Checking

Integrity Constraints in Deductive Systems

based on Production Rules and a Description Logic

Terminology

Jaime Ram

ırez and Ang

elica de Antonio

Technical University of Madrid

28660, Madrid, Spain

Abstract. The aim of this paper is to show a method that is able to detect a partic-

ular class of semantic inconsistencies in a deductive system (DS). A DS veriﬁed

by this method contains a set of ﬁrst order production rules, and a description

logic terminology that deﬁnes the domain problem. By building an ATMS-like

theory the method is able to give an speciﬁcation of all the initial Fact Bases

(FBs), and the rules that would have to be executed from these initial FBs to

produce an inconsistency.

1 Introduction

The purpose of this paper is to illustrate a method for verifying the semantic consis-

tency of deductive systems (DS) that deal with production rules and a description logic

(DL) terminology (also called TBox). In this sense, DSs that use a powerful language

for modelling the reasoning process and a DL language for modelling general knowl-

edge about the problem domain are often considered hybrid reasoning systems. Hybrid

reasoning systems were popular in the 1980’s; lately, the topic has regained attention,

focusing on knowledge bases with a DL component for concept deﬁnitions and a logic-

programming component for assertions about individuals [1] [2]. DLs are a family of

languages that are especially proper to model complex constraints related to concepts

structured in hierarchies. On the other hand, the production rules are more suitable to

represent the behaviour of a system, sometimes non-monotonically, expressing contin-

gent properties. DSs based on DLs have been employed successfully in conﬁguration

problems, and they constitute a promising area in the medicine and the database ﬁelds.

2 Scope of the method

Our method is able to verify a DS that is formed by a set of production rules and a Tbox.

The TBox is a set of DL formulas whose form can be C v D or C = D where C and

D are concepts. Moreover, the production rule form is: (x

∈ T

, x

∈ T

, ..., x

∈

) (l

, l

, ..., l

∨ l

, l

, ..., l

∨, ..., ∨ l

, l

, ..., l

) → a

, a

, ..., a

where

the antecedent part contains a declaration part for the variables used in the rule (x

Ramírez J. and de Antonio A. (2004).

Checking Integrity Constraints in Deductive Systems based on Production Rules and a Description Logic Terminology.

In Proceedings of the 2nd International Workshop on Veriﬁcation and Validation of Enterprise Information Systems, pages 84-86

DOI: 10.5220/0002673300840086

 SciTePress

and a disjunction of m conjunctions of literals (l

); and the consequent part contains

a list of actions (a

). Each variable used in the rule must be declared as an atomic

concept deﬁned in the TBox, so x

∈ T

is intended to mean that the variable x

only can be bound to an individual a s. t. T

(a) holds. A literal is either a ﬁrst-order

logic atom or a negated atom. The predicate of an atom corresponds to a DL atomic

concept or a DL atomic role, not necessarily deﬁned in the TBox. An assertion or fact

is a ground literal. Assertions in a DS are classiﬁed into two categories: a deducible

assertion is an assertion that is obtained from the execution of the DS; and an external

assertion is an assertion that cannot be deduced by the DS and can only be obtained

from an external source. We will see the Fact Base (FB) as a set of assertions. We

admit actions to be addition actions or deletion actions for assertions. DSs can use two

different approaches for the management of the negation: closed world assumption or

3-valued logic. Each semantic inconsistency that must be considered is represented by

means of an Integrity Constraint (IC). An IC deﬁnes a consistency criterion over input

data, output data or input and output data. The IC form is: ∃x

∈ T

...∃x

∈ T

(scope

) ∧ l

(scope

) ∧ ... ∧ l

(scope

)) ⇒⊥.

The DS is assumed to execute with forward chaining or backward chaining. Explicit

control mechanisms and meta-rules are not considered by the proposed method. The DS

can reason non-monotonically, even, the DS may employ rules of the form p → ¬p, so

we will situate ourselves quite far from the concept of logical inconsistency as deﬁned

in other works. Hence, we are going to clarify the meaning of logical inconsistency in

this work: a logical inconsistency is reached when, in order to deduce a pair of facts

F and F

, it is necessary to assume the truth of a set of contradictory external facts, or

when at any time in the process of deducing F (F

), the fact ¬F

(¬F ) is deduced, and

the fact F’(F) is not deduced later in the same deductive process. This notion of logical

inconsistency is explained with more depth in [3].

3 Contexts

As the output, the method, for each IC that can be violated, must generate a report

explaining the requirements that the initial FB must fulﬁll so that it is possible to ex-

ecute, starting from the initial FB, a certain deductive path (sequence of rule ﬁrings)

that causes the inconsistency described by the IC. The method will construct an object

called subcontext to specify how the initial FB must be and which deductive path must

be executed in order to cause an inconsistency. There can be different initial FBs and

different deductive paths that lead to the same inconsistency. An object called context

will specify all the different ways to violate a given IC. Consequently, a context will

be composed of n subcontexts. In turn, a subcontext is deﬁned as a pair (environment,

deductive path) where an environment is composed of a set of metaobjects, and a de-

ductive path is a sequence of rule ﬁrings. A metaobject describes the characteristics

that one object which can be present in a FB should have. We will distinguish three

kinds of objects in a FB: individuals, concepts and roles. Hence, we will consider the

following kinds of metaobjects: Metaindividual(identiﬁer, subinstance

of, roles), Meta-

concept(identiﬁer, individuals) and Metarole(identiﬁer, tuples). In order to describe a FB

object, a metaobject must include a set of constraints on the characteristics of the FB ob-

ject. These constraints will be included in the ﬁelds of the metaobjects. For instance, the

tuples ﬁeld will comprise the tuples that must belong or not belong (they will appear as

negated tuples) to the role described by the metarole. Each tuple, in turn, will include

a list of metaindividual names. Lets see an example of an environment describing a FB

in which the formula Owns(X, umbrella) ∧ P erson(X) is true. This environment is

{IND1(

, {Person}, {Owns}), IND2 (umbrella,

, {Owns}), CON1(Person, {IND1}),

ROLE1(Owns, {(IND1, IND2)})}. If there exists an object in the FB, for each metaob-

ject in the environment, that satisﬁes all the requirements imposed on it, then the given

formula will be true in the FB.

4 Operation of the method

In order to analyze the consistency of a DS, our method has to compute the context

associated to each IC. If this context results to be empty, that means that there is not any

valid initial FB that leads to the violation of the IC. The method implements a backward

chaining simulation of the real rule ﬁring. The recursive calls ﬁnish when, in the pro-

cess of computing the context associated with a fact, this fact is external. Basically, the

method can be divided into two phases. In the ﬁrst phase, the AND/OR deductive tree

associated with the IC is expanded. The leaves of this tree are rules that only contain ex-

ternal facts. During the ﬁrst phase, some metaobjects are associated to the object names

and the variables in the rules, and they are propagated from a rule to another one, and

updated in each rule. In this updating process some constraints derived from the literals

and variable declarations are inserted into the metaobjects. Moreover, in this updating

process, the satisfacibility of the constraints in the metaobjects is checked w.r.t. the

TBox. To carry out the satisﬁability test, our method relies on the calculus explained in

[4]. The computational properties of this calculus for the AL-languages family are also

discussed in [4].

In the second phase, the metaobjects associated to external facts are inserted in the sub-

contexts and the deductive tree is contracted by means of the context operations (these

operations are explained in [3]). Thus, all the metaobjects generated from external facts

are collected in the context associated with the IC.

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