Specification and Implementation of Abstract Data Types

Specification and Implementation of Abstract Data Types L4-5ADT

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. L4-5ADT

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 L4-5ADT

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 L4-5ADT

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) L4-5ADT

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 L4-5ADT

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 L4-5ADT

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 L4-5ADT

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 L4-5ADT

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 L4-5ADT

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.”) L4-5ADT

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. L4-5ADT

S-Exprs • car(a) • cons(a,nil) • car(cons(a,nil)) = • a • cons( car(cons(a,nil)), cdr(cons(a,a)) ) = • cons( a , a ) L4-5ADT

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)) = ... L4-5ADT

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”) L4-5ADT

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” L4-5ADT

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. L4-5ADT

(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). L4-5ADT

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 L4-5ADT

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. L4-5ADT

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. L4-5ADT

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 L4-5ADT

(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. L4-5ADT

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)), ... L4-5ADT

Algebraic Spec • constructors • nil, list, append • observer null(nil) = true null(list(a)) = false null(append(L1,L2)) = null(L1) /\ null(L2) L4-5ADT

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)) L4-5ADT

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 L4-5ADT

Writing ADT Specs • Idea: Specify “sufficient” axioms such that syntactically distinct terms (patterns) that denote the same value can be proven so. • Note: A term essentially records the detailed historyofconstruction of the value. L4-5ADT

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. L4-5ADT

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. L4-5ADT

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” \/ ... L4-5ADT

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] L4-5ADT

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] L4-5ADT

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] L4-5ADT

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 L4-5ADT

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. L4-5ADT

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) L4-5ADT

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)) L4-5ADT

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. L4-5ADT

Implementations refine algebraic spec; Equivalence/Substituitivity by abstraction ADT Spec Impl. 1 Impl. 2 L4-5ADT

Indirectly, ADT spec. gives us the ability to vary or substitute an implementation. L4-5ADT

(Cont’d) • ADT spec. are absolutely necessary to automate formal reasoning about programs. Theorem provers such as Boyer-Moore prover (NQTHM), LARCH, PVS, and HOL 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 L4-5ADT

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. L4-5ADT

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 L4-5ADT

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. L4-5ADT

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. L4-5ADT

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)) L4-5ADT

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 L4-5ADT

Scheme Implementation (define nullnull?) (define nil’()) (define inscons) (define (hd ol) *min* ) (define (tl ol) *list sans min* ) (define (order lis) *sorted list* ) L4-5ADT

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 L4-5ADT

Specification and Implementation of Abstract Data Types