Specifying algebraic data types by domain equations - IfIS

F. Gecseg, editor, Proc. FCT’81, LNCS 117, pages 120–129, Berlin, 1981. Springer–Verlag Specifying algebraic data types by domain equations

H.-D. Ehrich Abtolnformatik,Univ. 9ortmund, PF 500500,9-4600 Oortmund 50

ABSTRACT - The paper nrovides the theoretical foundation f o r a new algebraic specif i c a t i o n method, using parameterized specifications and algebraic domain equations, an algebraic analogon to the domain equations used in Scott's theory of data types. The main r e s u l t is that algebraic domain equations always have an i n i t i a l solution. Also, a parametric version of algebraic domain equations is investigated. In e i t h e r case, there is a simple syntactic solution method.

1. INTRODUCTION Scott's order theoretic approach to data types (SC 71 f f ) offers a characteristic s p e c i f i c a t i o n method: there are given basic types l i k e BOOL, INT, e t c . , and there is a fixed set of type constructors l i k e sum, product, function space, and mowerdomain. Data types are then defined by so-called domain equations that may be recursive~ e.g. ( c f . SC 72a, SP 77) N ~ N+I (natural numbers), L~(L~L)+D ( l i s t structures over D), F~[F--~F]

(a pure L-calculus model), S~S~O+I (stacks over D),

etc. In contrast, the algebraic approach to data types has hardly more to o f f e r than e x p l i c i t specifications: one has to give a l l sorts, a l l operations, and a l l axioms that describe the behaviour of the operations ( c f . ADJ 78). Some caution is required when comparing the order-theoretic and algebraic approaches to data types, since they do not model the same aspects and serve somewhat d i f f e r e n t purposes. I t may, however, be f r u i t f u l

to apply ideas from one approach to the f u r t h e r development of

the other one. The papers of Lehmann and Smyth (LS 77) and Kanda (KA 78) represent d i f f e r e n t attempts to do t h i s . Recently, an algebraic analogon to type constructors, called parameterized data types, has been investigated (BG 77, EH 79, EL 79a+b, BG 80, EKTWW 80a+b). So, in a sense, algebraic data types can be specified by expressions involving parameterized data types applied to actual parameters. The question n a t u r a l l y arises whether a meaningful and useful analogon to recursive domain equations can be established in the algebraic framework. This paper shows that the answer is p o s i t i v e . Our approach is inspired by the categorical versions of recursive domain equations due to Wand (WA 75), Lehmann (LE 76), and Sm~tth and Plotkin (SP 77). Thus, f i x p o i n t s

121 of functors play an important role. However, we do not need ~ - c o n t i n u i t y , and we have a simple syntactic method f o r solving domain equations. The solutions are algebraic data types, i . e . many-sorted algebras with a l l t h e i r operations. 2. FUNDAMENTAL NOTIONS A signature is a quadruple E=, where S and ~ are sets of sorts and operators ', respectively, and a r i t y : ~ --->S*, sort:~

~S are mappings. We w i l l v~ite

E= f o r short, assuming t a c i t l y the existence of the a r i t y and sort ma~ings. A signature morphism f:E

> E' is a pair of mappings f= S',f~:~

~'>

such

that a r i t y ( ~ f ) = a r i t y ( ~ ) f s and sort ( e l ) = s o r t ( ~ ) f s. For convenience, we often omit the index, w r i t i n g f f o r fs or f . Algebras are interpretations of signatures: a z-algebra A is an S-indexed family of sets, {SA}, the c a r r i e r of A, together with an ~-indexed family of mappings, { ~ A : a r i t y ( ~ ) A - - - - ~ s o r t ( ~ ) A } , the operations of A ( i f x=sls2...SnES * , x A denotes the cartesian product S1,A~...XSn,A}. A E-algebra morDhism m:A--~B is an S-indexed family of mappings @s:SA---~SB such t h a t , for each operator eE~ with a r i t y x and sort s, we have ~Ams =mx~B. Here, mx=mslx...XmSn

i f x=s1...s n. The class of a l l

z-algebras with a l l z-algebra morphisms forms a category z-alg. I t is well known that ~-alg has an i n i t i a l

algebra I~, having a unique morphism to any other algebra

in E-alg. If f:z---*E'

is a signature morphism, there is a corresponding forgetful functor

f - a l g : E ' - a l g -~->E-alg sending each ~'-algebra B to that E-algebra A such that SA=(Sf)B and ~A=(~f)B . LetE = be a signature . A ~-e~uation is a t r i p l e Xi

~i:XixXi---*B00L

P: ~ P : X I ~ . . . x X n ..... >p _(i)

: P ~X i : P~P --*B00L

eqs

x i ~ i xi = t r u e

p(i) = [i ( i ) = x i ~ p = true ~ = f a l s e ~ p = f a l s e

~ = Xl~X I . . . Xn~Xn

t24 Consider ( p , p - f r e e ) f o r n=2. Let D be a s p e c i f i c a t i o n of the integers (sort INT) and booleans (sort BOOL). Let f:X---~D be defined by XI~--~BOOL, X2~--~INT, XlW--~false, x2~---~O, mi~--~identity on BOOL or INT, r e s p e c t i v e l y , i=1,2. Then, the i n i t i a l boolean-integer algebra (D_-algebra) is sent to the algebra of ( b o o l e a n , i n t e g e r ) - p a i r s with one a d d i t i o n a l constant, a l l other operations of XP, and a l l boolean and i n t e ger operations on the components retained. The s p e c i f i c a t i o n of t h i s algebra is DP_P, the pushout object of p and f .

I t is obtained from XP by s u b s t i t u t i n g BOOL f o r X1,

false f o r X l ' INT f o r X2, 0 f o r x2,and the resoective i d e n t i t i e s f o r mi" Therefore, a suggestive notation f o r DP is BOOL×INT + 1. 3. ALGEBRAIC DOrIAIN EQUATIONS In the order t h e o r e t i c approach to data types, marameterized data types can be viewed as f u n c t o r s , domain equations as endofunctors, and t h e i r solutions as f i x p o i n t s of functors (WA 75, LE 76, SP 77). In the algebraic approach, parameterized data tvpes are e s s e n t i a l l y functors, too, but most often they are not endofunctors. T y p i c a l l y , the signatures of actual ~arameter and r e s u l t a n t algebras are d i f f e r e n t .

In order to get endofunctors, we define

algebraic domain equations to consist of a parameterized data type and a functor in the reverse d i r e c t i o n . We r e s t r i c t ourselves to free and strongly o e r s i s t e n t oarameterizations of the form ( p , p - f r e e ) and to algebraic reverse functors of the form e-alg f o r some spec morphism e. D e f i n i t i o n 3.1: An algebraic domain equation is a oair of smec morohisms (p,e)~ p,e:X---eXP, such that o is a s t r o n g l y p e r s i s t e n t parameterized s p e c i f i c a t i o n . Let P=p-free, P=p-alg, and E=e-alg. There are two endofunctors, namely PE on X-alg and EP on XP-alg. A f i x p o i n t is an ob.~ect that is sent to an isomormhic one. I t is immediate to see that the f i x p o i n t s of PE and EP are very c l o s e l y related: & is a f i x p o i n t of PE i f f

AP is a f i x p o i n t of EP, and vice versa.

For the d e f i n i t i o n of what we mean by a s o l u t i o n of an algebraic domain equation, we make use of the f o l l o w i n g r e s u l t . Let (q,Q) be the coequalizer in spec of D and e, ×

- -

P

q

,

e

and l e t Q=~-alg. Theorem 3.2: I f B is a f i x p o i n t of EP, then there is a unique (up to isomoruhism) ~-algebra C such that B=CQ. I t is convenient not tO take f i x p o i n t s of PE or EP as s o l u t i o n s , but these uniquely associated Q-algebras. D e f i n i t i o n 3.3: A s o l u t i o n of an algebraic domain equation (p,e) is a Q-algebra C such that CQ is a f i x p o i n t of EP.

125 The main r e s u l t can now be stated as f o l l o w s . Theorem3.4: The i n i t i a l

Q-algebra IQ is a solution of (p,e).

The proof is rather involved and requires some more technical machinery to be developed. I t w i l l be published elsewhere. Clearly, I~ is an i n i t i a l i t is i n i t i a l

in the f u l l

solution, i.e.

subcategory of Q-alg of a l l solutions of (p,e). A s p e c i f i -

cation of IQ is Q, the coequalizer ob,iect of p and e. There is a simple construction f o r Q, given p and e, based on the coequalizer construction in set applied to sorts and operators. Example 3.5: Consider the n-product with constant from example 2.1 f o r n=l. Let X -

P

~X+I

e

-

be defined by taking p as in example 2.1, and e sending sort X1 to P, Xl to p, and ~1 to m . Then the s o l u t i o n is IQ where Q is the f o l l o w i n g s p e c i f i c a t i o n obtained from X + 1 by i d e n t i f y i n g X1 and P, Xl and p, and ml and ~. For convenience, we rename P by N, ~, by O, by succ, and ( 1 ) sorts

by pred.

N

2P#

O: --*N succ: N---~N pred: N--->N : N-N --~BOOL pred(O) = 0

eqs

pred(succ(n)) = n OmO = true O~succ(n) = f a l s e succ(n)mO = false succ(n)msucc(m) = nmm This is a s p e c i f i c a t i o n of the natural numbers. Example 3.6: Let the above s p e c i f i c a t i o n be N. Let XxN + I be the parametric smecif i c a t i o n obtained from XI~X2 + 1 (example 2.1) by parameterized parameter massing with actual parameter ( f , ~ ) , where ~ : X •

>X + N and f sends XI to X ( f o r g e t t i n g the

index i ) , X2 to N, x2 to 0, and s 2 to m. Then, the algebraic domain equation

X -

P

• XxN+l

e

has stacks as s o l u t i o n s , specified by the f o l l o w i n g s p e c i f i c a t i o n (with obvious renamings):

126

sorts

S,N

ops

empty: --~S push : S , N - - - > S pop : S - - e S top : S---->N =

: SxS ---~BOOL

o.o (ops from N) eqs

pop(empty) = empty top(empty)= 0

pop(push(s,n))= s top(push(s,n))= n ...

(eqs from N and eqs f o r ~)

In a s i m i l a r way, we get trees w i t h a natural number attached to each node as s o l u t i o n s o f the domain equation X = XxXxN + 1, where = denotes an a p p r o p r i a t e mair (p,e) of morphisms, etc. 4. PARAMETERIZEDALGEBRAIC DOHAIN EQUATIONS Our theory so f a r gives a l g e b r a i c data types as s o l u t i o n s of a l g e b r a i c domain equations.

I t is natural to ask whether we can get parameterized data types as s o l u t i o n s

of parameterized a l g e b r a i c domain equations by a s i m i l a r method o f i m p l i c i t

soecifi-

c a t i o n and s y n t a c t i c s o l u t i o n . This works indeed, i f we proceed as f o l l o w s . D e f i n i t i o n 4.1:

A parameterized a l g e b r a i c domain equation is a t r i o l e

spec morphisms,

( r ; p , e ) of

P

y

r

> XY ~

>

~ e

XYP

such t h a t (1) (p,e) is an a l g e b r a i c domain equation, (2) r is a parameterized specification,

and (3) rp=re. ( r ; p , e )

is c a l l e d s t r o n g l ~ p e r s i s t e n t i f f

r has t h i s

property. Let (f,D) be an actual parameter f o r r . Then, using the mechanism of narameter passing as defined in the l a s t s e c t i o n , we can construct an a l g e b r a i c domain equation (p',e'), y

~

the (f_,D~-instance of ( r ; p , e ) ,

P

r

XY

(i)

Ifl ~

r ~ D .......

~

> XYP

(2) p

XD

> XDP e'

Fiiigure 4.

f2

as f o l l o w s ( c f . f i g u r e 4 . 1 ) :

127

i ) Let r ' , X_DD,fI be such that (1) is a oushout. 2) Let p',XDP_,f 2 be such that (2) is a pushout wrt p,p' 3) Define e' by r ' e ' are j o i n t l y D e f i n i t i o n 4.2:

= r ' p ' and f l e ' = e f 2 .

(e' is well defined s~nce r ' and f l

s u r j e c t i v e , and we e a s i l y prove that r f l e '

= fr'e'.)

A s o l u t i o n of a parameterized algebraic domain equation ( r ; p , e ) is

a parameterized data type (s,S) such t h a t , f o r each actual marameter ( f , ~ ) of r , (s,S) sends I D via f to the i n i t i a l

solution of the (f,D)-instance ( p ' , e ' )

of (r;p,e).

Of course, r and s must have the same source Y. Our main theorem 3.4 now extends to the parametric case as f o l l o w s . Theorem 4.3: Let: ( r ; p , e ) be a strongly persistent parameterized algebraic domain equation, and l e t (q,YQ) be the coequalizer of p and e. Then, i f s=rpq, ( s , s - f r e e ) is a s o l u t i o n of ( r ; p , e ) . Again, the proof is a l i t t l e

b i t lengthy and w i l l be published elsewhere. Examples

of i m p l i c i t parameteric s p e c i f i c a t i o n s are X=X×Y + i (stacks over Y), X=X~X~Y + I (trees over Y), etc. Here, = denotes a pair (p,e) of morphisms that are defined l i k e those in example 3.6. 5. CONCLUSIONS The theoretical results presented here provide a sound and consistent semantics f o r a new algebraic s p e c i f i c a t i o n method using parameterized s p e c i f i c a t i o n s and algebraic domain equations. The f e a s i b i l i t y

and usefulness of t h i s method f o r the development

of s p e c i f i c a t i o n methods and s p e c i f i c a t i o n languages should be subject to f u r t h e r study. Another possible area of a p p l i c a t i o n is the algebraic semantics of programming languages. In denotional semantics, domain equations are used e x t e n s i v e l y to specify the syntactic and semantic domains. Our theory can provide algebraic i n t e r p r e t a t i o n s f o r them. There i s , however, one d i f f i c u l t y : example ( i n i t i a l )

algebras of f i n i t e

we get only " f i n i t a r y "

sets or f i n i t e

solutions, for

functions. The central semantic

domains of environments and states usually are f i n i t a r y ,

so there seems to be no

problem. I t i s , however, not quite clear how to cope with cases l i k e procedure parameters. In p a r t i c u l a r , we cannot obtain a model f o r x-calculus with our method, l i k e Scott's r e f l e x i v e domain (SC 72b). For these and s i m i l a r cases, an extension of our theory to continuous algebras is necessary. This is subject to f u r t h e r study.

128 REFERENCES ADJ

77

Goguen,J.A./Thatcher,J.W./Wagner,E.G./Wright,J.B.: ! n i t i a l Algebra Semantics and Continuous Algebras. Journal AC~ 24,(1977), 68-95

ADJ

78

Goguen,J.A./ Thatcher,J.W./Wagner,E.G.: An I n i t i a l Algebra Agmroach to the SDecification, Correctness, and Implementation of Abstract Data Types.Current Trends in Programming Methodology,Vol IV (R.T. Yeh,ed.).Prentice Hall, Englewood C l i f f s 1978, 80-149

BG

77

Burstali,R.M./Goguen,J.A.: Putting Theories Together to Make Specifications. Proc. 5th Int. Joint Conf. on A r t i f i c i a l Intelligence, HIT, Cambridge (Mass.), 1977

BG

80

Burstall,R.N./Goguen,J.A.: The Semantics of CLEAR, a S~ecification Language. Proc. 1979 Copenhagen Winter School on Abstract Software Specifications (D. Bj6rner,ed.). LNCS 86, Springer-Verlag, Berlin i980, 292-331

EH

79

Ehrich,Ho-D.: On The Theory of Specification, Implementation, and Parameterization of Abstract ~ata Ty~es. Bericht Nr. 82/79, Ahtlg. Lnformatik,Univ. Dortmund 1979 (also to appear in Journal ACM)

EL

79a

Ehrich,H.-D./Lohberger,V.G.: Parametric Specification of Abstract Data Types, Parameter Substitution, and Graph Replacements. Gramhs, Data Structures, Algorithms (N.Nagl/H.-J. Schneider,eds.). Applied Computer Science 13,Hanser Verlag, MUnchen 1979, 169-182

EL

79b

Ehrich,H.-D./Lohberger,V.G.: Constructing Specifications of Abstract Data Types by Renlacements. Proc. Int. Workshop on Gramh Grammars and Their Application to Computer Science and Biology iV.Claus/ H.Ehrig/G.Rozenberg,eds.).LNCS 73,Springer-Verlag,Berlin 197- ,lo0-191

EKTWW

80a

Ehrig,H./Kreowski,H.-J./Thatcher,J.W./Wagner,E.G./Hright,J.B.: Parameterized Data Types in Algebraic Specification Languages. Proc.7th ICALP (J.~.deBakker/J.van Leeuwen,eds.) LNCS 85, Springer-Verlag, Berlin 1980, 157-168

EKTI:JW

80b

Ehrig,H./Kreowski,H.-J./Thatcher,J.W./Wagner,E.G./Wright,J.B.: Parameter Passing in Algebraic Specification Languages. Internal Re~ort, FB 20 TU Berlin, 1980

KA

78

Kanda,A.: Data Types as I n i t i a l Algebras: a Unification of Scottery and ADJery. Proc. 19th FOCS 1978, 221-230

LE

76

Lehmann,D.J.: Categories for Fixpoint Semantics. Proc.7th FOCS 1976, 122-126

LS

77

Lehmann,D.J./Smyth~M.B.:Data Types. Proc 18th FOCS 1977, 7-12

SC

71

Scott,D.S.:The Lattice of Flow Diagrams.Proc.Symp. on Semantics of Algorithmic Languages (E.Engeler,ed.).LNM 188, Smringer-Verlag, Berlin 1971, 311-372

SC

72a

Scott,D.S.: Lattice Theory,Data Types and Semantics.Formal Semanctics of Algorithmic Languages (R.Rustin,ed.).Prentice Hall, Englewood C l i f f s 1972, 65-106

SC

72b

Scott,D.S.: Continuous Lattices. Toposes, Algebraic Geometry and Logic (F.H.Lawvere,ed.).LNH 274,Springer-Verlag,Berlin 1972,97-136

129 SC

76

Scott,D.S.: Data Types as Lattices. SIAM Journal of Computing 5 (1976), 522-587

SP

77

Smyth~M.B./Plotkin,G.D.: The Category-Theoretic Solution of Recursive Domain Equations. Proc. 18th FOCS 1977, 13-17

WA

75

Wand,H.: On the Recursive Specification of Data Types. Proc. 1~t Int. Coil. on Category Theory A~plied to Computation and Control (E.G. Hanes,ed.). LNCS 25, Springer-Verlag, Berlin 1975, 214-217