EE1J2 – Discrete Maths Lecture 5 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)
Truth tables So far we have seen how to build a truth table T for a given formula f in propositional logic Today we’ll look at the opposite problem: Given a set of atomic propositions p1,…,pN and a truth table T, can we construct a formula f such that T is the truth table for f ?
Adequacy A set of propositional connectives is adequate if For any set of atomic propositions p1,…,pN and For any truth table for these propositions, There is a formula involving only the given connectives, which has the given truth table.
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
Some more definitions… f1, f2,…,fn a set of n formulae f1 f2 … fn is called the disjunction of f1, f2,…,fn f1 f2 … fn is 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
Disjunctive Normal Form A formula is in Disjunctive Normal Form (DNF) if it is a disjunction of conjunctions of literals. Examples: (p, q, r and s atomic propositions) p q (p) (q) (p q) (p r s) …
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) ….
Truth Functions A truth function is 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}.
Truth functions p, q and r atomic propositions Example truth function in {p, q} p q T F 22 rows
Truth functions p q r T F Example truth function in 3 atomic propositions {p, q, r} 23 rows
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 Corollary: Any formula is logically equivalent to a formula in disjunctive normal form Corollary: {, ,} is an adequate set of connectives
Proof of theorem Let p1, 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
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 fr takes the truth value T for the rth row of the truth table and F for all other rows.
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
Corollary 1 Any formula is logically equivalent to a formula in disjunctive normal form Any formula g defines a truth table By the above theorem there is a formula f in disjunctive normal form which has the same truth table as g Hence f is logically equivalent to g
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 .
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 g is logically equivalent to (f g) f g is logically equivalent to f g
Corollary 4 {, } and {, } are both adequate sets of connectives Proof – homework
The symbol The symbol means logical equivalence Next look at some standard equivalences using the set {, ,}
Standard equivalences f g g f, f g g f Commutativity (b) (f g) h f (g h) (f g) h f (g h) Associativity (c) f (g h) (f g) (f h) f (g h) (f g) (f h) Distributivity (d) (f g) (f) (g) (f g) (f) (g) De Morgan’s Laws (e) f f Rule of double negation (f) f f is a tautology f f is a contradiction
Theorem 2 Let be a truth function. Then there is a formula in Conjunctive Normal Form (CNF) whose truth table is given by
DNF - Example Let p, q and 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 theorem.
Truth table for f (p (q r)) ((p q) r) T F
Example (continued) From row 1: (pq r) From row 2: (pq r) Hence the desired formula is: (pq r)(pq r)(pqr)(pqr) (pqr)(pqr)
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
nand and nor gates Most common gates are nand gates and nor gates. Their truth tables are given by Truth tables for nand and nor p q p nand q p nor q T F
Theorem 3 Adequacy of nand and nor Theorem: The sets {nand} and {nor} are both adequate Proof {nand}: Since {, } is adequate, 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)
Proof (continued) For {nor}: It is enough to notice that: p p nor p p q (p nor p) nor (q nor q)
Summary of Lecture 5 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)