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Specification and Implementation of Abstract Data Types

Specification and Implementation of Abstract Data Types. Data Abstraction. Clients Interested in WHAT services a module provides, not HOW they are carried out. So, ignore details irrelevant to the overall behavior, for clarity. Implementors

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Specification and Implementation of Abstract Data Types

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  1. Specification and Implementation of Abstract Data Types L34ADT

  2. Data Abstraction • Clients • Interestedin WHAT services a module provides, not HOW they are carried out. So, ignore details irrelevant to the overall behavior, for clarity. • Implementors • Reserve the right to change the code, to improve performance. So, ensure that clients do not make unwarranted assumptions. L34ADT

  3. Specification of Data Types Type : Values + Operations Specify SyntaxSemantics Signature of Ops Meaning of Ops Model-basedAxiomatic(Algebraic) Description in terms of Give axioms satisfied standard “primitive” data types by the operations L34ADT

  4. Syntax of LISP S-expr • operations: nil, cons, car, cdr, null • signatures: nil: S-expr cons: S-expr*S-expr-> S-expr car: S-expr-> S-expr cdr: S-expr-> S-expr null: S-expr-> boolean for every atoma: a : S-expr L34ADT

  5. Signature tells us how to form complex terms from primitive operations. • Legal nil null(cons(nil,nil)) cons(car(nil),nil) • Illegal nil(cons) null(null) cons(nil) L34ADT

  6. Semantics of +: What to expect? + : NxN->N 1 + 2 = 3 zero + succ(succ(zero)) = succ(succ(zero)) x + 0 = x 2 * (3 + 4) = 2 * 7 = 14 = 6 + 8 x * ( y + z) = y * x + x * z L34ADT

  7. Semantics of S-Expr : What to expect? null(nil) = true car(cons(nil,nil)) = nil null(cdr(cons(nil,cons(nil,nil)))) = false • for all E,F in S-Expr car(cons(E,F)) = E null(cons(E,F)) = false L34ADT

  8. Formal Spec. of ADTs Characteristics of an “Adequate” Specification • Completeness (No “undefinedness”) • Consistency/Soundness (No conflicting definitions) • Minimality GOAL: Learn to write sound and complete algebraic(axiomatic) specifications of ADTs L34ADT

  9. Classification of Operations • Observers • generate a value outside the type • E.g., null in ADTS-expr • Constructors • required for representing values in the type • E.g., nil, cons, atoms a in ADTS-expr • Non-constructors • remaining operations • E.g., car, cdr in ADTS-expr L34ADT

  10. S-Expr in LISP a : S-Expr nil : S-Expr cons : S-Expr x S-Expr -> S-Expr car : S-Expr -> S-Expr cdr : S-Expr -> S-Expr null : S-Expr -> boolean Observers : null Constructors : a,nil, cons Non-constructors : car, cdr L34ADT

  11. Algebraic Spec • Write axioms (equations) that characterize the meaning of all the operations. • Describe the meaning of the observers and the non-constructors on all possible constructor patterns. • Note the use of typed variables to abbreviate the definition. (“Finite Spec.”) L34ADT

  12. for all S, T in S-expr cdr(nil) = ?error? cdr(a) = ?error? cdr(cons(S,T)) = T car(nil) = ?error? car(a) = ?error? car(cons(S,T)) = S null(nil) = true null(a) = false null(cons(S,T)) = false • Omitting the equation for “nil” implies that implementations that differ in the interpretation of “nil” are all equally acceptable. L34ADT

  13. S-Exprs • car(a) • cons(a,nil) • car(cons(a,nil)) = • a • cons( car(cons(a,nil)), cdr(cons(a,a)) ) = • cons( a , a ) L34ADT

  14. Motivation for Classification : Minimality • If car and cdr are also regarded as constructors (as they generate values in the type), then the spec. must consider other cases to guarantee completeness (or provide sufficient justification for their omission). • for all S in S-expr: null(car(S)) = ... null(cdr(S)) = ... L34ADT

  15. ADT Table (symbol table/directory) empty : Table update : Key x Info x Table ->Table lookUp: Key x Table -> Info lookUp(K,empty) = error lookUp(K,update(Ki, I, T)) = if K = Ki then I else lookUp(K,T) (“last update overrides the others”) L34ADT

  16. Tables • empty • update(5, “abc”, empty) • update(10, “xyz”, update(5, “abc”, empty)) • update(5, “xyz”, update(5, “abc”, empty)) (Search) • lookup (5,update(5, “xyz”, update(5, “abc”, empty)) ) • lookup (5,update(5, “xyz”, update(5, “xyz”, empty)) ) • lookup (5,update(5, “xyz”, empty) ) • “xyz” L34ADT

  17. Implementations • Array-based • Linear List - based • Tree - based • Binary Search Trees, AVL Trees, B-Trees etc • Hash Table - based • These exhibit a common Table behavior, but differ in performance aspects (search time). • Correctness of a program is assured even when the implementation is changed as long as the spec is satisfied. L34ADT

  18. (cont’d) • Accounts for various other differences (Data Invariants) in implementation such as • Eliminating duplicates. • Retaining only the final binding. • Maintaining the records sorted on the key. • Maintaining the records sorted in terms of the frequency of use (a la caching). L34ADT

  19. A-list in LISP a : A nil : A-list cons : A x A-list -> A-list car : A-list -> A cdr : A-list -> A-list null : A-list -> boolean • Observers : null, car • Constructors : nil, cons • Non-constructors : cdr L34ADT

  20. for all L in A-list cdr(cons(a,L)) = L car(cons(a,L)) = a null(nil) = true null(cons(a,L)) = false • Consciously silent about nil-list. L34ADT

  21. Natural Numbers zero : N succ : N-> N add : NxN-> N iszero : N->boolean observers : iszero constructors : zero, succ non-constructors : add Each number has a unique representation in terms of its constructors. L34ADT

  22. for all I,J in N add(I,J) = ? add(zero,I) = I add(succ(J), I) = succ(add(J,I)) iszero(I) = ? iszero(zero) = true iszero(succ(I)) = false L34ADT

  23. (cont’d) add(succ(succ(zero)), succ(zero)) = succ(succ(succ(zero))) • The first rule eliminates add from an expression, while the second rule simplifies the first argument to add. • Associativity, commutativity, and identity properties of add can be deduced from this definition through purely mechanical means. L34ADT

  24. A-list Revisted a : A nil : A-list list : A -> A-list append : A-list x A-list -> A-list null : A-list -> boolean • values • nil, list(a), append(nil, list(a)), ... L34ADT

  25. Algebraic Spec • constructors • nil, list, append • observer isnull(nil) = true isnull(list(a)) = false isnull(append(L1,L2)) = isnull(L1) /\ isnull(L2) L34ADT

  26. Problem : Same value has multiple representation in terms of constructors. • Solution : Add axioms for constructors. • Identity Rule append(L,nil) = L append(nil,L) = L • Associativity Rule append(append(L1,L2),L3) = append(L1, append(L2,L3)) L34ADT

  27. Intuitive understanding of constructors • The constructor patterns correspond to distinct memory/data patterns required to store/represent values in the type. • The constructor axioms can be viewed operationally as rewrite rules to simplify constructor patterns. Specifically, constructor axioms correspond to computations necessary for equality checking and aid in defining a normal form. • Cf. ==vsequal in Java L34ADT

  28. Writing ADT Specs • Idea: Specify “sufficient” axioms such that syntactically distinct terms (patterns) that denote the same value can be proven so. • Completeness • Define non-constructors and observers on all possible constructor patterns • Consistency • Check for conflicting reductions • Note: A term essentially records the detailed historyofconstruction of the value. L34ADT

  29. General Strategy for ADT Specs • Syntax • Specify signatures and classify operations. • Constructors • Write axioms to ensure that two constructor terms that represent the same value can be proven so. • E.g., identity, associativity, commutativity rules. L34ADT

  30. Non-constructors • Provide axioms to collapse a non-constructor term into a term involving only constructors. • Observers • Define the meaning of an observer on all constructor terms, checking for consistency. Implementation of a type An interpretation of the operations of the ADT that satisfies all the axioms. L34ADT

  31. Declarative Specification • Let *: N x N -> N denote integer multiplication. Equation: n * n = n Solution: n = 0 \/ n = 1. • Let f: N x N -> N denote a binary integer function. Equation: 0 f 0 = 0 Solution: f = “multiplication” \/ f = “addition” \/ f = “subtraction” \/ ... L34ADT

  32. delete : Set • for all n, m inN, s inSet delete(n,empty) = empty delete(n,insert(m,s)) = if (n=m) then delete(n,s) (invalid: s) else insert(m,delete(n,s)) delete(5, insert(5,insert(5,empty)) ) {5,5} == empty {} =/= insert(5,empty) [5,5] [] [5] L34ADT

  33. delete : List • Previous axioms capture “remove all occurrences” semantics. • For “remove last occurrence” semantics: for all n, m inN, s inList delete(n,empty) = empty delete(n,insert(m,s)) = if (n=m) then s else insert(m,delete(n,s)) delete(5, insert(5,insert(5,empty)) ) [5,5] == insert(5,empty) [5] L34ADT

  34. delete : List • Previous axioms capture “remove all / last occurrences” semantics. • For “remove first occurrence” semantics: for all n, m inN, s inList delete(n,empty) = empty delete(n,insert(m,s)) = if (n=m) andnot (n in s) then s else insert(m,delete(n,s)) delete(1, insert(1,insert(2,insert(1,insert(5,empty)))) ) [5,1,2,1] == insert(1,insert(2,insert(5,empty)))[5,2,1] L34ADT

  35. size: List vs Set • size(insert(m,l)) = 1 + size(l) • E.g., size([2,2,2]) = 1 + size([2,2]) • size(insert(m,s)) = if (m ins) then size(s) else 1 + size(s) • E.g., size({2,2,2}) = size({2,2}) = size ({2}) = 1 L34ADT

  36. Model-based vs Algebraic • A model-based specification of a type satisfies the corresponding axiomatic specification. Hence, algebraic spec. is “more abstract” than the model-based spec. • Algebraic spec captures the least common-denominator (behavior) of all possible implementations. L34ADT

  37. Axiomatization: Algebraic Structures • A set G with operation * forms a group if • Closure: a,b eG implies a*b eG. • Associativity: a,b,c eG implies a*(b *c) = (a*b)*c. • Identity: There exists i eG such that i*a = a*i = a for all a eG. • Inverses: For every a eG there exists an element ~a eG such that a * ~a = ~a * a = i. • Examples: • (Integers, +), but not (N, +) • (Reals - {0}, *), but not (Integers, *) • (Permutation functions, Function composition) L34ADT

  38. Example car( cons( X, Y) ) = X cdr( cons (X, Y) ) = Y (define (cons x y) (lambda (m) (cond ((eq? m ’first) x) (eq? m ’second) y) ) )) ; “closure” (define (car z) (z ’first)) (define (cdr z) (z ’second)) L34ADT

  39. Applications of ADT spec • Least common denominator of all possible implementations. • Focus on the essential behavior. • An implementation is a refinement of ADT spec. • IMPL. = Behavior SPEC + Rep “impurities” • To prove equivalence of two implementations, show that they satisfy the same spec. • In the context of OOP, a class implements an ADT, and the spec. is a class invariant. L34ADT

  40. (Cont’d) • Indirectly, ADT spec. gives us the ability to vary or substitute an implementation. • E.g., In the context of interpreter/compiler, a function definition and the corresponding calls (ADT FuncValues) together must achieve a fixed goal. However, there is freedom in the precise apportioning of workload between the two separate tasks: • How to represent the function? • How to carry out the call? L34ADT

  41. (Cont’d) • ADT spec. are absolutely necessary to automate formal reasoning about programs. Theorem provers such as Boyer-Moore prover (NQTHM), LARCH, PVS, HOL, etc routinely use such axiomatization of types. • Provides a theory of equivalence of values that enables design of a suitable canonical form. • Identity-> delete • Associativity-> remove parenthesis • Commutativity-> sort L34ADT

  42. Spec vs Impl The reason to focus on the behavioral aspects, ignoring efficiency details initially, is that the notion of a “best implementation” requires application specific issues and trade-offs. In other words, the distribution of work among the various operations is based on a chosen representation, which in turn, is dictated by the pragmatics of an application. However, in each potential implementation, there is always some operations that will be efficient while others will pay the price for this comfort. L34ADT

  43. Ordered Integer Lists null : oil->boolean nil : oil hd : oil->int tl : oil->oil ins : int x oil->oil order : int_list->oil Constructors: nil, ins Non-constructors: tl, order Observers: null, hd L34ADT

  44. Problem: • syntactically different, but semantically equivalent constructor terms ins(2,ins(5,nil)) = ins(5,ins(2,nil)) ins(2,ins(2,nil)) = ins(2,nil) • hd should return the smallest element. • It is not the case that for all I in int, L in oil, hd(ins(I,L)) = I. • This holds iff I is the minimum in ins(I,L). • Similarly for tl. L34ADT

  45. Axioms for Constructors • Idempotence • for all ordered integer lists L; for all I in int ins(I, ins(I,L)) = ins(I,L) • Commutativity • for all ordered integer lists L; for all I, J in int ins(I, ins(J,L)) = ins(J, ins(I,L)) Completeness : Any permutation can be generated by exchanging adjacent elements. L34ADT

  46. Axioms for Non-constructors tl(nil) = error tl(ins(I,L)) = ? tl(ins(I,nil)) = nil tl(ins(I,ins(J,L))) = I < J => ins( J, tl(ins(I,L)) ) I > J => ins( I, tl(ins(J,L)) ) I = J => tl( ins( I,L ) ) (cf. constructor axioms for duplicate elimination) order(nil) = nil order(cons(I,L)) = ins(I,order(L)) L34ADT

  47. Axioms for Observers hd(nil) = error hd(ins(I,nil)) = I hd(ins(I,ins(J,L))) = I < J => hd( ins(I,L) ) I > J => hd( ins(J,L) ) I = J => hd( ins(I,L) ) null(nil) = true null(ins(I,L)) = false L34ADT

  48. Scheme Implementation (define nullnull?) (define nil’()) (define inscons) (define (hd ol) *min* ) (define (tl ol) *list sans min* ) (define (order lis) *sorted list* ) L34ADT

  49. Possible Implementations • Representation Choice 1: • List of integers with duplicates • ins is cons but hd and tl require linear-time search • Representation Choice 2: • Sorted list of integers without duplicates • ins requires search but hd and tl can be made more efficient • Representation Choice 3: • Balanced-tree : Heap L34ADT

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