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1 Introduction
Answer set programming (ASP) is a form of declarative programming based on the stable model (answer set) semantics of logic programming.
Recall that stable semantics allows us to handle problems with default knowledge and produce non-monotonic reasoning using the concept of negation as failure.
It was originally defined for the class of theories called normal programs [11]. Currently, ASP is robust and mature enough, offering many important language constructs like aggregation, (weak) constraints, different types of negations, and optimization statements to mention a few, as high-performance solvers do.
An example of a state-of-the-art and award-winning ASP solver is clasp [9] demonstrating its competitiveness and versatility, by winning first places at various solver contests since 2011 (e.g., ASP, CASC, MISC, PB, and SAT competitions).
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, the term fuzzy logic emerged in the development of the theory of fuzzy sets by Zadeh in 1965, [14, 24].
Although the concept of uncertainty had been studied by philosophers, the significance of Zadeh’s work was that it challenged not only probability theory, as the sole agent for uncertainty, but also 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 [15].
Furthermore, fuzzy logic was introduced to study the question of uncertainty from a foundational point of view based on many-valued logics. In this sense, fuzzy logic can be considered as a degree-based approach to uncertainty.
Some systems like Łukasiewicz, product and Gödel logics are, just like fuzzy sets, valued over the real interval [0,1]. This supports the idea of fuzzy logic being as a kind of foundational counterpart of fuzzy set theory which is a discipline mainly devoted to engineering applications.
In particular, the infinite-valued Lukasiewicz logic Ł, introduced for philosophical reasons by Jan Lukasiewicz, is among the most important and widely studied of all non-classical logics. Later,
The logic Ł was introduced taking in mind 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, it was proposed to reject the law of Excluded Middle, that is, the assumption that every proposition is true or false. The
It is worth mentioning that initially Łukasiewicz proposed the
In this paper, we consider the Ł3 logic in order to show a non-standard application of fuzzy logic. Firstly, we show how Łukasiewicz logic can be used for knowledge representation based on logic programming.
In particular, we present a definition of stable propositional Łukasiewicz models using the
Moreover, we show that this equivalence fails when we consider the proposed definition of stable models for arbitrary propositional theories given in [23], see Remark 1. However, it is possible to reduce any propositional theory into the class of augmented programs [20].
Gelfond showed that the perfect models of stratified logic programs can be characterized in terms of extensions of the corresponding autoepistemic theory [10] . His characterization is based on the interpretation of
In this paper,
In section 2, we summarize some definitions, logics and semantics necessary to understand this work. Besides, we present a characterization of the stable models semantics for augmented programs in terms of the intermediate logic
Then, in section 3, we present our results; namely, we introduce the notion of Ł3-stable models. We show its correspondence with standard stable models for the class of augmented programs. Later on, we generalize theses results to many-valued Łukasiewicz logics. 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
– propositional symbols:
– constant symbols:
– binary connectives:
– unary connectives:
– auxiliary symbols: (, ).
where
If
A program is a finite theory and by default, we assume that our formulas and programs are constructed using the
A literal is either an atom
A
An augmented clause is a formula of the form:
Besides, we call
We define an augmented program
The body of an augmented formula could be just the
We write
Given a set of atoms
Given a program
Given the unary connective
Furthermore, it is worth observing that
Example 1. Let
And let
2.2 Logics
Now, we will review some logics that are relevant to this paper to characterize different semantics of normal and more general programs.
Moreover, we present definitions in terms of true values as well as Hilbert style definitions for most of these logics. The logics considered here have the modus ponens as a unique inference rule.
2.2.1 Łukasiewicz’s Logic
Łukasiewicz logic is presented in the language
The truth constants are defined as follows:
When
Afterwards, Łukasiewicz generalized his three-valued logic to
Recall that the propositional Łukasiewicz logic is defined as the following Hilbert style system of axioms and rule ([13, 3]):
– (Ł1)
– (Ł2)
– (Ł3)
– (Ł4)
– (MP) The rule of modus ponens:
Now, for every integer
A matrix for Łn, with
It is well-known that the class of MV-algebras is semantics for Łukasiewicz logic and its extension Łn has the class of MVn-algebras as algebraic semantics. For more details about the theory of MVn-algebras the reader can consult the book [3, Section 8.5].
2.2.2 Łukasiewicz’s 3-Valued Logic
The polish logician and philosopher Łukasiewicz began to create systems of multivalued logics in 1920. He developed, in particular, 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 Ł3, we first have to modify the syntax of our formulas to allow only as primitive connectives: the constant (0-place connective)
These connectives operate over a domain
Other connectives in Ł3 are introduced in terms of
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.
Table 1 Truth tables of connectives in
| 0 | 1 | 2 | ||||||
| 0 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | |
| 1 | 1 | 0 | 2 | 1 | 1 | 2 | 2 | |
| 2 | 0 | 2 | 2 | 2 | 0 | 1 | 2 |
A syntactic characterization of the modal content of Ł3 is studied in [17], where the behavior of modal operators is checked against some of the relevant modal principles.
It is important to note that the algebra
For more general results in Łukasiewicz logics, the reader can be referred to [3]. Another observation worth mentioning is that the modal operators
Furthermore, these operators was considered on Gödel-Dummmett logic by Baaz, [1]. Indeed, Baaz baptized this operator with
This operator was studied on fuzzy logics by Esteva, Godo, Hájek, Montagna and others. These authors presented an algebraic study of these modals operators over certain fuzzy logics, see for instance [6, 7].
2.2.3
Gödel defined, in fact, a family of many-valued logics
Table 2 illustrates the implication and negation connective for
2.3 Semantics
We assume that the reader is familiar with the notions of interpretations, models, and logical consequence in the usual way. Recall that our interest is in the above non classical logics.
Both have only
With respect to Logic Programming, we adopt the following conventions. Given a set of atoms
Notice that the values in
Given a program
By abuse of the language we may sometimes say that
A very important remark on notation is the following. When we write any of the following statements:
For instance
However, if one makes a particular connective such as
But, after all, it is also a non-primitive operator in Ł3 logic constructed using the primitive operators of the same logic.
2.4 Expressing Stable Semantics based on
We present a general characterization of stable models in terms of
This characterization allows us to understand what stable models are without having to appeal to the original definition.
Theorem 1. [21] Given a general logic program
Example 2. Let
And let
However, it is false that
Recall that in our programs we write
Example 3. Let
And let
2.5 Some Examples
We now provide some examples of augmented programs and their stable models. This with the purpose that the reader could check the models and thus verify if the previously presented theory has been well understood.
3 Stable Models in Łukasiewicz’s Logic Ł3
In this section, we present our main results, namely, we introduce Ł3-stable models and we show their correspondence with standard stable models. Then we generalize this results to consider many-valued logics.
3.1 3-Valued Logics
Theorem 2. Given an augmented theory
Proof. It is enough to take an augmented clause
Now, let
Then
But in neither of these two possibilities, the valuation takes true-valued 2. Consequently, if
We have showed that Łukasiewicz
Suppose that both implication and negation correspond to Łukasiewicz’s logic Ł3. Then clearly
It is well accepted to consider negation as a failure a kind of modality operator, see for instance [19], which is considered as follows:
Definition 1. Given a logic program
Here, we assume the use of Ł3 implication which behaves differently to the one of
Remark 1. Let us start by considering the following program
Theorem 3. Given an augmented logic program
Proof. Recall that
On the other hand,
We have already noticed that
Moreover,
Notice that in the examples provided in Section 2.5, the theories belong to the class of augmented programs and therefore the stable models correspond to the Ł3-stable models.
Lemma 1. Let
Then,
Proof. We will give the proof by induction on
Now, let us suppose
So, due to the induction hypothesis we have that
Finally, let
Theorem 4. Given an augmented logic program
where the connective of implication used in the above expression could be any of the two defined in our 3-valued logics considered in this paper.
Proof. For this proof, we will consider the connective of implication of Ł3 logic and let
From left-to-right, let us suppose that
In the following, we will consider an arbitrary, but fixed
–
Then,
–
Notice that
Recall the interpretation
Since
Consider an augmented clause
In this setting, we also have that
–
Then,
Therefore, every 3-valued interpretation maintains the desired equality. Then, the first condition is proved.
Now, let us prove the right-to-left direction. First, let
The value of the conjunction of all elements of
Consequently,
Let
This means that every formula
Let us observe that in the proof of Theorem 4, the implication of Ł3 logic has been considered. The following Corollary is an immediate consequence of the last Theorem:
Corollary 1. Given an augmented logic program
Is a tautology in any of our 3-valued logics considered in this paper.
3.2 Generalization
In this subsection, we will work on the family of logics
Recall that in the logic Łn, it is possible to define the binary operators
Furthermore, we can define the operator
It is not hard to see that
Theorem 5. Given an augmented theory
Proof. Let
Since
From the latter and the order defined through the implication on
Therefore, we can infer that that
Conversely, by an analogous argument but now using the definition of the implication on
Now, we are in conditions to define the notion Łn-stable model via the following definition in which the use of Łn implication is involved. Indeed:
Definition 2. Given a logic program
Here, we assume the use of Łn implication, which behaves differently to the one of
Theorem 6. Given an augmented logic program
Proof. Having in mind Theorem 1, we know that
By Definition 2, we have that
Taking into account Theorem 5, we obtain that
4 Conclusion and Future Work
ASP has proven to be a powerful tool to encode and solve problems in both academia and industry, becoming one of the most attractive KRR frameworks [8].
Therefore, ASP has applications in many fields, and it is widely used for knowledge representation due to its effectiveness.
In this paper, we have shown that Łukasiewicz logic can be used for knowledge representation based on logic programming.
Furthermore, we have also presented an alternative definition of stable models for general programs in terms of Ł3 logic and we have studied the circumstances in which they are equivalent to the standard stable models.
We have taken advantage of the fact that on Łukasiewicz’s logic Ł3 is possible to express the negation on
As future work, we will focus on the study of the correspondence of standard state models and Ł3 stable models when the theories do not belong to the class of augmented programs.
Remark 1 shows us that if we take more general programs, we can obtain more answers on Ł3. Moreover, a generalization of Theorem 4 for multivalued logics will also be studied.
Besides, the theory presented in this paper will be studied further with the aim of extending our characterization to model preferences in a natural way, following the work presented in [4].
Another important point that we are interested in is giving a notion of stable semantics in terms of MVn-algebras which would imply translating the result given here for generating algebra for a more general setting.
