A DPLL PROCEDURE FOR THE PROPOSITIONAL G

ODEL LOGIC

∗

Duˇsan Guller

Department of Applied Informatics, Comenius University, Mlynsk´a dolina, 842 15 Bratislava, Slovakia

Keywords:

The DPLL procedure, Automated deduction, G¨odel logic, Many-valued logics.

Abstract:

In the paper, we investigate the satisﬁability and validity problems of a formula in the propositional G¨odel

logic. Our approach is based on the translation of a formula to an equivalent CNF one which contains literals of

the augmented form: either a or a → b or (a → b) → b, where a, b are propositional atoms or the propositional

constants 0, 1. A CNF formula is further translated to an equisatisﬁable ﬁnite order clausal theory which

consists of order clauses, ﬁnite sets of order literals of the forms a ≖ b or a ≺ b. ≖ and ≺ are interpreted by

the equality and strict linear order on [0, 1], respectively. A variant of the DPLL procedure for deciding the

satisﬁability of a ﬁnite order clausal theory is proposed. The DPLL procedure is proved to be refutation sound

and complete. Finally, we reduce the validity problem of a formula (tautology checking) to the unsatisﬁability

of a ﬁnite order clausal theory.

1 INTRODUCTION

A noticeable effort has been made in the develop-

ment of SAT solvers (called SAT solvers for the

Boolean satisﬁability problem), especially in the last

decade. Roughly speaking, SAT solvers exploit

either complete solution methods, called complete

or systematic SAT solvers, or incomplete or hy-

brid ones. Complete SAT solvers are mostly based

on the Davis-Putnam-Logemann-Loveland procedure

(DPLL) (Davis, 1960; Davis, 1962) improved by var-

ious features. Some of the latest overviews of the de-

velopment of SAT solvers, with the underlying com-

plexity theory, may be found in (Dixon, 2004; Dixon,

2004; Kautz, 2007; Gomes, 2007; Biere, 2009).

The research in many-valued logics mainly concerns

ﬁnitely-valued ones. Thank to the ﬁniteness of truth

value sets of these logics, almost straightforward ex-

tensions of results achieved in classical logic are fea-

sible. The DPLL procedure has been ﬁrstly gener-

alised for regular clauses over a linearly ordered truth

value set (H¨ahnle, 1996). In ((any`a, 1998), it is de-

scribed an implementation of this regular DPLL pro-

cedure with the extended 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, 2000). It is based on a branch-

∗

Partially supported by the grants VEGA 1/0688/10 and

VEGA 1/0726/09.

ing rule forming branches for every truth value. So,

the branching 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 equiv-

alence. A slight modiﬁcation of that equivalence en-

ables a 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 arbitrary 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 a ﬁnite branching factor, may be consid-

ered as if a ”signed anti-hyperresolution rule”. The

procedure is refutation complete if the ﬁnitary dis-

junction condition for the set of signs occurring in the

input countable clausal theory is satisﬁed. Inﬁnitely-

valued logics 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. The solution of the SAT and VAL problems

strongly varies on a chosen inﬁnitely-valued logic.

The same holds for the translation of a formula to

clause form, the existence of which is not guaranteed

in general. The results in this area have been achieved

in several ways, since inﬁnite truth value sets form

distinct algebraic structures. One approach may be

based on the 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; Aguz-

Guller D..

A DPLL PROCEDURE FOR THE PROPOSITIONAL GÖDEL LOGIC.

DOI: 10.5220/0003061700310042

In Proceedings of the International Conference on Fuzzy Computation and 2nd International Conference on Neural Computation (ICFC-2010), pages

31-42

ISBN: 978-989-8425-32-4

 2010 SCITEPRESS (Science and Technology Publications, Lda.)

zoli, 2000). Another approach exploits the reduction

of the SAT problem to mixed integer programming

(MIP) (H¨ahnle, 1994a; H¨ahnle, 1997). (Baaz, 2001)

investigates the VAL problem in the prenex fragment

of the ﬁrst-order G¨odel logic enriched by the rela-

tivisation operator ∆, denoted as the prenex G

∆

∞

. At

ﬁrst, a variant of Herbrand’s Theorem for the prenex

∆

∞

is proved, which reduces the VAL problem of a

formula in the prenex G

∆

∞

to the VAL problem of an

open formula in G

∆

∞

. Then a chain normal form is

deﬁned using the formulae φ ⋖ ψ, as an abbreviation

for ¬∆(ψ → φ), and φ ≡

∆

ψ, as an abbreviation for

∆(φ → ψ) ∧ ∆(ψ → φ). These formulae express the

strict dense linear order with endpoints and equality

on [0, 1], which is not possible without ∆ in G

∞

. Fur-

ther, a meta-level logic of order clauses is deﬁned,

which is a fragment of classical one. An order clause

is a ﬁnite set of inequalities of the form either A < B

or A ≤ B where <, ≤ are meta-level predicate sym-

bols 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 interpretations on

[0, 1], varying on assigned (truth) values to atoms of

∆

∞

handled as 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 respect

to the semantics of the meta-level logic. The chaining

calculi in (Bachmair, 1994; Bachmair, 1998) may be

used for efﬁcient deduction over order clauses.

In the paper, we investigate SAT and VAL prob-

lems of a formula in the propositional G¨odel logic.

Our approach is based on the translation of a formula

to an equivalent CNF one, Lemma 3.1, Section 3,

which contains literals of the augmented form: ei-

ther a or a → b or (a → b) → b, where a, b are

propositional atoms or the propositional constants 0,

1. At this stage, unlike the chain normal form in

(Baaz, 2001), we do not need to express the linear

order of truth values by any formulae. We consider

a ground fragment of the ﬁrst-order two-valued logic

with equality and strict order. The syntax is given by

a class of order clausal theories. An order clause is a

ﬁnite set of order literals of the form either a ≖ b or

a ≺ b. The semantics is given by a class of order inter-

pretations. An order interpretationis a ﬁrst-order two-

valued interpretation such that its universum is [0, 1],

≖ is interpreted as =

[0,1]

, and ≺ as <

[0,1]

. For the

purpose of solving the SAT problem, a CNF formula

is translated to an equisatisﬁable ﬁnite order clausal

theory, Lemma 3.3, Section 3. The basis is the trans-

lation of a literal to an order clause: e.g. a → b is

translated to a ≺ b∨ a ≖ b∨ b ≖ 1 or (a → b) → b to

b ≺ a∨b ≖ 1. The trichotomy on order literals: either

a ≺ b or a ≖ b or b ≺ a, naturally invokes proposing

a variant of the DPLL procedure with a trichotomy

branching rule as an algorithm for deciding the sat-

isﬁability of a ﬁnite order clausal theory. The DPLL

procedure is proved to be refutation sound and com-

plete, Theorem 4.1, Section 4. The set of basic Rules

(37), (38), (39) may be augmented by the admissible

ones (50), (51), (52), (53), (54), (55), which are suit-

able for practical computing and considerably shorten

DPLL trees. In case of solving the VAL problem, we

exploit the fact that a formula φ is a tautology (valid)

if and only if the order formula φ ≺ 1 is unsatisﬁable,

Theorem 5.1, Section 5. At ﬁrst, φ is translated to an

equivalent CNF formula ψ =

i≤n

j≤m

, l

are lit-

erals. Hence, φ is a tautology if and only if the order

formula ψ ≺ 1 ≡

i≤n

j≤m

≺ 1 is unsatisﬁable.

Further, every order formula l

≺ 1 is translated to

an equisatisﬁable conjunction of disjunctions of order

literals: e.g. (a → b) ≺ 1 is translated to b ≺ a∧b ≺ 1

or ((a → b) → b) ≺ 1 to (a ≺ b∨a ≖ b)∧ b ≺ 1. This

yields an equisatisﬁable ﬁnite order clausal theory T

to ψ ≺ 1 and φ ≺ 1. So, φ is a tautology if and only if

is unsatisﬁable.

The paper is organised as follows. Section 2 gives

the basic notions, notation, and useful properties con-

cerning the propositional G¨odel 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 solves the VAL problem (tau-

tology checking).

2 PROPOSITIONAL G

ODEL

LOGIC

Throughout the paper, we shall use the common no-

tions of propositional many-valued logics. The set of

propositional atoms of G¨odel logic will be denoted

as PropAtom. By PropForm we designate the set

of all propositional formulae of G¨odel logic built up

from PropAtom using the propositional constants 0,

the false, 1, the true, and the connectives ¬, nega-

tion, ∧, conjunction, ∨, disjunction, →, implication.

We shall assume that G¨odel logic is interpreted by the

standard G-algebra

G = ([0, 1], ≤, ∨, ∧, ⇒

, 0, 1)

where ∨ and ∧ denote the respective supremum and

inﬁmum operators on [0, 1],

ICFC 2010 - International Conference on Fuzzy Computation

a ⇒

b =



1if a ≤ b,

belse,



1if a = 0,

0else.

We recall that G is a complete linearly ordered lat-

tice algebra; the supremum operator ∨ is commuta-

tive, associative, idempotent, monotone, 0 is its neu-

tral element; the inﬁmum operator ∧ is commutative,

associative, idempotent, monotone, 1 is its neutral el-

ement;

the residuum operator ⇒

of ∧ satisﬁes the

condition of residuation:

for all a, b, c ∈ G, a ∧b ≤ c ⇐⇒ a ≤ b ⇒

c; (1)

the G¨odel negation

satisﬁes the condition:

for all a ∈ G, a

= a ⇒

0; (2)

and the following properties, which will be exploited

later, hold:

For all a, b, c ∈ G,

a∨ b ∧c = (a∨ b) ∧ (a∨ c),

(3) (distributivity of ∨ over ∧)

a∧ (b ∨c) = a∧ b∨ a∧ c,

(4) (distributivity of ∧ over ∨)

a ⇒

(b∨ c) = a ⇒

b∨ a ⇒

b, (5)

a ⇒

b∧ c = (a ⇒

b) ∧ (a ⇒

b), (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 = a ⇒

b, (10)

(a ⇒

b) ⇒

c = ((a ⇒

b) ⇒

b) ∧ (b ⇒

c) ∨ c,

(11)

(a ⇒

b) ⇒

0 = ((a ⇒

0) ⇒

0) ∧ b ⇒

0. (12)

A valuation V of propositional atoms is a map-

ping V : PropAtom −→ [0, 1]. A partial valuation V

of propositional atoms with the domain dom(V ) ⊆

PropAtom is a mapping V : dom(V ) −→ [0, 1]. Let

atoms(φ), atoms(T) ⊆ dom(V ) in case of V being a

partial valuation. The truth value φ in V , in symbols

kφk

, is deﬁned by the standard way; the proposi-

tional constants 0, 1 are interpreted by 0, 1, respec-

tively, and the connectives by the respective operators

Using the commutativity, associativity, idempotence,

monotonicity, neutral elements of ∨ and ∧ will not be ex-

plicitly referred to.

We assume the decreasing operator priority sequence

, ∧, ⇒

, ∨, which enables writing order clauses without

parentheses.

on G. V is a (partial) propositional model of φ, in

symbols V |= φ, iff k φk

= 1. V is a (partial) propo-

sitional model of T, in symbols V |= T, iff for all

φ ∈ T, V |= φ. φ is a propositional consequence of T,

in symbols T |=

φ, iff for every propositional model

V of T, V |= φ. φ is equivalent to φ

′

, in symbols

φ ≡ φ

′

, ifffor everyvaluation V , kφk

= kφ

′

. φ | T

is satisﬁable iff there exists a propositional model of

φ | T. φ | T is equisatisﬁable to φ

′

| T

′

iff φ | T is

satisﬁable if and only if φ

′

| T

′

is satisﬁable.

Let X, Y, Z be sets, Z ⊆ X, and f : X −→ Y a

mapping. By X ⊆

Y we denote X is a ﬁnite subset

of Y. We designate P (X) = {x|x ⊆ X}, P (X) is the

power set of X; P

(X) = {x| x ⊆

X}, P

(X) is the

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

f[Z] is called the image of Z with respect to f; and

= {(z, f(z))|z ∈ Z}, f|

is the restriction of f onto

Z. f : ω −→ Y is a sequence of Y iff f is a bijection.

3 TRANSLATION TO CLAUSAL

FORM

We propose translation of a formula to an equivalent

CNF formula, Lemma 3.1. In contrast to two-valued

logic, we have to consider an augmented set of literals

appearing in CNF formulae. Let l, φ ∈ PropForm. l is

a literal iff either l = a or l = a → b or l = (a → b) → b

where a ∈ PropAtom and b ∈ PropAtom ∪ {0}. φ

is a conjunctive | disjunctive normal form, in sym-

bols CNF | DNF, iff either φ = 0 or φ = 1 or φ =

i≤n

j≤m

| φ =

i≤n

j≤m

where l

are liter-

als.

Lemma 3.1. Let φ ∈ PropForm. There exists a CNF

ψ ≡ φ.

Proof. It is straightforward to prove that there ex-

ists ϑ ≡ φ without any occurrence of ¬. The proof

is by induction on the structure of φ using (2); ev-

ery subformula of the form ¬ϕ of φ is replaced with

ϕ → 0 ≡ ¬ϕ. We further prove the statement:

There exists a CNF ψ ≡ ϑ. (13)

The proof is by induction on the structure of ϑ; all

the occurrences of → in ϑ are pushed down and the

resulting CNF ψ is recursively built up. The obvious

cases are ϑ ∈ PropAtom∪ {0, 1} and ϑ = ϑ

∧ ϑ

. In

the case ϑ = ϑ

∨ ϑ

, the distributivity of ∨ over ∧,

(3), is exploited.

Associativity of ∧, ∨ will not be explicitly referred to,

and hence,

i≤n

∈ PropForm are written without

parentheses.

A DPLL PROCEDURE FOR THE PROPOSITIONAL GÖDEL LOGIC

Let ϑ = ϑ

→ ϑ

. Then, by induction hypothesis,

there exist CNF’s ψ

≡ ϑ

, ψ

≡ ϑ

, and we distin-

guish three cases for ψ

, ψ

. Case 1: either ψ

= 0 or

= 1 is obvious; ψ

→ ψ

≡ 1. Case 2: ψ

= 1 is

also obvious; ψ

→ ψ

≡ ψ

. Case 3: neither ψ

= 0

nor ψ

= 1 nor ψ

= 1. Then ψ

i≤n

j≤m

are literals, and we get two cases for ψ

: either

r≤v

s≤u

, k

are literals, or ψ

= 0. Using

(6), (5), (8), (7), (3), in both the cases, there exists

θ≤Θ

ξ≤Ξ

→ κ

≡ ψ

→ ψ

(IH)

≡≡ ϑ

→ ϑ

= ϑ, (14)

are literals, either κ

are literals or κ

= 0. We

show that

for all θ ≤ Θ and ξ ≤ Ξ

, (15)

there exists a DNF δ

≡ λ

→ κ

Let θ ≤ Θ and ξ ≤ Ξ

. We then distinguish nine cases

for λ

and κ

. Case 3.1: λ

= a and κ

= b, a ∈

PropAtom, b ∈ PropAtom∪{0}. Hence, δ

= a→ b =

→ κ

is a DNF. Case 3.2: λ

= a → b and κ

= c,

a ∈ PropAtom, b, c ∈ PropAtom∪ {0}. Hence,

= ((a → b) → b) ∧ (b → c) ∨ c

(11)

≡≡ (a → b) → c

= λ

→ κ

is a DNF. Case 3.3: λ

= (a → b) → b and κ

= c,

a ∈ PropAtom, b, c ∈ PropAtom∪ {0}. Hence,

== (a → b) ∧ (b → c) ∨ c

(10)

≡≡ (((a → b) → b) → b) ∧ (b → c) ∨ c

(11)

≡≡ ((a → b) → b) → c = λ

→ κ

is a DNF. Cases 3.4 − 3.9: either λ

= a or λ

a → b or λ

= (a → b) → b, and κ

= ϕ → d where

either ϕ = c or ϕ = c → d, a, c ∈ PropAtom, b, d ∈

PropAtom∪ {0}. By Cases 3.1 − 3.3, there exists a

DNF λ

≡ λ

→ d, and

== λ

∨ ϕ → d ≡ λ

→ d ∨ ϕ → d

(8)

≡≡ λ

∧ ϕ → d

(9)

≡≡ λ

→ (ϕ → d) = λ

→ κ

is a DNF. So, the claim (15) holds. We get that there

exists a CNF

((3))

≡≡

θ≤Θ

ξ≤Ξ

(15)

≡≡

θ≤Θ

ξ≤Ξ

→ κ

(14)

≡≡ ϑ.

Thus, the claim (13) holds. The induction is com-

pleted. We conclude that there exists a CNF ψ

(13)

≡≡

ϑ ≡ φ.

Using Lemma 3.1, we translate (a → b) → ((b →

c) → (a → c)) ∈ PropForm, a, b, c ∈ PropAtom, to an

equivalent CNF:

(a → b) → ((b → c) → (a → c))

(9)

≡≡

(a → b) → (((b → c) ∧ a) → c)

(8)

≡≡

(5)

(a → b) → ((b → c) → c) ∨ (a → b) → (a → c)

(9)

≡≡

((a → b) ∧ (b → c)) → c∨ ((a → b) ∧ a) → c

(8)

≡≡

(a → b) → c∨ (b → c) → c∨

(a → b) → c∨ a → c ≡≡

(a → b) → c∨ (b → c) → c∨ a → c

(11)

≡≡

(((a → b) → b) ∧ (b → c)) ∨

c∨ (b → c) → c ∨ a → c

(3)

≡≡

((a → b) → b ∨ c∨ (b → c) → c∨ a → c) ∧

(b → c∨ c∨ (b → c) → c ∨ a → c).

In Lemma 3.1, we have laid no restrictions on the

use of the distributivity law, (3), during translation to

conjunctive normal form. Therefore the size of the

output CNF may be exponential in the size of an in-

put formula. To avoid this disadvantage, we propose

translation to CNF via interpolation using new atoms,

which produces CNF formulae in linear size. A sim-

ilar approach exploiting the renaming subformulae

technique can be found in (Plaisted, 1986; Boy, 1992;

H¨ahnle, 1994b; Nonnengart, 1998; Sheridan, 2004).

By p

∈ PropAtom we denote atoms not yet occur-

ring in the set of formulae in question. The empty se-

quence of symbols is denoted as ε. Let φ ∈ PropForm.

We deﬁne the size of φ by recursion on the structure

of φ:

|φ| =











1 if φ ∈ PropAtom∪ {0, 1},

|φ

| + 1 if φ = ¬φ

|φ

| + |φ

| + 1if φ = φ

⋄ φ

where ⋄ ∈ {∧, ∨, →}.

Let φ

∈ PropForm and p

∈ PropAtom. We denote



if φ

∈ PropAtom,

if φ

6∈ PropAtom;



ε if φ

∈ PropAtom,

→ φ

if φ

6∈ PropAtom;

−



ε if φ

∈ PropAtom,

→ p

if φ

6∈ PropAtom.

ICFC 2010 - International Conference on Fuzzy Computation

Table 1: Interpolation rules.

Case: Positive interpolation Laws

Size of antecedent

Maximum size of consequent

Negative interpolation

∧ φ

→ φ

∧ φ

→ φ

) ∧ (p

→ φ

)

(6)

|φ

| + |φ

| + 3

|φ

| + |φ

| + 5

(16)

∧ φ

→ p

(ϕ

→ p

∨ ϕ

→ p

) ∧

−

∧

−

(8)

|φ

| + |φ

| + 3

|φ

| + |φ

| + 13

(17)

∨ φ

→ (φ

∨ φ

)

→ ϕ

∨ p

→ ϕ

) ∧

∧

(5)

|φ

| + |φ

| + 3

|φ

| + |φ

| + 13

(18)

(φ

∨ φ

) → p

→ p

∧ φ

→ p

(7)

|φ

| + |φ

| + 3

|φ

| + |φ

| + 5

(19)

∧ φ

→ 0

→ (φ

∧ φ

→ 0)

→ 0∨ ϕ

→ 0) ∧

−

∧

−

(9), (8)

|φ

| + |φ

| + 5

|φ

| + |φ

| + 17

(20)

(φ

∧ φ

→ 0) → p

((φ

→ 0) → p

) ∧ ((φ

→ 0) → p

)

(8), (7)

|φ

| + |φ

| + 5

|φ

| + |φ

| + 9

(21)

(φ

∨ φ

) → 0

→ ((φ

∨ φ

) → 0)

→ (φ

→ 0)) ∧ (p

→ (φ

→ 0))

(7), (6)

|φ

| + |φ

| + 5

|φ

| + |φ

| + 9

(22)

((φ

∨ φ

) → 0) → p

((ϕ

→ 0) → 0∨ (ϕ

→ 0) → 0∨ p

) ∧

∧

(11), (7), (8)

|φ

| + |φ

| + 5

|φ

| + |φ

| + 19

(23)

(φ

∧ φ

→ 0) → 0

→ ((φ

∧ φ

→ 0) → 0)

→ ((φ

→ 0) → 0)) ∧ (p

→ ((φ

→ 0) → 0))

(8), (7), (6)

|φ

| + |φ

| + 7

|φ

| + |φ

| + 13

(24)

((φ

∧ φ

→ 0) → 0) → p

(ϕ

→ 0∨ ϕ

→ 0∨ p

) ∧

−

∧

−

(11), (10), (8)

|φ

| + |φ

| + 7

|φ

| + |φ

| + 15

(25)

((φ

∨ φ

) → 0) → 0

→ (((φ

∨ φ

) → 0) → 0)

→ 0∨ (ϕ

→ 0) → 0∨ (ϕ

→ 0) → 0) ∧

∧

(9), (8), (7), (8)

|φ

| + |φ

| + 7

|φ

| + |φ

| + 21

(26)

(((φ

∨ φ

) → 0) → 0) → p

(((φ

→ 0) → 0) → p

) ∧ (((φ

→ 0) → 0) → p

)

(7), (8), (7)

|φ

| + |φ

| + 7

|φ

| + |φ

| + 13

(27)

((φ

→ 0) → 0) → 0

→ (((φ

→ 0) → 0) → 0)

→ (φ

→ 0)

(10)

|φ

| + 8

|φ

| + 4

(28)

(((φ

→ 0) → 0) → 0) → p

(φ

→ 0) → p

(10)

|φ

| + 8

|φ

| + 4

(29)

((φ

→ φ

) → 0) → 0, φ

6= 0

→ (((φ

→ φ

) → 0) → 0)

→ 0∨ ϕ

→ 0∨ (ϕ

→ 0) → 0) ∧

−

∧

(9), (8), (12), (8), (10)

|φ

| + |φ

| + 7

|φ

| + |φ

| + 19

(30)

(((φ

→ φ

) → 0) → 0) → p

((φ

→ 0) → p

) ∧ (((φ

→ 0) → 0) → p

)

(12), (8), (10), (7)

|φ

| + |φ

| + 7

|φ

| + |φ

| + 11

(31)

(φ

→ φ

) → 0, φ

6= 0

→ ((φ

→ φ

) → 0)

→ ((φ

→ 0) → 0)) ∧ (p

→ (φ

→ 0))

(12), (6)

|φ

| + |φ

| + 5

|φ

| + |φ

| + 11

(32)

((φ

→ φ

) → 0) → p

(ϕ

→ 0∨ (ϕ

→ 0) → 0∨ p

) ∧

−

∧

(11), (12), (8), (10)

|φ

| + |φ

| + 5

|φ

| + |φ

| + 17

(33)

→ φ

, φ

6= 0

→ (φ

→ φ

)

→ ϕ

∨ ϕ

→ ϕ

) ∧

−

∧

(9), (8)

|φ

| + |φ

| + 3

|φ

| + |φ

| + 13

(34)

(φ

→ φ

) → p

((ϕ

→ ϕ

) → ϕ

∨ p

) ∧ (ϕ

→ p

) ∧

∧

−

(11), (3)

|φ

| + |φ

| + 3

|φ

| + |φ

| + 17

(35)

A DPLL PROCEDURE FOR THE PROPOSITIONAL GÖDEL LOGIC

Let φ

, φ

∈ PropForm and p

∈ PropAtom. In Ta-

ble 1, we introduce interpolation rules. Let φ ∈

PropForm. ψ is a CNF of φ iff ψ is a CNF ob-

tained from p

∧(p

→ φ) for some i by a ﬁnite deriva-

tion using the interpolation rules. We denote the set

of all CNF’s of φ as CNF(φ). Let f, g : M −→ N.

f ∈ O(g) iff there exists k such that for all m ∈ M,

f(m) ≤ k.g(m).

Lemma 3.2. Let φ ∈ PropForm. CNF(φ) 6=

0, and

for all ψ ∈ CNF(φ), ψ is equisatisﬁable to φ, |ψ| ∈

O(|φ|).

Proof. The proof of CNF(φ) 6=

0 is by induction on

the structure of φ. It is straightforward to prove that

∧(p

→ φ) is equisatisﬁable to φ; for every interpo-

lation rule, its antecedent is equisatisﬁable to its con-

sequent; if for every i, ψ

is equisatisﬁable to φ

, then

so is

; there exists k such that for every

interpolation rule, the size of its consequent is less

than or equal to k times the size of its antecedent. Let

ψ ∈ CNF(φ). Then there exist i, n, a ﬁnite derivation

= p

∧ (p

→ φ), . . . , ζ

= ψ, and k such that for all

j ≤ n, ζ

is equisatisﬁable to φ and |ζ

| ≤ k.|φ|. The

proof is by induction on n using the previous state-

ments.

Using Lemma 3.2, we translate (a → b) → ((b →

c) → (a → c)) ∈ PropForm, a, b, c ∈ PropAtom, to an

equisatisﬁable CNF:

∧ (p

→ (a → b

|{z }

) →

((b → c) → (a → c)

| {z }

)), (34)

∧ (p

→ p

∨ p

→ p

) ∧

((a → b) → p

) ∧

→ ((b → c

|{z}

) → (a → c

|{z}

))), (35), (34)

∧ (p

→ p

∨ p

→ p

) ∧

((a → b) → b ∨ p

) ∧ (b → p

) ∧

→ p

∨ p

→ p

) ∧

((b → c) → p

) ∧

→ (a → c)), (35), (34)

∧ (p

→ p

∨ p

→ p

) ∧

((a → b) → b ∨ p

) ∧ (b → p

) ∧

→ p

∨ p

→ p

) ∧

((b → c) → c ∨ p

) ∧ (c → p

) ∧

→ c ∨ a → c).

We further introduce a ground fragment of the

ﬁrst-order two-valued logic with equality and strict

order. The syntax is given by a class of order clausal

theories. We form order literals and clauses from

PropAtom∪ {0, 1}, regarded as constants, using bi-

nary predicates ≖, equality, and ≺, strict order. l is an

order literal iff either l = a ≖ b = b ≖ a; since equal-

ity is commutative by deﬁnition, we identify a ≖ b

and b ≖ a; or l = a ≺ b where a, b ∈ PropAtom ∪

{0, 1}. An order clause is a ﬁnite set of order lit-

erals. An order clause {l

, . . . , l

} is written in the

form l

∨ ··· ∨ l

. The order clause

0 is called the

empty clause and denoted as . An order clause {l}

is called a unit order clause and denoted as l if it does

not cause the ambiguitywith the denotationof the sin-

gle literal l in a given context. We designate the set

of order clauses as OrdCl. Let l, l

, . . . , l

be order lit-

erals and C,C

′

∈ OrdCl. By l ∨C we denote {l} ∪C

where l 6∈ C. Analogously, by

i=1

∨C we denote

} ∪ · ··∪{l

} ∪C where for all 1 ≤ i 6= i

′

≤ n, l

6∈ C

and l

6= l

′

. By C ∨C

′

we denote C ∪C

′

. C is a sub-

clause of C

′

, in symbols C ⊑ C

′

, iff C ⊆ C

′

. An order

clausal theory is a set of order clauses. A unit or-

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

T, T

′

⊆ OrdCl. By atoms(C)|atoms(T) ⊆ PropAtom

we denote the set of all the propositional atoms oc-

curring in C | T. The semantics is given by a class of

order interpretations. An order interpretation I with

the domain dom(I ) = PropAtom is a ﬁrst-order two-

valued interpretation such that U

= [0, 1], for all a ∈

PropAtom, a

∈ [0, 1], 0

= 0, 1

= 1, and ≖

[0,1]

≺

[0,1]

. A partial order interpretation I with the

domain dom(I ) ⊆ PropAtom is an order interpreta-

tion such that for all a ∈ dom(I ), a

∈ [0, 1]. An

(partial) order interpretation I is identiﬁed with the

(partial) valuation V

: dom(V

) −→ [0, 1], V

(a) =

. Let atoms(l), atoms(C), atoms(C

′

), atoms(T),

atoms(T

′

) ⊆ dom(I ). I is a (partial) model of l, in

symbols I |= l, iff either for l = a ≖ b, a

[0,1]

or for l = a ≺ b, a

[0,1]

. I is a (partial) model of

C, in symbols I |= C, iff there exists l ∈ C such that

I |= l. I is a (partial) model of T, in symbols I |= T,

iff for all C ∈ T, I |= C. Note that  and T such that

 ∈ T are unsatisﬁable by deﬁnition. C

′

is an order

consequence of C, in symbols C |=

′

, iff for every

model I of C, I |= C

′

. C is an order consequence of

T, in symbols T |=

C, iff for every model I of T,

I |= C. T

′

is an order consequence of T, in symbols

T |=

′

, iff for every model I of T, I |= T

′

. C | T

is satisﬁable iff there exists a model of C | T. C

′

| T

′

is equisatisﬁable to C | T iff C

′

| T

′

is satisﬁable if

ICFC 2010 - International Conference on Fuzzy Computation

and only if C | T is satisﬁable. By OrdPropForm we

designate the augmented set of all order propositional

formulae built up from PropAtom using 0, 1, ¬, ∧, ∨,

→, and ≺, ≖. Note that OrdPropForm ⊇ PropForm

by deﬁnition, and all the notions and notation con-

cerned with PropForm are straightforwardly extended

to OrdPropForm.

Lemma 3.3. Let φ be a conjunctive normal form.

There exists T

⊆

OrdCl such that T

is equisatis-

ﬁable to φ.

Proof. By the deﬁnition of CNF, we distinguish three

cases for φ. Case 1: φ = 0. Then φ is unsatisﬁable and

= {} ⊆

OrdCl is unsatisﬁable as well. So, the

claim holds. Case 2: φ = 1. Then φ is satisﬁable and

0 ⊆

OrdCl is satisﬁable as well. So, the claim

holds. Case 3: φ =

i≤n

j≤m

, l

are literals.

For all i ≤ n and j ≤ m

, there exists (36)

∈ OrdCl such that C

is equisatisﬁable to l

The proof is by deﬁnition. We get ﬁve cases for

. Case 3.1: l

= a, a ∈ PropAtom. Then C

a ≖ 1. Case 3.2: l

= a → 0, a ∈ PropAtom. Then

= a ≖ 0. Case 3.3: l

= a → b, a ∈ PropAtom,

b ∈ PropAtom. ThenC

= a ≺ b∨ a ≖ b∨b ≖ 1. Case

3.4: l

= (a → 0) → 0, a ∈ PropAtom. Then C

0 ≺ a. Case 3.5: l

= (a → b) → b, a ∈ PropAtom,

b ∈ PropAtom. Then C

= b ≺ a ∨ b ≖ 1. So, the

claim (36) holds. By (36), there exists T

⊆

OrdCl

such that T

= {

j≤m

|i ≤ n} is equisatisﬁable to

φ.

Using Lemma 3.3, we translate the CNF ((a → b) →

b∨c∨(b → c) → c∨a → c)∧ (b → c∨c∨(b → c) →

c ∨ a → c), a, b, c ∈ PropAtom, to an equisatisﬁable

T ⊆

OrdCl where

T = {b ≺ a ∨ b ≖ 1∨ c ≖ 1∨

c ≺ b ∨ c ≖ 1∨ a ≺ c∨ a ≖ c∨ c ≖ 1,

b ≺ c∨ b ≖ c∨ c ≖ 1 ∨ c ≖ 1 ∨

c ≺ b ∨ c ≖ 1∨ a ≺ c∨ a ≖ c∨ c ≖ 1}.

4 DPLL PROCEDURE

We devise a variant of the DPLL procedure over ﬁnite

order clausal theories. At ﬁrst, a minimal set of basic

rules is introduced. Let l, l

, l

be order literals. l

is a contradiction iff either l = 0 ≖ 1 or l = 0 ≺ 0 or

l = 1 ≺ 1 or l = a ≺ 0 or l = 1 ≺ a or l = a ≺ a where

a ∈ PropAtom. l is a tautology iff either l = 0 ≖ 0

or l = 1 ≖ 1 or l = 0 ≺ 1 or l = a ≖ a where a ∈

PropAtom. l

∨ l

is a general trichotomy iff l

a ≺ b, l

= a ≖ b, l

= b ≺ a where a, b ∈ PropAtom∪

{0, 1}. Let T ⊆ OrdCl. The basic rules are as follows:

(37) (One literal contradiction simpliﬁcation rule)

T ∪ {}

if T is a unit order clausal theory, l ∈ T, and

l is a contradiction;

(38) (One literal transitivity rule of ≖ and ≺)

T ∪ {a ⋄ c}

where ⋄ =



≖ if ⋄

= ⋄

=≖,

≺else,

if T is a unit order clausal theory,

a⋄

b, b⋄

c ∈ T, and ⋄

, ⋄

∈ {≖, ≺};

(39) (General trichotomy branching rule)

T − {l

∨C} ∪ {l

}



T − {l

∨C} ∪ {C} ∪ {l

}



T − {l

∨C} ∪ {C} ∪ {l

}

if l

∨C ∈ T, C 6= , and

∨ l

is a general trichotomy.

Rule (39) reﬂects the linearity of <

[0,1]

in terms of

general trichotomy. Rule (37) formalises its addi-

tional properties: the endpoints 0 <

[0,1]

1 and strict-

ness via contradictions. Rule (38) expresses the mu-

tual transitivity of =

[0,1]

together with <

[0,1]

. Rules

(37), (38), (39) are sound in view of satisﬁability:

T and T ∪ {} in the consequent of Rule (37) (40)

are both unsatisﬁable.

T is equisatisﬁable to T ∪ {a⋄ c} in the (41)

consequent of Rule (38).

Let I be a partial order interpretation, (42)

dom(I ) ⊇ atoms(T).

I |= T if and only if I |= T − {l

∨C} ∪ {l

} or

I |= T − {l

∨C} ∪ {C} ∪ {l

} or

I |= T − {l

∨C} ∪ {C} ∪ {l

}

in the consequent of Rule (39).

T is satisﬁable if and only if (43)

T − {l

∨C} ∪ {l

} or

T − {l

∨C} ∪ {C} ∪ {l

} or

T − {l

∨C} ∪ {C} ∪ {l

}

in the consequent of Rule (39) is satisﬁable.

A DPLL PROCEDURE FOR THE PROPOSITIONAL GÖDEL LOGIC

The proofis byassumption and deﬁnition. The refuta-

tional completeness argument of the basic rules, The-

orem 4.1(ii), may be provided using the excess lit-

eral technique (Anderson, 1970). From this point of

view, Rules (37) and (38) handle the base case: T is

a unit order clausal theory, while Rule (39) the induc-

tion one: it subtracts the excess literal measure of T

at least by 1 for every clausal theory in a branch of its

consequent.

T is closed under Rules (37) and (38) iff for every

application of Rules (37) and (38) of the form

′

= T. By trans(T) ⊆ OrdCl we denote the least set

such that trans(T) ⊇ T and trans(T) is closed under

Rules (37), (38).

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 applica-

tion of Rule (37) or (38) or (39) such that the parent

vertex is the theory in its antecedent and the children

vertices are the theories in its consequent. A branch

of a tree is closed iff it contains a vertex T

′

such that

 ∈ T

′

. A branch of a tree is open iff it is not closed.

A tree is closed iff everyits branchis ﬁnite and closed.

Note that 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 from its root

and ending in its only leaf.

Lemma 4.1. Let T ⊆ OrdCl.

(i) If T ⊆

OrdCl, then trans(T) ⊆

OrdCl.

(ii) If T is a unit order clausal theory and  6∈

trans(T), then trans(T) is a unit order clausal

theory.

(iii) atoms(trans(T)) = atoms(T).

(iv) T |=

trans(T).

(v) If T ⊆

OrdCl, then there exists a ﬁnite linear

tree with the root T and the leaf trans(T) con-

structed using Rules (37) and (38).

Proof. By assumption and deﬁnition.

The following lemma shows that Rules (37) and

(38) are refutation complete for a (countable) unit or-

der clausal theory, which will be exploited in the base

case of Theorem 4.1(ii).

Lemma 4.2. Let T = trans(T) ⊆ OrdCl be a count-

able unit order clausal theory. There exists a partial

model A of T, dom(A) = atoms(T).

Proof. By the theorem assumption that T is a unit or-

der clausal theory,  6∈ T = trans(T). In addition,

by the theorem assumption that T is a countable set,

there exists a sequence δ of atoms(T). At ﬁrst, we

deﬁne partial order interpretations Mod

by recursion

on α ≤ ω:

Mod

= Mod

α−1

∪ {(δ(α− 1), v

α−1

)} (0 < α < ω),

α−1

= {kak

Mod

α−1

|a ≖ δ(α− 1) ∈ T,

a ∈ dom(Mod

α−1

) ∪ {0, 1}},

α−1

= {Mod

α−1

(a)|a ≺ δ(α− 1) ∈ T,

a ∈ dom(Mod

α−1

)},

α−1

= {Mod

α−1

(a)|δ(α − 1) ≺ a ∈ T,

a ∈ dom(Mod

α−1

)},

α−1



α−1

, M

α−1

, M

α−1

Mod

[

α<ω

Mod

It is straightforward to prove the following state-

ments:

For all α ≤ ω, Mod

is a partial order (44)

interpretation, dom(Mod

) = δ[α], and

for all β ≤ α, Mod

⊆ Mod

For all α ≤ ω and l ∈ T such that (45)

atoms(l) ⊆ dom(Mod

), Mod

|= l.

For all α ≤ ω and a ∈ dom(Mod

), (46)

if Mod

(a) = 0, then a ≖ 0 ∈ T.

For all α ≤ ω and a ∈ dom(Mod

), (47)

if Mod

(a) = 1, then a ≖ 1 ∈ T.

The proofs are by induction on α ≤ ω. We put

A = Mod

. By (44), A = Mod

is a partial or-

der interpretation, dom(A) = dom(Mod

)

(44)

== δ[ω] =

atoms(T). Let l ∈ T. Then atoms(l) ⊆ atoms(T) =

dom(Mod

) = dom(A) and A = Mod

(45)

|==== 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 4.1 (Refutational Soundnessand Complete-

ness of the DPLL Procedure). Let T ⊆

OrdCl.

(i) If there exists a closed tree Tree with the root T

constructed using Rules (37), (38), (39), then T

is unsatisﬁable.

ICFC 2010 - International Conference on Fuzzy Computation

(ii) There exists a ﬁnite tree Tree with the root T

constructed using Rules (37), (38), (39) with

the following properties:

If T is unsatisﬁable, then Tree is closed. (48)

If T is satisﬁable, then Tree is open and (49)

there exists a partial model A of T,

dom(A) = atoms(T), related to Tree.

Proof. (i) The proof is by induction on the structure

of Tree using (40), (41), (42).

(ii) We exploit the excess literal technique to con-

struct a ﬁnite tree Tree with the root T using Rules

(37), (38), (39). Let T

⊆

OrdCl. We deﬁne

elmeasure(T

) = (

∑

C∈T

|C|)− |T

|. We proceed by

induction on elmeasure(T).

Let elmeasure(T) = 0. We distinguish two cases:

either  ∈ T or  6∈ T.

Case 1:  ∈ T. Then T is unsatisﬁable and Tree =

T is a closed tree with the root T. So, (48) holds and

(49) holds trivially.

Case 2:  6∈ T. Then, by the denotation of

elmeasure(T), T is a unit order clausal theory. By

Lemma 4.1(v), there exists a ﬁnite linear tree Tree

with the root T and the leaf trans(T) constructed us-

ing Rules (37) and (38). We get two cases:

either  ∈ trans(T) or  6∈ trans(T).

Case 2.1:  ∈ trans(T). Then Tree is a closed tree

with the root T; its only branch from T to trans(T) is

closed. Hence, by (i), T is unsatisﬁable. So, (48)

holds and (49) holds trivially.

Case 2.2:  6∈ trans(T). Then Tree is an open

tree with the root T; its only branch from T to

trans(T) is open. Since T is a unit order clausal the-

ory, by Lemma 4.1(ii), we get trans(T) is a unit or-

der clausal theory, and by Lemma 4.2 for trans(T),

there exists a partial model A of trans(T), dom(A) =

atoms(trans(T)). Hence, A is a partial model of T ⊆

trans(T), dom(A) = atoms(trans(T))

Lemma 4.1(iii)

==========

atoms(T), related to Tree and T is satisﬁable. So, (49)

holds and (48) holds trivially.

Let elmeasure(T) = n > 0 and the statement hold

for all T

⊆

OrdCl such that elmeasure(T

) <

n. Since elmeasure(T) > 0, by the denotation of

elmeasure(T), there exists l

∨C ∈ T such that C 6= .

Let l

, l

be order literals such that l

∨ l

is a

general trichotomy. This yields the application of

Rule (39)

(T − { l

∨C}) ∪ {l

}



(T − { l

∨C}) ∪ {C} ∪ {l

}



(T − { l

∨C}) ∪ {C} ∪ {l

}

We denote T

= (T − {l

∨ C}) ∪ {l

}, T

= (T −

∨ C}) ∪ {C} ∪ {l

}, T

= (T − {l

∨ C}) ∪

{C} ∪ {l

}. Then elmeasure(T

) < elmeasure(T),

elmeasure(T

) < elmeasure(T), elmeasure(T

) <

elmeasure(T), and by induction hypothesis, there ex-

ist ﬁnite trees Tree

with the root T

, Tree

with the

root T

, Tree

with the root T

constructed using Rules

(37), (38), (39) such that (48) and (49) hold for

Tree

, Tree

. This yields

Tree =

Tree



Tree



Tree

is a ﬁnite tree with the root T constructed using Rules

(37), (38), (39). We get two cases:

either T is unsatisﬁable or T is satisﬁable.

Case 4: T is unsatisﬁable. Then, by (42), T

, T

are unsatisﬁable, and by (48) for Tree

, Tree

Tree

, Tree

are closed trees. Hence,

Tree is a closed tree. So, (48) holds and (49) holds

trivially for Tree.

Case 5: T is satisﬁable. Then, by (42), there ex-

ists 1 ≤ i ≤ 3 such that T

is satisﬁable. Hence, by (49)

for Tree

, Tree

is an open tree and there exists a par-

tial model A

of T

, dom(A

) = atoms(T

), related to

Tree

. By the deﬁnition of T

, T

T. As {l

, l

}

is a trichotomy, atoms(l

) = atoms(l

)

and atoms(T

) ⊆ atoms(T). We get Tree is an open

tree and A = A

∪{(p, 0)| p ∈ atoms(T)−atoms(T

)},

dom(A) = atoms(T), is a partial model of T related to

Tree. So, (49) holds and (48) holds trivially for Tree.

The induction is completed.

The set of basic rules has been proposed as a min-

imal one, which is suitable for theoretical purposes;

e.g. not to have a too complicated completeness ar-

gument. For practical computing, it may be aug-

mented by additional rules. Let l, l

, l

be or-

der literals and C ∈ OrdCl. l

∨ l

is a dichotomy

iff either l

= 0 ≖ a and l

= 0 ≺ a or l

= a ≺ 1

and l

= a ≖ 1 where a ∈ PropAtom. l

∨ l

a trichotomy iff l

= a ≺ b, l

= a ≖ b, l

= b ≺ a

where a, b ∈ PropAtom. C is a tautology iff there ex-

ists C

′

∈ OrdCl such that C

′

⊑ C and either C

′

= {l}

where l is a tautology or C

′

is a dichotomy or C

′

is a

trichotomy.

(50) (Contradiction simpliﬁcation rule)

(T − {l ∨C}) ∪ {C}

if l ∨C ∈ T and l is a contradiction;

(51) (Tautology simpliﬁcation rule)

A DPLL PROCEDURE FOR THE PROPOSITIONAL GÖDEL LOGIC

T − {C}

(52)

if C ∈ T and C is a tautology;

(53) (One literal positive propagation rule)

T − {C}

if l,C ∈ T, l ∈ C, and l is a literal;

(54) (One literal negative propagation rule)

(T − { l

∨C}) ∪ {C}

if l

, l

∨C ∈ T and there exists l

such that

∨ l

is a general trichotomy;

(55) (Dichotomy branching rule)

(T − { l

∨C}) ∪ {l

}



(T − {l

∨C}) ∪ {C} ∪ {l

}

if l

∨C ∈ T, C 6= , and l

∨ l

is a dichotomy;

(56) (Trichotomy branching rule)

(T − { l

∨C}) ∪ {l

}



(T − { l

∨C}) ∪ {C} ∪ {l

}



(T − { l

∨C}) ∪ {C} ∪ {l

}

if l

∨C ∈ T, C 6= , and l

∨ l

is a trichotomy.

Rules (50), (51), (53), (54), (55), (56) are obvi-

ously sound and helpful for constructing more com-

pact DPLL trees in many cases, however, superﬂuous

for the completeness argument. Concerning the SAT

problem of a formula, we conclude.

Corollary 4.1. Let φ ∈ PropForm. There exist an eq-

uisatisﬁable T

⊆

OrdCl to φ and a ﬁnite tree Tree

with the root T

constructed using Rules (37), (38),

(39) with the following properties:

If φ is unsatisﬁable, then Tree

is closed. (57)

If φ is satisﬁable, then Tree

is open and (58)

there exists a partial model A

of φ,

dom(A

) = atoms(φ).

Proof. An immediate consequence of Lemma 3.3 and

Theorem 4.1.

Note that the SAT problem of a ﬁnite theory can

be reduced to the SAT one of a formula in the usual

manner. Let T = {φ

|i ≤ n} ⊆

PropForm. Then

φ =

i≤n

∈ PropForm is equisatisﬁable to T.

5 TAUTOLOGY CHECKING

One application of the DPLL procedure may be to

tautology checking. Let φ ∈ PropForm. φ is a tau-

tology (valid) iff for every valuation V , V |= φ. As

explained in Introduction, the VAL problem of a for-

mula φ can be reduced to the unsatisﬁability of the

order formula φ ≺ 1 consequently translated to an eq-

uisatisﬁable ﬁnite order clausal theory T

. Then the

unsatisﬁability of T

is decided by the DPLL proce-

dure. This section provides technical details of the

reduction, Theorem 5.1. In addition to the properties

stated in Section 2, the following ones hold:

For all φ

, φ

∈ PropForm and ψ

, ψ

∈

OrdPropForm,

(φ

∧ φ

) ≺ 1 ≡ φ

≺ 1∨ φ

≺ 1, (59)

(φ

∨ φ

) ≺ 1 ≡ φ

≺ 1∧ φ

≺ 1, (60)

∨ ψ

∧ ψ

= (ψ

∨ ψ

) ∧ (ψ

∨ ψ

). (61)

Theorem 5.1 (Reduction Theorem). Let φ ∈

PropForm. There exists T

⊆

OrdCl such that T

is unsatisﬁable if and only if φ is a tautology.

Proof. By Lemma 3.1, there exists a conjunctive nor-

mal form ψ such that ψ ≡ φ and we distinguish tree

cases:

either ψ = 0 or ψ = 1 or ψ =

i≤n

j≤m

, l

are literals.

Case 1: φ ≡ ψ = 0. Then φ is not a tautology and

0 ⊆

OrdCl is satisﬁable. So, the claim holds.

Case 2: φ ≡ ψ = 1. Then φ is a tautology and

= { } ⊆

OrdCl is unsatisﬁable. So, the claim

holds.

Case 3: φ ≡ ψ =

i≤n

j≤m

, l

are literals.

Then

φ is a tautology if and only if (62)

φ ≺ 1 ∈ OrdPropForm is unsatisﬁable;

φ ≺ 1 ≡ ψ ≺ 1 = (

i≤n

j≤m

) ≺ 1

(59)

≡≡

(60)

i≤n

j≤m

≺ 1.

(63)

For all i ≤ n and j ≤ m

, there exists (64)

a conjunction of disjunctions of order literals

∈ OrdPropForm such that

is equisatisﬁable to l

≺ 1.

The proof is by deﬁnition. We get ﬁve cases for l

Case 3.1: l

= a, a ∈ PropAtom. Then δ

= a ≺ 1.

Case 3.2: l

= a → 0, a ∈ PropAtom. Then δ

= 0 ≺ a.

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Case 3.3: l

= a → b, a ∈ PropAtom, b ∈ PropAtom.

Then δ

= b ≺ a∧ b ≺ 1. Case 3.4: l

= (a → 0) → 0,

a ∈ PropAtom. Then δ

= a ≖ 0. Case 3.5: l

= (a →

b) → b, a ∈ PropAtom, b ∈ PropAtom. Then δ

(a ≺ b∨ a ≖ b) ∧ b ≺ 1. So, the claim (64) holds. By

(64) and (63),

i≤n

j≤m

is equisatisﬁable to (65)

i≤n

j≤m

≺ 1 and φ ≺ 1.

Hence, there exists ϕ ∈ OrdPropForm such that

ϕ =

r≤v

s≤u

((61))

≡≡

i≤n

j≤m

(66)

where κ

are order literals. By (66) and (65), there

exists T

⊆

OrdCl such that

= {

s≤u

|r ≤ v} is equisatisﬁable to ϕ, (67)

i≤n

j≤m

, and φ ≺ 1.

We close that T

is unsatisﬁable

(67)

⇐⇒ φ ≺ 1 is unsatis-

ﬁable

(62)

⇐⇒ φ is a tautology.

Let φ = (a → b) → ((b → c) → (a → c)) ∈ PropForm,

a, b, c ∈ PropAtom. Using Theorem 5.1, we show that

φ is a tautology. At ﬁrst, using Lemma 3.1, we trans-

late φ to an equivalent CNF

ψ = ((a → b) → b∨ c∨ (b → c) → c∨ a → c) ∧

(b → c∨ c∨ (b → c) → c ∨ a → c),

cf. the example after Lemma 3.1. Then, using (59)

and (60), ψ ≺ 1 is equivalent to

ξ = ((a → b) → b) ≺ 1∧ c ≺ 1∧

((b → c) → c) ≺ 1∧ (a → c) ≺ 1∨

(b → c) ≺ 1∧ c ≺ 1∧

((b → c) → c) ≺ 1∧ (a → c) ≺ 1.

Hence, using (64) and (67), ξ is equisatisﬁable to

⊆

OrdCl where

= {a ≺ b∨ a ≖ b∨ c ≺ b, [1]

a ≺ b ∨ a ≖ b∨ c ≺ 1, [2]

a ≺ b ∨ a ≖ b∨ b ≺ c∨ b ≖ c, [3]

a ≺ b ∨ a ≖ b∨ c ≺ a, [4]

b ≺ 1 ∨ c ≺ b, [5]

4.1

5.1

6.1.1

 ∈ T

7.1.1

6.1.2

 ∈ T

7.1.2

6.1.3

 ∈ T

7.1.3

4.2

5.2

6.2.1

 ∈ T

7.2.1

6.2.2

 ∈ T

7.2.2

6.2.3

 ∈ T

7.2.3

4.3

 ∈ T

5.3

Figure 1: Closed tree Tree

b ≺ 1∨ c ≺ 1, [6]

b ≺ 1∨ b ≖ c, [7]

b ≺ 1∨ c ≺ a, [8]

c ≺ 1∨ c ≺ b, [9]

c ≺ 1, [10]

c ≺ 1∨ b ≖ c, [11]

c ≺ 1∨ c ≺ a, [12]

b ≺ c∨ b ≖ c∨ c ≺ b,[13]

b ≺ c∨ b ≖ c∨ c ≺ 1,[14]

b ≺ c∨ b ≖ c, [15]

b ≺ c∨ b ≖ c∨ c ≺ a,[16]

c ≺ a∨ c ≺ b, [17]

c ≺ a∨ c ≺ 1, [18]

c ≺ a∨ b ≖ c, [19]

c ≺ a}. [20]

Finally, using the DPLL procedure rules, we can con-

struct a closed tree Tree

with the root T

, outlined in

Figure 1.

We close that T

is unsatisﬁable, and by Theorem

5.1, φ is a tautology.

6 CONCLUSIONS

We have investigated the satisﬁability and validity

problems of a formula in the propositional G¨odel

logic. The satisﬁability problem has been solved via

the translation of a formula to an equivalent CNF

one, containing literals of the forms a, a → b, or

(a → b) → b. A CNF formula has further been trans-

lated to an equisatisﬁable ﬁnite order clausal theory,

which consists of order clauses with order literals of

the forms a ≖ b or a ≺ b. ≖ and ≺ are interpreted

by the equality and strict linear order on [0, 1], respec-

tively. The trichotomy on order literals: either a ≺ b

A DPLL PROCEDURE FOR THE PROPOSITIONAL GÖDEL LOGIC

or a ≖ b or b ≺ a, has naturally led to a variant of the

DPLL procedure with a trichotomy branching rule,

which is refutation sound and complete. We have re-

duced the validity problem of a formula to the unsat-

isﬁability of a ﬁnite order clausal theory.

REFERENCES

Aguzzoli, S. and Ciabattoni, A. Finiteness of inﬁnite-valued

Łukasiewicz logic. Journal of Logic, Language and

Information, 9:5-29, 2000.

Anderson, R. and Bledsoe, W. W. A linear format for res-

olution with merging and a new technique for estab-

lishing completeness. Journal of the ACM, 17(3):525-

534, 1970.

Baaz, M., Ferm¨uller, C. G. and Ciabattoni, A. Herbrand’s

theorem for prenex G¨odel logic and its consequences

for theorem proving. Proceedings of the LPAR con-

ference, LNCS vol. 2250, Springer-Verlag, 201-215,

2001.

Bachmair, L. and Ganzinger, H. Rewrite-based equa-

tional theorem proving with selection and simpliﬁca-

tion. Journal of Logic and Computation, 4(3):217-

247, 1994.

Bachmair, L. and Ganzinger, H. Ordered chaining calculi

for ﬁrst-order theories of transitive relations. Journal

of the ACM, 45(6):1007-1049, 1998.

Beckert, B., H¨ahnle, R. and Many`a, F. The SAT problem of

signed CNF formulas. In Labelled Deduction, Basin,

D., D’Agostino, M., Gabbay, D., Matthews, S. and

Vigan´o, L., eds., Applied Logic Series, vol. 17, Kluwer

Academic Publishers, 61-82, 2000.

Biere, A., Heule, M., van Maaren, H. and Walsh, T., eds.

Handbook of Satisﬁability. Frontiers in Artiﬁcial

Intelligence and Applications Series, vol. 185, IOS

Press, 2009.

Boy de la Tour, T. An optimality result for clause

form translation. Journal of Symbolic Computation,

14(4):283-301, 1992.

Davis, M. and Putnam, H. A computing procedure for quan-

tiﬁcation theory. Communications of the ACM, 7:201-

215, 1960.

Davis, M., Logemann, G. and Loveland, D. A machine

program for theorem-proving. Communications of the

ACM, 5:394-397, 1962.

Dixon, H. E., Ginsberg, M. L., Luks, E. M. and Parkes, A. J.

Generalizing Boolean satisﬁability II: Theory. Journal

of Artiﬁcial Intelligence Research, 22:481-534, 2004.

Dixon, H. E., Ginsberg, M. L. and Parkes, A. J. General-

izing Boolean satisﬁability I: Background and survey

of existing work. Journal of Artiﬁcial Intelligence Re-

search, 21:193-243, 2004.

Gomes, C. P., Kautz, H., Sabharwal, A. and Selman, B. Sat-

isﬁability solvers. In Handbook of Knowledge Rep-

resentation, Harmelen, F. v., Lifschitz, V. and Porter,

B., eds., Elsevier Science Publishers, Part I, Chap. 3,

2007.

Guller, D. On the refutational completeness of signed bi-

nary resolution and hyperresolution. Fuzzy Sets and

Systems, 160(8):1162-1176, 2009.

H¨ahnle, R. Many-valued logic and mixed integer program-

ming. Annals of Mathematics and Artiﬁcial Intelli-

gence, 12(3,4):231-264, 1994.

H¨ahnle, R. Short conjunctive normal forms in ﬁnitely-

valued logics. Journal of Logic and Computation,

4(6):905-927, 1994.

H¨ahnle, R. Exploiting data dependencies in many-valued

logics. Journal of Applied Non-Classical Logics,

6(1):49-69, 1996.

H¨ahnle, R. Proof theory of many-valued logic - linear opti-

mization - logic design: Connections and interactions.

Soft Computing - A Fusion of Foundations, Method-

ologies and Applications, 1(3):107-119, 1997.

Kautz, H. and Selman, B. The state of SAT. Discrete Ap-

plied Mathematics, 155(12):1514-1524, 2007.

Many`a, F., B´ejar, R. and Escalada-Imaz, G. The satisﬁabil-

ity problem in regular CNF-formulas. Soft Computing

- A Fusion of Foundations, Methodologies and Appli-

cations, 2(3):116-123, 1998.

Mundici, D. Satisﬁability in many-valued sentential logic is

NP-complete. Theoretical Computer Science, 52:145-

153, 1987.

Nonnengart, A., Rock, G. and Weidenbach, Ch. On gener-

ating small clause normal forms. Proceedings of the

CADE conference, LNAI vol. 1421, Springer-Verlag,

397-411, 1998.

Plaisted, D. A. and Greenbaum, S. A structure-preserving

clause form translation. Journal of Symbolic Compu-

tation, 2(3):293-304, 1986.

Sheridan, D. The optimality of a fast CNF conver-

sion and its use with SAT. Online Proceed-

ings of International Conference on the The-

ory and Applications of Satisﬁability Testing,

www.satisﬁability.org/SAT04/programme/114.pdf,

2004.

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