1. School of Computing and Mathematics, University of Huddersfield CIA2326: WEEK ?? LECTURE: Introduction to Algebras SUPPORTING NOTES: See chapters 8,9,10 of my ‘online book’ on my homepage TUTORIAL: hand out exercises “Telescopes are to Astronomy what Computers are to Computer Science” (Edgar Dijstra)

2. School of Computing and Mathematics, University of Huddersfield Algebras Q1. What is an algebra? Q2.What has it to do with computing?

3. School of Computing and Mathematics, University of Huddersfield Algebras A1. Roughly, an algebra is A SET OF VALUES + the specification of some OPERATIONS on those values A2. Roughly, a data type is A SET OF VALUES + the implementation of some OPERATIONS on those values. Hence we can give a computer-independent meaning to data types using algebras.

4. School of Computing and Mathematics, University of Huddersfield Abstraction in Programming Algebras inform us in both the THEORY and PRACTICE of programming: Good modularity of software is ensured to a large degree by procedural and data abstractions Object technologies owe much of their success to the fact that their structure results in a high degree of procedural and data abstraction. Data abstraction or “abstract data types” abstract away the implementation of data types’ values and operations - just like ALGEBRAS!

5. School of Computing and Mathematics, University of Huddersfield Abstract Data Types A very abstract way of defining data types is as follows: abstract away the implementation of data types’ values and operations in two ways: define the operations in terms of each other; don’t represent values AT ALL except in terms of the operations that construct them This give us an ‘abstract algebra’…….

6. School of Computing and Mathematics, University of Huddersfield Homogenous algebras: Formal Definition (A,O) is a Homogenous algebra if A is a non-empty set which contains values of the algebra and is known as the carrier set. O is a set of closed, total operations defined over the carrier set which may include nullary operations (constants). EXAMPLE: (Natural Numbers, {“+”}), EXAMPLE: ({True,False}, {and,not}) COUNTER-EX : (Natural Numbers, {“-”}) COUNTER-EX : (Real Numbers, {“/”})

7. School of Computing and Mathematics, University of Huddersfield heterogeneous algebras (A,O) is a heterogeneous algebra if A is a SET of carrier sets and O is a set of closed and total operations over these sets. The semantics of data types are often given by heterogeneous algebras as we shall see...

8. School of Computing and Mathematics, University of Huddersfield  Equational Specification of Algebras The most abstract form of “Presentation” (ie how to define them) is NOT TO GIVE ANY NAMES to their values. A very abstract way to specify a (family of) Algebras/Data Types is to give an Equational Specification.

9. School of Computing and Mathematics, University of Huddersfield Example - Equational Presentation of a Homogenous Algebra SPEC Boolean SORT bool OPS true : -> bool false : -> bool not : bool -> bool and : bool bool -> bool AXIOMS: FORALL b : bool (1) not(true) = false (2) not(false) = true (3) and(true,b) = b (4) and(b,true) = b (5) and(false,b) = false (6) and(b,false) = false ENDSPEC

10. School of Computing and Mathematics, University of Huddersfield Syntax for Equational Specs of Algebras (ADTs) SPEC % Name of Specification SORT % ‘Type of interest’ - algebra being defined OPS % Signature of operations FORALL % Universally defined variables AXIOMS % Equations defining MEANING of operations ENDSPEC

11. School of Computing and Mathematics, University of Huddersfield Another Example SPEC Natural SORT nat OPS zero : -> nat succ : nat -> nat add : nat nat -> nat AXIOMS: FORALL m, n : nat (1) add(zero, n) = n (2) add(succ(m), n) = succ(add(m, n)) ENDSPEC

12. School of Computing and Mathematics, University of Huddersfield SPEC A_State_Machine SORT state OPS S1 : -> state S2 : -> state S3 : -> state a : state -> state b : state -> state c : state -> state AXIOMS: FORALL m, n : nat (1) a(S1) = S3 (6) c(S2) = S1 (2) b(S1) = S1 (7) a(S3) = S3 (3) c(S1) = S2 (8) b(S3) = S1 (4) a(S2) = S2 (9) c(S3) = S3 (5) b(S2) = S2 ENDSPEC

13. School of Computing and Mathematics, University of Huddersfield Another Example... SPEC Stack USING Natural + Boolean % Carriers are Stack, Natural SORT stack% and Boolean OPS % Type of Interest = Stack init : -> stack % Signature of each Operation push : stack nat -> stack pop : stack -> stack top : stack -> nat is-empty? : stack -> bool stack-error : -> stack nat-error : -> nat FORALL s : stack, n : nat % universally quantified vars AXIOMS for is-empty?: (1) is-empty?(init) = true % this part gives `meaning' (2) is-empty?(push(s,n)) = false % to each operation AXIOMS for pop: (3) pop(init) = stack-error (4) pop(push(s,n)) = s AXIOMS for top: (5) top(init) = nat-error (6) top(push(s,n)) = n ENDSPEC

14. School of Computing and Mathematics, University of Huddersfield Conclusions ALGEBRA gives us an ABSTRACT way to specify DATA TYPES NEXT week we will examine how to REASON with algebraic expressions.