Chapter 7 Reasoning about Knowledge

1. Chapter 7Reasoning about Knowledge by Neha Saxena Id: 13 CS 267

2. 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

3. 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

4. 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 ( ≡ )

5. 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 |= 

6. 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

7. 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)

8. 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

9. 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

10. 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

11. 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)

12. Example Consider the following KR-system

13. Deduction in Decision Logic (cont.) • Suitable axioms and inferences rules are needed to prove the equivalence of formulas in a formal way • (a,v)  (a,u) ≡ 0 for any a  A v,u  V and v ≠ u • (a,v) ≡ 1, for every a  A • ~(a,v) ≡ (a,u), for every a  A • Preposition • |=s ∑(P) ≡ 1, for any P  A

14. 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

15. 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

16. 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 |- 

17. 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

18. 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

19. 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

20. 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

21. Example Example • Consider the following KR-system

22. 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’