1. CS344: Artificial Intelligence Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture 20,21: Application of Predicate Calculus

2. Application of Predicate Calculus Systematic Inferencing Knowledge Representation Puzzles -- Circuit Verification -- Robotics -- Intelligent DB

3. Circuit Verification Does the circuit meet the specs? Are there faults? are they locatable?

4. Example : 2-bit full adder C1X2X1YC2 00000 00110 01010 01101 10010 10101 11001 11111 X 1, X 2 : inputs; C 1 : prev. carry; C 2 : next carry; Y: output

5. K-Map 0101 1010 0001 11 10 x2x1 c1 Y 0 1

6. K-Map (contd..) 0010 0111 0001 11 10 x2x1 c1 0 1 C2

7. Circuit

8. Verification First task (most difficult)  Building blocks : predicates  Circuit observation : Assertion on terminals

9. Predicates & Functions Function–1signal(t)t is a terminal ; signal takes the value 0 or 1 Function–2type(x)x is a circuit element; type(x) takes the value AND, OR, NOT, XOR Predicate – 3connected(t1,t2)t1 is an output terminal and t2 is an input terminal Function-3In(n,x)n th input of ckt element x Function-4Out(x)Output of ckt element x

10. Alternate Full Adder Circuit

11. Functions type(X) : takes values AND, OR NOT and XOR, where X is a gate. in(n, X) : the value of signal at the n th input of gate X. out(X) : output of gate X. signal(t) : state at terminal t = 1/0 Predicates connected(t1,t2): true, if terminal t1 and t2 are connected

12. General Properties Commutativity: ∀ t 1,t 2 [connected(t 1,t 2 ) → connected(t 2,t 1 )] By definition of connection: ∀ t 1,t 2 [connected(t 1,t 2 ) → { signal(t 1 ) = signal(t 1 )}]

13. Gate properties 1. OR definition: 2. AND definition:

14. Gate properties contd… 1. XOR definition: 2. NOT definition:

15. Some necessary functions a. no_of_input(x), takes values from N. b. Count_ls(x), returns #ls in the input of X

16. Circuit specific properties Connectivity: connected(x 1, in(1,A 1 )), connected(x 1, in(2, A 1 )), connected(out(A 1 ), in(1, A 2 )), connected(c1, in(2, A 2 )), connected(y, out(A 2 )) … Circuit elements: type(A 1 ) = XOR, type(A 2 ) = XOR, type(A 3 ) = AND …