Outline Mousavi: ADT

Transcription

1 Outline Abstract Data Types Mohammad Mousavi Eindhoven University of Technology, The Netherlands Requirement Analysis and Design Verification (RADV)

2 Outline Outline Basic Concepts Booleans Standard Data Types Numbers Lists Sets and Bags

3 Outline Outline Basic Concepts Booleans Standard Data Types Numbers Lists Sets and Bags

4 Outline Outline Basic Concepts Booleans Standard Data Types Numbers Lists Sets and Bags

5 Outline Outline Basic Concepts Booleans Standard Data Types Numbers Lists Sets and Bags

6 Outline Outline Basic Concepts Booleans Standard Data Types Numbers Lists Sets and Bags

7 Outline Outline Basic Concepts Booleans Standard Data Types Numbers Lists Sets and Bags

8 System Design 1. State (data): Algebraic specification, Abstract Data Types (ADT) Examples: Clear, CASL 2. Behavior (process): Labelled Transition Systems (LTS) Examples: CCS, ACP mcrl2: the best of the two worlds!

9 ADTs Sorts: non-empty sets of data elements Examples: Bool, Nat, List(Nat), Bag(Bool) Constructors: denotation for all elements of the sort Examples: sort Nat; const zero : Nat; one : Nat; two : Nat;.

10 ADTs Sorts: non-empty sets of data elements Examples: Bool, Nat, List(Nat), Bag(Bool) Constructors: denotation for all elements of the sort Examples: sort Nat; const zero : Nat; one : Nat; two : Nat;.

11 ADTs Sorts: non-empty sets of data elements Constructors: denotation for all elements of the sort Examples: sort Nat; const zero : Nat; successor : Nat Nat;

12 ADTs Sorts: non-empty sets of data elements Constructors: denotation for all elements of the sort Examples: sort Dummy; const f : Dummy Dummy; Error: sort Dummy is empty. But constructors may be left out altogether... Constructor sorts: sorts containing at least one constructor.

13 ADTs Sorts: non-empty sets of data elements Constructors: denotation for all elements of the sort Examples: sort Dummy; const f : Dummy Dummy; Error: sort Dummy is empty. But constructors may be left out altogether... Constructor sorts: sorts containing at least one constructor.

14 ADTs Sorts: non-empty sets of data elements Constructors: denotation for all elements of the sort Examples: sort Dummy; const f : Dummy Dummy; Error: sort Dummy is empty. But constructors may be left out altogether... Constructor sorts: sorts containing at least one constructor.

15 ADTs Sorts: non-empty sets of data elements Constructors: denotation for all elements of the sort Maps: useful transformations on data const 0 : Nat; 1 : Nat; s : Nat Nat; maps add : Nat Nat Nat; p : Nat Nat; Equations: definition of maps, equalities among constructor terms

16 ADTs Sorts: non-empty sets of data elements Constructors: denotation for all elements of the sort Maps: useful transformations on data const 0 : Nat; 1 : Nat; s : Nat Nat; maps add : Nat Nat Nat; p : Nat Nat; Equations: definition of maps, equalities among constructor terms

17 ADTs Equations: definition of maps, equalities among constructor terms Examples: const 0 : Nat; 1 : Nat; s : Nat Nat; maps add : Nat Nat Nat; p : Nat Nat; var m, n : Nat; eqn 1 = s(0); add(n, 0) = n; add(n, s(m)) = s(add(n, m));

18 Exercises Equations: definition of maps, equalities among constructor terms Exercises: 1. Define the sort Bool; 2. (3.2.1) and the concept of,, < and > between natural numbers. Conditional equations: cond t = t, where cond is a term of type B (boolean).

19 Induction Principle Idea: to prove a theorem, you need to prove it for all members of the sorts, i.e., closed terms containing only constructors Example; φ(x) holds for x of sort Nat if 1. φ(0) holds; 2. given that φ(n) holds, then φ(s(n)) holds.

20 Induction Principle Idea: to prove a theorem, you need to prove it for all members of the sorts, i.e., closed terms containing only constructors Example; φ(x) holds for x of sort Nat if 1. φ(0) holds; 2. given that φ(n) holds, then φ(s(n)) holds.

21 Induction Principle Principle: φ(x) with x of sort S holds if 1. φ(c) holds, for each constructor c : S; 2. given that φ(c i ) holds, for each 0 i n and each term c i of sort S, then φ(c(c 0,..., c n )) holds, for each constructor c : S... S S ;

22 Induction Principle Exercise: prove for Nat that 1. add(0, m) = m, 2. add(s(n), m) = s(add(n, m)), and 3. m + n = n + m. const 0 : Nat; s : Nat Nat; maps add : Nat Nat Nat; var m, n : Nat; eqn add(n, 0) = n; add(n, s(m)) = s(add(n, m));

23 One Point Models Equations: definition of maps, equalities among constructor terms Is 0 different from s(0)? Not necessarily! Then, how can we enforce that?

24 One Point Models Equations: definition of maps, equalities among constructor terms Is 0 different from s(0)? Not necessarily! Then, how can we enforce that?

25 Booleans sort B; const true, false : B; maps : B B;,, : B B B; var b : B; eqn true = false; false = true; b = b; b true = b; b false = false;... Important assumption: true is different from false. Thus, any derivation leading to true = false is unsound.

26 Exercise Given that true and false are different, prove in the theory of natural numbers that it 0 = s(0) is unsound.

27 Booleans Quantification is also possible: maps Fermat : B var b : B; eqn Fermat = n : N.n 3 a, b, c : N. a / 0 b / 0 c / 0 a n + b n c n ;

28 Standard Data Types Commonly used data types: Booleans (sort B), natural numbers (sort N), etc. Each sort S is quipped with the following maps: maps, / : S S B; if : B S S S; var b : B; x, y : S; eqn x x = true; x / y = (x y); if (true, x, y) = x; if (false, x, y) = y; if (b, x, x) = x; if (x y, x, y) = y;

29 = Vs. eqn x x = true; x / y = (x y); if (true, x, y) = x; if (false, x, y) = y; if (b, x, x) = x; if (x y, x, y) = y; Theorem. For any such sort S, x y = true if and only if x = y. Proof. Assume that x y = true. x = if (true, x, y) = if (x y, x, y) = y. Assume that x = y. x y = x x = true.

30 = Vs. eqn x x = true; x / y = (x y); if (true, x, y) = x; if (false, x, y) = y; if (b, x, x) = x; if (x y, x, y) = y; Theorem. For any such sort S, x y = true if and only if x = y. Proof. Assume that x y = true. x = if (true, x, y) = if (x y, x, y) = y. Assume that x = y. x y = x x = true.

31 = Vs. eqn x x = true; x / y = (x y); if (true, x, y) = x; if (false, x, y) = y; if (b, x, x) = x; if (x y, x, y) = y; Theorem. For any such sort S, x y = true if and only if x = y. Proof. Assume that x y = true. x = if (true, x, y) = if (x y, x, y) = y. Assume that x = y. x y = x x = true.

32 Numbers Natural numbers (sort N), positive natural numbers (sort N + ), positive natural numbers (sort Z), and reals (sort R). Automatic type conversion (type cast) among numbers (a positive natural number is a natural number, an integer and a real number). Commonly used operators and relations (+,,, etc.) are defined among all appropriate sorts.

33 Positive Natural Numbers Efficient implementation: using constructors 1. 1 and 2. cdub(b, p), for boolean b and positive natural number p, where { 2n + 1 if b = true cdub(b, p) = 2n if b = false sort N + ; const 1 : N + ; cdub : B N + N + ; maps succ : N + N +.

34 Positive Natural Numbers Efficient implementation: using constructors 1. 1 and 2. cdub(b, p), for boolean b and positive natural number p, where { 2n + 1 if b = true cdub(b, p) = 2n if b = false sort N + ; const 1 : N + ; cdub : B N + N + ; maps succ : N + N +.

35 Positive Natural Numbers var b : B; p, q : N + ; eqn 1 cdub(b, p) = false; cdub(b, p) cdub(b, q) = p q; cdub(true, p) cdub(false, q) = false; succ(1) = cdub(false, 1);

36 Natural Numbers sort N; const c0 : N + ; cnat : N + N; maps Pos2Nat : N + N Nat2Pos : N N + succ : N N +.

37 Natural Numbers var b : B; p, q : N + ; eqn c0 cnat(p) = false; cnat(p) cnat(q) = p q; Pos2Nat = cnat; Nat2Pos(p) = p; succ(c0) = cnat(1); Exercise (3.2.6): prove that c0 cannot be equal to 1.

38 Natural Numbers var b : B; p, q : N + ; eqn c0 cnat(p) = false; cnat(p) cnat(q) = p q; Pos2Nat = cnat; Nat2Pos(p) = p; succ(c0) = cnat(1); Exercise (3.2.6): prove that c0 cannot be equal to 1.

39 Lists List of elements of sort S: List(S) Signature: const [] : List(S); : S List(S) List(S); maps in : S List(S) Bool # : List(S) N : List(S) S List(S); head : List(S) S; tail : List(S) List(S);.

40 Lists List enumeration: [s 0, s 1,...] is an acronym for s 0 s 1... []. Exercise: give the equations for.

41 Sets and Bags Set of elements of sort S: Set(S) (N.B. sets can be infinite) 1. enumeration: {s 0, s 1,...} 2. comprehension: {x : S φ} Example: {n : N m : N.m + m = n}. 3. membership s 0 set Bag of elements of sort S: Bag(S) 1. enumeration: {s 0 : n 0, s 1 : n 1,...} 2. count: count(s 0, bag) Exercise (3.2.10): Specify the set of all prime numbers (given the maps mod and ).