On parametric algebraic specifications with clean error ... - Springer Link

ON PARAMETRIC ALGEBRAIC SPECIFICATIONS WITH CLEAN ERROR HANDLING

martin gogolla I n f o r m a t i k B, TU Braunschweig P o s t f a c h 3329,

D-3300 Braunschweig

ABSTRACT Usual a l g e b r a i c s p e c i f i c a t i o n partially

ordered s o r t s .

s u p e r s o r t s as w e l l tures

(e.g.

t e c h n i q u e s can be e x t e n d e d t o t r e a t

This a l l o w s t h e i n t r o d u c t i o n of

as o v e r l o a d e d o p e r a t o r s ,

existence

of

initial

a l g e b r a s and

a l g e b r a i c and o p e r a t i o n a l s e m a n t i c s ) o f tion

method

handling is

are preserved. studied.

On t h i s

proach tric

1.

equivalence

of

the equational specifica-

basis error

and

exception

a l g e b r a s which a r e o k / e r r o r -

and o k / e r r o r - c o m p l e t e ) a r e

considered.

a l l o w s t o p r o v e an e x t e n s i o n lemma f o r

specifications

fea-

For each s o r t an ok and an e r r o r s u b s o r t i s

i n t r o d u c e d and c l e a n a l g e b r a s ( i . e . consistent

sub- and

while pleasant

T h i s new

ap-

p e r s i s t e n t parame-

which p e r m i t e r r o r h a n d l i n g .

INTRODUCTION

During

the

promising

last

method f o r

programming approaches

tics

and p h i l o s o p h i e s f o r

final

[ 8 8 M 76,

opment

of

[EFH 8 3 ] ,

Partially treated etc.].

the specification

Among them a r e i n i t i a l [Wa 79, WPPDB 83, Ga 83] ST 8 5 ] .

specification

proved t o

be

There a r e

many such

[ADJ 76, ADJ 81, EKMP 82, and o b s e r v a t i o n a l seman-

Research i n t h e f i e l d languages l i k e

a

a b s t r a c t data types in

the a l g e b r a i c semantics of

led to the

OBJ [FGJM 8 5 ] ,

develACT

ONE

ASL [SW 83] and many o t h e r s .

ordered

sorts first

introduced in

i n a s e r i e s o f p a p e r s EGo 83, They a r e t h e b a s i s f o r

handling,

of

languages and s o f t w a r e e n g i n e e r i n g .

specifications. K1 84]~

years a l g e b r a i c s p e c i f i c a t i o n s

a t o p i c which i s

Po 84,

[Go 78] 8M

84,

have G~M

been 85,

our approach t o e r r o r and e x c e p t i o n

studied extensively in the literature

82 [ADJ 76, etc.].

Go 77, The

clean

Go 78,

BGP 82,

GDLE 82, Bi 84~ Po 84, BBC 8b,

fundamental new n o t i o n s i n t r o d u c e d h e r e a r e t h a t

a l g e b r a s and c l e a n s p e c i f i c a t i o n s ,

ok/error-consistency allows

and

where c l e a n r e f e r s

ok/error-completeness.

t h e use o f pure e r r o r v a r i a b l e s ,

before.

In

the

specifications

literature

o n l y [Po

This

approach possible

84]

considers

i m p o r t a n t because s p e c i a l problems a r i s e h e r e .

to

the

exception

extension

lemma

By t h i s

[Eh 81] and use i t

approach, guaranteeing the w e l l

parametric

which i s q u i t e

[Po 84] works w i t h

whereas we c a r r y o v e r p e r s i s t e n c y

h a n d l i n g case.

of

to

which was n o t

in connection with e r r o r handling,

non p e r s i s t e n t s p e c i f i c a t i o n s ,

of

we can a p p l y

for

definedness of

our

the

clean

R-

algebra

the a p p l i c a t i o n of

parametric specifications.

The paper i s o r g a n i z e d as f o l l o w s . i d e a s by means o f tal

definitions

fications.

some examples.

Chapter 2 i n t r o d u c e s t h e b a s i c Chapter 3 r e v i e w s t h e fundamen-

and f a c t s c o n c e r n i n g s u b s o r t s i n a l g e b r a i c s p e c i -

Chapter

4 t r e a t s c l e a n a l g e b r a s and c l e a n s p e c i f i c a -

tions.

Chapter

5

d i s c u s s e s p a r a m e t r i z a t i o n and

lemma.

Chapter

6

g i v e s some s h o r t c o n c l u d i n g remarks.

space l i m i t a t i o n s

2.

all

our

extension Due

to

p r o o f s are omitted.

THE BASIC IDEA

Our

main new c o n c e p t f o r

a clean algebra. for

the

ok

e r r o r and e x c e p t i o n h a n d l i n g i s

T h i s means t h a t our a l g e b r a s have two

and e r r o r p a r t o f each s o r t and

ok/error-consistent

(there

the

error).

The approach i s e x p l a i n e d b e s t by an example. Here i s our of

t h e n a t u r a l numbers.

s o r t s Nat opns

0 : - > Nat-Ok Succ: Error

Nat-Ok - > Nat-Ok : ->

Nat-Error

ok

and

and

spe~ N a t u r a l N u m b e r s W i t h E r r o r H a n d l i n g =

either

ok

are

error)

specification

( e v e r y element i s

subsorts

carriers

i s no element which i s b o t h

ok/error-complete

t h a t of

or

83

Succ,

Pred

:

Nat - > Nat

Plus,

Times :

Nat N a t - > Nat

vats

n : N a t n+,m+:Nat-Ok n - : N a t - E r r o r

e~

Succ ( n - ) Pred(O)

= n= Error

Pred (Succ ( n + ) ) Pred Entry-Error

Tree :

has

(2)

function

The

The f i r s t

occurrence such

one.

Entry-Ok ->

Tree-Ok

to the

corresponding to

(4)

important to

It

is

= O, o t h e r w i s e t h i s

(6)

and

axiom

for

derived

variable

the function

essential

use

can be c l a s s i -

The e r r o r

propagation for

Entry

Leaf

the

variables

as t h e n e x t e x a m p l e shows.

NoEntry :

in Nat

the opns-part)

~e~ec P a r a m e t r i c B i n a r y T r e e s = parm s o r t s

dis-

The s o r t

The f u n c t i o n s

t h e o~pD_s-part).

pure error

specifications,

one

Succ may a l s o be a p p l i e d

t h e axiom Times(O,n+)

constructors

n-

that

and s u b s o r t s a r e u s e d .

c a u s e an e r r o r

use

(I)

car-

makes no s t a t e m e n t a b o u t t h e n a t u r e o f

Three d i f f e r e n t

an ok v a r i a b l e

fied

Nat-Error

an ok v a l u e when a p p l i e d t o

The

would

an a l g e b r a h a v i n g as

T h e r e a r e some p e c u l i a r i t i e s

above w o r t h t o

that

result.

is

t h e s u b s o r t s N a t - O k and N a t - E r r o r .

is

assures

the specification

in

Suet.

parametric

84 Node :

Tree-Ok Tree-Ok - > Tree-Ok

NoTree : - > T r e e - E r r o r Leaf : E n t r y -> Tree Node : T r e e T r e e - > T r e e GetEntry :

Tree -> Entry

GetRight, GetLeft :

Tree -> Tree

vare

e+:Entry-Ok e-:Entry-Error

e_~ns

Leaf(e-)

t:Tree

tl+,t2+:Tree-Ok

= NoTree

N o d e ( N o T r e e , t > = N o d e ( t , N o T r e e ) = NoTree G e t E n t r y ( L e a f ( e + ) ) = e+ G e t E n t r y ( N o d e ( t l + , t 2 + > ) = NoEntry G e t E n t r y ( N o T r e e ) = NoEntry GetRight(Leaf(e+)) = GetLeft(Leaf(e+)) GetRight(Node(tl+,t2+))

= NoTree

= GetLeft(Node(t2+,tl+))

= t2+

G e t R i g h t ( N o T r e e ) = G e t L e f t ( N o T r e e ) = NoTree end spec The

specification

leaves sists

when it in

function t h e use o f If

is

builds

binary trees with given entries

applied.

the resulting GetEntry is the error

The g i v e n p a r a m e t e r s o r t

specification,

well

defined.

variable

e-

struction

the

T h i s can o n l y be a c h i e v e d by

in

would n o t be p e r s i s t e n t

per-

because

t h e axiom L e a f ( e - )

one would s p e c i f y o n l y L e a f ( N o E n t r y ) = N o T r e e ,

more e x c e p t i o n s t h a n t h e s i n g l e of

especially

the

at

Entry

for

error

=

NoTree.

then the con-

parameter algebras NoEntry.

Again,

having

lines

i-3

t h e 9 ~ D ~ - p a r t can be c o n s i d e r e d as t h e s i g n a t u r e s p e c i f i c a t i o n

for

t h e c o n s t r u c t o r s and l i n e s

ideas

4-7 f o r

the derived functions.

s k e t c h e d above a r e now made p r e c i s e i n

the (ollowing

The

chap-

ters.

3.

REVIEW OF ALGEBRAIC SPECIFICATIONS WITH SUBSORTS

The facts

following

remarks

r e v i e w the fundamental

and o u r n o t a t i o n c o n c e r n i n g a l g e b r a i c

subsorts.

Readers

etc.]

find

will

~amilar with

[Go 78,

many common d e t a i l s .

definitions

and

speci÷ications

Go 83,

Po 84,

and

SM 84,

85 3.1De÷inition A

signature

partial

(S,~,E)

c o n s i s t s of

(1) a s e t S of

sorts,

(2)

a

o r d e r ~ on S and (3) a f a m i l y Z=w~S* ssS of Bets of

function ~:v->r.

(Signature~ Algebra~ Morphism)

symbols Name(E)

such

that

(4) ~:w->s,

v~w

and

r~s

implies

= {~w'sl~eEw~s} denotes t h e f u n c t i o n names and

Symb(Z) = {~Id~Zw, s} the f u n c t i o n symbols of Z. A E - a l g e b r a (A,F) c o n s i s t s of that

(1) a f a m i l y A=s~S of s e t s such

(2) s~r i m p l i e s Asia r and (3) a f a m i l y F-< _ ~A w , s>~W'SsName(Z)

of f u n c t i o n s w i t h ~ ' S : A w - > As such that~

(4) i f

~:w->s,

~:v->r

and a~AwnAv, then ~ ' s ( a ) --~A v'r(a) A

Z-morphism

s~S

of

f:A->B

mappings such t h a t

a~Aw and (2) a~AsnAt

3.2 D e f i n i t i o n The

E-term

seS

of

s1...sn->s

between

E - a l g e b r a s A and B (i)

algebra

and

(I)

least

family

the for

d

:

functions ~:->s

and

f o r ~ : s l . . . s n - > s and t i E T s i .

of the term a l g e b r a )

E-term a l g e b r a TZ i s i n i t i a l

i n t h e c a t e g o r y ALGZ of a l l

E-

Z-morphisms between them.

(Congruence, Q u o t i e n t ) s on a E - a l g e b r a A i s a f a m i l y seS of

t i o n s ~s on As such t h a t implies

for

d : - > s i m p l i e s ~ T s and (2)

are determined by (3) ~ , s : = ~

(Initiality

Z-congruence

the

t i ~ T s i i m p l i e s d ( t l . . . t n ) ~ T s and

algebras with a l l

A

fs(~'S(a))=~'S(fw(a))

(Tz,F Z) has as c a r r i e r s

sets s a t i s f y i n g

3.4 D e f i n i t i o n

family

implies fs(a)=ft(a).

(4) ~ ? 1 " ' ' s n ' s ( t l . . . t n ) : = ~ ( t l . o . t n )

The

a

(Term a l g e b r a )

K~'S>dw,SEName(Z)

3.3 Fact

is

rela-

(1) ~s = [~EQ n AsXAs ] and (2) ai ~EQ bi

~ l"''sn's(al"''an)

~EQ ~ i ' ' ' u n ' r ( b l " ' ' b n )

for alibi

E

AsinAui ,

d : s l . . . s n - > s and d : u 1 . . . u n - > r , where ~EQ i s the e q u i v a u lence on sasAs generated by ~.

The q u o t i e nt A/~ of a E - a l g e b r a A by a E-congruence ~ has (1) the c a r r i e r s A/E s = { [ a ] l a e A s } ~ the

where [ a ] = { b ~ U-A la~-~b}

f u n c t i o n s ~w,S~Name(Z)

[~l'''sn's(bl...bn)],

and

(2)

sl"''sn,s([al]...Zan]):= w i t h ~A/~

where [ a i ] ~ A / ~ s i ~ [ a i ] = [ b i ]

and b i ~ A s i .

86 3.5 D e f i n i t i o n

(Equation~ S a t i s f a c t i o n ,

A

L=R i s a p a i r o f

~zeguation

signature Z Z-algebra cide.

Z(V)-terms,

having a d d i t i o n a l l y

A satisfies

A specification

L=R~ (Z,E)

Specification)

if

where

Z(V)

is

the

t h e v a r i a b l e s V as c o n s t a n t s . all

e v a l u a t i o n s o f L and R

A

coin-

c o n s i s t s o f a s i g n a t u r e Z and a s e t E

of E - e q u a t i o n s .

3 . 6 Fact A

(Induced Congruence)

s e t of E - e q u a t i o n s E i n d u c e s u n i q u e l y a s e t o f c o n s t a n t

tions

E(Tz) ,

which

a g a i n i n d u c e s a l e a s t congruence ~E

equaon

TZ

c o n t a i n i n g E(Tz)°

3 . 7 Fact

(Initiality

of

t h e q u o t i e n t term a l g e b r a )

The q u o t i e n t term a l g e b r a TZ/E E i s of a l l

(Z,E)-algebras satisfying

3 . 8 Example ( B i t s t r i n g s The

h a v i n g as s u b s o r t s

and s i n g l e b i t s

i n t h e c a t e g o r y ALGz, E

t h e e q u a t i o n s E.

handling)

error

following lines define bitstrings

String*)

~

avoiding

initial

of

arbitrary

non empty b i t s t r i n g s

length

(sort

( s o r t S t r i n g +)

(sort Bit).

B i t S t r i n g s A v o i d i n g E r r o r H a n d l i n g= sorts Bit

< String+ < String*

opns

: -> B i t

0,1

: -> S t r i n g

.

.l.

: S t r i n g * S t r i n g * -> S t r i n g *

.I.

: Bit

.l.

: String* Bit

First,

S t r i n g * -> S t r i n g + -> S t r i n g +

L a s t : S t r i n g + -> B i t

vats

b:Bit

e~ns

s l l ( s 2 1 s 3 ) = ( s l l s 2 ) Is3

e,sl~s2,s3:String*

sl~ = his = s First(bls)

= Last(slb)

= b

end spec Please

note

t h a t the s p e c i f i c a t i o n

s o r t s and v a t s has n o t r e a l l y determines

p a r t between t h e

t o be a s i g n a t u r e ,

but i t

a s i g n a t u r e i n t h e sense o f our d e f i n i t i o n .

more t h e f u n c t i o n s F i r s t

key

and L a s t r e t u r n i n g t h e f i r s t

words

uniquely Further-

respective-

87 ly

last

cally

bit

are well

defined,

because a l l

a l l o w e d by t h e s i g n a t u r e e i t h e r

applications syntacti-

yield

0 or

1.

3 . 9 Remark ( D e c l a r a t i o n s ) One can a l s o use so c a l l e d d e c l a r a t i o n s i n s p e c i f i c a t i o n s Go

83].

A d e c l a r a t i o n c o n s i s t s of

that

the

term w i l l

sort

(e.g.

a term and a

sort,

a l w a y s e v a l u a t e t o an element o f

i * i : N o n N e g a t i v e ~ where i

is

a v a r i a b l e of

[Go 787 assuring

the sort

given int).

4. CLEAN SPECIFICATIONS

4.1 D e f i n i t i o n A

signature

(Clean, o k / e r r o r - c o n s i s t e n t , (S,~,Z)

S-OKuS-ERROR,

is called

ok/error-complete)

(ok/error-)clean,

S-OK={s-OklsES-MAIN},

if

S=S-MAINu

S-ERROR={s-Error~sES-MAIN}

and ~ = { s ~ s l s s S } u { s - O k ~ s , s - E r r o r ~ s l s s S - M A I N } . A Z - a l g e b r a A w i t h Z a

clean s i g n a t u r e i s c a l l e d

As_Error and (3) A

= 0,

A is

specification is

ok~error-consistent,

(2~ o k / e r r o r - c o m p l e t e ,

clean, if

algebra

(I)

A

eeE(T~) i m p l i e s e i t h e r s u i t a b l e s o r t s.

As_Ok n

As_ok u As_Error

=

As,

o k / e r r o r - c o n s i s t e n t and o k / e r r o r - c o m p l e t e .

(Z,E)

clean.

if

if

is

called clean,

if

the

s e t E of e q u a t i o n s i s

initial

called

(E,E)-

clean,

eeTs_okXTs_Ok o r eeTs_ErrorXTs_Error

ALGz,E,CLEA N d e n o t e s t h e c a t e g o r y o f

(Z,E)-algebras with all

all

if

for

a

clean

morphisms between them.

4 . 2 C h a r a c t e r i s a t i o n ( S p e c i f i c a t i o n s w i t h c l e a n term a l g e b r a s ) Given the

a specification specification

(Z,E)

(~,E)

is

w i t h a c l e a n term a l g e b r a clean,

if

and o n l y i f

TE,

then

the set E

of

equations is clean.

4 . 3 C h a r a c t e r i s a t i o n (Clean s p e c i f i c a t i o n s ) A specification (I)

T~, E i s

(2)

there is

(E,E)

is clean, if

and o n l y i f

o k / e r r o r - c o n s i s t e n t and a subspecification

(a) ZG c o n t a i n i n g a l l ok o r e r r o r

result

(b) EG c o n t a i n i n g a l l

(ZG,EG)~(Z~E)

s o r t s and s u b s o r t s and s o r t s and c l e a n e q u a t i o n s o f E and

with all

operations with

88 (c)

there is

a unique s u r j e c t i v e

morphism f:~G,EG->U~_>EG(T%,E ) .

4 . 4 Remark ( S u r j e c t i v e morphiem i n If

t h e morphism f

of

(~B,EG)

then t h e r e a r e terms t l

that

TZG,EG ~ [ t l ] ~ [ t 2 ]

maximal s e t o f additional ok

is also injective,

above) then

(Z,E)

i s an e n r i c h m e n t

: T~G,ES and U~_>EB(T~, E) a r e i s o m o r p h i c . I f

injective, such

(c)

:

identification

and TZ, E ~ [ t l ] = [ t 2 ] .

in

t3eTs-(Ts_okUTs_Error), can

also

tl=t3

be done c h o o s i n g

involving

o n l y ok o r e r r o r t e r m s . this

establish

not

error

But EG i s

T~, E i s done v i a a term t 3

rule

out

is

a

e q u a t i o n s a p p l i c a b l e t o ok and e r r o r t e r m s , so t h e

identification

nor e r r o r

it

and t 2 b o t h ok o r b o t h

It

is

and

t3=t2.

different

a l s o much

neither

equations

smoother

case from a m e t h o d o l o g i c a l p o i n t o f view

a clear distinction

between ok and e r r o r

This

and

to to

constructors

and d e r i v e d f u n c t i o n s .

4 . 5 Concept ( P r a g m a t i c s f o r A

clean

specification

clean s p e c i f i c a t i o n s )

(Z,E)

should

(ZG,EG) w i t h T~B and EG c l e a n such t h a t

have

a

subspecification

(Z,E)

is

an e n r i c h m e n t o f

(ZG,EG).

4 . 6 Example ( B i t s t r i n g s This

with error handling)

clean s p e c i f i c a t i o n

defines bitstrings

of

arbitrary

E r r o r s a r e i n t r o d u c e d by t h e f u n c t i o n s Head and T a i l t o t h e empty s t r i n g . BitStrings = sorts Bit, cons

0,1

String : -> B i t - O k

NoHead : -> B i t - E r r o r : -> S t r i n g - O k • t.

: S t r i n g - O k B i t - O k -> S t r i n g - O k

NoTail : -> S t r i n g - E r r o r runes . I .

: String Bit

-> S t r i n g

Head : S t r i n g - > B i t Tail

: S t r i n g -> S t r i n g

vats

s:String s+:String-Ok b:Bit

eqns

N o T a i l l b = sINoHead = N o T a i l

length.

when a p p l i e d

b+,b1+,b2+:Bit-Ok

89 H e a d ( s + I b l + I b 2 + ) = He~d(s+Ib1+) Head(~ib+) = b+ Head(~) = Head(NoTail) = NoHead Tail(s+Ibl+lb2+) = Tail(s+Ibl+)ib2+ Tail(~Ib+) = T a i l ( ~ ) = T a i l ( N o T a i l ) = NoTail end spec The

p a r t s f o r t h e ok and e r r o r = o n s t r u c t o r s and f o r t h e

functions general

are

indicated

there will

by t h e keywords cons

be an e q u a t i o n p a r t f o r t h e

and

derived

funcs.

In

constructors

as

w e l l . For t h e s u b s o r t s t h e f o l l o w i n g e q u a t i o n s h o l d : T~,E,Bit_Ok ~{0,1},

T E , E , B i t _ E r r o r ~ { N o H e a d } v TE,E,String_Ok ~ ( { 0 ~ i } ) *

and

T E , E , S t r i n g _ E r r o r ~ { N o T a i l } . On t h i s b a s i s t h e f u n c t i o n s Head and T a i l a r e d e f i n e d such t h a t t h e s u b s o r t s a r e r e s p e c t e d .

5. CLEAN PARAMETRIC SPECIFICATIONS

5.1 D e f i n i t i o n A

( S i g n a t u r e morphism, s p e c i f i c a t i o n morphism)

s i g n a t u r e morphism f : Z 1 - > Z 2 between s i g n a t u r e s ( S I , ~ $ I , [ 1 ) and

($2,~$2,E2) Symb(E2)

consists

implies

forgetful

mappings

f:SI->S2

such t h a t s ~ S l r i m p l i e s f ( s ) ~ s 2 f ( r )

f ( ~ ) e £ 2 f ( w ) , f ( s ). s

s i g n a t u r e morphism f

c a l l e d s p e c i f i c a t i o n morphism from (Z1,EI) t o

specification

and

(E2,E2),

if

f

a is

every

b e l o n g s t o E2 : f ( E I ) ~ E 2 . A

i s c a l l e d simple,

if

$1~$2,

Symb(El)~

Symb(E2) and f : S I - > S 2 and f:Symb(El)->Symb(E2) a r e i n c l u s i o n s .

5.2 D e f i n i t i o n A

(Parametric s p e c i f i c a t i o n ~ p e r s i s t e n t )

p a r a m e t r i c s p e c i f i c a t i o n c o n e i s t s of a p a r a m e t e r s p e c i f i c a t i o n

{ZP,EP) EP~EB.

and The

a body s p e c i f i c a t i o n

(ZB~EB) such

c o n s t r u c t i o n F:ALGzp,Ep->ALGEB~EB [ADJ 78, specification rally"

that

EP~ZB and

semantics of a p a r a m e t r i c s p e c i f i c a t i o n i s t h e

is called persistent~

[WE 85] i s o m o r p h i c f o r

all

if

Po 8 4 ] .

free

A parametric

A and U(F(A)) a r e " n a t u -

(ZP~EP)-algebras A,

where U i s

90 the f o r g e t f u l

f u n c t o r U:ALGEB->ALGzp induced by t h e s i g n a t u r e s EP

and EB.

5.3 D e f i n i t i o n The

( A p p l i c a t i o n of a p a r a m e t r i c s p e c i f i c a t i o n )

r e s u l t of a p p l y i n g a p a r a m e t r i c s p e c i f i c a t i o n w i t h parameter

(ZP,EP) means

and body (~B,EB) of

a

t o an a c t u a l s p e c i f i c a t i o n

s p e c i f i c a t i o n morphism

specification

(~R,ER),

IF ssSP THEN h(s)

(ZA,EA)

h:(~P,EP)->(~A~EA)

is

by the

where ER=EA+hR(EB), ER=EA+hR(EB), hR(s) =

ELSE s FI

and hR(d) = IF ~sSymb(EP) THEN h(~)

ELSE d F I . (ZP~EP) . . . . ~ - - - >

(EB~EB)

I I h

I I

I

I hR

I

I

(ZA,EA) The

sR

> (ER~ER)

r e s u l t s p e c i f i c a t i o n i s t h e pushout of t h e a c t u a l s p e c i f i c a -

tion the

(ZA~EA) and t h e body s p e c i f i c a t i o n

(ZB~EB) w i t h r e s p e c t

parameter (EP,EP) and t h e s p e c i f i c a t i o n morphisms h

where

s

is

t h e s i m p l e s p e c i f i c a t i o n morphiem

induced

and by

to s, the

i n c l u s i o n of t h e parameter i n t h e body.

5.4 D e f i n i t i o n

(Clean p a r a m e t r i c s p e c i f i c a t i o n )

A

specification

parametric

(ZB,EB) the free

is called clean,

if

with

parameter

(EP~EP) and

f r e e c o n s t r u c t i o n F i s p e r s i s t e n t on ALBEp~EP~CLEAN and construction

i m p l i e s F(A)

body

t h e s i g n a t u r e s ZP and ZB a r e c l e a n ,

F p r e s e r v e s cleanness :

A

~

the

ALSEp,EP,CLEAN

E ALGEB,EB,CLEAN.

5 . 5 E x t e n s i o n Lemma ( f o r c l e a n p a r a m e t r i c s p e c i f i c a t i o n s ) Let

t h e r e be g i v e n a c l e a n p a r a m e t r i c s p e c i f i c a t i o n w i t h parame-

ter

(EP~EP) and body (~B,EB)~

(EA,EA)~

a

strict

actual

clean

specification

s p e c i f i c a t i o n morphism h: (EP~EP)->(EA~EA) and

the r e s u l t s p e c i f i c a t i o n (I)

an

(ER,ER) as d e f i n e d above.

The r e s u l t i n g p a r a m e t r i c s p e c i f i c a t i o n w i t h p a r a m e t e r (EA,EA) and body (ZR,ER) i s c l e a n : FR i s p e r s i s t e n t on ALGEA,EA~CLEAN and i t

preserves cleanness.

91 (2) F o Uh = UhR o FR. . . . . E___> ALG~p,EP,CLEAN ALGZB,EB,CLEAN

t

I

l

Uh ;

i UhR

l i

i I

ALGEA,EA,CLEAN . . . . ~ - - >

ALG~R,ER,CLEAN

5 . 6 Remark ( c o n c e r n i n g t h e e x t e n s i o n lemma) The p r o o f o f [Eh

81].

our e x t e n s i o n lemma a p p l i e s t h e R - e x t e n s i o n lemma o f

The

restriction

of ALBzp,EP t o c l e a n a l g e b r a s can

e x p r e s s e d as p r e d i c a t e f o r m u l a r e q u i r e m e n t s . clean

algebras

method,

because

neither

ok

morphism

the

underlying

The s t r i c t n e s s

o f t h e parameter

i m p l i e s t h a t ok o r e r r o r o p e r a t i o n s will

also

be ok o r e r r o r o p e r a t i o n s

to

specification

one does n o t want t o c a r e a b o u t e l e m e n t s

nor e r r o r .

h

parameter

is essential for

This r e s t r i c t i o n

be

being

passing

of

the

formal

in

the

actual

parameter.

5 . 7 Concept

(Pragmatics f o r

Analogously

to

clean parametric s p e c i f i c a t i o n s )

t h e case w i t h o u t p a r a m e t e r s a c l e a r

distinction

between ok and e r r o r c o n s t r u c t o r s and d e r i v e d f u n c t i o n s s h o u l d be established.

Therefore

parameter (ZP,EP) (ZP,EP)

a

clean parametric

~ (ZB,EG) ~ (ZB,EB)

all

w i t h T~G(A) and EG c l e a n such t h a t G

A~ALGEp,EP,CLEAN,

where G i s

duced by t h e p a r a m e t r i c s p e c i f i c a t i o n body

is

an e n r i c h m e n t o f B(A)

the f r e e c o n s t r u c t i o n w i t h p a r a m e t e r (EP,EP)

inand

(ZG,EG).

5 . 8 Example ( P a r a m e t r i c s t r i n g s

with error handling)

This clean parametric s p e c i f i c a t i o n trary

with

and body (ZB~EB) s h o u l d have a s u b s p e c i f i c a t i o n

i s p e r s i s t e n t on ALGzp,EP,CLEAN and F(A) for

specification

parameter

s o r t Char.

f u n c t i o n s Head and T a i l

o v e r an a r b i -

Again e r r o r s a r e i n t r o d u c e d by

when a p p l i e d t o t h e empty s t r i n g .

s pec P a r a m e t r i c S t r i n g s = parm s o r t s Char o~ons

defines strings

NoHead : -> C h a r - E r r o r

the

92 bod~ s o r t s S t r i n g cons

A : -> S t r i n g - O k .I.

: S t r i n g - O k Char-Ok -> S t r i n g - O k

NoTail : -> S t r i n g - E r r o r funcs

.I.

: S t r i n g Char - > S t r i n g

Head : S t r i n g -> Char Tail vats

: S t r i n g -> S t r i n g

s:String

s+:String-Ok

c:Char c + ~ c l + , c 2 + : C h a r - O k t - : C h a r - E r r o r eqns

NoTaillc = slc-

= NoTail

Head(s+Icl+Ic2+) = Head(s+Icl+) Head(Ale+) = c+ Head(A) = H e a d ( N o T a i l ) = NoHead Tail(s+Icl+Ic2+)

= Tail(s+Icl+)Ic2+

Tail(AIc+) = A Tail(A)

= T a i l ( N o T a i l ) = NoTail

e n d sp,ec The

parts

keywords for

the

it

is

for

the

parameter

p a r m and b o d ~ , parameter

and t h e

essential for

Char-Error algebra

In

and t h e

general

body are

there

constructors

will

indicated

be an e q u a t i o n

as w e l l .

= NoTail.

algebra

F(A)Char_Ok

F(A)

will

have

the

the part that sort

parameter

given~

following

note

c - of

I÷ a c l e a n

A w i t h s e t s AChar_Ok and AChar_Error i s

resulting

Please

p e r s i s t e n c y t o use t h e v a r i a b l e

in the equation s l c -

by

then t h e

carriers

:

~

AChar_Ok, F(A)Char-Error ~ AChar_Error , )* F ( A ) s t r i n g _ Ok ~ (AChar_Ok and F ( A ) s t r i n g _ E r r o r ~ { N o T a i l } . F u r t h e r m o r e t h e c o r r e s p o n d i n g f r e e c o n s t r u c t i o n i s n o t p e r s i s t e n t on ALG~p,Ep,

if

no r e s t r i c t i o n

t o c l e a n a l g e b r a s i s made.

5 . 9 Remark ( P o i n t e d a l g e b r a s and s p e c i f i c a t i o n s ) All

c o n s i d e r a t i o n s p r e s e n t e d here can be s p e c i a l i z e d t o

a l g e b r a s [Go 8 6 ] , sort.

In t h i s

pointed

where t h e r e i s o n l y one e r r o r element f o r

case e r r o r r e c o v e r y i s n o t s u p p o r t e d t o o w e l l ,

e s p e c i a l l y e r r o r p r o p a g a t i o n can be done a u t o m a t i c a l l y .

each but

93 6.

CONCLUSION

The

notion

algebra

of

a

clean algebra i s

satisfying

certain

a special

case

of

s o r t e q u a t i o n s which e s p e c i a l l y

sense i n t h e c o n t e x t o f p a r t i a l l y considered

just

an make

o r d e r e d s o r t s and which can

as a n o t h e r c o n s t r u c t f o r

algebraic specification

guages. For example i n c l e a n a l g e b r a s t h e s o r t

be lan-

equations

s-Ok n ~ - E r r o r = ~ and s-Ok u s - E r r o r = s are v a l i d f o r sort like

all

s o r t s s.

terms b u i l t union,

over the g i v e n s e t of

intersection,

An a l g e b r a s a t i s f i e s u a t i o n s of

A s o r t equation c o n s i s t s of a p a i r of s o r t s and s e t

operations

d i f f e r e n c e ~ complement and empty s e t .

a sort equation,

if

the set theoretic eval-

t h e two e x p r e s s i o n s w i t h r e s p e c t t o t h e g i v e n

coincide. This t o p i c

is

algebra

subject to f u t u r e research.

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