EE1J2 - Slide 1 EE1J2 – Discrete Maths Lecture 5 Adequacy of a set of connectives Disjunctive and...
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Transcript of EE1J2 - Slide 1 EE1J2 – Discrete Maths Lecture 5 Adequacy of a set of connectives Disjunctive and...
EE1J2 - Slide 1
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)
EE1J2 - Slide 2
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 ?
EE1J2 - Slide 3
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.
EE1J2 - Slide 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
EE1J2 - Slide 5
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
EE1J2 - Slide 6
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)
…
EE1J2 - Slide 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)….
EE1J2 - Slide 8
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}.
EE1J2 - Slide 9
Truth functions p, q and r atomic propositions Example truth function in {p, q}
p q
T T F
T F T
F T T
F F F
22 rows
EE1J2 - Slide 10
Truth functions
Example truth function in 3 atomic propositions {p, q, r}
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
23 rows
EE1J2 - Slide 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
Corollary: Any formula is logically equivalent to a formula in disjunctive normal form
Corollary: {, ,} is an adequate set of connectives
EE1J2 - Slide 12
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
EE1J2 - Slide 13
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.
by rowin F value theassigned is if
by rowin T value theassigned is if )(
rpp
rpprf
ii
iii
irf
EE1J2 - Slide 14
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
Rrrr fff ...21
EE1J2 - Slide 15
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
EE1J2 - Slide 16
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 .
EE1J2 - Slide 17
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
EE1J2 - Slide 18
Corollary 4 {, } and {, } are both adequate sets of
connectives Proof – homework
EE1J2 - Slide 19
The symbol The symbol means logical equivalence Next look at some standard equivalences
using the set {, ,}
EE1J2 - Slide 20
Standard equivalences
(a) 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 tautologyf f is a contradiction
EE1J2 - Slide 21
Theorem 2
Let be a truth function. Then there is a formula in Conjunctive Normal Form (CNF) whose truth table is given by
EE1J2 - Slide 22
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.
EE1J2 - Slide 23
Truth table for f(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
EE1J2 - Slide 24
Example (continued) From row 1: (pq r) From row 2: (pq r) From row 3: (pqr) From row 4: (pqr) From row 5: (pqr) From row 7: (pqr) Hence the desired formula is:(pq r)(pq r)(pqr)(pqr)
(pqr)(pqr)
EE1J2 - Slide 25
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
EE1J2 - Slide 26
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 T F F
T F T F
F T T F
F F T T
EE1J2 - Slide 27
Theorem 3Adequacy 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)
EE1J2 - Slide 28
Proof (continued) For {nor}: It is enough to notice that:
p p nor p
p q (p nor p) nor (q nor q)
EE1J2 - Slide 29
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)