This paper develops a declarative language, P-log, that combines logical and probabilistic arguments in its reasoning. Answer Set Prolog is used as the logical foundation, while causal Bayes nets serve as a probabilistic foundation. We give several non-trivial examples and illustrate the use of P-log for knowledge representation and updating of knowledge. We argue that our approach to updates is more appealing than existing approaches. We give sufficiency conditions for the coherency of P-log programs and show that Bayes nets can be easily mapped to coherent P-log programs.
Deep Dive into Probabilistic reasoning with answer sets.
This paper develops a declarative language, P-log, that combines logical and probabilistic arguments in its reasoning. Answer Set Prolog is used as the logical foundation, while causal Bayes nets serve as a probabilistic foundation. We give several non-trivial examples and illustrate the use of P-log for knowledge representation and updating of knowledge. We argue that our approach to updates is more appealing than existing approaches. We give sufficiency conditions for the coherency of P-log programs and show that Bayes nets can be easily mapped to coherent P-log programs.
The goal of this paper is to define a knowledge representation language allowing natural, elaboration tolerant representation of commonsense knowledge involving logic and probabilities. The result of this effort is a language called P -log.
By a knowledge representation language, or KR language, we mean a formal language L with an entailment relation E such that (1) statements of L capture the meaning of some class of sentences of natural language, and (2) when a set S of natural language sentences is translated into a set T (S ) of statements of L, the formal consequences of T (S ) under E are translations of the informal, commonsense consequences of S .
One of the best known KR languages is predicate calculus, and this example can be used to illustrate several points. First, a KR language is committed to an entailment relation, but it is not committed to a particular inference algorithm. Research on inference mechanisms for predicate calculus, for example, is still ongoing while predicate calculus itself remains unchanged since the 1920’s.
Second, the merit of a KR language is partly determined by the class of statements representable in it. Inference in predicate calculus, e.g., is very expensive, but it is an important language because of its ability to formalize a broad class of natural language statements, arguably including mathematical discourse.
Though representation of mathematical discourse is a problem solved to the satisfaction of many, representation of other kinds of discourse remains an area of active research, including work on defaults, modal reasoning, temporal reasoning, and varying degrees of certainty.
Answer Set Prolog (ASP) is a successful KR language with a large history of literature and an active community of researchers. In the last decade ASP was shown to be a powerful tool capable of representing recursive definitions, defaults, causal relations, special forms of self-reference, and other language constructs which occur frequently in various non-mathematical domains (Baral 2003), and are difficult or impossible to express in classical logic and other common formalisms. ASP is based on the answer set/stable models semantics (Gelfond et al. 1988) of logic programs with default negation (commonly written as not ), and has its roots in research on non-monotonic logics. In addition to the default negation the language contains “classical” or “strong” negation (commonly written as ¬) and “epistemic disjunction” (commonly written as or).
Syntactically, an ASP program is a collection of rules of the form: l 0 or . . . or l k ← l k +1 , . . . , l m , not l m+1 , . . . , not l n where l ’s are literals, i.e. expressions of the form p and ¬p where p is an atom. A rule with variables is viewed as a schema -a shorthand notation for the set of its ground instantiations. Informally, a ground program Π can be viewed as a specification for the sets of beliefs which could be held by a rational reasoner associated with Π. Such sets are referred to as answer sets. An answer set is represented by a collection of ground literals. In forming answer sets the reasoner must be guided by the following informal principles:
One should satisfy the rules of Π. In other words, if one believes in the body of a rule, one must also believe in its head.
One should not believe in contradictions.
One should adhere to the rationality principle, which says: “Believe nothing you are not forced to believe.”
An answer set S of a program satisfies a literal l if l ∈ S ; S satisfies not l if l ∈ S ; S satisfies a disjunction if it satisfies at least one of its members. We often say that if p ∈ S then p is believed to be true in S , if ¬p ∈ S then p is believed to be false in S . Otherwise p is unknown in S . Consider, for instance, an ASP program P 1 consisting of rules:
p(a).
q(c) ← not p(c), not ¬p(c). 4. ¬q(c) ← p(c).
The first two rules of the program tell the agent associated with P 1 that he must believe that p(a) is true and p(b) is false. The third rule tells the agent to believe q(c) if he believes neither truth nor falsity of p(c). Since the agent has reason to believe neither truth nor falsity of p(c) he must believe q(c). The last two rules require the agent to include ¬q(c) in an answer set if this answer set contains either p(c) or ¬p(c). Since there is no reason for either of these conditions to be satisfied, the program will have unique answer set S 0 = {p(a), ¬p(b), q(c)}. As expected the agent believes that p(a) and q(c) are true and that p(b) is false, and simply does not consider truth or falsity of p(c).
If P 1 were expanded by another rule: 6. p(c) or ¬p(c) the agent will have two possible sets of beliefs represented by answer sets S 1 = {p(a), ¬p(b), p(c), ¬q(c)} and S 2 = {p(a), ¬p(b), ¬p(c), ¬q(c)}. Now p(c) is not ignored. Instead the agent considers two possible answer sets, one containing p(c) and another containing ¬p(c). Both, of course, contain ¬q(c).
The exampl
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