A characterisation of computable data types by means of a finite ...

A CHARACTERISATION OF COMPUTABLE DATA TYPES BY MEANS OF A FINITE EQUATIONAL SPECIFICATION METHOD

J.A. BERGSTRA

Department of Computer Science, University of Leiden, Wassenaarseweg 80, 2300 RA LEIDEN, The Netherlands J.V. TUCKER

Department of Computer Science, Mathematical Centre, 2e Boerhaavestraat 49, 1091 AL AMSTERDAM, The Netherlands

INTRODUCTION By redefining the construction of finite equational hidden function specifications of data types, as these are made with the initial algebra methodology of the ADJ Group, we are able to give an algebraic characterisation of the computable data types and data structures. Our technical motivation are the simple notions of strong and weakly normalised Church-Rosser replacement systems studied in the l-Calculus, and in plain mathematical terms the theorem we prove is this. THEOREM. Let A be a many-sorted

signature.

algebra finitely generated by elements named in its

Then the following statements are equivalent:

i. A is computable. 2. A possesses a finite, equational which is Church-Rosser 3. A possesses a finite, which is Church-Rosser

hidden enrichment replacement

system specification

and strongly normalising° equational hidden enrichment replacement

system specification

and weakly normalising.

The unexplained concepts are carefully defined in Section 2, on replacement system specifications, and in Section 3, on computable algebras. In Sections 4 and 5 we prove the theorem. Section i explains in detail the theoretical issues to do with data types which the theorem attempts to resolve. This paper continues our studies on the adequacy and power of definition of algebraic specification methods for data t~rpes which we began in [i~, see also [53.

(It

is an edited version of [2]; subsequent papers are [3,43.) Here the reader is assumed well versed in the initial algebra specification methods of the ADJ Group [63, see also KAMIN [73; knowledge of our [i~ is desirable but not, strictly speaking, essential. Prior exposure to the l-calculus is no~ required, of course, but hopefully the reader is acquainted with the Church-Rosser property from ROSEN [ii]. i. INITIAL ALGEBRA SEMANTICS AND DATA TYPES

A data structure

is defined to be a many-sorted algebra A finitely generated

77

from initial v a l u e s a l,...,a n ~ A n a m e d in its signature Z. A data type is d e f i n e d to be any class K of such d a t a structures of common signature. A t the h e a r t of the ADJ Group's theory of d a t a types is the idea that the semantics w h i c h e h a r a c t e r i s e a data type K should be i n v e s t e d in the c o n s t r u c t i o n of an initial a l g e b r a I K for K w i t h the result that every data structure A 6 K is u n i q u e l y definable, up to isomorphism,

as an

e p i m o r p h i c image of IK. In its turn, this initial a l g e b r a IK is u n i q u e l y definable, a g a i n up to isomorphism, as a factor a l g e b r a of the s y n t a c t i c a l g e b r a T(Z) of all terms over Z b e c a u s e T(Z)

is initial for the class ALG(Z) of all E-algebras. L e t IK ~ T ( Z ) / ~ K

w h e r e zK is a c o n g r u e n c e for w h i c h t £K t' m e a n s t h a t the terms t a n d t ' a r e

identical

syntactic e x p r e s s i o n s as far as the semantics of K is concerned. O b s e r v e that in these c i r c u m s t a n c e s we may p l a u s i b l y call a d a t a type

(semantics for) K computable w h e n

K is decidable on T(Z). And the p r o b l e m of s y n t a c t i c a l l y s p e c i f y i n g the data type K can

be i n v e s t i g a t e d through the p r o b l e m of s p e c i f y i n g the congruence =K" The p r e f e r r e d m e t h o d of p r e s c r i b i n g ~K is to use a finite set of equations or c o n d i t i o n a l e q u a t i o n s E over T(Z) T(Z)

to e s t a b l i s h a b a s i c set of i d e n t i f i c a t i o n s D E c

x T(Z) and to take z K as the s m a l l e s t c o n g r u e n c e =E on T(Z)

containing DE

With

reference to our [13, it is k n o w n that this m e t h o d w i l l not define all C o m p u t a b l e data types, b u t that it is able to define m a n y n o n - c o m p u t a b l e ones. E n r i c h i n g the m e t h o d to a l l o w the use of a finite n u m b e r of h i d d e n functions does enable it to s p e c i f y any c o m p u t a b l e data type, b u t can be shown to expand the n u m b e r of n o n - c o m p u t a b l e data types it defines. Our p r o p o s a l h e r e is to d e t e r m i n e the c o n g r u e n c e s for initial algebras b y m e a n s of replacement systems. A r e p l a c e m e n t system is i n t e n d e d to formalise a system of deductions,

g o v e r n e d b y simple algebraic s u b s t i t u t i o n rules, w i t h i n w h i c h a d e d u c t i o n

t ÷ t' says that the rSle of t can be p l a y e d b y t' as far as the semantics of K is concerned, m e a n i n g t + t' implies t =K - t' ' b u t n o t conversely. W h a t w e do is to m a k e an

analysis o f the c o n g r u e n c e =K t h r o u g h the s t r u c t u r a l b e h a v i o u r of a l g e b r a i c a l l y styled "proof systems" for it a n d f r o m this, a n d an a p p r o p r i a t e s p e c i f i c a t i o n machinery, w e are able to g u e s s the c l a s s i f i c a t i o n t h e o r e m for c o m p u t a b l e data types. T h a t this is

precisely the t h e o r e m stated in the I n t m o d ~ c t i o n comes from the r e f l e c t i o n that the semantics of a type K is supposed to be u n i q u e l y d e t e r m i n e d up to i s o m o r p h i s m w i t h an initial a l g e b r a IK, and n o t b y a p a r t i c u l a r s y n t a c t i c a l construction.

Since the comput-

ability of =K means the c o m p u t a b i l i t y of the algebra T(Z)/Z K u n d e r our d e f i n i t i o n and, in particular,

since this notion of an algebra's c o m p u t a b i l i t y is an i s o m o r p h i s m in-

variant, we can erase all m e n t i o n of syntax in the semantical c o n c e p t of a c o m p u t a b l e data type a n d i d e n t i f y these w i t h the c o m p u t a b l e algebras. So w i t h r e g a r d to the c o n t e n t of our theorem, the r e a d e r m a y care to c o n s i d e r the ease w i t h w h i c h s t a t e m e n t

(3) implies

(I) is p r o v e d as e v i d e n c e for the n a t u r a l signi-

ficance 6f strong and w e a k l y n o r m a l i s e d C h r u c h - R b s s e r r e p l a c e m e n t s y s t e m s p e c i f i c a t i o n s w h i l e the i m p l i c a t i o n

(1) implies

(2) m a y b e c o n s i d e r e d as the h a r d w o n answer to the

q u e s t i o n a b o u t adequacy: Do these specifications define all the data types one wants? B e c a u s e of the n o v e l t y of the s p e c i f i c a t i o n technique and the i n v o l v e d p r o o f of

78

the t h e o r e m we shall w o r k Q~t the m a t e r i a l which

it b e c o m e s

m u c h easier

of the t h e o r e m in the m a n y - s o r t e d In w h a t follows

2. R E P L A C E M E N T

The technical relation.

~ denotes

SYSTEMS

in the case of a single-sorted

for us to explain,

and the reader

algebra after

to understand, the p r o o f

case.

the set of n a t u r a l

numbers.

AND T H E I R S P E C I F I C A T I O N

p o i n t of d e p a r t u r e

is the idea of a t r a v e r s a l

L e t A b e a set a n d ~ an e q u i v a l e n c e

relation

for an e q u i v a l e n c e

on A. A traversal for ~ is a set

J c A wherein (i)

for e a c h a 6 A there is some t £ J such t h a t t H a; and

(ii)

if t,t'

Consider

6 J and t ~ t' then t ~ t'.

an initial

signature

algebra

specification

for a d a t a type K w h e r e

of K and E is some f o r m u l a or other for a x i o m a t i s ~ n g

the d e f i n i n g

congruence

H K is =E" The choice of a t r a v e r s a l

tional v i e w of the type as it is specified: imagines

(Z,E)

having

completing

to use E to c a l c u l a t e

these d e d u c t i o n s

theory about algebraic to a b s t r a c t

its p r o p e r t i e s

so that

J for H E fixes an opera-

g i v e n t,t'

E T(Z),

to decide

their p r e s c r i b e d

"normal

forms"

- t' one t =E

n,n' e J a n d on

t +E n, t' + E n' one checks n = n'. T h e f o l l o w i n g b i t of

replacement

the b a r e e s s e n t i a l s

systems

is m a d e up w i t h this in m i n d and is m e a n t

of such an o p e r a t i o n a l

L e t A b e a set a n d let R b e a reflexive, R as ÷ R so to d i s p l a y m e m b e r s h i p to b or t h a t b is a reduct

Z gives the

(a,b)

v i e w o f d a t a type specification.

transitive

binary relation

e R b y a + R b and say a reduces

(under R or +R)

on A. We write

(under R or ÷ R )

of a; and we shall call R and + R a reduction

system or a replacement system on the set A. F o l l o w i n g the t e r m i n o l o g y of t h e l-Calculus we m a k e these distinctions: An e l e m e n t

a c A is a normal form for + R if there is no b ~ A so t h a t a ~ b and

a + R b; the set of all normal The reduction

forms

for + R is d e n o t e d NF(R).

s y s t e m + R is Church-Rosser

if for a n y a ~ A if there are b l , b 2 6 A

so t h a t a + R bl a n d a + R b2 then there is c e A so t h a t b I + R c a n d b 2 + R csystem ÷ is weakly normalising if for e a c h a £ A there is some R f o r m b e A so t h a t a ÷ R b.

The reduction normal

The r e d u c t i o n

s y s t e m ÷ R is strongly normalising if there does n o t exist an infi-

nite chain

a0 + ~ a l ÷~ . . . . wherein

R an

+~ ...

for i e ~, a i # ai+ I.

A reduction system is Church-Rosser and weakly normalising if, and only if, every element reduces to a unique normal form. C l e a r l y s t r o n g n o r m a l i s a t i o n

ent&ilsweak

nor-

malisation. L e t HR d e n o t e exercise

the s m a l l e s t

to show that for a,a'

a HR a' ~=~ there

equivalence

is a s e q u e n c e

there exists

relation

on A c o n t a i n i n g

+R"

It is an e a s y

~ A a = bl,...,bk

a c o m m o n r e d u c t ci,

= a' such t h a t for each p a i r b i , b i + 1 i ~ i ~ k-l.

79

Schematically: a : bI

b2

cI

b3

c2

bk_ 1

b k = a'

Ck-i

U s i n g this c h a r a c t e r i s a t i o n of ~R it is s t r a i g h t f o r w a r d to p r o v e these facts. 2.1. LEPTA. a,a'

The r e p l a c e m e n t s y s t e m ÷ R on A is C h u r c h - R o s s e r if, and only if, for any

{ A i f a ~R a' then there is c { A so that a ÷ R c and a' "+R c.

2.2. LEPTA. Let ÷ the set of normal

b e a C h u r c h - R o s s e r weakly n o r m a l i s i n g r e p l a c e m e n t system on A. Then

R

forms NF(R)

is a traversal for =R"

Suppose n o w t h a t A is an algebra. T h e n b y an a l g e b r a i c r e p l a c e m e n t s y s t e m + R on the algebra A we m e a n a r e p l a c e m e n t system ÷ R on the domain of A w h i c h is closed u n d e r its o p e r a t i o n s in the sense that for each k - a r y o p e r a t i o n c of A,

al +R bl . . . . . ak+R bk

implies g(al . . . . . ak) +R g 0 and assume as i n d u c t i o n h y p o t h e s i s

that if r l , s l , u I are s t r o n g l y n o r m a l i s i n g and X(rl,s I) < n then _l(rl,sl,u I) is strongly normalising. equation

C o n s i d e r a r e d u c t i o n s e q u e n c e from t in w h i c h the f i r s t a p p l i c a t i o n of

(0) p r o d u c e s l(S(r'),s',_g(S(r'),s'))

from _l(r',s',S(T)) . B y our a s s u m p t i o n s

and the m a i n i n d u c t i o n h y p o t h e s i s all subterms of the n e w r e d u c t are s t r o n g l y n o r m a l ising. Moreover,

x(S(r'),s')

< x(r',s')

= X(r,s) = n and t h e r e f o r e b y the l a t e s t in-

d u c t i o n h y p o t h e s i s _i(S (r') ,s' ,g (S (r') ,s' ) ) is s t r o n g l y n o r m a l i s i n g a n d the r e d u c t i o n s e q u e n c e m u s t terminate. CASE 6. 1(x) = f(x) . This is, by now, obvious. H a v i n g c o n c l u d e d the p r o o f of L e m m a 4.3 we have also c o n t l u d e d the a r g u m e n t for L e m m a 4.2.

Q.E.D.

5. T H E M A N Y SORTED CASE We assume the r e a d e r t h o r o u g h l y a c q u a i n t e d w i t h the t e c h n i c a l foundations of the a l g e b r a of m a n y - s o r t e d structures for w h i c h no r e f e r e n c e can b e t t e r s u b s t i t u t e for the A D J ' s b a s i c p a p e r [6~. In n o t a t i o n c o n s i s t e n t w i t h our [IX, we assume A to be a m a n y - s o ~ t e d a l g e b r a w i t h domains A I , . . . , A n + m and o p e r a t i o n s of the form ~,~

= ~ l ' ' ' ' ' X k ;~ : A ~ I × . . . × A I k ,~,~ A

w h e r e I . ~ £ {l,...,n+m}, I ~ i ~ k. 1 The c o n c e p t s and m a c h i n e r y of Section 2 m u s t be reformulated, difficult:

b u t this is not

An algebraic replacement system R on A consists of a c o l l e c t i o n of set-

theoretic r e p l a c e m e n t systems R I , . . . , R n on its domains w h i c h s a t i s f y the p r o p e r t y that I,~ 0Z A, w i t h arguments a l l , . . . , a l k and b11,...,blk, w h e r e a ,b e A , If a ÷ b , ,a ÷ b then ~ l,~ (a , .. ,a ) ÷ ~i li li " 11 R1 11 "'" ~k Rk ~k 11 • Ik R~ I,~ (bll,...,blk). The c l a s s i f i c a t i o n of r e p l a c e m e n t systems a n d the d e f i n i t i o n s of for each o p e r a t i o n ~

the a s s o c i a t e d congruence, o n e - s t e p r e d u c t i o n s and so on as families of single sorted

88

relations p r o c e e d along the lines established single-sorted mdchanisms

to m a n y - s o r t e d

for specifying

To lift Section

algebras;

replacement

for generalising

algebraic

this is true of their properties

ideas from and of the

systems.

3 to computable m a n y - s o r t e d

algebras

is also quite straight-

forward and, in fact, has been virtually written out already in our [13. Those lemmas pertaining

to replacement

tion of sort indices

system specifications

require only the appropriate

introduc-

into their proofs.

Up to and including the proofs that full theorem in its m a n y - s o r t e d

(2) implies

(3), and

(3) implies

(i), for the

case, it m a y be truly said that no new ideas or tech-

niques are required. Consider

the proof that

ject of this section)

(i) implies

(2). With the help of a trick

(the real sub-

we are able to construct this proof with the toolkit of Section

4. Dispensing with an easy case where all the domains of A are finite, we assume A to be a m a n y - s o r t e d

computable

algebra with at least one domain infinite.

W i t h o u t loss of generality we can take these domains to be A I , . . . , A n, BI,...,B m where the A i are infinite and the B.l are finite of cardinality b.l + I. The generalised Lemma 3.1 provides

us with a recursive m a n y - s o r t e d

algebra of numbers R with domains

~l,...,~n and FI,...,F m where ~l = w for i ~ i ~ n, F i = {0,1,...,b i} for i -< i -< m, and R is isomorphic

to A. When not interested

in the cardinality

of a domain of R we

refer to it as R., i ~ i ~ n+m. The aim is to give R a finite equational I tion replacement system specification.

hidden func-

The first task is to build a recursive number algebra R 0 by adding to R new constants and functions. first infinite ing functions

The main idea is to code the many-sorted

sort ~I by means of functions

R i ÷ ~1 and ~i ÷ Ri affd recursive

on ~I associated to the m u l t i s o r t e d

we shall dissolve

algebra R into its

operations

track-

of R. At the same time

the finite sorts b y adding them as sets of constants.

Here is the

formal construction. For each infinite and as

the successor

new

sort i we add as a n e w constant of sort i the number 0 E ~i

function x+1. For each finite sort i we add

all

the elements of F. l

constants.

Each domain R. is coded into ~1 by adding the function foldl(x) = x, and is re1 covered b y adding the function unfoldl: ~ + R , defined for infinite sorts i b y 1 unfoldl(x) = x, and for finite sorts i b y

unfoldl(x)

= ix

if x ~ b i

Ib i

otherwise.

Next we add for each operation k

~i w h i c h commutes

f = fl,~ of R a recursive tracking function

the following diagram: f

X1

fold x...×fold

Xk

Rxlx'''XRxk

~ i~

1 ~ix...X~l

~ .....

~ el

unfoldp

f:

89

And,

just as in the s i n g l e - s o r t e d case, we factorise f into functions t , h , g and add

these a l o n g w i t h all the p r i m i t i v e recursive functions a r i s i n g from the p r i m i t i v e recursive d e f i n i t i o n s of h and g. T h a t is all. Observe R0] ~ = ~ = R, so it remains to give a finite e q u a t i o n a l r e p l a c e m e n t s y s t e m s p e c i f i c a t i o n for R 0 w h i c h is ChurchRosser and strongly normalising. Let Z0 be the signature of R 0 in w h i c h 12, iS, FOLD l, i UNFOLD name the zero, successor function, and c o d i n g maps a s s o c i a t e d to sort i; for c o n v e n i e n c e we d r o p the sort s u p e r s c r i p t in case i = i. Here are the requisite set of e q u a t i o n s E0, b e g i n n i n g w i t h the o p e r a t i o n s of R. Let f = fi,Z be an o p e r a t i o n of R n a m e d b y function symbol f { ~ c Z0 and let be its a s s o c i a t e d t r a c k i n g m a p on ~i n a m e d b y _ f e Z0" First,

f o l l o w i n g the p r o c e d u r e

of Section 4, w r i t e out all the equations a s s i g n e d to f and its factorisation. ly, add this e q u a t i o n to "eliminate"

f 1

f(Xll ..... Xlk)

~

Second-

UNFOLD~(~(FOLD

l(Xll) ....

,FOLD

1 k(Xlk))

w h e r e Xli is a v a r i a b l e of sort I i. D o this for e v e r y o p e r a t i o n of R. T u r n i n g to the coding machinery,

c o n s i d e r first the f o l d i n g functions.

For each

infinite sort i add the equations,

FOLDI(IO)

~ 0

FOLml(iS(Xi )) ~ S(FOLDI(Xi )) where X. is a v a r i a b l e of sort i. 1 For e a c h finite sort i, if i n E ~0

is a n e w c o n s t a n t of sort i d e n o t i n g num-

b e t c ~ r. then a d d 1 • FOLD i (iC)

~ SO(O).

S e c o n d l y c o n s i d e r the u n f o l d i n g functions. For e a c h i n f i n i t e sort i add the equations,

UNFOLD 1 (0) UNFOLD

i

~ i0

(S(X)) ~

iS(UNFOLDi(X) )

w h e r e X is a v a r i a b l e of sort i. For e a c h finite sort i, if

UNFOLD i (Sc (0))

i

=c is as b e f o r e then a d d the equations

> ie -

if c < b.

-> b. =l

if c ~ b. l

-

UNFOLD±(sC(x))

l

w h e r e b I• is the last e l e m e n t of F i and is n a m e d in ~ 0 - E b y b.=l; and X is a v a r i a b l e of sort I. A n d f i n a l l y we c o n s i d e r the equations for the constants. For each infinite sort i, if

ic_

~ E denotes the n u m b e r c { ~i then add ic >- isC(u). ~ For each finite sort i,

if ic 6 ~ denotes the n u m b e r e £ F. and ic { Z0 i remove the d u p l i c a t i o n b y adding ic > c. _ -

Z is its n e w c o n s t a n t symbol then we

T h i s completes the c o n s t r u c t i o n of E 0. W h a t remains of the p r o o f follows c l o s e l y the arguments of Section 4. Here the

90

sets of normal forms are, of course,

{isC (i0) : c ~ }

when i is an infinite sort, and

{ic:cEFi} when i is a finite sort. And the arguments which lift Lemma 4.1 and 4.2 are in all essential

respects the same. Given, then, that

(Z0,E0) is Church-Rosser

and

strongly normalising, the normal forms being a traversal for =E0, we can prove ~ T(E^, E ^) by using the mappings ~ i defined ~ i (c) = [ isC (0) i ] f or i an infinite R^• = U

U

,U

.

sort and @l(c) = lc for i a finite sort. REFERENCES [13

BERGSTRA, J.A. & J.V. TUCKER, Algebraic specifications of computable and semicomputable data structures, Mathematical Centre, Department of Computer Science Research Report IW 115, Amsterdam, 1979.

[23

~

, A characterisation of computable data types by means of a finite, equational specification method, Mathematical Centre, Department of Computer Science Research Report IW 124, Amsterdam, 1979.

[33

~

, Equational specifications for computable data types: six hidden functions suffice and other sufficiency bounds, Mathematical Centre, Department of Computer Science Research Report IW 128, Amsterdam, 1980.

[4]

- -

, On bounds for the specification of finite data types by means of equations and conditional equations, Mathematical C e n ~ e , Department of Computer Science Research Report IW 131, Amsterdam, 1980.

[5]

- -

, On the adequacy of finite equational methods for data type specification, ACM-SIGPLAN Notices 14 (11) (1979) 13-18.

[6]

GOGUEN, J.A., J.W. THATCHER & E.G. WAGNER, An initial algebra approach to the specification, correctness and implementation of abstract data types, in R.T. YEH (ed.) Current trends in programming methodology IV, Data structuring, Prentice-Hall, Engelwood Cliffs, New Jersey, 1978, 80-149.

[7]

KAMIN, S., Some definitions for algebraic data type specifications, Notices 14 (3) (1979) 28-37.

[8]

MACHTEY, M. & P. YOUNG, An introduction North-Holland, New York, 1978.

[9]

MAL'CEV,

A.I., Constructive

algebras,

SIGPLAN

to the general theory of algorithms,

I., Russian Mathematical

Surveys,

16 (1961)

77-129. [i03 RABIN, M.O., Computable algebra, general theory and the theory of computable fields, Transactions American Mathematical Society, 95 (1960) 341-360. [i13 ROSEN, B.K., Tree manipulating systems and Church-Rosser tion Computing Machinery, 20 (1973) 160-187.

theorems,

J. Associa-