=Paper=
{{Paper
|id=Vol-2585/paper7
|storemode=property
|title=An extension of the stable semantics via Lukasiewicz logic
|pdfUrl=https://ceur-ws.org/Vol-2585/paper7.pdf
|volume=Vol-2585
|authors=Mauricio Osorio,José Luis Carballido
|dblpUrl=https://dblp.org/rec/conf/lanmr/0001C19
}}
==An extension of the stable semantics via Lukasiewicz logic==
https://ceur-ws.org/Vol-2585/paper7.pdf
An extension of the stable semantics via
Lukasiewicz logic
Mauricio Osorio1 and José Luis Carballido2
1
Universidad de las Américas-Puebla,
osoriomauri@gmail.com
2
Benemérita Universidad Atónoma de Puebla
Abstract. Logic Programming and fuzzy logic are active areas of re-
search,and their scopes in terms of applications are growing fast. Fuzzy
logic is a branch of many-valued logic based on the paradigm of inference
under vagueness. In this work we recall some of the interplay between
three 3-valued logics that are relevant in these areas: The Lukasiewicz
logic, the intermediate logic G3 and the paraconsistent logic G′3 , and we
present a contribution to the area of answer sets that consists in extend-
ing a definition of stable model based on proof theory in logic G3 , to a
more general definition that can be based on any of the more expressive
logics G′3 or Lukasiewicz.
Keywords: Knowledge representation, stable semantics, paraconsistency.
1 Introduction
The stable semantics allows us to handle problems with default knowledge and
produce non-monotonic reasoning using the concept of negation as failure. The
p-stable semantics is an alternative semantics, except that in some cases offers
models where the stable semantics has none. There are two popular software im-
plementations to compute the stable models: dlv3 and smodels4 . The efficiency
of such programs has increased the list of practical applications in the areas
of planning, logical agents and artificial intelligence. On the other hand, there
exist different approaches for knowledge representation based on the p-stable
semantics, such as, updates [14], preferences [15], and argumentation [1]. Cur-
rently, in [16], a schema for the implementation of the p-stable semantic using
two well known open source tools: Lparse and Minisat is described. The authors
also present a prototype5 written in Java of a tool based on that schema.
The term fuzzy logic emerged in the development of the theory of fuzzy sets
by Lotfi A. Zadeh (1965) [3]. It is generally agreed that an important point in
the evolution of the modern concept of uncertainty was the publication of the
3
http://www.dbai.tuwien.ac.at/proj/dlv/
4
http://www.tcs.hut.fi/Software/smodels/
5
http://cxjepa.googlepages.com/home
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0)
73
seminal paper by Lotfi A. Zadeh in 1965 [20], where he introduced a theory whose
objects, called fuzzy-sets, have boundaries that are not precise. The membership
in a fuzzy-set is not a matter of affirmation or denial, but rather a matter of a
degree. Although the concept of uncertainty had been studied by philosophers,
the significance of Zadeh’s paper was that it challenged not only probability
theory as the sole agent for uncertainty, but the very foundations upon which
probability theory is based: Aristotelian two-valued logic. When A is a fuzzy-set
and x is a relevant object, the proposition “x is a member of A” is not necessarily
either true or false, as required by two-valued logic, but it may be true to some
degree, the degree at which x is actually a member of A [4].
We can distinguish two main directions in fuzzy logic [3]. The first one corre-
sponds to fuzzy logic in the broad sense, it serves mainly as apparatus for fuzzy
control, analysis of vagueness in natural language and several other application
domains. The second one corresponds to fuzzy logic in the narrow sense which is
symbolic logic with a comparative notion of truth developed fully in the spirit of
classical logic. So, this fuzzy logic has syntax, semantics, axiomatization, truth-
preserving deduction, completeness, etc. It is a branch of many-valued logic based
on the paradigm of inference under vagueness. This last direction in fuzzy logic
is a relatively recent discipline, both serving as a foundation for the fuzzy logic in
a broad sense and of independent logical interest, since it turns out that strictly
logical investigation of this kind of logical calculi can go rather far (interested
readers can see [2,9]). Currently it is possible to find strong formal systems of
fuzzy logic, such as, Lukasiewicz, Gödel and product logic [3].
In particular, Lukasiewicz and Post gave the first published systematic de-
scriptions of many-valued logical systems in the modern era [19]. Lukasiewicz
argued that if statements about future events are already true or false, then the
future is as much determined as the past and differs from the past only in so
far as it has not yet come to pass. In order to avoid the situations in which
further development is impossible, he proposed to reject the law of excluded
middle, that is, the assumption that every proposition is true or false. Moreover,
he proposed a logic system where a third truth-value is added, which is read as
possible. The Lukasiewicz logic [19] is a non-classical, many-valued logic. It was
originally defined as a three-valued logic, denoted by L3 , and as we mentioned,
it belongs to the classes of fuzzy logics. Afterwards, Lukasiewicz generalized his
three-valued logic to n values and also to an infinite-valued system [19]. In this
paper, we consider the L3 logic in order to show a non-standard application of
fuzzy logic. We show how Lukasiewicz logic can be used for knowledge represen-
tation based on logic programming. Our results are based on the fact that the
stable semantics and the p-stable semantics can be expressed in terms of similar
expressions involving the G3 logic and the G′3 logic respectively, and the fact
that the Lukasiewicz logic can express these two logics.
Our paper is structured as follows. In section 2, we summarize some defini-
tions, logics and semantics necessary to understand this work. In section 3, we
show how to express the stable and the p-stable semantics of normal programs
in terms of Lukasiewicz logic. We present a definition of stable model for more
74
general programs in terms of the intermediate logic G3 , and then we extend such
definition, in a conservative way, in terms of the paraconsistent logic G′3 . Finally,
in section 4, we present some conclusions.
2 Background
In this section we summarize some basic concepts and definitions necessary to
understand this paper.
2.1 Logic programs
A signature L is a finite set of elements that we call atoms, or propositional
symbols. The language of a propositional logic has an alphabet consisting of
propositional symbols: p0 , p1 , . . . ;
connectives: ∧, ∨, ←, ¬; and
auxiliary symbols: (, ),
where ∧, ∨, ← are 2-place connectives and ¬ is a 1-place connective. Formulas
are built up as usual in logic. A literal is either an atom a, called positive literal ;
or the negation of an atom ¬a, called negative literal.
A normal clause is a clause of the form
a ← b1 ∧ . . . ∧ bn ∧ ¬bn+1 ∧ . . . ∧ ¬bn+m
where a and each of the bi are atoms for 1 ≤ i ≤ n + m. We define a normal
program P , as a finite set of normal clauses.
The body of a normal clause could be empty, in which case the clause is
known as a fact and can be denoted just by: a ←.
We write LP , to denote the set of atoms that appear in the clauses of P .
Given a set of atoms M and a signature L, we define ¬M f = {¬a | a ∈ L\M }.
Since we shall restrict our discussion to propositional programs, we take for
granted that programs with predicate symbols are only an abbreviation of the
ground program.
2.2 Logics
We review some logics that are relevant in this paper to characterize different
semantics of normal and more general programs.
We present definitions in terms of true values as well as Hilbert style defini-
tions for most of these logics. The logics considered here have the modus ponens
as a unique inference rule.
75
Lukasiewicz’s 3-valued logic The polish logician and philosopher Jan Lukasiewicz
began to create systems of multivalued logics in 1920. He developed, in particu-
lar, a system with a third value to denote “possible” that could be used to express
the modalities “it is necessary that” and “it is possible that”. To construct this
logic, denoted by L3 , we first have to modify the syntax of our formulas to
allow, as primitive connectives, only: the 0-place connective ⊥ (failure) and the
2-place connective → (implication). These connectives operate over a domain
D = {0, 1, 2}, with 2 as the unique designated value, and are defined as follows:
– ⊥ = 0,
– x → y = min(2, 2 − x + y).
Other connectives in L3 are introduced in terms of ⊥ and → as follows:
¬A := A → ⊥ ⊤ := ¬⊥
A ∨ B := (A → B) → B A ∧ B := ¬(¬A ∨ ¬B)
A := ¬(A → ¬A) ♢A := ¬A → A
The truth tables of most connectives are shown in Table 1, the conjunction and
disjunction connectives (not shown) coincide with the min and max functions
respectively. A syntactic characterization of the modal content of L3 is studied in
[8], where the behavior of modal operators is checked against some of the relevant
modal principles. Let us observe that, in logic L3 , the formula (a ∧ ¬a) → b is
x ¬x x ♢x →012
0 2 0 0 0 222
1 1 0 2 1 122
2 0 2 2 2 012
Table 1. Truth tables of connectives in L3 .
not a tautology, which implies a paraconsistent behavior.
Now we present an axiomatization of L3 [19]:
(1) (p → q) → ((q → r) → (p → r))
(2) (¬p → ¬q) → (q → p)
(3) ((p → ¬p) → p) → p,
using the rules of substitution and modus ponens.
Lukasiewicz generalized his three-valued logic to n values and also to an
infinite-valued system [19]. The matrix for the infinite-valued system is defined
on the rational numbers in the unit interval from 0 to 1. For x, y in the interval,
we have: x → y = min(1, 1 − x + y), ¬x = 1 − x. If instead of the whole rational
interval, a finite subset closed under the above functions is chosen, the result is a
set of n−valued Lukasiewicz connectives. For more general results in Lukasiewicz
logics, including the case where a different set of designated values is adopted in
Lm , the reader is referred to [18].
76
G′3 logic It is defined as a three-valued logic with truth values in the domain
D = {0, 1, 2} where 2 is the designated value. The evaluation functions of the
logic connectives are then defined as follows: x∧y = min(x, y); x∨y = max(x, y);
and the ¬ and → connectives are defined according to the truth tables given in
Table 2. An axiomatization of G′3 is given in [10].
x ¬x →012
0 2 0 222
1 2 1 022
2 0 2 012
Table 2. Truth tables of connectives in G′3 .
G3 logic Gödel defined, in fact, a family of many-valued logics Gi with truth
values over the domain D = {0, 1, . . . , i − 1} and with i − 1 as the unique desig-
nated value. Logic connectives are defined as:
– ⊥ = 0, x ∧ y = min(x, y), x ∨ y = max(x, y), and
– x → y = i − 1 if x ≤ y and y otherwise.
The only difference between the true tables of G3 and G′3 is the negation of
the value 1 that is 0 in G3 .
A Hilbert style version of this logic G3 is obtained from intuitionistic logic
[7] by adding the following axiom:(¬b → a) → (((a → b) → a) → a). Therefore the
set of theorems in this logic is the set of tautologies of Gödel’s 3-valued logic G3 .
From now on we will denote by ¬ the negation of G′3 and by ∼ the negation
of G3 .
Classical logic Classical logic, C, is obtained from intuitionistic logic [7] by
adding the following axiom: (¬a → a) → a. This axiom enables any sort of proofs
by contradiction, and thus gives to the negation connective its full deduction
power. Classical logic, of course, coincides with the well known standard ”truth
table” logic of two values [6].
2.3 Semantics
From now on, we assume that the reader is familiar with the notion of classical
minimal model [6].
We present the characterization of the stable semantics for normal programs
in terms of the G3 logic. A similar characterization exists for p-stable semantics
in terms of logic G′3 .
We use the notation X P to indicate that formula P is a theorem or a
tautology in logic X depending on logic X. For a finite family of formulas Q =
77
q1 , q2 , ..., qn and a set of atoms M , the expression Q X M will mean that M is
a classical model of Q and that X q1 → (q2 → (...(qn → m)...) for each m in
M
Stable semantics From now on, we assume that the reader is familiar with
the notion of classical minimal model [6].
The characterization of the stable semantics for normal programs in terms
of logic G3 is given in the following definition.
Definition 1. [17] Given a normal program P , a set of atoms M ⊆ LP is a
f G M.
stable model of P if P ∪ ∼ M 3
Let us observe that, in particular, M is also a classical model of P as we men-
tioned.
Example 1. Let P be the following normal program:
{a ←∼ b, b ←∼ a}
and let M1 = {a} and M2 = {b}, according to the definition of stable semantics,
since P ∪ {∼ b} G3 {a} and P ∪ {∼ a} G3 {b}, then M1 and M2 are stable
models of P as the reader can easily check.
2.4 Defining G3 and G′3 via L3
One of our motivations to study Lukasiewicz’s L3 logic is the fact that it is able
to express the semantics of other logics such as Gödel’s G3 and G′3 .
This subsection presents some results from [12]. We first define, in Table 3,
the implication and negation connectives for G3 and G′3 via L3 (connectives
that are not subscripted correspond to L3 ). Table 4 shows the truth tables of
these connectives for the G3 and G′3 logics. The reader can easily verify that the
definitions here given reproduce the values shown in the tables. Conjunction and
disjunction are defined, just as in all other logics considered, as the min and max
functions respectively. Hence, these two connectives have the same semantics in
L3 , G3 , and G′3 logics.
∼G3 x := ¬x
x →G3 y := (x → y)∧ ∼G3 ∼G3 (∼G3 ∼G3 x → y)
¬G′3 x := ¬x
x →G′3 y := x →G3 y
Table 3. Connectives of G3 and G′3 in L3 .
78
x ∼G3 x ¬G′3 x →012
0 2 2 0 222
1 0 2 1 022
2 0 0 2 012
Table 4. Truth tables: connectives in G3 and G′3 .
2.5 The X − or operator
An interesting property we try to express in terms of our logics, is a characteriza-
tion of the X − or operator endowed with an encoding-decoding property. In its
original form, this operator works as an exclusive disjunction in two variables: 0
(false) and 1 (true). When implementing this particular generalization to three
true values we obtain the arrangement given in table 5.
Let us denote the new X − or operator by the symbol ⊖. We can characterize
this operator by the following three properties in order to obtain the values we
need. Observe that the last equation reflects the encoding-decoding property and
also that the commutativity of the operator follows from the equations shown
below.
x⊖0=x
x⊖x=0
(x ⊖ y) ⊖ y = x
According to these properties we have from (1⊖1)⊖1 = 1 and (2⊖2)⊖2 = 2,
that 0 ⊖ x = x, furthermore from (1 ⊖ 2) ⊖ 2 = 1 it follows that 1 ⊖ 2 = 1. A
similar argument shows that 2 ⊖ 1 = 2.
Therefore we get the Table 5 for the X − or operator in three truth values.
⊖012
0 012
1 101
2 220
Table 5. The X − or operator.
What we want next is to express this truth table as a function of two variables,
however this is not possible in logic G3 , since as it is well known, any function f
of two variables defined in this logic has the property that f (2, 2) = 2 whenever
f (1, 2) = 1 [13], and according to the table f (2, 2) must be 0.
Let us remember that the symbol ¬ is the G′3 -negation, the symbol ∼ is the
G3 -negation, and that ∼ x = x → (¬x ∧ ¬¬x), then in G′3 the X − or operator
is expressed by the following formula:
(x ∨ y) ∧ (¬x ∨ ¬y) ∧ (x∨ ∼ x) ∧ ((¬¬x∨ ∼ x) ∨ (¬¬y∨ ∼ y))
79
3 Expressing p-stable and stable semantics based on L3
Here, we show how to express stable semantics via L3 logic. This is possible due
to two facts: L3 logic is able to express the semantics of G3 logic, and the stable
semantics is defined in terms of this logic.
We first define a function that obtains the clause that results when we sub-
stitute the G3 connectives by the L3 connectives from a given clause.
Given a clause r expressed in terms of the G3 connectives, we define T radG3 toL3 (r)
as the clause that results when we substitute the G3 connectives for the L3 con-
nectives according to Table 3. Given a normal program P , we define T radG3 toL3 (P )
as the set {T radG3 toL3 (r) | r ∈ P }.
Example 2. Let us consider the program P1 = P ∪{∼ b} where P is the program
of Example 1, namely the program
a ←∼ b.
b ←∼ a.
∼ b.
We can see that L1 = T radG3 toL3 (P1 ) is the following program:
(¬b → a) ∧ ¬¬(¬¬¬b → a).
(¬a → b) ∧ ¬¬(¬¬¬a → b).
¬b.
Now we present the definition of the stable semantics based on the definition
of G3 via L3 . Similar results are obtained for the p-stable semantics and logic
G′3 .
Theorem 1. [12] Given a normal program P , a set of atoms M ⊆ LP is a
f) L M .
stable model of P if T radG3 toL3 (P ∪ ∼ M 3
Proof. According to the relations between the L3 connectives and the G3 con-
nectives, and our interpretation of the symbol X , we have that the relation
P∪ ∼ M f G M is equivalent to T radG toL (P ∪ ∼ M f) L M .
3 3 3 3
Example 3. Let us consider the program P of Example 1. Let M1 = {a} and
M2 = {b}. Then we have that T radG3 toL3 (P ∪{∼ b}) L3 {a} and T radG3 toL3 (P ∪
{∼ a}) L3 {b} where T radG3 toL3 (P ∪ {∼ b}) corresponds to the program L1
of Example 2. Hence, M1 and M2 are stable models of P as we obtained in
Example 1.
Next we take advantage of the fact that logic G3 can be expressed in terms
of paraconsistent logic G′3 to provide a definition, by way of paraconsistency,
that extends the concept of stable model to general programs. In order to do
this, we state a general definition of stable model in terms of G3 logic, which in
turn generalizes the definition we have for normal programs. It is important to
observe that in the following formula we can use intuitionistic logic instead of
G3 logic.
80
Definition 2. [11] Given a general logic program P , a set of atoms M ⊆ LP is
f∪ ∼∼ M G M .
a stable model of P if and only if P ∪ ∼ M 3
As expected, this definition is a generalization of definition 1 for normal
programs; it uses the fact that the double negation of the atoms in the set M
are added as premises in the equation.
As it is well known, it is desirable that replacing equivalent formulas as parts
of programs would leave the same stable models as in the original program. This
property does not hold if we start with two programs that have the same stable
models. For example, take the two programs P1 = {a ←∼ b} and P2 = {a} with
the same stable models, and now consider the two programs Q1 = {a ←∼ b, b}
and Q2 = {a, b} originated from the previous two programs after adding the same
atom to both of them, these two programs are no longer equivalent according to
the next definition.
Definition 3. Two programs P1 and P2 are equivalent if they have the same
stable models.
In [5] the authors present the following convenient definition which is stronger
than the previous one.
Definition 4. Two programs P1 and P2 are strongly equivalent if for every pro-
gram P , P1 ∪ P and P2 ∪ P have the same stable models.
In order to present the generalization of the stable semantics in terms of
paraconsistency, we remind the reader that the symbol ∼ that appears in the
next definition is the negation of G3 and the program P is expressed in terms
of logic G′3 .
Definition 5. Let P be a program defined in the language of G′3 and M ⊆ LP
be a set of atoms. We say that M is an L-stable model of P if T radG3 toL3 (P ∪ ∼
f∪ ∼∼ M ) L M
M 3
We observe that this definition is a conservative extension of Definition 2
since the language of logic G3 is fully expressed in terms of the paraconsistent
logic G′3 as previously noted. Also observe that the G′3 -negation appears only in
the program P .
As a consequence of this definition we have the following result for L-stable
models of programs expressed in the G′3 -language.
Theorem 2. Let P and Q programs expressed in the language of logic G′3 . If
P ≡G′3 Q, then they are strongly equivalent in the context of L-stable models.
f∪ ∼∼ M )
Proof. The result follows from Definition 5 since T radG3 toL3 (P ∪ ∼ M
f
and T radG3 toL3 (Q∪ ∼ M ∪ ∼∼ M ) are equivalent in L3 .
As a natural consequence that follows from our definitions we have:
81
Theorem 3. Let P be a program expressed in the original language of the stable
semantics, then Definitions 2 and 5 provide the same sets as stable and L-stable
models respectively.
Proof. We only need to observe that a program in the original language of the
answer set semantics can be interpreted as written in terms of the G3 language,
then the conclusion follows from the relation that exists between G3 and L3 .
Finally, as an example let us compute the L-stable models of the program
given by the formula x ⊖ y. This program is expressed below in terms of its
clauses
x∨y
¬x ∨ ¬y
x∨ ∼ x
(¬¬x∨ ∼ x) ∨ (¬¬y∨ ∼ y)
The language of the program is {x, y}. According to Definition 5, a set of
atoms M is a L-stable model of P if T radG3 toL3 (P ∪ ∼ Mf∪ ∼∼ M ) L M . We
3
′
work with logic G3 whose language is more natural. Note also that the expression
Q X M as defined in the background, is equivalent to (q1 ∧ q2 ∧ ...qn ) → m for
each m in M when X is G′3 or L3 .
Let us propose M = {x}, then ∼ M f =∼ y and ∼∼ M =∼∼ x
The left hand side of this implication is a conjunction of the four rules that
express x ⊖ y plus the two rules or facts: ∼ y and ∼∼ x
Let us assume that for certain valuation x takes the value 1, then the third
rule of the antecedent takes the value 1 and therefore the antecedent cannot take
the value 2. Now if we assume that the atom x takes the value 0, then the last
rule of the antecedent takes the value 0 and so does the antecedent. We conclude
that the implication is a tautology in G′3 and the set M = {x} is a L-stable
model.
Now, if we try M = {y}, then the last two rules of the implication we are
working with, become: ∼ x and ∼∼ y.
Let us assume that for certain valuation y takes the value 1, then the first
and fifth rules of the antecedent cannot be 2 at the same time. In the case the
atom y takes the value 0 the last rule of the antecedent is 0 too. Therefore the
implication is a tautology and the set {y} is a L-stable model too.
In a similar way we can see that the valuation x = 2, y = 1 shows that the
set M = {x, y} is not a L-stable model.
4 Conclusions
We show that Lukasiewicz logic can be used for knowledge representation based
on logic programming. We review how two useful semantics to represent knowl-
edge, stable and p-stable, are characterized via L3 logic. In particular we take
advantage of the fact that Lukasiewicz’s logic L3 is able to express G3 and G′3
82
logics, which characterize those two semantics for normal programs respectively.
We also present the definition of stable model for general programs in terms of
logic G3 and extend it for programs in the languages of G′3 and L3 . Finally we
present the concept of strong equivalence in the new context of paraconsistency.
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