Lecture notes for ESSLLI-97

begin
let r be any rule in Q remove r from Q if c(r) 2 Cn(S ) then =
begin end
S := S fc(r)g for every rule s such that p(s) 2 Cn(S ) do add s to Q;
end
Figure 1: Algorithm to compute a basis for CnB (W )
Then, for every set of formulas W there is a least set T closed under PC + B and containing W .
Proposition 2.1 Let B be a set of inference rules (defaults without justi cations).
1
2 Introduction to Default Logic
Default logic is a knowledge representation mechanism allowing for reasoning in the presence of incomplete information. It handles the logical aspects of modalities such as \normally", \usually", etc. Syntactically, default logic extends the rst order logic (we will be treating propositional case almost exclusively) by introducing new entities called default rules or, simply, defaults. A default rule is a construct of the form : r = ' : M 1 ;# : : ; M m where '; 1; : : : ; k ; # are formulas of the language. The formula ' is called the premise or prerequisite of r and is denoted by p(r). The set f 1 ; : : : ; k g is called the set of justi cation of r and is denoted by j (r). The formula # is called the conclusion or consequent of r and is denoted c(r). Justi cations are used in default logic to explicitly represent conditions blocking applicability of defaults. That is, application of a rule of proof is quali ed by the absence of explicit information that would implying inconsistency of one of the justi cations of the rule. Put in yet another way, a default is applicable if its premise has been already established and all its justi cations are consistent, that is, their negations are not provable. It is precisely that presence of justi cations that allows us to model modalities such as \normally" and \usually" within default logic. In our format, a default rule has just one premise. This is an immaterial restriction since we will be assuming the usual rules of logic anyway. Default logic deals with default theories, that is, pairs (D; W ), where D is a collection of defaults and W is a collection of formulas. Default logic subsumes standard proof systems. It turns out that usual inference rules of the form ' # can simply be considered as defaults with empty set of justi cations. All major approaches to semantics of default logic are based on the natural semantics for a proof system in which propositional logic (PC) is extended by a collection of such standard inference rules (say B ). We will denote such systems by PC + B . We will now describe some properties of systems PC + B . A set of formulas T is closed under an inference rule r if the fact that ' 2 T implies that # 2 T . Similarly, T is closed under a set of rules B is it is closed under all rules in B . A set of formulas T is closed under inference in PC + B if T is closed under B and under propositional provability. We have the following fact. 2
1 Introduction
Nonmonotonic logics were introduced in the late 70s as knowledge representation formalisms. Default logic Rei80], circumscription McC80], autoepistemic logic Moo85, Lev90] and logic programming with negation ABW88, GL88, Apt90] turned out over the years to be most widely studied and most in uential in the development of the area. To serve as a knowledge representation tool, a formal system must o er an expressive language with well-understood semantics, and a computational mechanism supporting e ective reasoning. In this tutorial we will present theoretical foundations for default logic and logic programming. We will emphasize results that have bearing on algorithms, computational complexity and implementations. We will discuss methodology of programming with nonmonotonic knowledge representation systems and demonstrate an implementation of default logic, Default Reasoning System (or DeReS), developed in the University of Kentucky. This reader contains all necessary de nitions, key results, some proofs and an exhaustive list of references.
Nonmonotonic rห้องสมุดไป่ตู้asoning | computational perspective
Lecture notes for ESSLLI-97
fmarek,mirekg@
Victor W. Marek Miroslaw Truszczynski Department of Computer Science University of Kentucky Lexington, KY 40506-0027
Input: a nite collection of inference rules B and a nite collection of formulas W Returns: a nite basis for CnB (W )
dl basis(B; W )
Q := ;; (will be used as a list of rules) S := ;; (will represent a basis) for every rule r 2 B do if the premise of r is provable from W , add r to Q; while Q 6= ; do
We will denote the unique least set closed under PC + B by CnB (W ). Since the set CnB (W ) is closed under consequence, it contains all tautologies and, consequently, it is in nite. Yet, when W is nite we can nd a nite basis for CnB (W ). Given a theory S , any set U of formulas such that Cn(U ) = S is called a basis for S . An algorithm to nd a nite basis for CnB (W ), in the case when both W and B are nite, is given in Figure 1. This algorithm requires O(n2) calls to propositional provability procedure. We are now ready to analyze rules of the form ' : M 1; : : : ; M m : # Let us look at a simple example of a rule of the form: p : Mq : r 3