Formal Program Specification: Logic, Sets, Relations, Functions, and Predicate Calculus - , Study notes of Engineering

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Formal Program Specification

Prepared by Stephen M. Thebaut, Ph.D.

University of Florida

Software Testing and Verification

Lecture 16

Overview •^

Review of Basics –^ Propositions, propositional logic,predicates, predicate calculus

-^ Sets, Relations, and Functions -^ Specification via pre- and post-conditions -^ Specifications via functions

Propositions and Propositional Logic(cont’d) • Propositional Logic

is a formal language

that allows us to reason aboutpropositions. The alphabet of thislanguage is:

{P, Q, R, …

, V,

where P, Q, R, … are propositions, andthe other symbols, usually referred to as^ connectives

, provide ways in which

compound propositions can be built fromsimpler ones.

Truth Tables • Truth tables

provide a concise way of

giving the meaning of compound formsin a tabular form.Example: construct a truth table to showall possible interpretation for thefollowing sentences:

A V B, A

⇒^

B, and A

⇔^

B

Equivalence^ •^ Two sentences are said to be

equivalent

if and only if their truth values are thesame under every interpretation. • If A is equivalent to B, we write

A^ ≡

B.

Exercise: Use a truth table to show:

(P^ ⇒

Q)

≡^ (Q V

¬P)

Equivalence (cont’d) •^ Many users of logic slip into the habit ofusing

⇔^

and

≡^ interchangeably.

-^ However,

A⇔

B^ is written down in the full

knowledge that it may denote either true

or^

false

in some interpretation,

whereas

A≡

B is an expression of fact

(i.e., the writer thinks it is true).

Predicates (cont’d) •^ Note that a predicate itself has

no

truth

value; it expresses a property or relationusing

variables.

Predicates (cont’d) •^ There are TWO ways in which predicatescan give rise to propositions:

-^ As illustrated above, their freevariables may be

instantiated

with the

names of specific objects, and – They may be

quantified

Quantification introduces twoadditional symbols:

∀^ and

Predicates (cont’d) •^ For a predicate with two free variables,quantifying over one of them yieldsanother predicate with one freevariable, as in

∀x • Q(x,y)

or

∃x • Q(x,y)

Predicates (cont’d)^ •^ Where appropriate, a

domain of interest

may be specified which contains the objectsfor which the quantifier applies. Forexample,

∀i∈

{1,2,…,N} • A[i]>

represents the predicate “the first Nelements of array A are all greater than 0.”

Predicate Calculus (cont’d) •^ A

rule of inference

is expressed in the

form:

A, A^1

, …, A 2

n C

and is interpreted to mean

(A^1

A

2

…^

A)n

⇒^

C

Predicate Calculus (cont’d) •^ Examples of deductive rules:

A^

B A

A A V B

A A

A, A

⇒^ B B

Sets and Relations •^ A

set

is any well-defined collection of objects, called members or elements. • The relation of

membership

between a

member, m, and a set, S, is written:

m^ ∈

S

-^ If m is not a member of S, we write:

m^ ∉

S

Sets and Relations (cont’d) •^ A

relation

, r, is a set whose members (if

any) are all ordered pairs. • The set composed of the first member ofeach pair is called the

domain

of r and is

denoted D(r). Members of D(r) arecalled

arguments

of r.

-^ The set composed of the secondmember of each pair is called the rangeof r and is denoted R(r). Members ofR(r) are called

values

of r.