Michael Hanus and I have been having some discussions recently on the current
Curry design document and, in particular, have been comparing the way Curry
and Escher handle various things. While there has been a substantial
convergence of Curry and Escher on certain matters, there are still have some
fundamental differences.
In the hope of getting some more debate going on the Curry design,
I enclose at the end of this message the latest Escher paper which summarises
the main aspects of the language. It seems to me to be important when
designing Curry to have available for comparion some alternative ways of doing
things. I hope the Escher paper will serve this purpose at least.
To be more specific, I would like to raise the following issues.
1. A longstanding difference between Curry and Escher is that the operational
semantics of Curry is based on narrowing (plus a little more to subsume
what Escher does), while Escher is based on rewriting. I think there is
agreement that the Escher model is simpler, but it's been argued that it
is not sufficiently flexible for various reasons. My experience in
writing a lot of Escher code is that it is flexible enough as it is. Since
narrowing and rewriting are the two obvious computational models for a
declarative language, perhaps some debate on this might somehow clarify
which is best or at least what the trade-offs are.
2. Escher emphasises set processing partly because I think set processing
facilities have been sadly neglected in language design (usually
programmers are expected to do everything with lists even when they are
not at all appropriate) and partly because I use sets to "encapsulate"
logic programming computations. I would like to see set processing added
to Curry. What do you think?
3. Curry will have concurrency. My preferred model for this is based on the
concurrent Haskell proposal which uses monads (Peyton Jones, Gordon,
Finne, POPL'96), except I've replaced their forkIO function by
something more "symmetric". Unfortunately, the write-up of this isn't
in the Escher paper, but one can see enough from the POPl paper to know
what it would look like. Is this the way to do concurrency in Curry?
If not, what should we do?
4. The Curry document introduces the type Constraint and the value Valid.
I do not understand the need for these. Certainly, one can write
constraint systems in the usual mathematical way and solve them without the
need for such things and, logically, Constraint is just Bool. In my view,
Constraint solving is primarily about building clever constraint solvers -
it shouldn't have a significant effect on the look or facilities of the
language. Is the current Curry proposal the best way to handle constraints
or can we find a way to make constraint solving blend more seamlessly into
the language?
I'm sure there are other important issues that could also be debated.
John Lloyd
*******************************************************************************
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Fl(if)i(the)f(\014rst)150 3275 y(argumen)n(t)f(is)h(an)g(in)n(teger,)h
(the)g(second)e(is)i(a)e(string,)i(the)g(third)f(is)h(an)f(asso)r
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(the)150 3474 y(\014rst)i(and)g(second)g(argumen)n(ts)f(if)h(and)g
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(otherwise,)h Fi(lookup)e Fl(maps)j(a)f(quadruple)g(to)g
Fi(False)p Fl(.)275 3673 y(F)-7 b(or)27 b(the)g(mo)r(dule)h
Fi(AssocList)p Fl(,)c(the)k(term)150 3825 y Fi(lookup\(5,)40
b(value,)h([\(4,)h("How"\),)f(\(5,)h("You"\)],)e(list\))150
3977 y Fl(reduces)27 b(to)150 4129 y Fi(\(value)41 b(==)i("You"\))e(&&)
i(\(list)e(==)i([\(4,)f("How"\),)f(\(5,)h("You"\)]\))150
4281 y Fl(and)27 b(the)h(term)150 4433 y Fi(lookup\(5,)40
b(value,)h([\(4,)h("How"\),)f(\(5,)h("You"\),)f(\(5,)h("Then"\)],)e
(list\))150 4585 y Fl(reduces)27 b(to)150 4737 y Fi(\(\(value)41
b(==)h("You"\))g(&&)g(\(list)g(==)h([\(4,)f("How"\),)e(\(5,)j("You"\),)
d(\(5,)j("Then"\)]\)\))c(||)150 4837 y(\(\(value)i(==)h("Then"\))f(&&)i
(\(list)e(==)i([\(4,)f("How"\),)f(\(5,)h("You"\),)f(\(5,)h
("Then"\)]\)\).)275 4989 y Fl(The)27 b(G\177)-42 b(odel)28
b(b)r(o)r(ok)f([9)o(])h(con)n(tains)f(man)n(y)g(programs)e(whic)n(h)j
(use)f(conditionals.)150 5218 y Fd(3.5)112 b(Higher-Order)37
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Fi(Relational)31 b Fl(b)r(elo)n(w)k(is)g(the)g(Esc)n(her)f(v)n(ersion)f
(of)i(a)g Fh(\025)p Fl(Prolog)e(program)h([17)o(].)59
b(The)36 b(function)150 5471 y Fi(mappred)20 b Fl(is)i(a)g(relational)f
(v)n(ersion)g(of)i(the)f(usual)h Fi(map)e Fl(function.)36
b(The)22 b(function)h Fi(forevery)c Fl(maps)k(a)f(predicate)150
5571 y(and)j(a)g(list)h(to)f Fi(True)f Fl(if)i(the)g(predicate)f(is)g
(true)h(for)f(eac)n(h)f(elemen)n(t)i(of)f(the)h(list;)h(otherwise,)e
(it)h(maps)f(to)g Fi(False)p Fl(.)275 5670 y(F)-7 b(or)27
b(the)g(mo)r(dule)h Fi(Relational)p Fl(,)c(the)k(term)p
eop
%%Page: 13 13
13 12 bop 150 100 a Fk(3)83 b(PR)n(OGRAMMING)27 b(EXAMPLES)2143
b Fl(13)p 150 324 3511 10 v 150 490 a Fi(module)41 b(ListProcessing\(m)
o(emb)o(er)o(,)d(append,)i(permute,)h(delete,)f(sorted\))h(where)g({)
150 690 y(member)g(::)i(\(a,)f([a]\))g(->)h(Bool;)150
789 y(member\(x,)d([]\))i(=)324 889 y(False;)150 989
y(member\(x,)e(y)j(:)g(z\))g(=)324 1088 y(\(x)g(==)g(y\))g(||)f
(member\(x,)e(z\);)150 1287 y(append)h(::)i(\([a],)e([a],)h([a]\))g(->)
h(Bool;)150 1387 y(append\(u,)d(v,)j(w\))f(=)324 1487
y(\(u)h(==)g([])g(&&)f(v)i(==)e(w\))h(||)g(exists)e([r,x,y])g(\(u)h(==)
h(r)g(:)g(x)h(&&)e(w)i(==)e(r)i(:)f(y)g(&&)g(append\(x,)c(v,)k(y\)\);)
150 1686 y(permute)e(::)h(\([a],)g([a]\))g(->)h(Bool;)150
1786 y(permute\([],)c(x\))k(=)324 1885 y(x)h(==)e([];)150
1985 y(permute\(x)e(:)j(y,)g(w\))g(=)324 2084 y(exists)e([u,v,z])g(\(w)
i(==)g(u)g(:)g(v)g(&&)g(delete\(u,)d(x)j(:)g(y,)g(z\))g(&&)f
(permute\(z,)e(v\)\);)150 2284 y(delete)h(::)i(\(a,)f([a],)g([a]\))g
(->)h(Bool;)150 2383 y(delete\(x,)d([],)i(y\))h(=)324
2483 y(False;)150 2583 y(delete\(x,)d(y)j(:)g(z,)g(w\))g(=)324
2682 y(\(x)g(==)g(y)g(&&)g(w)g(==)g(z\))f(||)324 2782
y(exists)f([v])i(\(w)g(==)f(y)h(:)h(v)f(&&)g(delete\(x,)c(z,)k(v\)\);)
150 2981 y(sorted)e(::)i([Int])e(->)i(Bool;)150 3081
y(sorted\([]\))c(=)324 3180 y(True;)150 3280 y(sorted\(x)h(:)j(y\))g(=)
324 3380 y(if)g(y)g(==)g([])324 3479 y(then)412 3579
y(True)324 3678 y(else)412 3778 y(exists)e([u,)h(v])h(\(y)f(==)h(u)g(:)
h(v)f(&&)g(x)g(<=)f(u)i(&&)e(sorted\(y\)\);)150 3977
y(})p 150 4143 V 150 4235 V 150 4401 a(module)f(AssocList\(lookup)o(\))
d(where)j({)150 4600 y(lookup)g(::)i(\(Int,)e(String,)g([Int,)h
(String],)e([Int,)h(String]\))g(->)h(Bool;)150 4700 y(lookup\(key,)d
(value,)i(assoc_list,)e(new_assoc_list\))f(=)324 4799
y(if)43 b(exists)e([v])i(member\(\(key,)38 b(v\),)43
b(assoc_list\))324 4899 y(then)412 4998 y(value)e(==)i(v)g(&&)412
5098 y(new_assoc_list)37 b(==)43 b(assoc_list)324 5198
y(else)412 5297 y(new_assoc_list)37 b(==)43 b(\(\(key,)e(value\))g(:)i
(assoc_list\);)150 5497 y(})p 150 5663 V eop
%%Page: 14 14
14 13 bop 150 100 a Fk(3)83 b(PR)n(OGRAMMING)27 b(EXAMPLES)2143
b Fl(14)p 150 317 3511 10 v 150 483 a Fi(module)41 b(Relational\(Perso)
o(n\(.)o(.\))o(,)d(parent,)i(age,)i(mappred,)e(forevery\))g(where)i({)
150 682 y(data)g(Person)f(=)87 b(Bob)42 b(|)h(John)f(|)h(Mary)f(|)i
(Sue)e(|)h(Dick)f(|)h(Kate)f(|)h(Ned;)150 881 y(parent)e(::)i
(\(Person,)d(Person\))h(->)i(Bool;)150 981 y(parent)e(=)324
1080 y({\(Bob,)g(John\),)368 1180 y(\(Bob,)g(Dick\),)368
1280 y(\(John,)g(Mary\),)368 1379 y(\(Sue,)g(Dick\),)368
1479 y(\(Dick,)g(Kate\)};)150 1678 y(age)h(::)h(\(Person,)d(Int\))i(->)
h(Bool;)150 1778 y(age)f(=)324 1877 y({\(Bob,)f(24\),)368
1977 y(\(John,)g(7\),)368 2077 y(\(Mary,)g(13\),)368
2176 y(\(Sue,)g(23\),)368 2276 y(\(Dick,)g(53\),)368
2376 y(\(Kate,)g(11\),)368 2475 y(\(Ned,)g(23\)};)150
2674 y(mappred)g(::)h(\(\(\(a,)g(b\))h(->)f(Bool\),)g([a],)g([b]\))f
(->)i(Bool;)150 2774 y(mappred\(p,)c([],)k(z\))g(=)324
2874 y(z)h(==)e([];)150 2973 y(mappred\(p,)d(x)44 b(:)f(xs,)f(z\))h(=)
324 3073 y(exists)e([y,ys])h(\(p\(x,)f(y\))i(&&)g(mappred\(p,)c(xs,)j
(ys\))h(&&)f(z)i(==)e(y)i(:)f(ys\);)150 3272 y(forevery)d(::)j(\(\(a)f
(->)h(Bool\),)e([a]\))h(->)h(Bool;)150 3372 y(forevery\(p,)c([]\))j(=)
324 3471 y(True;)150 3571 y(forevery\(p,)d(x)k(:)g(y\))g(=)324
3671 y(p\(x\))f(&&)324 3770 y(forevery\(p,)d(y\);)150
3970 y(})p 150 4136 V 150 4383 a(mappred\(age,)g([Bob,)i(Sue],)h(x\))
150 4548 y Fl(reduces)27 b(to)150 4713 y Fi(x)43 b(==)g([24,)f(23].)150
4977 y Fl(The)28 b(term)150 5142 y Fi(mappred\(parent,)37
b([Bob,)42 b(Dick],)f(x\))150 5306 y Fl(reduces)27 b(to)150
5471 y Fi(\(x)43 b(==)g([John,)e(Kate]\))g(||)150 5571
y(\(x)i(==)g([Dick,)e(Kate]\).)p eop
%%Page: 15 15
15 14 bop 150 100 a Fk(3)83 b(PR)n(OGRAMMING)27 b(EXAMPLES)2143
b Fl(15)150 390 y(The)28 b(term)150 556 y Fi(mappred\(r,)39
b([Bob,)j(Sue],)f([24,)h(23]\))150 722 y Fl(reduces)27
b(to)150 888 y Fi(r\(Bob,)41 b(24\))h(&&)h(r\(Sue,)e(23\).)150
1154 y Fl(The)28 b(term)150 1320 y Fi(mappred\(\\z)39
b(->)k(\(exists)e([x,y])g(\(z)i(==)g(\(x,)f(y\))h(&&)g(age\(y,)e
(x\)\)\),)g([24,)h(23],)g(w\))150 1486 y Fl(reduces)27
b(to)150 1652 y Fi(\(w)43 b(==)g([Bob,)e(Sue]\))h(||)150
1752 y(\(w)h(==)g([Bob,)e(Ned]\).)150 2017 y Fl(The)28
b(term)150 2183 y Fi(\(\\x)42 b(->)h(age\(x,)e(24\)\)\(Bob\))150
2350 y Fl(reduces)27 b(to)150 2516 y Fi(True.)150 2781
y Fl(The)h(term)150 2947 y Fi(forevery\(\\x)39 b(->)k(age\(x,)e(y\),)h
([Ned,)g(Bob,)g(Sue]\))150 3113 y Fl(reduces)27 b(to)150
3279 y Fi(False.)150 3545 y Fl(The)h(term)150 3711 y
Fi(forevery\(\\x)39 b(->)k(age\(x,)e(y\),)h([Ned,)g(Sue]\))150
3877 y Fl(reduces)27 b(to)150 4043 y Fi(y)43 b(==)g(23.)150
4309 y Fl(The)28 b(term)150 4475 y Fi(forevery\(\\x)39
b(->)k(\(exists)d([y])j(age\(x,)e(y\)\),)h([Ned,)f(Bob,)h(Sue]\))150
4641 y Fl(reduces)27 b(to)150 4807 y Fi(True.)p eop
%%Page: 16 16
16 15 bop 150 100 a Fk(3)83 b(PR)n(OGRAMMING)27 b(EXAMPLES)2143
b Fl(16)150 390 y Fd(3.6)112 b(Databases)150 543 y Fl(The)35
b(next)f(example)h(illustrates)f(t)n(ypical)g(database)f(querying)h(in)
h(Esc)n(her.)56 b(The)35 b(mo)r(dule)g Fi(England)c Fl(b)r(elo)n(w)150
643 y(con)n(tains)c(an)g(\(incomplete\))h(database)f(ab)r(out)g(cities)
h(and)f(coun)n(ties)g(in)h(England.)p 131 842 3511 10
v 150 988 a Fi(module)41 b(England\(County\(.)o(.\),)c(City\(..\),)j
(neighbours,)f(distance,)h(isin\))h(where)h({)150 1187
y(data)86 b(County)41 b(=)i(Avon)f(|)h(Bedfordshire)c(|)k(Berkshire)d
(|)j(Buckinghamshire)37 b(|)804 1287 y(Cambridgeshire)h(|)43
b(Cornwall)d(|)j(Devon)f(|)h(Dorset)e(|)i(Essex)f(|)804
1386 y(Gloucestershire)37 b(|)43 b(Hampshire)d(|Herefordshire)e(|)43
b(Hertfordshire)38 b(|)804 1486 y(Kent)k(|)h(London)e(|)i
(Northamptonshire)37 b(|)43 b(Oxfordshire)c(|)44 b(Somerset)c(|)804
1586 y(Surrey)h(|)i(Sussex)e(|)i(Warwickshire)c(|)k(Wiltshire)d(|)j
(Worcestershire;)150 1785 y(data)f(City)g(=)h(Bath)f(|)h(Bournemouth)c
(|)k(Bristol)e(|)i(Cheltenham)d(|)j(Cirencester)c(|)673
1885 y(Dorchester)g(|)44 b(Exeter)d(|)i(Gloucester)c(|)44
b(Penzance)c(|)j(Plymouth)d(|)673 1984 y(Salisbury)g(|)j(Shaftesbury)c
(|)k(Sherbourne)d(|)j(Taunton)e(|)i(Torquay)d(|)673 2084
y(Truro)i(|)h(Winchester;)150 2383 y(neighbours)c(::)k(\(County,)d
(County\))h(->)i(Bool;)150 2582 y(neighbours)c(=)324
2682 y({\(Devon,)i(Cornwall\),)368 2781 y(\(Devon,)g(Dorset\),)368
2881 y(\(Devon,)g(Somerset\),)368 2980 y(\(Avon,)g(Somerset\),)368
3080 y(\(Avon,)g(Wiltshire\),)368 3180 y(\(Avon,)g(Gloucestershire\))o
(,)368 3279 y(\(Dorset,)f(Wiltshire\),)368 3379 y(\(Somerset,)f
(Wiltshire\),)368 3479 y(\(Gloucestershire)o(,)e(Wiltshire\),)368
3578 y(\(Dorset,)j(Somerset\),)368 3678 y(\(Dorset,)g(Hampshire\),)368
3778 y(\(Hampshire,)f(Wiltshire\),)368 3877 y(\(Hampshire,)g
(Berkshire\),)368 3977 y(\(Hampshire,)g(Sussex\),)368
4076 y(\(Hampshire,)g(Surrey\),)368 4176 y(\(Sussex,)h(Surrey\),)368
4276 y(\(Sussex,)g(Kent\),)368 4375 y(\(London,)g(Surrey\),)368
4475 y(\(London,)g(Kent\),)368 4575 y(\(London,)g(Essex\),)368
4674 y(\(London,)g(Hertfordshire\),)368 4774 y(\(London,)g
(Buckinghamshire\),)368 4873 y(\(Surrey,)g(Buckinghamshire\),)368
4973 y(\(Surrey,)g(Kent\),)368 5073 y(\(Surrey,)g(Berkshire\),)368
5172 y(\(Oxfordshire,)e(Berkshire\),)368 5272 y(\(Oxfordshire,)g
(Wiltshire\),)368 5372 y(\(Oxfordshire,)g(Gloucestershire\),)368
5471 y(\(Oxfordshire,)g(Warwickshire\),)368 5571 y(\(Oxfordshire,)g
(Northamptonshire\))o(,)368 5670 y(\(Oxfordshire,)g(Buckinghamshire\),)
p eop
%%Page: 17 17
17 16 bop 150 100 a Fk(3)83 b(PR)n(OGRAMMING)27 b(EXAMPLES)2143
b Fl(17)368 390 y Fi(\(Berkshire,)39 b(Wiltshire\),)368
490 y(\(Berkshire,)g(Buckinghamshire\))o(,)368 589 y(\(Gloucestershire)
o(,)e(Worcestershire\),)368 689 y(\(Worcestershire,)g(Herefordshire\),)
368 789 y(\(Worcestershire,)g(Warwickshire\),)368 888
y(\(Bedfordshire,)h(Buckinghamshire\))o(,)368 988 y(\(Bedfordshire,)g
(Northamptonshire)o(\),)368 1088 y(\(Bedfordshire,)g(Cambridgeshire\),)
368 1187 y(\(Bedfordshire,)g(Hertfordshire\),)368 1287
y(\(Hertfordshire,)f(Essex\),)368 1386 y(\(Hertfordshire,)g
(Cambridgeshire\),)368 1486 y(\(Hertfordshire,)g(Buckinghamshire\),)368
1586 y(\(Buckinghamshire)o(,)g(Northamptonshire\))o(};)150
1885 y(distance)j(::)j(\(City,)e(City,)h(Int\))g(->)g(Bool;)150
2084 y(distance)e(=)324 2183 y({\(Plymouth,)f(Exeter,)i(42\),)368
2283 y(\(Exeter,)f(Bournemouth,)f(82\),)368 2383 y(\(Bristol,)h
(Taunton,)g(43\),)368 2482 y(\(Bristol,)g(Gloucester,)f(35\),)368
2582 y(\(Torquay,)h(Exeter,)g(23\),)368 2682 y(\(Plymouth,)f(Torquay,)i
(24\),)368 2781 y(\(Bristol,)f(Bath,)h(13\),)368 2881
y(\(Exeter,)f(Taunton,)g(34\),)368 2980 y(\(Penzance,)f(Plymouth,)h
(78\),)368 3080 y(\(Taunton,)g(Bournemouth,)e(70\),)368
3180 y(\(Bournemouth,)g(Salisbury,)i(28\),)368 3279 y(\(Taunton,)g
(Salisbury,)f(64\),)368 3379 y(\(Salisbury,)g(Bath,)i(40\),)368
3479 y(\(Bath,)g(Gloucester,)e(39\),)368 3578 y(\(Bournemouth,)f(Bath,)
k(65\),)368 3678 y(\(Truro,)f(Penzance,)e(26\),)368 3778
y(\(Plymouth,)g(Truro,)i(52\),)368 3877 y(\(Shaftesbury,)d(Salisbury,)i
(20\),)368 3977 y(\(Sherbourne,)f(Shaftesbury,)f(16\),)368
4076 y(\(Dorchester,)h(Bournemouth,)f(28\),)368 4176
y(\(Salisbury,)h(Winchester,)g(24\),)368 4276 y(\(Exeter,)h
(Sherbourne,)f(53\),)368 4375 y(\(Sherbourne,)g(Taunton,)h(29\),)368
4475 y(\(Bath,)h(Cirencester,)e(32\),)368 4575 y(\(Cirencester,)f
(Cheltenham,)h(16\),)368 4674 y(\(Cheltenham,)g(Gloucester,)g(9\),)368
4774 y(\(Dorchester,)g(Sherbourne,)g(19\),)368 4873 y(\(Bath,)i
(Shaftesbury,)e(33\),)368 4973 y(\(Winchester,)g(Bournemouth,)f(41\),)
368 5073 y(\(Exeter,)i(Dorchester,)f(53\)};)150 5372
y(isin)j(::)h(\(City,)e(County\))f(->)j(Bool;)150 5571
y(isin)f(=)324 5670 y({\(Bristol,)e(Avon\),)p eop
%%Page: 18 18
18 17 bop 150 100 a Fk(3)83 b(PR)n(OGRAMMING)27 b(EXAMPLES)2143
b Fl(18)368 390 y Fi(\(Taunton,)40 b(Somerset\),)368
490 y(\(Salisbury,)f(Wiltshire\),)368 589 y(\(Bath,)i(Avon\),)368
689 y(\(Bournemouth,)d(Dorset\),)368 789 y(\(Gloucester,)h
(Gloucestershire)o(\),)368 888 y(\(Torquay,)h(Devon\),)368
988 y(\(Penzance,)f(Cornwall\),)368 1088 y(\(Plymouth,)g(Devon\),)368
1187 y(\(Exeter,)h(Devon\),)368 1287 y(\(Winchester,)f(Hampshire\),)368
1386 y(\(Dorchester,)g(Dorset\),)368 1486 y(\(Cirencester,)f
(Gloucestershire\),)368 1586 y(\(Truro,)j(Cornwall\),)368
1685 y(\(Cheltenham,)e(Gloucestershire)o(\),)368 1785
y(\(Shaftesbury,)f(Dorset\),)368 1885 y(\(Sherbourne,)h(Dorset\)};)150
2084 y(})p 150 2250 3511 10 v 275 2449 a Fl(The)28 b(predicates)g
Fi(neighbours)p Fl(,)c Fi(distance)p Fl(,)i(and)i Fi(isin)f
Fl(ha)n(v)n(e)g(b)r(een)i(written)g(in)f(the)h(most)f(general)f(form)
150 2549 y(p)r(ossible)f(to)h(allo)n(w)e(\015exible)i(querying)e(of)i
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2748 y(only)33 b(ev)n(er)g(in)n(v)n(olv)n(e)f(asking)h(ab)r(out)h
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(it)g(w)n(ould)f(b)r(e)i(p)r(ossible)e(to)150 2848 y(replace)27
b(the)h(predicate)f Fi(distance)d Fl(in)k(the)g(mo)r(dule)g
Fi(England)c Fl(b)n(y)k(a)f(\(more)g(e\016cien)n(t\))h(function)150
2947 y Fi(distance)40 b(::)j(\(City,)e(City\))h(->)g(Int)p
Fl(.)275 3047 y(Note)30 b(the)i(use)e(of)h(underscores)e(in)j(some)e
(queries.)46 b(An)31 b(underscore)e(\()p Fi(_)p Fl(\))j(stands)e(for)h
(a)f(unique)h(v)-5 b(ariable)150 3147 y(whic)n(h)41 b(is)f
(\(implicitly\))i(existen)n(tially)e(quan)n(ti\014ed)h(at)f(the)h(fron)
n(t)f(of)h(the)g(smallest)f(subterm)h(of)f(the)h(form)150
3246 y Fh(f)9 b Fl(\()p Fh(t)262 3258 y Fg(1)299 3246
y Fh(;)14 b(:)g(:)g(:)g(;)g(t)514 3258 y Ff(n)559 3246
y Fl(\))37 b(whic)n(h)h(has)f(t)n(yp)r(e)g Fi(Bool)f
Fl(and)h(con)n(tains)f(the)i(underscore.)65 b(So,)39
b(for)e(example,)i(the)f(subterm)150 3346 y Fi(isin\()p
375 3346 27 4 v 29 w(,)43 b(x\))25 b Fl(in)g(query)f(8)g(is)h(an)f
(abbreviation)f(for)i Fi(exists)41 b([w])h(isin\(w,)f(x\))p
Fl(,)25 b(where)f Fi(w)g Fl(is)h(a)f(new)h(v)-5 b(ariable.)150
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b(x)h(is)g(a)g(variable;)d(and)i(y)h(is)g(not)f(a)i(variable.)150
2631 y(\(x1,...,xn\))39 b(==)k(\(y1,...,yn\))82 b(=)87
b(\(x1)43 b(==)f(y1\))h(&&)f(...)h(&&)f(\(xn)h(==)g(yn\);)150
2830 y({x)g(|)g(u})g(==)f({y)h(|)g(v})87 b(=)f(\({x)43
b(|)g(u})g(subset)e({y)i(|)g(v}\))f(&&)h(\({y)f(|)h(v})g(subset)e({x)i
(|)g(u}\);)150 3129 y(\(/=\))f(::)h(a)g(->)g(a)g(->)g(Bool;)150
3328 y(x)g(/=)g(y)87 b(=)f(not)43 b(\(x)g(==)f(y\);)150
3627 y(\(&&\))g(::)h(Bool)f(->)g(Bool)g(->)h(Bool;)150
3827 y(True)f(&&)h(x)86 b(=)h(x;)150 4026 y(x)43 b(&&)g(True)85
b(=)i(x;)150 4225 y(False)42 b(&&)g(x)87 b(=)g(False;)150
4424 y(x)43 b(&&)g(False)85 b(=)i(False;)150 4624 y(\(x)43
b(||)g(y\))f(&&)h(z)87 b(=)g(\(x)42 b(&&)h(z\))g(||)g(\(y)f(&&)h(z\);)
150 4823 y(x)g(&&)g(\(y)g(||)f(z\))87 b(=)g(\(x)42 b(&&)h(y\))g(||)g
(\(x)f(&&)h(z\);)150 5022 y(x)g(&&)g(\(exists)e([x1,...,xn])e(v\))j(&&)
h(y)87 b(=)g(exists)41 b([x1,...,xn])e(\(x)k(&&)f(v)i(&&)e(y\);)150
5122 y(--)150 5221 y(--)86 b(where)42 b(each)g(xi)h(is)f(not)h(free)f
(in)g(x)i(and)e(y;)h(and)f(x)h(or)g(y)g(may)f(be)h(absent.)150
5421 y(y)g(&&)g(\(x)g(==)f(u\))h(&&)g(z)87 b(=)f(y{x/u})42
b(&&)g(\(x)h(==)g(u\))g(&&)f(z{x/u};)150 5520 y(--)150
5620 y(--)86 b(where)42 b(x)h(is)g(a)g(variable;)d(x)j(is)g(not)f(free)
g(in)h(u;)g(x)g(is)f(free)g(in)h(y)g(or)g(z;)p eop
%%Page: 28 28
28 27 bop 150 100 a Fk(C)83 b Fi(BOOLEANS)24 b Fk(MODULE)2548
b Fl(28)150 390 y Fi(--)86 b(u)44 b(is)e(free)g(for)h(x)g(in)f(y)i(and)
e(z;)h(and)f(y)h(or)g(z)g(may)g(be)f(absent.)150 689
y(\(||\))g(::)h(Bool)f(->)g(Bool)g(->)h(Bool;)150 888
y(True)f(||)h(x)86 b(=)h(True;)150 1088 y(x)43 b(||)g(True)85
b(=)i(True;)150 1287 y(False)42 b(||)g(x)87 b(=)g(x;)150
1486 y(x)43 b(||)g(False)85 b(=)i(x;)150 1685 y(--)f(Rewrites)41
b(for)h(the)g(conditional)d(syntactic)h(sugar.)150 1885
y(if)j(exists)e([x1,...,xn])e(True)j(then)g(x)h(else)f(y)87
b(=)f(exists)41 b([x1,...,xn])f(x;)150 2084 y(if)j(exists)e
([x1,...,xn])e(False)i(then)h(x)i(else)d(y)87 b(=)g(y;)150
2283 y(if)43 b(exists)e([x1,...,xn])e(\(x)k(&&)f(\(xi)h(==)g(u\))f(&&)h
(y\))g(then)f(z)h(else)f(v)87 b(=)324 2383 y(if)43 b(exists)e
([x1,...,xi-1,xi+1)o(,.)o(..,)o(xn)o(])c(\(x{xi/u})k(&&)h(y{xi/u}\))f
(then)h(z{xi/u})e(else)i(v;)150 2482 y(--)150 2582 y(--)86
b(where)42 b(xi)h(is)f(not)h(free)f(in)g(u;)h(u)g(is)g(free)f(for)g(xi)
h(in)g(x,)g(y)g(and)f(z;)h(and)f(x)h(or)g(y)150 2682
y(--)86 b(\(or)43 b(both\))e(may)i(be)f(absent.)150 2881
y(if)h(exists)e([x1,...,xn])e(\(x)k(||)f(\(xi)h(==)g(u\))f(||)h(y\))g
(then)f(z)h(else)f(v)87 b(=)324 2980 y(exists)41 b([x1,...,xn])e(\(\(x)
k(||)g(\(xi)f(==)h(u\))f(||)h(y\))g(&&)g(z\);)150 3080
y(--)150 3180 y(--)86 b(where)42 b(x)h(or)g(y)g(\(or)f(both\))g(may)g
(be)h(absent.)150 3479 y(not)f(::)h(Bool)f(->)h(Bool;)150
3678 y(not)f(False)85 b(=)i(True;)150 3877 y(not)42 b(True)86
b(=)h(False;)150 4076 y(not)42 b(not)h(x)86 b(=)h(x;)150
4276 y(not)42 b(\(x)h(||)g(y\))86 b(=)h(\(not)42 b(x\))h(&&)f(\(not)g
(y\);)150 4475 y(not)g(\(x)h(&&)g(y\))86 b(=)h(\(not)42
b(x\))h(||)f(\(not)g(y\);)150 4774 y(sigma)g(::)g(\(a)h(->)g(Bool\))e
(->)i(Bool;)150 4973 y(exists)e([x1,...,xn])e(True)86
b(=)g(True;)150 5172 y(exists)41 b([x1,...,xn])e(False)85
b(=)i(False;)150 5372 y(exists)41 b([x1,...,xn])e(\(x)k(||)g(y\))86
b(=)h(\(exists)40 b([x1,...,xn])f(x\))k(||)g(\(exists)e([x1,...,xn])e
(y\);)150 5571 y(exists)i([x1,...,xn])e(\(x)k(&&)g(\(xi)f(==)h(u\))g
(&&)f(y\))87 b(=)324 5670 y(exists)41 b([x1,...,xi-1,xi+1)o(,..)o(.,)o
(xn])c(\(x{xi/u})j(&&)j(y{xi/u}\);)p eop
%%Page: 29 29
29 28 bop 150 100 a Fk(C)83 b Fi(BOOLEANS)24 b Fk(MODULE)2548
b Fl(29)150 390 y Fi(--)150 490 y(--)86 b(where)42 b(xi)h(is)f(not)h
(free)f(in)g(u;)h(u)g(is)g(free)f(for)g(xi)h(in)g(x)g(and)f(y;)h(and)g
(x)g(or)f(y)i(\(or)e(both\))150 589 y(--)86 b(may)43
b(be)f(absent.)f(If)i(both)f(x)h(and)f(y)i(are)e(absent,)f(then)g(the)i
(RHS)f(is)h(True.)150 888 y(pi)g(::)g(\(a)f(->)h(Bool\))f(->)g(Bool;)
150 1088 y(forall)f([x1,...,xn])e(\(False)i(-->)i(u\))86
b(=)h(True;)150 1287 y(forall)41 b([x1,...,xn])e(\(\(x1)j(==)h(t1\))f
(&&)h(...)f(&&)h(\(xn)f(==)h(tn\))f(-->)h(u\))86 b(=)h
(u{x1/t1,...,xn/t)o(n})o(;)150 1386 y(--)150 1486 y(--)f(where)42
b(xi)h(is)f(not)h(free)f(in)g(tj,)h(for)f(i,j=1,...,n.)150
1685 y(forall)f([x1,...,xn])e(\(\(v)j(||)h(w\))g(-->)f(u\))87
b(=)324 1785 y(forall)41 b([x1,...,xn])e(\(v)k(-->)g(u\))f(&&)h(forall)
e([x1,...,xn])e(\(w)k(-->)f(u\);)150 2084 y(union)g(::)g(\(a)h(->)g
(Bool\))e(->)i(\(a)g(->)g(Bool\))e(->)i(\(a)g(->)f(Bool\);)150
2283 y(s)h(`union`)e(t)87 b(=)f({x)43 b(|)g(\(x)g(`in`)f(s\))h(||)g
(\(x)f(`in`)g(t\)};)150 2383 y(--)150 2482 y(--)86 b(where)42
b(x)h(is)g(not)f(free)g(in)h(s)g(or)g(t.)150 2781 y(inters)e(::)i(\(a)g
(->)f(Bool\))g(->)h(\(a)f(->)h(Bool\))f(->)g(\(a)h(->)g(Bool\);)150
2980 y(s)g(`inters`)d(t)87 b(=)g({x)43 b(|)g(\(x)g(`in`)f(s\))g(&&)h
(\(x)g(`in`)f(t\)};)150 3080 y(--)150 3180 y(--)86 b(where)42
b(x)h(is)g(not)f(free)g(in)h(s)g(or)g(t.)150 3479 y(minus)f(::)g(\(a)h
(->)g(Bool\))e(->)i(\(a)g(->)g(Bool\))e(->)i(\(a)g(->)f(Bool\);)150
3678 y(s)h(`minus`)e(t)87 b(=)f({x)43 b(|)g(\(x)g(`in`)f(s\))h(&&)g
(\(not)e(\(x)i(`in`)f(t\)\)};)150 3778 y(--)150 3877
y(--)86 b(where)42 b(x)h(is)g(not)f(free)g(in)h(s)g(or)g(t.)150
4176 y(subset)e(::)i(\(a)g(->)f(Bool\))g(->)h(\(a)f(->)h(Bool\))f(->)g
(Bool;)150 4375 y({})h(`subset`)d(s)87 b(=)g(True;)150
4575 y({u})42 b(`subset`)f(s)86 b(=)h(u)43 b(`in`)f(s;)150
4774 y({x)h(|)g(u)g(||)g(v})g(`subset`)d(s)87 b(=)f(\({x)43
b(|)g(u})g(`subset`)d(s\))j(&&)f(\({x)h(|)g(v})g(`subset`)d(s\);)150
5073 y(superset)g(::)j(\(a)g(->)g(Bool\))e(->)i(\(a)g(->)f(Bool\))g(->)
h(Bool;)150 5272 y(s)g(`superset`)d({})86 b(=)h(True;)150
5471 y(s)43 b(`superset`)d({u})86 b(=)g(u)43 b(`in`)f(s;)150
5670 y(s)h(`superset`)d({x)i(|)i(u)f(||)f(v})87 b(=)g(\(s)42
b(`superset`)e({x)j(|)g(u}\))f(&&)h(\(s)g(`superset`)c({x)k(|)g(v}\);)p
eop
%%Page: 30 30
30 29 bop 150 100 a Fk(C)83 b Fi(BOOLEANS)24 b Fk(MODULE)2548
b Fl(30)150 589 y Fi(power)42 b(::)g(\(a)h(->)g(Bool\))e(->)i(\(a)g(->)
g(Bool\))e(->)i(Bool;)150 689 y(--)150 789 y(--)86 b(Power)42
b(set)g(function.)150 988 y(power)g({})86 b(=)h({{}};)150
1187 y(power)42 b({u})85 b(=)i({{},)42 b({u}};)150 1386
y(power)g({x)g(|)h(u)h(||)e(v})87 b(=)324 1486 y({s)43
b(|)g(exists)e([l,r])h(\(l)h(`in`)f(\(power)f({x)h(|)i(u}\))e(&&)h(r)g
(`in`)f(\(power)f({x)i(|)g(v}\))f(&&)629 1586 y(s)i(==)e(l)i(`union`)c
(r\)};)150 1885 y(mapset)h(::)i(\(a)g(->)f(b\))h(->)g(\(a)g(->)f
(Bool\))g(->)h(\(b)f(->)h(Bool\);)150 1984 y(--)150 2084
y(--)86 b(Analogue)41 b(of)h(map)h(for)f(set)g(processing.)150
2283 y(mapset)f(f)i({})87 b(=)f({};)150 2482 y(mapset)41
b(f)i({u})86 b(=)h({f)43 b(u};)150 2682 y(mapset)e(f)i({x)g(|)g(u)g(||)
g(v})86 b(=)h(\(mapset)41 b(f)i({x)g(|)g(u}\))f(`union`)f(\(mapset)g(f)
i({x)g(|)g(v}\);)150 2980 y(in)g(::)g(a)g(->)f(\(a)h(->)g(Bool\))e(->)i
(Bool;)150 3080 y(--)150 3180 y(--)g(Set)f(membership.)150
3379 y(y)h(`in`)f(s)87 b(=)g(s)43 b(y;)150 3479 y(--)150
3578 y(--)86 b(where)42 b(s)h(is)g(not)f(a)h(variable.)150
3877 y(--)86 b(Beta-reduction)150 4076 y(\(\\x)42 b(->)h(y\))g(z)87
b(=)f(y{x/z};)150 4176 y(--)150 4276 y(--)g(where)42
b(z)h(is)g(free)f(for)g(x)h(in)g(y.)150 4475 y({x)g(|)g(y})g(z)86
b(=)h(y{x/z};)150 4575 y(--)150 4674 y(--)f(where)42
b(z)h(is)g(free)f(for)g(x)h(in)g(y.)150 4873 y(})p 150
5039 3511 10 v eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF
Received on Di Jul 01 1997 - 17:53:00 CEST