EE1J2 – Discrete Maths Lecture 6

1. EE1J2 – Discrete Maths Lecture 6 • Adequacy of a set of connectives • Disjunctive and conjunctive normal form • Adequacy of {, , , }, {, , }, {, } and {, } • Every formula is logically equivalent to one in conjunctive normal form (or disjunctive normal form)

2. Truth tables • So far we have seen how to build a truth table Tfor a given formula fin propositional logic • Today we’ll look at the opposite problem: Given a set of atomic propositions p1,…,pNand a truth table T, can we construct a formula f such that T is the truth table for f?

3. Adequacy • A set of propositional connectives is adequate if • For any set of atomic propositions p1,…,pNand • For any truth table for these propositions, • There is a formula involving only the given connectives, which has the given truth table.

4. Adequacy • The goal of today’s lecture is to show that the set {, , , } is adequate and contains redundancy, in the sense that it contains subsets which are themselves adequate • We shall also introduce other sets of adequate connectives

5. Some more definitions… • f1, f2,…,fn a set of n formulae • f1 f2… fnis called the disjunction of f1, f2,…,fn • f1 f2… fnis called the conjunction of f1, f2,…,fn • Let p be an atomic proposition. A formula of the form p or p is called a literal

6. Disjunctive Normal Form • A formula is in Disjunctive Normal Form(DNF) if it is a disjunction of conjunctions of literals. • Examples: (p, q, rand s atomic propositions) p  q (p)  (q) (p  q) (p  r  s) …

7. Conjunctive Normal Form • A formula is in Conjunctive Normal Form (CNF) if it is a conjunction of disjunctions of literals • Examples: p  q (p)  (q) (p  q)  (p) ….

8. Truth Functions • A truth functionis a function  which assigns to a set of atomic propositions {p1,…,pN} a truth table (p1,…,pN) in which one of the truth values T or F is assigned to each possible assignment of truth values to the atomic propositions {p1,…,pN}.

9. p q T T F T F T F T T F F F Truth functions • p, q and r atomic propositions • Example truth function in {p, q} 22 rows

10. p q r T T T T T T F F T F T T T F F T F T T F F T F T F F T F F F F F Truth functions • Example truth function in 3 atomic propositions {p, q, r} 23 rows

11. First Theorem (Disjunctive Normal Form) • Theorem: Let  be a truth function. Then there is a formula in disjunctive normal form whose truth table is given by 

12. The symbol  • Recall that the symbol  means logical equivalence • Two formulae are logically equivalent if they have the same truth table

13. Corollaries to First Theorem • Corollary 1: Any formula is logically equivalent to a formula in disjunctive normal form • To see this: • Let f be a formula • Construct the truth table for f • By the First Theorem there is a formula g in DNF which has this truth table • Then fg by definition

14. Corollaries to First Theorem • Corollary 2: {, ,} is an adequate set of connectives • To see this, note: • Given any truth table we can construct a formula f in DNF which has that truth table • By definition of DNF, f only involves the connectives , ,

15. Proof of First Theorem • Letp1, p2,…,pn be the atomic propositions • Want a formula  in disjunctive normal form whose truth table is given by  • If  assigns the value F to every row of the truth table, just choose  =  • Otherwise, there will be at least one row for which the truth value is T. Let that row be row r

16. Proof (continued) • let be the formula defined by: • Let fr be the conjunction f(r)1 f(r)2f(r)3…f(r)n • frtakes the truth value Tfor the rth row of the truth table and F for all other rows.

17. Proof (continued) • Suppose that there are R rows r1,…,rR for which the truth value is T. • Define = • Clearly  is in disjunctive normal form • By construction  has the truth table defined by 

18. DNF - Example Let p, qand r be atomic propositions Consider f = (p(q  r))  ((p  q)  r) How do we put this in disjunctive normal form? Use the construction from the proof of the First Theorem (DNF) from lecture 5.

19. (p  (q  r))  ((p  q)  r) T T T T T T T T T T T T F T F F T T T T F F T T F T T T T F F T T T T F T F T T F F T F F T T T T T F T T T T F T T F F F F T T F F F T F T T T F T F T T F T F T F F F T F F F Truth table for f

20. Example (continued) • First identify the rows in the truth table for which f is true • Then use these rows to construct the components of the DNF version of f: • From row 1: (p  q  r) • From row 2: (p  q  r) • From row 3: (p  q  r) • From row 4: (p  q  r) • From row 5: (p  q  r) • From row 7: (p  q  r)

21. Example (continued) • Now combine these using ‘or’ symbols to obtain the desired formula in DNF: • (pq r)  (pqr)  (pqr)  (pqr)  (pqr)  (pqr)

22. Corollary 1 • Any formula is logically equivalent to a formula in disjunctive normal form • Any formula gdefines a truth table • By the above theorem there is a formula f in disjunctive normal form which has the same truth table as g • Hencef is logically equivalent to g

23. Corollary 2 • {, ,} is an adequate set of connectives • From the theorem, any truth table can be satisfied by a formula in disjunctive normal form. • By definition, such a formula only employs the connectives ,  and .

24. Corollary 3 • {, } is an adequate set of connectives • Enough to show that  and  can both be expressed in terms of the symbols  and . • To see this, note that if f and g are formulae in propositional logic: f  gis logically equivalent to (f g) f  g is logically equivalent to f  g

25. Corollary 4 • {, } and {, } are both adequate sets of connectives • Proof – homework

26. Theorem 2 • Let  be a truth function. Then there is a formula in Conjunctive Normal Form (CNF) whose truth table is given by 

27. Switching Circuits • Connections between propositional logic and switching circuits • Can think of a truth table as indicating the ‘output’ of a particular circuit once its inputs have been set to ‘On’ or ‘Off’ • Now know that any desired behaviour can be obtained provided that the gates of the circuit can instantiate the connectives ,  and 

28. Truth tables for nand and nor p q p nand q pnorq T T F F T F T F F T T F F F T T nand and nor gates • Most common gates are nand gates and nor gates. Their truth tables are given by

29. Theorem 3Adequacy of nand and nor • Theorem: The sets {nand} and {nor} are both adequate • Proof {nand}: Since {, } is adequate, it is enough to show that  and  can be expressed in terms of nand. Let p and q be atomic propositions. Then: pp nand p and p  q (p nand q) nand (p nand q)

30. Proof (continued) • For {nor}: It is enough to notice that: pp nor p p  q (p nor p) nor (q nor q)

31. Summary of Lecture 6 Adequacy of a set of connectives defined • Disjunctive and conjunctive normal form defined • Adequacy of {, , , }, {, , }, {, }, {, }, {nand} and {nor} • Every formula is logically equivalent to one in disjunctive normal form (DNF)