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1 Introduction
Deductive databases are an important aspect in the convergence of artificial intelligence and databases [6].
Currently it is necessary to have complex reasoning tasks to deal with great amounts of data. Logic based systems are an option to provide such complex reasoning capabilities.
Specifically, Deductive Database Systems are forms of database management systems whose storage structures are designed around a logical model of data and at the same time, inference modules for the Deductive Database Systems are designed on logic programming systems.
The Deductive Database Systems are based on deductive database theories that always have associated a semantics. In general, a deductive database theory may give different answers to a query depending on the semantics used.
Two of the semantics that a database theory can be based on, are the stable logic programming semantics (stable semantics) as well as the p-stable logic programing semantics (p-stable semantics).
The mathematical formalism to support those semantics is the theory of intermediate and paraconsistent logics; thus, intuitionism helps to express the stable semantics and the logic
In this work we study a new 4-valued logic called
Furthermore,
Our paper is structured as follows. In section 2, we summarize some definitions and logics necessary to understand this paper. In section 3, we introduce the new
This new logic satisfies a substitution theorem, and can express the stable semantics as well as the p-stable semantics. 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 Logics
We present several logics that are useful to define and study the new
2.1.1 Hilbert Style Proof Systems
One way of defining a logic is by means of a set of axioms together with the inference rule of Modus Ponens. As examples we offer two important logics defined in terms of axioms, which are related to the logics we study later.
The first eight axioms of the list define positive logic. Note that these axioms somewhat constraint the meaning of the
It is a well known result that in any logic satisfying axioms Pos1 and Pos2, and with modus ponens as its unique inference rule, the Deduction Theorem holds [3]. We present a Hilbert-style axiomatization of
We present this logic, since it will be extended to a new logic called
GLukG logic has all the axioms of
– E1.
– E2.
– E3.
– E4.
– E5.
Note that Classical logic is obtained from GLukG by adding to the list of axioms any of the following formulas:
On the other hand,
In this paper we consider the standard substitution, here represented with the usual notation:
Recall the recursive definition: if
2.1.2 GLukG as a Multi-Valued Logic
It is very important for the purposes of this work to note that GLukG can also be presented as a multi-valued logic. Such presentation is given in [2], where GLukG is called
In this form it is defined through a 3-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:
Table 1 Truth tables: connectives in
| 0 | 1 | 2 | |||||
| 0 | 2 | 2 | 0 | 2 | 2 | 2 | |
| 1 | 0 | 2 | 1 | 0 | 2 | 2 | |
| 2 | 0 | 0 | 2 | 0 | 1 | 2 |
We write
In this paper we keep the notation
There is a small difference between the definitions of
In particular, the formula
Theorem 1. [5] For every formula
Theorem 2. (Substitution theorem for
Corollary 1. [5] Let
3 The Logic M4M4
Next we introduce the following 4-valued logic, which we will call
The weak or standard negation is denoted by
The third negation is denoted by the symbol
The double implication is defined as usual, as the conjunction of two opposite conditionals. There is only one designated value, which is 3.
With the three negations logic
We define the bottom particle by the formula
As mentioned before the first interesting property of
Theorem 3. The restriction of logic
Proof. It is enough to observe that the connectives of
In fact, we have a converse for the second part of the previous result, according to which, we obtain equivalence between logics
Theorem 4. For any formula
Proof. We use the fact that
Such axiomatic system has Modus Ponens as its unique inference rule. It is not difficult to check that each axiom that defines logic
Let
According to the truth tables of
As a consequence of the previous two results we have that for any given formula
Corollary 2. For any formula
We state the next result as another corollary.
Corollary 3. The formula
Proof. According to Theorem 3 any formula in one atom using exclusively the connectives
3.1 Some Other Properties of Logic
Theorem 5. The following formulas are tautologies in
–
–
–
–
–
–
–
–
It is worth noting that the reciprocal of the second formula in this theorem is not a tautology. As mentioned before, we provide a form of the substitution theorem.
In order to do that, we need to define a strong equivalence connective since the regular biconditional does not satisfy the substitution property, as shown in the following example.
Last formula in the previous theorem is a tautology, however for the valuation
To proceed with our plan we define a new non-primitive connective, which we call the strong equivalence. See Tables 3 and 4:
Table 3 Truth table: standard bi-conditional connective in
| 0 | 0 | 3 |
| 0 | 1 | 3 |
| 0 | 2 | 0 |
| 0 | 3 | 0 |
| 1 | 0 | 3 |
| 1 | 1 | 3 |
| 1 | 2 | 1 |
| 1 | 3 | 1 |
| 2 | 0 | 0 |
| 2 | 1 | 1 |
| 2 | 2 | 3 |
| 2 | 3 | 2 |
| 3 | 0 | 0 |
| 3 | 1 | 1 |
| 3 | 2 | 2 |
| 3 | 3 | 3 |
Table 4 Truth table: strong bi-conditional connective in
| 0 | 0 | 3 | 3 | 3 |
| 0 | 1 | 3 | 2 | 2 |
| 0 | 2 | 0 | 1 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 1 | 0 | 3 | 2 | 2 |
| 1 | 1 | 3 | 3 | 3 |
| 1 | 2 | 1 | 1 | 1 |
| 1 | 3 | 1 | 0 | 0 |
| 2 | 0 | 0 | 1 | 0 |
| 2 | 1 | 1 | 1 | 1 |
| 2 | 2 | 3 | 3 | 3 |
| 2 | 3 | 2 | 3 | 2 |
| 3 | 0 | 0 | 0 | 0 |
| 3 | 1 | 1 | 0 | 0 |
| 3 | 2 | 2 | 3 | 2 |
| 3 | 3 | 3 | 3 | 3 |
Definition 1.
In the case
For the first case we define:
For the second case we define:
Finally, we present a weak version of the substitution theorem for
Theorem 6.
Proof. The proof is done by induction on the length of
If
If
If
According to the strong bi-conditional truth table, we know that any interpretation gives the same truth values to:
And we also know that the truth values of:
Depend on those truth values solely, hence the result follows.
4 Conclusions and Future Work
In this paper, we introduce a new 4-valued logic called
These logic programming semantics could be used to define the semantics of deductive databases. As future work we propose the study of properties of this logic and the comparison of it to some well known paraconsistent logics.
