is proved to be
refutation sound and complete in the finite case, The-
orem 4.2, Section 4. The set of basic rules may be
augmented by some admissible ones, which are suit-
able for practical computing and considerably shorten
DPLL trees. For solving the deduction problem, we
exploit the fact that a formula φ is a propositional con-
sequence of a finite theory T in Product logic if and
only if their translation to a finite order clausal the-
ory S
φ
T
is unsatisfiable, and the DPLL procedure pro-
duces a closed DPLL tree with the root S
φ
T
in this case,
Corollary 4.3, Section 4.
The paper is organised as follows. Section 2 gives
the basic notions, notation, and useful properties con-
cerning the propositional Product logic. Section 3
deals with clause form translation. In Section 4,
we propose a variant of the DPLL procedure with a
trichotomy branching rule and prove its refutational
soundness, completeness. Section 5 brings conclu-
sions.
2 PROPOSITIONAL PRODUCT
LOGIC
Throughout the paper, we shall use the common no-
tions of propositional many-valued logics. The set of
propositional atoms of Product logic will be denoted
as PropAtom. By PropForm we designate the set of
all propositional formulae of Product logic built up
from PropAtom using the propositional constants 0,
false, 1, true, and the connectives: ¬, negation, ∧,
conjunction, ∨, disjunction, &, strong conjunction,
→, implication. In addition, we introduce new bi-
nary connectives P, equality, and ≺, strict order. By
OrdPropForm we designate the set of all so-called or-
der propositional formulae of Product logic built up
from PropAtom using the propositional constants 0,
1, and the connectives: ¬, ∧, ∨, &, →, P, ≺.
1
In the
paper, we shall assume that PropAtom is a countable
set. Let ε
i
, 1 ≤ i ≤ n, be either an expression or a set of
expressions or a set of sets of expressions, in general.
By atoms(ε
1
, . . . , ε
m
) ⊆ PropAtom we denote the set
of all propositional atoms of Product logic occurring
in ε
1
, . . . , ε
m
.
Let X, Y , Z be sets, Z ⊆ X; f : X −→ Y be a map-
ping. By kX k we denote the set-theoretic cardinal-
ity of X. X being a finite subset of Y is denoted as
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
sequence δ of X is a bijection δ : γ −→ X. X is
countable if and only if there exists a sequence of
X. N | R designates the set of natural | real num-
bers and ≤, < the standard, standard strict order on
N | R, respectively. We denote R
+
0
= {c |0 ≤ c ∈ R},
R
+
= {c |0 < c ∈ R}, [0, 1] = {c |0 ≤ c ≤ 1, c ∈ R};
[0, 1] is the unit interval. Let c ∈ R
+
. log c denotes the
binary logarithm of c. Let f , g : N −→ R
+
0
. f is of the
order of g, in symbols f ∈ O(g), iff there exist n
0
∈ N
and c
∗
∈ R
+
0
such that for all n ≥ n
0
, f (n) ≤ c
∗
· g(n).
Let φ ∈ OrdPropForm and T ⊆
F
OrdPropForm. The
size of φ, in symbols |φ| > 0, is defined as the number
of nodes of its standard tree representation. We define
the size of T as |T | =
∑
φ∈T
|φ|.
Product logic is interpreted by the standard Π-
algebra augmented by binary operators P
P
P and ≺
≺
≺ for
P and ≺, 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;
a P
P
P b =

1 if a = b,
0 else;
a ≺
≺
≺ b =

1 if a < b,
0 else.
We recall that Π is a complete linearly ordered lattice
algebra; ∨
∨
∨ | ∧
∧
∧ is commutative, associative, idempo-
tent, monotone; 0 | 1 is its neutral element; · is com-
1
We assume a decreasing connective precedence: ¬, &,
∧, →, P, ≺, ∨.
IJCCI2013-InternationalJointConferenceonComputationalIntelligence
214

mutative, associative, monotone; 1 is its neutral ele-
ment; the residuum operator ⇒
⇒
⇒ of · satisfies the con-
dition of residuation:
for all a, b, c ∈ Π, a ·b ≤ c ⇐⇒ a ≤ b⇒
⇒
⇒c; (1)
Product (G
¨
odel) negation satisfies the condition:
for all a ∈ Π, a = a⇒
⇒
⇒0; (2)
the following properties, which will be exploited later,
hold:
2
for all a, b, c ∈ Π,
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,
(distributivity of · over ∨
∨
∨) (5)
a⇒
⇒
⇒(b∨
∨
∨c) = a⇒
⇒
⇒b∨
∨
∨a⇒
⇒
⇒c, (6)
a⇒
⇒
⇒b∧
∧
∧c = (a⇒
⇒
⇒b)∧
∧
∧(a⇒
⇒
⇒c), (7)
(a∨
∨
∨b)⇒
⇒
⇒c = (a⇒
⇒
⇒c)∧
∧
∧(b⇒
⇒
⇒c), (8)
a∧
∧
∧b⇒
⇒
⇒c = a⇒
⇒
⇒c∨
∨
∨b⇒
⇒
⇒c, (9)
a⇒
⇒
⇒(b⇒
⇒
⇒c) = a ·b⇒
⇒
⇒c, (10)
((a⇒
⇒
⇒b)⇒
⇒
⇒b)⇒
⇒
⇒b = a⇒
⇒
⇒b. (11)
A propositional theory is a set of propositional
formulae of Product logic. An order propositional
theory is a set of order propositional formulae of
Product logic. A valuation V is a mapping V :
PropAtom −→ [0, 1]. A partial valuation V with
the domain dom(V ) ⊆ PropAtom, is a mapping V :
dom(V ) −→ [0, 1]. Let V be a (partial) valua-
tion; φ, φ
0
∈ OrdPropForm, T ⊆ OrdPropForm. Let
atoms(φ), atoms(T) ⊆ dom(V ) in case of V being a
partial valuation. The truth value of φ in V , in sym-
bols kφk
V
, is defined by the standard way; the propo-
sitional constants 0, 1 are interpreted by 0, 1, respec-
tively, and the connectives by the respective operators
on Π. V is a (partial) propositional model of φ, in
symbols V |= φ, iff kφk
V
= 1. V is a (partial) propo-
sitional 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 ORDER
CLAUSAL FORM
We now describe some translation of a formula to a
finite order clausal theory. To have the output theory
of polynomial size, our translation exploits interpola-
tion using new atoms. The output theory will be of
linearithmic size at the cost of being only equivalent
satisfiable to the input formula. A similar approach
exploiting the renaming subformulae technique can
be found in (Plaisted and Greenbaum, 1986; de la
Tour, 1992; H
¨
ahnle, 1994b; Nonnengart et al., 1998;
Sheridan, 2004; Guller, 2010). At first, we intro-
duce notions of a to the power of n and of conjunc-
tion of propositional atoms. Let a ∈ PropAtom and
n > 0. a to the power of n is the pair (a, n), written
as a
n
. The power a
1
is denoted as a; if it does not
cause the ambiguity with the denotation of the sin-
gle propositional atom a in given context. We de-
fine the size of a
n
as |a
n
| = n > 0. A conjunction
Cn of propositional atoms is a non-empty finite set
of powers such that for all a
m
, b
n
∈ Cn, a 6= b. A
conjunction {a
m
0
0
, . . . , a
m
n
n
} of propositional atoms is
written in the form a
m
0
0
&·· ·&a
m
n
n
. A conjunction
{p} of propositional atoms is called a unit conjunc-
tion of propositional atoms and denoted as p; if it
does not cause the ambiguity with the denotation of
the single power p in given context. The set of all
conjunctions of propositional atoms is designated as
PropConj. Let V be a (partial) valuation; p be a
power, Cn ∈ PropConj, Cn
1
, Cn
2
∈ PropConj ∪ {
/
0}.
Let atoms(Cn) ⊆ dom(V ) in case of V being a par-
tial valuation. The truth value 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| > 0.
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
and m ≤
n. We define Cn
1
u Cn
2
= {a
min(m,n)
|a
m
∈ Cn
1
, a
n
∈
Cn
2
} ∈ PropConj ∪ {
/
0}. Cn
1
and Cn
2
are disjoint iff
Cn
1
u Cn
2
=
/
0. We finally introduce order clauses in
Product logic. l is an order literal of Product logic
iff l = ε
1
 ε
2
where either ε
1
∈ PropAtom ∪ {0, 1},
ε
2
∈ {0, 1}, or ε
1
∈ {0, 1}, ε
2
∈ PropAtom ∪ {0, 1},
or ε
i
∈ PropConj, ε
1
u ε
2
=
/
0,  ∈ {P, ≺}. The set
of all order literals of Product logic is designated as
OrdLit. Let l = ε
1
 ε
2
∈ OrdLit. We define the size
of l as |l| = 1 + |ε
1
| + |ε
2
| > 0. An order clause of
Product logic is a finite set of order literals of Prod-
uct logic; since = is commutative, we identify the
order literals ε
1
P ε
2
and ε
2
P ε
1
. An order clause
ADPLLProcedureforthePropositionalProductLogic
215

{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 or-
der literal l in given context. We designate the set
of all order clauses of Product logic as OrdCl. Let
l, l
0
, . . . , l
n
∈ OrdLit and C,C
0
∈ OrdCl
L
. We define
the size of C as |C| =
∑
l∈C
|l|. By l ∨C we denote
{l} ∪C where l 6∈ C. Analogously, by l
0
∨ · ·· ∨ l
n
∨C
we denote {l
0
} ∪ ··· ∪ {l
n
} ∪C where, for all i, i
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 is a set of order clauses. A
unit order clausal theory is a set of unit order clauses.
Let φ, φ
0
∈ PropOrdForm, T, T
0
⊆ PropOrdForm,
S, S
0
⊆ OrdCl; V be a (partial) valuation. Let
atoms(l), atoms(C), atoms(S) ⊆ dom(V ) in case of V
being a partial valuation. Note that V |= l if and only
if either 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
.
V is a (partial) propositional model of C, in sym-
bols V |= C, iff there exists l
∗
∈ C such that V |= l
∗
.
V is a (partial) propositional model of S, in symbols
V |= S, iff, for all C ∈ S, V |= C. φ
0
| T
0
| C
0
| S
0
is a
propositional consequence of φ | T | C | S, in symbols
φ|T |C | S |=
P
φ
0
|T
0
|C
0
|S
0
, iff, for every propositional
model V of φ | T | C | S, V |= φ
0
|T
0
|C
0
|S
0
. φ | T | C |
S is satisfiable iff there exists a propositional model of
φ | T | C | S. Note that both  and  ∈ S are unsatisfi-
able. 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
satisfiable. Let S ⊆
F
OrdCl. We define the size of S as
|S| =
∑
C∈S
|C|. Let l ∈ OrdLit. l is a simplified order
literal of Product logic iff if l = ε
1
ε
2
, ε
i
∈ PropConj,
then either ε
1
= a, ε
2
= b, or ε
1
= a, ε
2
= b&c, or
ε
1
= a & b, ε
2
= c. The set of all simplified order lit-
erals of Product logic is designated as SimOrdLit. We
denote SimOrdCl = {C |C ∈ OrdCl,C ⊆ SimOrdLit}.
Let I = N × N; I is an infinite countable set of in-
dices. Let
˜
A = { ˜a | ∈ I} ⊆ PropAtom;
˜
A is an in-
finite countable set of new propositional atoms. Let
A ⊆
˜
A. We denote E
A
= {ε|ε ∈ E , atoms(ε) ∩
˜
A ⊆
A}, E = PropForm | E = PropConj | E = OrdLit |
E = OrdCl | E = SimOrdLit | E = SimOrdCl. From a
computational point of view, the worst case time and
space complexity will be estimated using the logarith-
mic cost measurement. Let A be an algorithm. #O
A
denotes the number of all elementary operations exe-
cuted by A. The translation to order clausal form is
based on the following lemma.
Lemma 3.1. Let φ ∈ PropForm
/
0
, T ⊆
F
PropForm
/
0
;
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
⊆ {(i, j)|i ≥ n
0
}, J
φ
∩F =
/
0, and S
φ
⊆
F
SimOrdCl
{ ˜a | ∈J
φ
}
such that
(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 partial valuation V ,
dom(V ) = atoms(φ), and V |= φ if
and only if there exists a partial valuation
V
0
, dom(V
0
) = atoms(φ) ∪ { ˜a | ∈ J
φ
},
and V
0
|= S
φ
, satisfying V = V
0
|
atoms(φ)
;
(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|φ|);
(e) if S
φ
6=
/
0 and S
φ
6= {}, then J
φ
6=
/
0; for all
C ∈ S
φ
,
/
0 6= atoms(C) ∩
˜
A ⊆ { ˜a | ∈ J
φ
}.
(ii) There exist J
T
⊆
F
{(i, j)| i ≥ n
0
}, J
T
∩ F =
/
0,
and S
T
⊆
F
SimOrdCl
{ ˜a | ∈J
T
}
such that
(a) kJ
T
k ≤ 2 · |T |;
(b) either J
T
=
/
0, S
T
= {} or J
T
= S
T
=
/
0 or
J
T
6=
/
0,  6∈ S
T
6=
/
0;
(c) there exists a partial valuation V ,
dom(V ) = atoms(T ), and V |= T if
and only if there exists a partial valuation
V
0
, dom(V
0
) = atoms(T ) ∪ { ˜a | ∈ J
T
},
and V
0
|= S
T
, satisfying V = V
0
|
atoms(T )
;
(d) |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 +
|T |));
(e) if S
T
6=
/
0 and S
T
6= {}, then J
T
6=
/
0; for all
C ∈ S
T
,
/
0 6= atoms(C) ∩
˜
A ⊆ { ˜a | ∈ J
T
}.
Proof. Technical using interpolation.
Let θ ∈ PropForm
/
0
. There exists θ
0
∈
PropForm
/
0
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
.
(12)
The proof is by induction on the structure of θ.
IJCCI2013-InternationalJointConferenceonComputationalIntelligence
216

Let θ ∈ PropForm
/
0
− {0, 1}; (12c–e) hold for
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, j)| i ≥ n
1
}, ˜a ∈
˜
A, 6∈ G. There exist
J = {(n
θ
, j)| j + 1 ≤ j ≤ n
J
} ⊆ {(i, j)|i ≥
n
1
}, j ≤ n
J
, J ∩ (G ∪ { }) =
/
0, and S
s
⊆
F
SimOrdCl
{ ˜a }∪{ ˜a | ∈J}
, s = +, −, such that for
both s,
(a) kJk ≤ |θ| − 1;
(b) there exists a partial valuation V ,
dom(V ) = atoms(θ) ∪ { ˜a }, and
V |= ˜a → θ ∈ PropForm
{ ˜a }
if and
only if there exists a partial valuation V
0
,
dom(V
0
) = atoms(θ) ∪ { ˜a } ∪ { ˜a | ∈ J},
and V
0
|= S
+
, satisfying V =
V
0
|
atoms(θ)∪{ ˜a }
;
(c) there exists a partial valuation V ,
dom(V ) = atoms(θ) ∪ { ˜a }, and
V |= θ → ˜a ∈ PropForm
{ ˜a }
if and
only if there exists a partial valuation V
0
,
dom(V
0
) = atoms(θ) ∪ { ˜a } ∪ { ˜a | ∈ J},
and V
0
|= S
−
, satisfying V =
V
0
|
atoms(θ)∪{ ˜a }
;
(d) |S
s
| ≤ 20 ·|θ|, S
s
can be built up from θ via
a preorder traversal of θ with #O ∈ O(|θ|);
(e) for all C ∈ S
s
,
/
0 6= atoms(C) ∩
˜
A ⊆ { ˜a } ∪
{ ˜a | ∈ J}; ˜a P 1, ˜a ≺ 1 6∈ S
s
.
(13)
The proof is by induction on the structure of θ using
the interpolation rules in Table 1.
(i) By (12) for φ, there exists φ
0
∈ PropForm
/
0
such
that (12a–e) hold for φ
0
. We then distinguish three
cases for φ
0
.
Case 1: φ
0
= 0. We put J
φ
=
/
0 ⊆ {(i, j)| i ≥ n
0
},
J
φ
∩ F =
/
0, and S
φ
= {} ⊆
F
SimOrdCl
/
0
.
Case 2: φ
0
= 1. We put J
φ
=
/
0 ⊆ {(i, j)| i ≥ n
0
},
J
φ
∩ F =
/
0, and S
φ
=
/
0 ⊆
F
SimOrdCl
/
0
.
Case 3: φ
0
6= 0, 1. We have n
φ
≥ n
0
. We put
= (n
φ
, 0) ∈ {(i, j)| i ≥ n
0
}. Then ˜a ∈
˜
A. We
get by (13) for φ
0
, F, n
0
, n
φ
, , ˜a that there ex-
ist J = {(n
φ
, j)| 1 ≤ j ≤ n
J
} ⊆ {(i, j)|i ≥ n
0
}, J ∩
(F ∪ { }) =
/
0, S
+
⊆
F
SimOrdCl
{ ˜a }∪{ ˜a | ∈J}
, and
(13a–e) hold for φ
0
, ˜a , J, S
+
. We put n
J
φ
= n
J
,
J
φ
= { } ∪ J ⊆ {(i, j) |i ≥ n
0
}, J
φ
∩ F =
/
0, and S
φ
=
{ ˜a P 1} ∪ S
+
⊆
F
SimOrdCl
{ ˜a | ∈J
φ
}
.
(ii) straightforwardly follows from (i).
We conclude this section by the following theorem.
Theorem 3.2. Let φ ∈ PropForm
/
0
, T ⊆
F
PropForm
/
0
;
F ⊆ I such that there exists n
0
and F ∩ {(i, j)|i ≥
n
0
} =
/
0. There exist J
φ
T
⊆
F
{(i, j)| i ≥ n
0
}, J
φ
T
∩ F =
/
0, and S
φ
T
⊆
F
SimOrdCl
{ ˜a | ∈J
φ
T
}
such that
(i) T |=
P
φ if and only if S
φ
T
is unsatisfiable;
(ii) kJ
φ
T
k ∈ O(|T | + |φ|); |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 | + |φ|) · log(|T | +
|φ|)).
Proof. (i) We put J
n
0
= {(n
0
, j)| } ⊆ {(i, j)| i ≥ n
0
}
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
⊆
F
{(i, j)| i ≥ n
0
+ 1},
J
T
∩G =
/
0, S
T
⊆
F
SimOrdCl
{ ˜a | ∈J
T
}
, and 3.1(ii a–e)
hold for T , J
T
, S
T
. By (12) for φ, there exists φ
0
∈
PropForm
/
0
such that (12a–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
⊆
F
{(i, j)| i ≥ n
0
},
J
φ
T
∩ F =
/
0, and S
φ
T
= S
T
⊆
F
SimOrdCl
{ ˜a | ∈J
φ
T
}
.
Case 2: φ
0
= 1. We put J
φ
T
=
/
0 ⊆
F
{(i, j)| i ≥ n
0
},
J
φ
T
∩ F =
/
0, and S
φ
T
= {} ⊆
F
SimOrdCl
/
0
.
Case 3: φ
0
6= 0, 1. We put = (n
0
, 0) ∈ {(i, j)|i ≥
n
0
}. Then ˜a ∈
˜
A. We get by (13) for φ
0
, F,
n
0
, n
0
, , ˜a that there exist J = {(n
0
, j)| 1 ≤ j ≤
n
J
} ⊆ {(i, j) |i ≥ n
0
}, J ∩ (F ∪ { }) =
/
0, S
−
⊆
F
SimOrdCl
{ ˜a }∪{ ˜a | ∈J}
, and (13a–e) hold for φ
0
, ˜a ,
J, S
−
. We put J
φ
T
= J
T
∪ { } ∪ J ⊆
F
{(i, j)| i ≥
n
0
}, J
φ
T
∩ F =
/
0, and S
φ
T
= S
T
∪ { ˜a ≺ 1} ∪ S
−
⊆
F
SimOrdCl
{ ˜a | ∈J
φ
T
}
.
(ii) straightforwardly follows. The theorem is
proved.
4 DPLL PROCEDURE
We devise a variant of the DPLL procedure
over finite order clausal theories. Let a, . . . , f ∈
PropAtom, Cn, Cn
1
, . . . , Cn
4
∈ PropConj, 
1
, 
2
∈
{P, ≺}, l, l
1
, l
2
, l
3
∈ OrdLit, C ∈ OrdCl, T ⊆ OrdCl.
l is a contradiction iff either l = 0 P 1 or l = 0 ≺ 0 or
l = 1 ≺ 0 or l = 1 ≺ 1 or l = a ≺ 0 or l = 1 ≺ a or
l = Cn ≺ Cn. l is a tautology iff either l = 0 P 0 or
l = 1 P 1 or l = 0 ≺ 1 or l = Cn P Cn. 0 P a ∨ 0 ≺ a
is a 0-dichotomy. a ≺ 1 ∨ a P 1 is a 1-dichotomy.
Cn
1
≺ Cn
2
∨ Cn
1
P Cn
2
∨ Cn
2
≺ Cn
1
is a trichotomy.
Some auxiliary operations are defined in Table 2. We
define a transitivity operation in Table 3. For exam-
ple,
(a&b ≺ c & e) I (c &d P a & f ) =
(a&b & d ≺ c & d & e) I (c&d P a & f ) =
a&b & d ≺ a & e& f =
b&d ≺ e & f .
ADPLLProcedureforthePropositionalProductLogic
217

Table 1: Interpolation rules for ∧, ∨, &, →.
Case: Laws
θ = θ
1
∧ θ
2
Positive interpolation
˜a → θ
1
∧ θ
2
n
˜a ≺ ˜a
1
∨ ˜a P ˜a
1
, ˜a ≺ ˜a
2
∨ ˜a P ˜a
2
, ˜a
1
→ θ
1
, ˜a
2
→ θ
2
o
(7) (14)
|Consequent| = 12 + | ˜a
1
→ θ
1
| + | ˜a
2
→ θ
2
| ≤ 20 + | ˜a
1
→ θ
1
| + | ˜a
2
→ θ
2
|
Negative interpolation
θ
1
∧ θ
2
→ ˜a
n
˜a
1
≺ ˜a ∨ ˜a
1
P ˜a ∨ ˜a
2
≺ ˜a ∨ ˜a
2
P ˜a , θ
1
→ ˜a
1
, θ
2
→ ˜a
2
o
(9) (15)
|Consequent| = 12 + |θ
1
→ ˜a
1
| + |θ
2
→ ˜a
2
| ≤ 20 + |θ
1
→ ˜a
1
| + |θ
2
→ ˜a
2
|
θ = θ
1
∨ θ
2
Positive interpolation
˜a → (θ
1
∨ θ
2
)
n
˜a ≺ ˜a
1
∨ ˜a P ˜a
1
∨ ˜a ≺ ˜a
2
∨ ˜a P ˜a
2
, ˜a
1
→ θ
1
, ˜a
2
→ θ
2
o
(6) (16)
|Consequent| = 12 + | ˜a
1
→ θ
1
| + | ˜p
2
→ θ
2
| ≤ 20 + | ˜a
1
→ θ
1
| + | ˜p
2
→ θ
2
|
Negative interpolation
(θ
1
∨ θ
2
) → ˜a
n
˜a
1
≺ ˜a ∨ ˜a
1
P ˜a , ˜a
2
≺ ˜a ∨ ˜a
2
P ˜a , θ
1
→ ˜a
1
, θ
2
→ ˜a
2
o
(8) (17)
|Consequent| = 12 + |θ
1
→ ˜a
1
| + |θ
2
→ ˜a
2
| ≤ 20 + |θ
1
→ ˜a
1
| + |θ
2
→ ˜a
2
|
θ = θ
1
&θ
2
Positive interpolation
˜a → θ
1
&θ
2
n
˜a ≺ ˜a
1
& ˜a
2
∨ ˜a P ˜a
1
& ˜a
2
, ˜a
1
→ θ
1
, ˜a
2
→ θ
2
o
(18)
|Consequent| = 8 + | ˜a
1
→ θ
1
| + | ˜a
2
→ θ
2
| ≤ 20 + | ˜a
1
→ θ
1
| + | ˜a
2
→ θ
2
|
Negative interpolation
θ
1
∧ θ
2
→ ˜a
n
˜a
1
& ˜a
2
≺ ˜a ∨ ˜a
1
& ˜a
2
P ˜a , θ
1
→ ˜a
1
, θ
2
→ ˜a
2
o
(19)
|Consequent| = 8 + |θ
1
→ ˜a
1
| + |θ
2
→ ˜a
2
| ≤ 20 + |θ
1
→ ˜a
1
| + |θ
2
→ ˜a
2
|
θ = θ
1
→ 0
Positive interpolation
˜a → (θ
1
→ 0)
n
˜a P 0 ∨ ˜a
1
P 0, θ
1
→ ˜a
1
o
(10) (20)
|Consequent| = 6 + |θ
1
→ ˜a
1
| ≤ 20 + |θ
1
→ ˜a
1
|
Negative interpolation
(θ
1
→ 0) → ˜a
n
0 ≺ ˜a
1
∨ ˜a = 1, ˜a
1
→ θ
1
o
(21)
|Consequent| = 6 + | ˜a
1
→ θ
1
| ≤ 20 + | ˜a
1
→ θ
1
|
θ = θ
1
→ θ
2
, θ
2
6= 0
Positive interpolation
˜a → (θ
1
→ θ
2
)
n
˜a & ˜a
1
≺ ˜a
2
∨ ˜a & ˜a
1
P ˜a
2
, θ
1
→ ˜a
1
, ˜a
2
→ θ
2
o
(10) (22)
|Consequent| = 8 + |θ
1
→ ˜a
1
| + | ˜a
2
→ θ
2
| ≤ 20 + |θ
1
→ ˜a
1
| + | ˜a
2
→ θ
2
|
Negative interpolation
(θ
1
→ θ
2
) → ˜a
n
˜a
1
≺ ˜a
2
∨ ˜a
1
P ˜a
2
∨ ˜a
2
≺ ˜a
1
& ˜a ∨ ˜a
2
P ˜a
1
& ˜a , ˜a
2
≺ ˜a
1
∨ ˜a = 1, ˜a
1
→ θ
1
, θ
2
→ ˜a
2
o
(23)
|Consequent| = 20 + | ˜a
1
→ θ
1
| + |θ
2
→ ˜a
2
| ≤ 20 + | ˜a
1
→ θ
1
| + |θ
2
→ ˜a
2
|
IJCCI2013-InternationalJointConferenceonComputationalIntelligence
218

Table 2: Auxiliary operations.
Cn
1
 Cn
2
= {a
m+n
|a
m
∈ Cn
1
, a
n
∈ Cn
2
} ∪ {a
m
|a
m
∈ Cn
1
, a 6∈ atoms(Cn
2
)} ∪
{a
n
|a
n
∈ Cn
2
, a 6∈ atoms(Cn
1
)} ∈ PropConj ∪ {
/
0},
Cn
1
⇓Cn
2
= {a
m−n
|a
m
∈ Cn
1
, a
n
∈ Cn
2
, m > n} ∪ {a
m
|a
m
∈ Cn
1
, a 6∈ atoms(Cn
2
)} ∈ PropConj ∪ {
/
0}
if Cn
2
v Cn
1
,
Cn
1
B Cn
2
= {a
n−m
|a
m
∈ Cn
1
, a
n
∈ Cn
2
, n > m} ∪ {a
n
|a
n
∈ Cn
2
, a 6∈ atoms(Cn
1
)} ∈ PropConj ∪ {
/
0}
Cn
1
, Cn
2
∈ PropConj ∪ {
/
0}.
Table 3: Transitivity operation.
(Cn
1

1
Cn
2
) I (Cn
3

2
Cn
4
) =













1  1 if Cn
7
= Cn
8
=
/
0,
 if Cn
7
=
/
0, Cn
8
6=
/
0,
 if Cn
7
6=
/
0, Cn
8
=
/
0,  =P,
1 P 1 if Cn
7
6=
/
0, Cn
8
=
/
0,  =≺,
Cn
7
 Cn
8
if Cn
7
6=
/
0, Cn
8
6=
/
0,
Cn
5
= (Cn
1
 (Cn
2
B Cn
3
)),
Cn
6
= (((Cn
2
 (Cn
2
B Cn
3
))⇓Cn
3
)  Cn
4
),
Cn
7
= (Cn
5
⇓(Cn
5
u Cn
6
)),
Cn
8
= (Cn
6
⇓(Cn
5
u Cn
6
)),
 =

P if 
1
= 
2
=P,
≺ else,
(Cn
1

1
Cn
2
) I (Cn
3

2
Cn
4
) ∈ OrdCl
Cn
1
, . . . , Cn
4
∈ PropConj, 
1
, 
2
∈ {P, ≺}.
An auxiliary simplification function is defined in Ta-
ble 4. Basic rules are defined as follows:
(Contradiction simplification rule) (24)
T
T − {l ∨C}∪ {C}
l ∨C ∈ T, l is a contradiction.
(One literal 0-simplification rule) (25)
T
T − {l ∨C}∪ simpl(a P 0, l ∨C)
a P 0, l ∨C ∈ T, a ∈ atoms(l).
(One literal 1-simplification rule) (26)
T
T − {l ∨C}∪ simpl(a P 1, l ∨C)
a P 1, l ∨C ∈ T, a ∈ atoms(l).
(0-dichotomy branching rule) (27)
T
T ∪ {l
1
}


T ∪ {l
2
}
l
1
∨ l
2
is a 0-dichotomy, atoms(l
1
∨ l
2
) ⊆ atoms(T ).
(1-dichotomy branching rule) (28)
T
T ∪ {l
1
}


T ∪ {l
2
}
l
1
∨ l
2
is a 1-dichotomy, atoms(l
1
∨ l
2
) ⊆ atoms(T ).
ADPLLProcedureforthePropositionalProductLogic
219

Table 4: Auxiliary simplification function.
simpl(a P 0, a  ε ∨C) = {0  ε ∨C} if a P 0 6= a  ε ∨C,
simpl(a P 0, ε  a ∨C) = {ε  0 ∨C} if a P 0 6= ε  a ∨C,
simpl(a P 0, Cn
1
P Cn
2
∨C) = {
_
b∈atoms(Cn
2
)
b P 0 ∨C} if a ∈ atoms(Cn
1
),
simpl(a P 0, Cn
1
≺ Cn
2
∨C) = {0 ≺ b ∨C |b ∈ atoms(Cn
2
)} if a ∈ atoms(Cn
1
),
simpl(a P 0, Cn
1
≺ Cn
2
∨C) = {C} if a ∈ atoms(Cn
2
);
simpl(a P 1, a  ε ∨C) = {1  ε ∨C} if a P 1 6= a  ε ∨C,
simpl(a P 1, ε  a ∨C) = {ε  1 ∨C} if a P 1 6= ε  a ∨C,
simpl(a P 1, Cn
1
P Cn
2
∨C) = {(Cn
1
− {a
n
}) P Cn
2
∨C} if {a} ⊂ atoms(Cn
1
), a
n
∈ Cn
1
,
simpl(a P 1, Cn
1
P Cn
2
∨C) = {b P 1 ∨C |b ∈ atoms(Cn
2
)} if {a} = atoms(Cn
1
),
simpl(a P 1, Cn
1
≺ Cn
2
∨C) = {(Cn
1
− {a
n
}) ≺ Cn
2
∨C} if {a} ⊂ atoms(Cn
1
), a
n
∈ Cn
1
,
simpl(a P 1, Cn
1
≺ Cn
2
∨C) = {C} if {a} = atoms(Cn
1
),
simpl(a P 1, Cn
1
≺ Cn
2
∨C) = {Cn
1
≺ (Cn
2
− {a
n
}) ∨C} if {a} ⊂ atoms(Cn
2
), a
n
∈ Cn
2
,
simpl(a P 1, Cn
1
≺ Cn
2
∨C) = {
_
b∈atoms(Cn
1
)
b ≺ 1 ∨C} if {a} = atoms(Cn
2
);
simpl(l,C) ⊆
F
OrdCl
a ∈ PropAtom, ε ∈ {0, 1}, Cn
1
, Cn
2
∈ PropConj, l ∈ {a P 0, a P 1}, C ∈ OrdCl.
(One literal transitivity rule) (29)
T
T ∪ {(Cn
1

1
Cn
2
) I (Cn
3

2
Cn
4
)}
T is a unit order clausal theory,
Cn
1

1
Cn
2
, Cn
3

2
Cn
4
∈ T,
for all a ∈ atoms(Cn
1
, . . . , Cn
4
), 0 ≺ a, a ≺ 1 ∈ T .
(Trichotomy branching rule) (30)
T
T − {l
1
∨C}∪ {l
1
}


T − {l
1
∨C}∪ {C} ∪ {l
2
}


T − {l
1
∨C}∪ {C} ∪ {l
3
}
l
1
∨C ∈ T, C 6= , l
1
∨ l
2
∨ l
3
is a trichotomy,
for all a ∈ atoms(l
1
, l
2
, l
3
), 0 ≺ a, a ≺ 1 ∈ T .
Rules (24)–(30) are sound in view of satisfiabil-
ity. The proof is straightforward. The refutational
completeness argument of the basic rules, Theo-
rem 4.2(ii), can be provided using the excess literal
technique (Anderson and Bledsoe, 1970). From this
point of view, Rules (24) and (29) handle the base
case: T is a unit order clausal theory; while Rule (30)
handles the induction one: it subtracts the excess lit-
eral measure of T at least by 1 for the clausal theory
in every branch of its consequent.
T is closed under Rules (24) and (29) iff for every
application of Rules (24) and (29) of the form
T
T
0
,
T
0
= T . By trans(T ) ⊆ OrdCl we denote the least set
such that trans(T ) ⊇ T and trans(T ) is closed under
Rules (24), (29).
Using the basic rules, one can construct a finitely
generated tree with the input theory as the root in
the usual manner, so as the classical DPLL procedure
does; for every parent vertex, there exists an appli-
cation of Rule (24)–(30) such that the theory in its
antecedent is in the parent vertex and the theories in
its consequent are in the children vertices. A branch
of a tree is closed iff it contains a vertex T
0
such that
 ∈ T
0
. A branch of a tree is open iff it is not closed.
A tree is closed iff every branch of it is finite and
closed. A closed tree is finite by K
¨
onig’s Lemma. A
tree is open iff it is not closed. A tree is linear iff it
consists of only one branch, beginning in its root and
ending in its only leaf.
The following lemma shows that Rules (24) and
(29) are refutation complete for a special kind of
(countable) unit order clausal theory, which will be
exploited in the base case of Theorem 4.2(ii).
Lemma 4.1. Let T = trans(T ) ⊆ OrdCl be a count-
IJCCI2013-InternationalJointConferenceonComputationalIntelligence
220

able unit order clausal theory such that for all a ∈
atoms(T ), either there exists a P ε ∈ T , ε ∈ {0, 1},
satisfying, for all C ∈ T and C 6= a P ε, a 6∈ atoms(C);
or 0 ≺ a, a ≺ 1 ∈ T . There exists a partial model A of
T , dom(A) = atoms(T ).
Proof. By the lemma assumption that T is a unit or-
der clausal theory,  6∈ T = trans(T ). In addition,
by the lemma assumption that T is a countable set,
there exist γ ≤ ω and a sequence δ : γ −→ atoms(T )
of atoms(T ). At first, we define a partial valuation V
α
by recursion on α ≤ γ in Table 5. It is straightforward
to prove the following statements:
For all α ≤ γ, V
α
is a partial valuation,
dom(V
α
) = δ[α]; and for all α ≤ α
0
≤ γ, V
α
⊆
V
α
0
.
(31)
The proof is by induction on α ≤ γ.
For all α ≤ γ and l ∈ T such that atoms(l) ⊆
dom(V
α
), V
α
|= l.
(32)
The proof is by induction on α ≤ γ.
We put A = V
γ
. By (31), A is a partial valua-
tion, dom(A)
(31)
== δ[γ] = atoms(T ). Let l ∈ T . Then
atoms(l) ⊆ atoms(T ) = dom(A) and A |=
(32)
== l. So,
A |= T . We conclude that A is a partial model of T ,
dom(A) = atoms(T ).
The DPLL procedure is refutation sound and com-
plete.
Theorem 4.2 (Refutational Soundness and Complete-
ness of the DPLL Procedure). Let S ⊆
F
OrdCl.
(i) If there exists a closed tree Tree with the root
S constructed using Rules (24)–(30), then S is
unsatisfiable.
(ii) There exists a finite tree Tree with the root S con-
structed using Rules (24)–(30) with the follow-
ing properties:
if S is unsatisfiable, then Tree is closed; (33)
if S is satisfiable, then Tree is open
and there exists a partial model A of S,
dom(A) = atoms(S), related to Tree.
(34)
Proof. (i) The proof is by induction on the structure
of Tree using Rules (24)–(30).
(ii) In the first phase, we can construct a finite tree
Tree
∗
with leaves S
i
, i ≤ n, using Rules (24)–(28) such
that for all i ≤ n, atoms(S
i
) ⊆ atoms(S), S
i
|=
P
S; for
all a ∈ atoms(S
i
), either there exists a P ε ∈ S
i
, ε ∈
{0, 1}, satisfying, for all C ∈ S
i
and C 6= a P ε, a 6∈
atoms(C); or 0 ≺ a, a ≺ 1 ∈ S
i
; S is satisfiable if and
only if there exists i
∗
≤ n such that S
i
∗
is satisfiable.
The proof is by induction on katoms(S)k.
In the second phase, we exploit the excess lit-
eral technique. Let S
F
⊆
F
OrdCl. We define
elmeasure(S
F
) = (
∑
C∈S
F
kCk) − kS
F
k. For all i ≤ n,
there exists a finite tree Tree
i
with the root S
i
con-
structed using Rules (24), (29), (30) with the follow-
ing properties:
if S
i
is unsatisfiable, then Tree
i
is closed; (35)
if S
i
is satisfiable, then Tree
i
is open and there
exists a partial model A
i
of S
i
, dom(A
i
) =
atoms(S
i
), related to Tree
i
.
(36)
Let i ≤ n. We proceed by induction on elmeasure(S
i
).
Case 1: elmeasure(S
i
) = 0. We distinguish two
cases.
Case 1.1:  ∈ S
i
. We put Tree
i
= S
i
. Then S
i
is
unsatisfiable; Tree
i
is a closed tree with the root S
i
;
(35) holds and (36) holds trivially.
Case 1.2:  6∈ S
i
. Then S
i
is a unit order clausal
theory; there exists a finite linear tree Tree
i
with
the root S
i
and the leaf trans(S
i
) constructed using
Rules (24) and (29). We get two cases.
Case 1.2.1:  ∈ trans(S
i
). Then Tree
i
is closed;
its only branch from S
i
to trans(S
i
) is closed; by (i)
for Tree
i
, S
i
is unsatisfiable; (35) holds and (36) holds
trivially.
Case 1.2.2:  6∈ trans(S
i
). Then Tree
i
is open;
its only branch from S
i
to trans(S
i
) is open; trans(S
i
)
is a unit order clausal theory; we have, for all a ∈
atoms(S
i
), either there exists a P ε ∈ S
i
, ε ∈ {0, 1},
satisfying, for all C ∈ S
i
and C 6= a P ε, a 6∈ atoms(C);
or 0 ≺ a, a ≺ 1 ∈ S
i
; for all C ∈ trans(S
i
) − S
i
, for all
a ∈ atoms(C), 0 ≺ a, a ≺ 1 ∈ S
i
⊆ trans(S
i
); the proof
is by induction on ktrans(S
i
) − S
i
k using Rule (29);
for all a ∈ atoms(S
i
) = atoms(trans(S
i
)), either there
exists a P ε ∈ S
i
⊆ trans(S
i
), ε ∈ {0, 1}, satisfying, for
all C ∈ trans(S
i
) and C 6= a P ε, a 6∈ atoms(C); or 0 ≺
a, a ≺ 1 ∈ S
i
⊆ trans(S
i
); by Lemma 4.1 for trans(S
i
),
there exists a partial model A
i
of trans(S
i
), dom(A
i
) =
atoms(trans(S
i
)); A
i
, dom(A
i
) = atoms(trans(S
i
)) =
atoms(S
i
), is a partial model of S
i
⊆ trans(S
i
) related
to Tree
i
; S
i
is satisfiable; (36) holds and (35) holds
trivially.
Case 2: elmeasure(S
i
) > 0. Then there exist
l
1
, l
2
, l
3
∈ OrdLit,  6= C ∈ OrdCl, and l
1
∨C ∈ S
i
, l
1
∨
l
2
∨ l
3
is a trichotomy. We put S
1
i
= (S
i
− {l
1
∨C}) ∪
{l
1
} ⊆
F
OrdCl, S
2
i
= (S
i
−{l
1
∨C})∪{C} ∪ {l
2
} ⊆
F
OrdCl, S
3
i
= (S
i
− {l
1
∨C}) ∪ {C} ∪ {l
2
} ⊆
F
OrdCl.
Then
S
i
S
1
i


S
2
i


S
3
i
is an application of Rule (30); for all 1 ≤ j ≤ 3,
elmeasure(S
j
i
) < elmeasure(S
i
); for all 1 ≤ j ≤ 3, by
induction hypothesis for S
j
i
, there exists a finite tree
ADPLLProcedureforthePropositionalProductLogic
221

Table 5: V
α
.
V
0
=
/
0;
V
α
= V
α−1
∪ {(δ(α − 1), λ
α−1
)} (1 ≤ α ≤ γ is a successor ordinal),
E
α−1
=
(
kCn
1
k
V
α−1
kCn
2
k
V
α−1
!
1
n




Cn
1
P δ(α − 1)
n
&Cn
2
∈ T,Cn
1
, Cn
2
∈ PropConj, atoms(Cn
1
, Cn
2
) ⊆ dom(V
α−1
)
)
∪
(

kCn
1
k
V
α−1

1
n




Cn
1
P δ(α − 1)
n
∈ T,Cn
1
∈ PropConj, atoms(Cn
1
) ⊆ dom(V
α−1
)
)
∪
n
kεk
V
α−1
|δ(α − 1) P ε ∈ T, ε ∈ {0, 1}
o
,
D
α−1
=
(
kCn
1
k
V
α−1
kCn
2
k
V
α−1
!
1
n




Cn
1
≺ δ(α − 1)
n
&Cn
2
∈ T,Cn
1
, Cn
2
∈ PropConj, atoms(Cn
1
, Cn
2
) ⊆ dom(V
α−1
)
)
∪
(

kCn
1
k
V
α−1

1
n




Cn
1
≺ δ(α − 1)
n
∈ T,Cn
1
∈ PropConj, atoms(Cn
1
) ⊆ dom(V
α−1
)
)
,
U
α−1
=
(
kCn
1
k
V
α−1
kCn
2
k
V
α−1
!
1
n




δ(α − 1)
n
&Cn
2
≺ Cn
1
∈ T,Cn
1
, Cn
2
∈ PropConj, atoms(Cn
1
, Cn
2
) ⊆ dom(V
α−1
)
)
∪
(

kCn
1
k
V
α−1

1
n




δ(α − 1)
n
≺ Cn
1
∈ T,Cn
1
∈ PropConj, atoms(Cn
1
) ⊆ dom(V
α−1
)
)
,
λ
α−1
=



W
W
W
D
α−1
+
V
V
V
U
α−1
2
if E
α−1
=
/
0,
W
W
W
E
α−1
else;
V
γ
=
[
α<γ
V
α
(γ is a limit ordinal)
Tree
j
i
with the root S
j
i
constructed using Rules (24),
(29), (30), and (35), (36) hold for Tree
j
i
. We put
Tree
i
=
S
i
Tree
1
i


Tree
2
i


Tree
3
i
.
Then Tree
i
is a finite tree with the root S
i
constructed
using Rules (24), (29), (30). We get two cases.
Case 2.1: S
i
is unsatisfiable. Then, for all 1 ≤
j ≤ 3, S
j
i
is unsatisfiable; by (35) for Tree
j
i
, Tree
j
i
is
closed; Tree
i
is closed; (35) holds and (36) holds triv-
ially.
Case 2.2: S
i
is satisfiable. Then there exists
1 ≤ j
∗
≤ 3 and S
j
∗
i
is satisfiable; by (36) for Tree
j
∗
i
,
Tree
j
∗
i
is open, there exists a partial model A
j
∗
i
of S
j
∗
i
, dom(A
j
∗
i
) = atoms(S
j
∗
i
), related to Tree
j
∗
i
;
Tree
i
is open; we have l
1
∨ l
2
∨ l
3
is a trichotomy;
atoms(l
1
) = atoms(l
2
) = atoms(l
3
), atoms(S
j
∗
i
) ⊆
atoms(S
i
), S
j
∗
i
|=
P
S
i
. We put A
i
= A
j
∗
i
∪ {(a, 0) |a ∈
atoms(S
i
) − atoms(S
j
∗
i
)}, dom(A
i
) = atoms(S
i
), a
partial valuation. Then A
i
|
atoms(S
j
∗
i
)
= A
j
∗
i
|= S
j
∗
i
,
A
i
|= S
i
, A
i
, dom(A
i
) = atoms(S
i
), is a partial model
of S
i
, related to Tree
i
; (36) holds and (35) holds triv-
ially. The induction is completed.
We construct Tree from Tree
∗
by replacing the leaf
S
i
with Tree
i
for every i ≤ n. We have Tree
∗
, for all
i ≤ n, Tree
i
are finite. Hence, Tree is finite. It remains
to prove (33) and (34).
Let S be unsatisfiable. We have S is satisfiable if
and only if there exists i
∗
≤ n such that S
i
∗
is satisfi-
able. Then, for all i ≤ n, S
i
is unsatisfiable; by (35)
for Tree
i
, Tree
i
is closed; Tree is closed; (33) holds.
Let S be satisfiable. We have S is satisfiable if
and only if there exists i
∗
≤ n such that S
i
∗
is sat-
isfiable. Then, there exists i
∗
≤ n and S
i
∗
is satisfi-
able; by (36) for Tree
i
∗
, Tree
i
∗
is open, there exists
a partial model A
i
∗
of S
i
∗
, dom(A
i
∗
) = atoms(S
i
∗
),
related to Tree
i
∗
; Tree is open; we have, for all
i ≤ n, atoms(S
i
) ⊆ atoms(S), S
i
|=
P
S; atoms(S
i
∗
) ⊆
atoms(S), S
i
∗
|=
P
S. We put A = A
i
∗
∪ {(a, 0)| a ∈
atoms(S) − atoms(S
i
∗
)}, dom(A) = atoms(S), a par-
tial valuation. Then A|
atoms(S
i
∗
)
= A
i
∗
|= S
i
∗
, A |= S,
A, dom(A) = atoms(S), is a partial model of S related
to Tree; (34) holds. The theorem is proved.
The set of basic rules has been proposed as a min-
imal one, which is suitable for theoretical purposes;
i.e. not to get complicated soundness and complete-
ness arguments. For practical computing, it may be
IJCCI2013-InternationalJointConferenceonComputationalIntelligence
222

Table 6: Translation of φ to S
φ
.
φ = a → 0 ∨ (a → a& b) → b
{ ˜a
0
≺ 1,

a → 0
|{z}
˜a
1
∨(a → a & b) → b
| {z }
˜a
2

→ ˜a
0
} (17)
{ ˜a
0
≺ 1, ˜a
1
≺ ˜a
0
∨ ˜a
1
P ˜a
0
, ˜a
2
≺ ˜a
0
∨ ˜a
2
P ˜a
0
, ( a
|{z}
˜a
3
→ 0) → ˜a
1
, ((a → a& b
| {z }
˜a
4
) → b
|{z}
˜a
5
) → ˜a
2
} (21), (23)
{ ˜a
0
≺ 1, ˜a
1
≺ ˜a
0
∨ ˜a
1
P ˜a
0
, ˜a
2
≺ ˜a
0
∨ ˜a
2
P ˜a
0
, 0 ≺ ˜a
3
∨ ˜a
1
P 1, ˜a
3
≺ a ∨ ˜a
3
P a, ˜a
4
≺ ˜a
5
∨ ˜a
4
P ˜a
5
∨ ˜a
5
≺ ˜a
4
& ˜a
2
∨ ˜a
5
P ˜a
4
& ˜a
2
, ˜a
5
≺ ˜a
4
∨ ˜a
2
P 1,
b ≺ ˜a
5
∨ b P ˜a
5
, ˜a
4
→ ( a
|{z}
˜a
6
→ a&b
|
{z}
˜a
7
)} (22)
{ ˜a
0
≺ 1, ˜a
1
≺ ˜a
0
∨ ˜a
1
P ˜a
0
, ˜a
2
≺ ˜a
0
∨ ˜a
2
P ˜a
0
, 0 ≺ ˜a
3
∨ ˜a
1
P 1, ˜a
3
≺ a ∨ ˜a
3
P a, ˜a
4
≺ ˜a
5
∨ ˜a
4
P ˜a
5
∨ ˜a
5
≺ ˜a
4
& ˜a
2
∨ ˜a
5
P ˜a
4
& ˜a
2
, ˜a
5
≺ ˜a
4
∨ ˜a
2
P 1,
b ≺ ˜a
5
∨ b P ˜a
5
, ˜a
4
& ˜a
6
≺ ˜a
7
∨ ˜a
4
& ˜a
6
P ˜a
7
, a ≺ ˜a
6
∨ a P ˜a
6
, ˜a
7
→ a
|{z}
˜a
8
& b
|{z}
˜a
9
} (18)
S
φ
= { ˜a
0
≺ 1 [1] ˜a
1
≺ ˜a
0
∨ ˜a
1
P ˜a
0
[2] ˜a
2
≺ ˜a
0
∨ ˜a
2
P ˜a
0
[3] 0 ≺ ˜a
3
∨ ˜a
1
P 1 [4]
˜a
3
≺ a ∨ ˜a
3
P a [5] ˜a
4
≺ ˜a
5
∨ ˜a
4
P ˜a
5
∨ ˜a
5
≺ ˜a
4
& ˜a
2
∨ ˜a
5
P ˜a
4
& ˜a
2
[6] ˜a
5
≺ ˜a
4
∨ ˜a
2
P 1 [7] b ≺ ˜a
5
∨ b P ˜a
5
[8]
˜a
4
& ˜a
6
≺ ˜a
7
∨ ˜a
4
& ˜a
6
P ˜a
7
[9] a ≺ ˜a
6
∨ a P ˜a
6
[10] ˜a
7
≺ ˜a
8
& ˜a
9
∨ ˜a
7
P ˜a
8
& ˜a
9
[11] ˜a
8
≺ a ∨ ˜a
8
P a [12]
˜a
9
≺ b ∨ ˜a
9
P b [13]}
augmented by additional admissible rules, which do
not change the semantics of the DPLL procedure. For
example, we can add a rule:
(Tautology simplification rule) (37)
T
T − {l ∨C}
l ∨C ∈ T, l is a tautology.
We can strengthen Rule (29), denoted as (29)
#
, by
omitting the application condition: T is a unit or-
der clausal theory. Such admissible rules are obvi-
ously sound and helpful for constructing more com-
pact DPLL trees in many cases, however, superfluous
for the completeness argument. Concerning the de-
duction problem of a formula from a finite theory, we
conclude.
Corollary 4.3. Let φ ∈ PropForm
/
0
and T ⊆
F
PropForm
/
0
. There exist A
φ
T
⊆
F
˜
A, S
φ
T
⊆
F
SimOrdCl
A
φ
T
, a finite tree Tree with the root S
φ
T
con-
structed using Rules (24)–(30) with the following
properties:
if T |=
P
φ, then Tree is closed; (38)
if T 6|= φ, then Tree is open and there exists a
partial model A of T , dom(A) = atoms(T, φ),
related to Tree such that A 6|= φ.
(39)
Proof. An immediate consequence of Theorems 3.2
and 4.2.
Let φ = a → 0 ∨ (a → a & b) → b ∈ PropForm
/
0
,
a, b ∈ PropAtom
/
0
. Using Corollary 4.3, we show
that φ is a tautology. At first, we translate φ to
S
φ
⊆
F
SimOrdCl in Table 6. Before we start DPLL
derivation, it is suitable to investigate several cases
when the input atoms a, b get the truth values 0, 1.
Case 1: kak = 0. Then kφk = 1. Case 2: kak = 1.
Then kφk = kbk⇒
⇒
⇒kbk = 1. Hence, in all the cases,
kφk = 1, and it remains to investigate whether kφk = 1
for the case 0 ≺ a, a ≺ 1, 0 ≺ b, b ≺ 1 by the DPLL
procedure.
Case 3: We add 0 ≺ a [14], a ≺ 1 [15], 0 ≺ b [16],
b ≺ 1 [17]. Primarily using Rules (27) and (28), we
can derive a branch in the constructed tree such that
for all i ≤ 9, 0 ≺ ˜a
i
, ˜a
i
≺ 1; the other branches are
closed, ending in . We then lengthen this branch by
deriving
˜a
5
≺ ˜a
4
[18] : [7]
˜a
5
≺ ˜a
4
& ˜a
2
∨ ˜a
5
P ˜a
4
& ˜a
2
[19] : [6] [18]
b ≺ ˜a
4
& ˜a
2
∨ b P ˜a
4
& ˜a
2
[20] : [19] [8]
˜a
7
≺ a &b ∨ ˜a
7
P a &b [21] : [11] [12] [13]
˜a
4
&a ≺ ˜a
7
∨ ˜a
4
&a P ˜a
7
[22] : [9] [10]
˜a
4
≺ b ∨ ˜a
4
P b [23] : [22] [21]
 [24] : [20] [23] (29)
#
;
˜a
2
≺ 1.
Hence, all the cases-branches of the constructed tree
are closed; we have reached  in all of them. We get
the constructed tree by the DPLL procedure is closed.
ADPLLProcedureforthePropositionalProductLogic
223

So, we have proved
/
0 |=
P
φ and φ is a tautology.
5 CONCLUSIONS
We have investigated the deduction problem of a for-
mula from a finite theory in the propositional Prod-
uct logic. The deduction problem has been solved
via translation of a formula to an equivalent satisfi-
able finite order clausal theory, consisting of order
clauses. An order clause is a finite set of order liter-
als of the form ε
1
 ε
2
where ε
i
is either a conjunction
of propositional atoms or the propositional constant
0 (false) or 1 (true), and  is a connective either P
or ≺. P and ≺ are interpreted by the equality and
standard strict order on [0, 1], respectively. The tri-
chotomy over order literals: either ε
1
≺ ε
2
or ε
1
P ε
2
or ε
2
≺ ε
1
, has naturally led to a variant of the DPLL
procedure with a trichotomy branching rule, which is
refutation sound and complete in the finite case.
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