Foundations of Programming Languages and Software Engineering

Transcription

1 Foundations of Programming Languages and Software Engineering Jan-Georg Smaus (Peter Thiemann) Universität Freiburg July 2011 Abstract Data Types Foundations of Programming Languages and Software Engineering 1 / 27

2 Abstract Data Types We have learned about different datastructures, e.g. for dictionaries: Search trees Lists Tables with hashing Implementations of these concepts may have different characteristics: Memory usage Efficiency Implementations should be exchangeable Abstract over the concepts, use ADTs! Functional specification Implementation independent Different implementations of a single ADT are possible Foundations of Programming Languages and Software Engineering 2 / 27

3 ADTs are Special Signatures Definition Let Σ be a signature. A Σ-identity is a pair (s, t) T (Σ, X) T (Σ, X). We write a Σ-identity as s t for emphasis. An ADT is a pair (Σ, E) where Σ is a signature, E T (Σ, X) T (Σ, X) is a set of Σ-identities. Foundations of Programming Languages and Software Engineering 3 / 27

4 Examples An ADT for natural numbers Σ nat = {zero (0), succ (1) } E nat = Foundations of Programming Languages and Software Engineering 4 / 27

5 Examples An ADT for natural numbers Σ nat = {zero (0), succ (1) } E nat = An ADT for integers Σ int = {zero (0), pred (1), succ (1) } E int = {pred(succ(x)) x, succ(pred(x)) x} Foundations of Programming Languages and Software Engineering 4 / 27

6 Datatypes are Σ-Algebras Definition An identity s t is valid in a Σ-algebra A = (A, J ) iff J α (s) = J α (t) for all variable assignments α : X A. Foundations of Programming Languages and Software Engineering 5 / 27

7 Datatypes are Σ-Algebras Definition An identity s t is valid in a Σ-algebra A = (A, J ) iff J α (s) = J α (t) for all variable assignments α : X A. A datatype is a Σ-algebra D. A datatype D implements the ADT (Σ, E) iff every identity s t E is valid in D. (Note: We shall refine this definition later.) Foundations of Programming Languages and Software Engineering 5 / 27

8 Implementations of the ADT for Naturals Implementation 1 Dnat = (N, J ), J (zero) = 0, J (succ)(x) = x + 1 (note that x is meta-notation!). No identities, so all are valid. The function J is bijective. Foundations of Programming Languages and Software Engineering 6 / 27

9 Implementations of the ADT for Naturals Implementation 1 Dnat = (N, J ), J (zero) = 0, J (succ)(x) = x + 1 (note that x is meta-notation!). No identities, so all are valid. The function J is bijective. Implementation 2 Dnat = ({0, 1, 2, 3}, J ), J (zero) = 0, J (succ)(x) = (x + 1) mod 4. No identities, so all are valid. The function J is not injective (but surjective). J (zero) = 0 = J (succ(succ(succ(succ(zero))))) Foundations of Programming Languages and Software Engineering 6 / 27

10 Implementations of the ADT for Integers (1) Implementation 1 Dint = (Z, J ), J (zero)() = 0 J (succ)(x) = x + 1 J (pred)(x) = x 1 For arbitrary α : {x} Z we have J α(pred(succ(x))) = (α(x) + 1) 1 = J α(x) J α(succ(pred(x))) = (α(x) 1) + 1 = J α(x) J is surjective but not injective. Consider J (zero) = 0 = J (succ(pred(zero))). Foundations of Programming Languages and Software Engineering 7 / 27

11 Implementations of the ADT for Integers (2) Implementation 2 Dint = ({0, 1, 2, 3}, J ), J (zero)() = 0 J (succ)(x) = x + 1 mod 4 J (pred)(x) = x 1 mod 4 For arbitrary α : {x} Z we have J α (pred(succ(x))) = J α (x) J α (succ(pred(x))) = J α (x) J is surjective but not injective. Foundations of Programming Languages and Software Engineering 8 / 27

12 Implementations of the ADT for Integers (3) A Non-implementation Dint = (N, J ) J (zero)() = 0 J (succ)(x) = { x + 1 J x 1 x > 0 (pred)(x) = 0 x = 0 Not an implementation: For α : X N with α(x) = 0 we have J α (succ(pred(x))) = 1 0 = J α (x) Foundations of Programming Languages and Software Engineering 9 / 27

13 Fixing the Problems Want to rule out implementations such as Dnat and Dint Definition of implementation is too weak Needed: restriction on function J J is not necessarily injective (see Dint ) Idea: J must be injective on the equivalence classes induced by the identities of an ADT. In other words: J should make those terms equal that are equal according to the identities of the ADT, but not more! Foundations of Programming Languages and Software Engineering 10 / 27

14 Equivalence Classes Definition Suppose R is an equivalence relation on some set M. The set [x] R := {y M x R y} is called the equivalence class of x. y [x] R is called a representative of [x] R. The quotient of M with respect to R is the set of equivalence classes induced by R, written M/ R := {[x] R x M}. Note: For equivalence classes [x] R and [y] R we have either [x] R = [y] R or [x] R [y] R =. Foundations of Programming Languages and Software Engineering 11 / 27

15 Congruence Relations Definition Suppose Σ is a signature and let R be an equivalence relation on T (Σ, X). R is a congruence relation iff R is closed under Σ-operations, i.e. t i R t i implies f (t 1,..., t i,..., t n ) R f (t 1,..., t i,..., t n ) for any n 0, f Σ (n), and t 1,..., t n, t i T (Σ, X). Foundations of Programming Languages and Software Engineering 12 / 27

16 Syntactic Quotient Algebras Definition Let Σ be a signature and R be a congruence on T (Σ, X). For all n 0, f Σ (n), and t 1,..., t n T (Σ, X), define J R as follows: Note J R (f )([t 1 ] R,..., [t n ] R ) = [f (t 1,..., t n )] R (T (Σ, X)/ R, J R ) is a Σ-algebra. Its carrier elements are sets of terms. The representatives are arbitrary: Let n 0, f Σ (n), and s 1, t 1,..., s n, t n T (Σ, X). If s 1 R t 1,..., s n R t n then f (s 1,..., s n ) R f (t 1,..., t n ). Hence, [f (s 1,..., s n )] R = [f (t 1,..., t n )] R, as R is a congruence. Foundations of Programming Languages and Software Engineering 13 / 27

17 Equational Theory Definition Let (Σ, E) be an ADT. We define a relation E on T (Σ, X) as the smallest relation such that E is a congruence relation; E contains E, i.e. s t E implies s E t; E is closed under substitutions, i.e. s E t implies σ(s) E σ(t) for any substitution σ and all s, t T (Σ, X). Foundations of Programming Languages and Software Engineering 14 / 27

18 Example Congruence classes of Eint Σ int = {zero (0), pred (1), succ (1) } E int = {pred(succ(x)) x, succ(pred(x)) x} [zero] Eint = {zero, } succ(pred(zero)), pred(succ(zero)), succ(succ(pred(pred(zero)))),... Foundations of Programming Languages and Software Engineering 15 / 27

19 Revised Definition for ADT Implementations Definition A datatype D = (M, J ) implements ADT (Σ, E) with constructors Γ Σ if (M, J ) is a Σ-algebra (as before) All identities from E are valid in M (as before) For all s, t T (Γ, ): s E t iff J (s) = J (t) (new!) Foundations of Programming Languages and Software Engineering 16 / 27

20 Syntactic Implementations of ADTs Theorem Let (Σ, E) be an ADT with constructors Γ Σ. Then D = (T (Σ, )/ E, J E ) is an implementation of (Σ, E). Proof. Omitted Foundations of Programming Languages and Software Engineering 17 / 27

21 Example Nat as a constructor-based ADT (CADT) CADT: Σ = {zero, succ}, E = {}, Γ = Σ Implementation: (N, J 1 ) with J 1 (zero)() = 0 and J 1 (succ)(x) = x + 1 (N, J 1 ) is Σ-algebra No identities to check Since E =, E is =. Suppose s, t T (Γ, ). If s = t then J 1 (s) = J 1 (t) Suppose s t. Then s = succ n (t) with n > 0. Hence, J 1 (s) = J 1 (t) + n J 1 (t). Foundations of Programming Languages and Software Engineering 18 / 27

22 Example Dint is not an implementation of the natural numbers CADT: Σ = {zero, succ}, E = {}, Γ = Σ ({0, 1, 2, 3}, J ) with J (zero)() = 0, J (succ)(x) = (x + 1) mod 4 is not an implementation. ({0, 1, 2, 3}, J ) is Σ-algebra No identities to check Since E =, E is =. We have zero succ 4 (zero) but J (zero) = 0 = J (succ 4 (zero)). Foundations of Programming Languages and Software Engineering 19 / 27

23 Example Alternative implementation of the natural numbers CADT: Σ = {zero, succ}, E = {}, Γ = Σ Implementation: ({a}, J ) with J (zero)() = ɛ, J (succ)(w) = aw ({a}, J ) is Σ-algebra No identities to check Since E =, E is =. If s = t then J (s) = J (t) Suppose s t. Then s = succ n (t) with n > 0. Hence, J (s) = J (t) } a. {{.. a } J (t). n Foundations of Programming Languages and Software Engineering 20 / 27

24 Equivalence Classes for Terms Representing Integers Suppose Γ = Σ = {zero, succ, pred}, E = {succ(pred(x)) = x, pred(succ(x)) = x} Question: What is T (Σ, )/ E? Answer: Give a representative for every equivalence class. Lemma For every term t T (Σ, ), exactly one of the following propositions holds A There exists n > 0 such that t [succ n (zero)] E. B t [zero] E. C There exists n > 0 such that t [pred n (zero)] E. Foundations of Programming Languages and Software Engineering 21 / 27

25 Proof The proof is by term induction over t. Base case: t = zero. Then B holds. Foundations of Programming Languages and Software Engineering 22 / 27

26 Proof (cont.) Induction Step for t = succ(t ). By the IH, one of the following holds for t : A t E succ (n) (zero) for n > 0: then succ(t ) E succ(succ (n) (zero)) = succ (n+1) (zero). Since n + 1 > 0 we have case A. B t E zero: then succ(t ) E succ(zero). We have case A with n = 1. C t E pred (n) (zero) for n > 0: Then succ(t ) E succ(pred (n) (zero)). If n = 1 then succ(pred(zero)) E zero so case B holds. If n > 1 then succ(pred(pred (n 1) (zero))) E pred (n 1) (zero), so case C holds. Foundations of Programming Languages and Software Engineering 23 / 27

27 Proof (cont.) Induction step for pred(t) analogous. Foundations of Programming Languages and Software Engineering 24 / 27

28 Equivalence Classes for Terms Representing Integers Lemma Suppose n > 0, m > 0. Then we have zero E succ n (zero), zero E pred n (zero), succ n (zero) E pred m (zero), succ n (zero) E succ m (zero) provided n m, and pred n (zero) E pred m (zero) provided n m. If follows that {succ n (zero) n > 0} {zero} {pred n (zero) n > 0} is a set of representatives for T (Σ, )/ E. Foundations of Programming Languages and Software Engineering 25 / 27

29 Example Integers as a CADT CADT: Γ = Σ = {zero, succ, pred}, E = {succ(pred(x)) = x, pred(succ(x)) = x} Implementation: (Z, J ) with J (z) = 0, J (succ)(x) = x + 1, J (pred)(x) = x 1 (Z, J ) is a Σ-algebra All identities are valid (as seen before) An easy term induction shows for all t T (Σ, ) that if t E zero then J (t) = 0, if t E succ n (zero) then J (t) = n, and if t E pred n (zero) then J (t) = n. Hence, if s E t then J (s) = J (t). Conversely, if s E t then J (s) J (t) because J maps different representatives to different integers. Foundations of Programming Languages and Software Engineering 26 / 27

30 Summary Slogan Calculating with ADT = applying term operations + determining set of representatives. Foundations of Programming Languages and Software Engineering 27 / 27