1INM175 Topic 8

Module INM175Discrete Mathematics

Topic 8

Algebraic Theories

2INM175 Topic 8

Natural Numbers RevisitedThe operators

zero: N a nullary or constant operator

and succ: N N a unary operator

are constructors of N , that is, every value of N may be denoted by a term involving only these operators. So the following axioms hold for N

1. zero N

2. x:Ny:Ny=succ x

3INM175 Topic 8

Peano’s Axioms

Now, let us add the following axioms:

3. x:Nsucc x zero

4. x,y:Nsucc x = succ y x=y

5. if, for any property,Q, Q(zero) (x:NQ(x) Q(succ x),

then x:NQ(x).

Axiom 5 is the Principle of Mathematical Induction.

These five predicates are Peano’s axioms [Giuseppe Peano (1858-1932)]

They define certain properties of N, but do not refer to any representation of it (except that it contain a distinguished element called ‘zero’).

4INM175 Topic 8

Theories and ModelsThis form of definition is called a theory, whose subject is a sort, in this case, N.

A model of a theory has some set of values, for its sort, and some functions, for its operators, that satisfy its axioms.

The set NAT = {0,1,2, ...} is a model of N.

The term algebra of a theory consists of the language (i.e. the set of strings of symbols) generated by considering the theory’s signature as a grammar.

The term algebra of N is the set {‘zero’, ‘succ zero’, ‘succ succ zero’, ...} where zero is modelled by the string ‘zero’ and

succ by the function that appends the ‘succ‘ to the front of its argument.

The constructors of a theory are the operators (including the constants) that together can generate all the terms.

5INM175 Topic 8

Theory Extensions

When we define a new binary operator on N

plus: N, N Nwe extend the term algebra with an infinite number of new terms such as

‘plus(zero,zero)’, ‘plus(succ zero,zero)’, ‘succ plus (zero,zero)’,

But some of these terms denote the same value of N, so we must add the following (recursive)

Axioms

plus (zero, n) = n

plus ((succ m), n) = succ (plus (m,n))

6INM175 Topic 8

Calculation

Plus is defined in terms of itself, but the recursion always terminates.

Let 1=‘succ zero’, 2=‘succ succ zero’, 3=‘succ succ succ zero’

and use the axioms to compute 1+2

plus(succ zero,succ succ zero)

= succ(plus(zero,succ succ zero) by axiom 2

[Axiom 6 is inapplicable here because, by axiom 1, succ zero zero]

= succ succ succ zero by axiom 1

[Axiom 7 is inapplicable here because, by axiom 1, succ zero zero]

Since the final expression does not match the left-hand-side of any of our axioms, our computation stops, and the result denotes the value 3.

7INM175 Topic 8

Quotient Algebra

and so on ...

zero plus(zero,zero)

plus(zero, plus(zero,zero)) and so on ...

succ(zero) plus(zero, succ(zero))

plus(succ(zero),zero))and so on ...

The set of all termspartitioned by theaxioms

Every term in the original term algebra is in one and only one equivalence class.

8INM175 Topic 8

Conservative Extension

Our extension is conservative because it introduces:

no junk, because every term involving plus is provably equivalent, under the axioms, to one involving only zero and succ, so in every model which satisfies the axioms, the result of adding any two values of the type N must be in N; and

no confusion, because no pair of distinct terms (that is, terms which are not defined to be equivalent under the original axioms) are equated by the extension.

9INM175 Topic 8

Algebraic Proof Rules

Whenever an abstract data type is extended, the extension must be proved to be conservative.

Testing is not adequate because these are infinite structures.

The proof rules involved are equational reasoning and structural induction.

These were just the rules that we used in our proofs using the Principle of Mathematical Induction.

10INM175 Topic 8

Multiplication in N

Multiplication () in N is an extension of the theory N with plus:

Signature mult: N, N N

Axioms m,n:Nmult (zero, n) = zero

mult ((succ m), n) = plus (n,mult(m,n))Note that this refers to plus which was defined earlier.

Conservative extension is transitive.

11INM175 Topic 8

Exponentiation in NThe exponentiation operator, mn extends N with plus and multiply

Signature exp: N, N NIf 0n = 0 and m0 =1 for all m and n,

then 00 = 0 and 00 = 1

so, by equational reasoning, 1 = 0 a clear case of confusion!

By convention, 00 is an illegal operation i.e. exp(zero,zero) is undefined

Axioms m,n:N

exp (zero,succ n) = zero

exp(succ m,zero) = succ zero

exp (succ m,succ n) = mult(succ m,exp(succ m,n))

12INM175 Topic 8

The Theory BOOL

The simplest theory, TRIV, has just one set in its quotient algebra, i.e. all terms in the theory are equal to each other. It is of no interest!

The Boolean logic that we studied in Topic 2 is a model of the theory whose quotient algebra has exactly two partitions so we call it

Sort BOOL

Signature T: BOOL

F: BOOL

This has two constants, T and F, which we modelled as true and false, respectively. We assert that these terms are not equal to each other simply by not providing an axiom that equates them.

13INM175 Topic 8

Extensions to BOOL

We extend BOOL by adding the usual operators:

Signatures not: BOOL BOOL

and: BOOL, BOOL BOOL

or: BOOL, BOOL BOOL

Axioms x:BOOLnot T = F

not F = T

and(T,x) = x

and(F,x) = F

or(T,x) = T

or(F,x) = x

14INM175 Topic 8

Combining N and BOOL

To define operators whose models are relations on the Natural Numbers, we combine the theories N and BOOL and extend them together.

For example, ‘less-than’ (modelled by the relation <) may be defined as a binary function from N to BOOL:

Sorts N, BOOL

Signature less-than: N, N BOOL

Axioms m,n:Nless-than(zero, zero) = F

less-than(zero, succ n) = T

less-than(succ m, zero) = F

less-that(succ m,succ n) = less-than(m,n)