This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: CSE541 Midterm 1 SOLUTIONS Fall 2010 L semantics for L { , , , } is defined as follows L Negation F T T F L Conjunction F T F F F F F T F T L Disjunction F T F F T T T T T T LImplication F T F T T T T T T F T QUESTION 1 (1) Use the fact that v : V AR { F, ,T } be such that v * (( a b ) b ) = under L semantics to evaluate v * ((( b a ) ( a b )) ( a b )). Use shorthand notation. (1) Solution : (( a b ) b ) = in two cases. C1 ( a b ) = and b = F . C2 ( a b ) = T and b = . Case C1: b = F , i.e. b = T , and hence ( a T ) = iff a = . We get that v is such that v ( a ) = and v ( b ) = T . We evaluate: v * ((( b a ) ( a b )) ( a b )) = ((( T ) ( T )) ( T )) = (( ) T ) = T . Case C2: b = , i.e. b = , and hence ( a ) = T what is impossible, hence v from case C1 is the only one. (2) Prove that in classical semantics L { , } L { , , } . 1 We define the EQUIVALENCE of LANGUAGES as follows: Given two languages: L 1 = L CON 1 and L 2 = L CON 2 , for CON 1 6 = CON 2 . We say that they are logically equivalent , i.e. L 1 L 2 if and only if the following conditions C1, C2 hold. C1: For every formula A of L 1 , there is a formula B of L 2 , such that A B, C2: For every formula C of L 2 , there is a formula D of L 1 , such that C D. (2) Solution: (Classical case) C1 holds because any formula of L { , } is a formula of L { , , } . C2 holds due to the following definability of connectives equivalence ( A B ) ( A B ) . (3) Prove that the equivalence defining in classical logic does not hold under L semantics, but nevertheless L { , } L { , , } ....
View
Full
Document
This note was uploaded on 02/12/2011 for the course CSE 541 taught by Professor Bachmair,l during the Spring '08 term at SUNY Stony Brook.
 Spring '08
 Bachmair,L
 Computer Science

Click to edit the document details