1. Chapter 7 Reasoning 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 Uabc 1102 2203 3111 4111 5213 6103

13. 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

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  Consider the following KR-system Example Uabcde 110211 221010 321202 412211 512002

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’