Propositional Logic-Software Engineering-Lecture 10 Slides-Computer Science, Slides of Software Engineering

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LECTURE 10: FIRST-ORDER LOGIC

Software Engineering

1 Why not Propositional Logic?

• Consider the following statements: - all monitors are ready; - X12 is a monitor.
• We saw in an earlier lecture that these statements are propositions: their meaning is either true or false.
• Propositional logic is the most abstract level at which we can study logic.
• As we shall say, it is too coarse grained to allow us to represent and reason about the kind of statement we need to write in formal specification.

2 First-Order Logic: Syntax

• We shall now introduce a generalisation of propositional logic called first-order logic (FOL). This new logic affords us much greater expressive power.
• First, we shall look at how the language of first-order logic is put together.

2.1 Terms

• The basic components of FOL are called terms.
• Essentially, a term is an object that denotes some object other than true or false.
• The simplest kind of term is a constant.
• A value such as 8 is a constant; the denotation of this term is the number 8 — a value that is contained in the sets IN and Z.
• We often use constants in maths; we introduce them by writing things like

Let S be the set { 1 , 2 , 3 }.

In this case, we have introduced a constant and made its denotation clear; we have given it an interpretation.

• We can have constants that stand for any kind of object; for example, we could have a constant that stood for (denoted) the individual ‘Michael Wooldridge’.

• We can now introduce a more complex class of terms — functions.
• The idea of functional terms in logic is similar to the idea of a function in programming: recall that in programming, a function is a procedure that takes some arguments, and returns a value. In Modula-2:

PROCEDURE f(a1:T1; ... an:Tn) : T;

this function takes n arguments; the first is of type T1, the second is of type T2, and so on. The function returns a value of type T.

• In FOL, we have a set of function symbols; each symbol corresponds to a particular function. (It denotes some function.)
• Each function symbol is associated with a natural number called its arity. This is just the number of arguments it takes.

• Each function symbol has a return-type associated with it... -... and each function symbol has an argument type associated with it.
• A functional term is then built up by applying a function symbol to the appropriate number of terms, of the appropriate type.
• Formally...

Definition: Let f be an arbitrary function symbol of type T, with arity n ∈ IN, taking arguments of type T 1 ,... , Tn respectively. Also, let τ 1 ,... , τn be terms of type T 1 ,... , Tn respectively. Then f (τ 1 ,... , τn) is a functional term.

• In maths, we have many functions; the obvious ones are + − / ∗ √^ sin cos...
• The fact that we write

2 + 3 instead of something like plus(2, 3) is merely a matter of convention, and is not relevant from the point of view of logic; all these are functions in exactly the way we have defined.

• Using functions, constants, and variables, we can build up expressions, e.g.: (x + 3) ∗ sin 90 (which might just as well be written times(plus(x, 3), sin(90)) for all it matters.)

2.2 Predicates

• In addition to having terms, FOL has relational operators, which capture relationships between objects.
• The language of FOL contains a stock of predicate symbols.
• These symbols stand for relationships between objects.
• Again, each predicate symbol has an associated arity... -... and each argument has a type.
• Definition: Let P be a predicate symbol of arity n ∈ IN, which takes arguments of types T 1 ,... , Tn. Then if τ 1 ,... , τn are terms of type T 1 ,... , Tn respectively, then P(τ 1 ,... , τn) is a predicate, which will either be true or false under some interpretation.

• So a predicate just expresses a relationship between some values.
• What happens if a predicate contains variables: can we tell if it is true or false? Not usually; we need to know an interpretation for the variables.
• A predicate that contains no variables is a proposition.
• Predicates of arity 1 are called properties.
• EXAMPLE. The follolwing are properties:

Man(x) Mortal(x) Malfunctioning(x).

• Predicate that have arity 0 (i.e., take no arguments) are called primitive propositions.

3 Quantifiers

• We now come to the central part of first order logic: quantification.
• Consider trying to represent the following statements: - all men have a mother; - every natural number has a prime factor.
• We can’t represent these using the apparatus we’ve got so far; we need quantifiers.

• The simplest form of quantified formula in Z is as follows: quantifier signature • predicate where - quantifier is one of ∀, ∃, ∃ 1 ; - signature is of the form variable : type - and predicate is a predicate.

• EXAMPLES.

- ∀x : Man • Mortal(x) ‘For all x of type Man, x is mortal.’ (i.e. all men are mortal) - ∀x : Man • ∃ 1 y : Woman • MotherOf (x, y) ‘For all x of type Man, there exists a unique y of type Woman, such that y is the mother of x.’ - ∃m : Monitor • MonitorState(m, ready) ‘There exists a monitor that is in a ready state.’ - ∀r : Reactor • ∃ 1 t : 100.. 1000 • Temp(r) = t ‘Every reactor will have a temperature in the range 100 to 1000.’

4 Comments

• Note that universal quantification is similar to conjunction: ∀n : { 2 , 4 , 6 } • Even(n) is the same as Even(2) ∧ Even(4) ∧ Even(6).
• In the same way, existential quantification is the same as disjunction: ∃n : { 7 , 8 , 9 } • Prime(n) is the same as Prime(7) ∨ Prime(8) ∨ Prime(9).

• The universal and existential quantifiers are in fact duals of each other: ∀x : T • P(x) ⇔ ¬∃x : T • ¬P(x) Saying that everything has some property is the same as saying that there is nothing that does not have the property. ∃x : T • P(x) ⇔ ¬∀x : T • ¬P(x) Saying that there is something that has the property is the same as saying that its not the case that everything doesn’t have the property.