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<Paper uid="W04-2503">
  <Title>Using answer set programming to answer complex queries</Title>
  <Section position="6" start_page="0" end_page="0" type="evalu">
    <SectionTitle>
4 Syntax and Semantics of AnsProlog
</SectionTitle>
    <Paragraph position="0"> An AnsProlog knowledge base consists of rules of the form:</Paragraph>
    <Paragraph position="2"> where each of the lis is a literal, i.e. an atom, a, or its classical negation, -a and not is a logical connective called negation as failure or default negation. While -a states that a is false, an expression not l says that there is no reason to believe in l.</Paragraph>
    <Paragraph position="3"> The answer set semantics of a logic program P assigns to P a collection of answer sets - consistent sets of ground literals corresponding to beliefs which can be built by a rational reasoner on the basis of rules of P. In the construction of these beliefs the reasoner is guided by the following informal principles: + He should satisfy the rules of P, understood as constraints of the form: If one believes in the body of a rule one must belief in its head.</Paragraph>
    <Paragraph position="4"> + He should adhere to the rationality principle which says that one shall not believe anything he is not forced to believe.</Paragraph>
    <Paragraph position="5"> The precise definition of answer sets is first given for programs whose rules do not contain default negation. Let P be such a program and X a consistent set of ground literals. Set X is closed under P if, for every rule (4.1) of P, l0 2 X whenever for every 1 * i * m, li 2 X and for every m+1 * j * n, lj 62 X.</Paragraph>
    <Paragraph position="6"> Definition 1 (Answer set - part one) A state X of (P) is an answer set for P if X is minimal (in the sense of set-theoretic inclusion) among the sets closed under P.</Paragraph>
    <Paragraph position="7"> To extend this definition to arbitrary programs, take any program P, and consistent set X of ground literals. The reduct, PX, of P relative to X is the set of rules l0 ^ l1;:::;lm for all rules (4.1) in P such that lm+1;:::;ln 62 X. Thus PX is a program without default negation.</Paragraph>
    <Paragraph position="8"> Definition 2 (Answer set - part two) X is an answer set for P if X is an answer set for PX. Definition 3 (Entailment) A program P entails a literal l (P j= l) if l belongs to all answer sets of P.</Paragraph>
    <Paragraph position="9"> The P's answer to a query l is yes if P j= l, no if P j= l, and unknown otherwise.</Paragraph>
  </Section>
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