Chapter 7Reasoning about Knowledge

by

Neha Saxena

Id: 13

CS 267

Topics Covered

Introduction Language of Decision logic Semantics of Decision Logic Language Deduction in Decision Logic Normal Forms Decision Rules and Decision Algorithms Truth and Indiscernibility

Introduction

Knowledge is represented as value-attribute table, called Knowledge Representation System (KR-system)

The data table is viewed as a model for decision logic

It is used to derive conclusions from data available in KR-system

Fundamental notion of decision logic is decision algorithm: a set of decision rules

Language of Decision Logic

Alphabets of the language are A – set of attribute constants V = - set of attribute value constants Set {~, , , , } of propositional connectives

The set of formulas of DL-language is the least set satisfying following conditions

Expressions of the form (a,v), in short , called elementary (atomic) formulas, are formulas for DL-language for any a A and v Va

If and are formulas of DL –language, then so are ~ ,( ), ( ), ( → ) and ( ≡ )

Semantics of DL-Language

Concept of satisfiability of a formula by an object: An object x U satisfies a formula in S = (U,A),

denoted x |=s or x |= , if S is understood, if and only if the following conditions are satisfied: x |= (a,v) iff a(x) = v x |= ~ iff non x |= x |= Ψ iff x |= or x |= x |= Ψ iff x |= and x |=

Semantics of DL-Language (cont.)

As a corollary from the above conditions we get x |= → iff x |= ~ x |= ≡ iff x |= → and x |= →

If is a formula then the set ||s defined as

||s = {x U : x |=s }

will be called the meaning of the formula in S

Semantics of DL-language (cont.)

Following proposition explains meaning of an arbitrary formula |(a,v)|s = {x U : a(x) = v} |~|s = -||s | |s = ||s ||s | |s = ||s ||s | → |s = -||s ||s | ≡ |s = (||s ||s) (-||s -||s)

Semantics of DL-language (cont.)

Notion of truth A formula is said to be true in KR-system S, |

=s , iff ||s = U

Formulas and are equivalent in S iff ||s = ||s

Semantics of DL-language (cont.)

Following proposition give simple properties of the introduced notions |=s iff ||s = U |=s~ iff ||s = 0 (empty set) |=s → iff ||s ||s |=s ≡ iff ||s = ||s

Deduction in Decision Logic

Formula of the form where , P = {a1,a2,.. an} and P A, will be called P-basic formula, in short P-formula.

A-basic formulas will be called basic formulas Let P A, be a P-formula and x U

If x |= , then is called P-description of x in S Set of all basic formulas satisfiable in KR-system S =

(U,A) will be called basic knowledge in S

Deduction of Decision Logic (cont.)

Formula ∑(P) is disjunction of all P-formulas satisfied in S

If P = A then ∑(A) will be called characteristic formula of KR-system S = (U,A)

Example

Consider the following KR-systemU a b c

1 1 0 2

2 2 0 3

3 1 1 1

4 1 1 1

5 2 1 3

6 1 0 3

Deduction in Decision Logic (cont.)

Suitable axioms and inferences rules are needed to prove the equivalence of formulas in a formal way

1. (a,v) (a,u) ≡ 0 for any a A v,u V and v ≠ u

2. (a,v) ≡ 1, for every a A

3. ~(a,v) ≡ (a,u), for every a A

Preposition |=s ∑(P) ≡ 1, for any P A

Deductions of Decision Logic (cont.)

Basic concepts Formula Φ is derivable from a set of formulas Ω,

denoted Ω |- Φ, iff it is derivable from the axioms and formulas of Ω, by finite application of the inference rule (modus ponens)

Formula Φ is a theorem of DL, |- Ω, if it derivable from the axioms only

A set of formulas Ω is consistent iff formula Φ ~Φ is not derivable from Ω

Normal Forms

Formulas in KR-system can be presented in normal form

Let P A be subset of attributes and let be a formula in KR-language. Then is in P-normal form in S, iff either = 0 or 1 or is a disjunction of non empty P-basic formulas in S

A-normal form is referred as normal form

Normal Forms (cont.)

An important property of formulas in DL-language is Let be a formula in DL-language and let P

contain all the attributes occurring in . Also assume axioms 1 – 3 and the formula s(A). Then, there is a formula in the P–normal form such that |-

Decision Rules and Decision Algorithms

Decision Rules Any implication will be called a decision rule in KR-system; and

are referred to as the predecessor and successor of respectively If a decision rule is true in S, we say that it is consistent; otherwise it is

inconsistent in S If is a decision rule and and are P-basic and Q-basic formulas,

then the decision rule will be called PQ-basic decision rule, in short PQ-rule, or basic rule when PQ is known

If 1 , 2 ,…, n are basic decision rules then the decision rule 1 2 … n will be called combination of basic decision rules , in short combined decision rule

A PQ-rule is admissible in S if is satisfiable in S

Decision Rules and Decision Algorithm (cont.)

The following preposition can be used to find if a PQ-rule is true or not

A PQ-rule is consistent in S, iff all {P Q}-basic formulas which occur in the {P Q}normal form of the predecessor of the rule also occur in the {P Q}-normal form of the successor of the rule; otherwise the rule is false (inconsistent) in S

Decision Rules and Decision Algorithm (cont.)

Decision Algorithms Any finite set of decision rules in a DL-language, is referred to as a

basic decision algorithm If all decision rules are PQ-decision rules, then the algorithm is PQ-

decision algorithm, in shot PQ-algorithm, and denoted as (P,Q) PQ-algorithm is admissible in S, if it the set of all the PQ-rules

admissible in S A PQ-algorithm is complete in S, if for every x U there exists a

PQ-decision rule in the algorithm such that x |= in S; otherwise the algorithm is incomplete in S

PQ-algorithm is consistent in S, iff all its decision rules are consistent (true) in S; otherwise the algorithm is inconsistent in S

Decision Algorithm (cont.)

Given a KR-system, any two non empty subsets of attributes P, Q determine uniquely a PQ-algorithm – and a decision table with P and Q as conditions and decision attributes respectively

Hence PQ-algorithm and PQ-decision table may be considered equivalent concepts

Example

Consider the following KR-system

Example

U a b c d e

1 1 0 2 1 1

2 2 1 0 1 0

3 2 1 2 0 2

4 1 2 2 1 1

5 1 2 0 0 2

Truth and Indiscernibility

To check if a decision algorithm is consistent we need to check if all its decision rules are true

The preposition given in previous slide does this, but the following proposition gives a simple method to do the same

A PQ-decision rule Φ→Ψ in a PQ-decision algorithm is consistent (true) in S, iff for any PQ-decision rule Φ’→Ψ’ in PQ-decision algorithm, Φ = Φ’ implies Ψ = Ψ’