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y(The)8 b(second)h(class,)g Ff(symbolic)f(variables)p
Fk(,)g(is)g(more)h(general)f(than)f(the)h(\256rst)g(one.)14
b(A)8 b(symbolic)262 531 y(v)o(ariable)f(can)h(take)g(a)g(set)f(of)h
(arbitrary)e(v)o(alues.)15 b(F)o(or)7 b(e)o(xample,)j(a)e(v)o(ariable)f
(that)g(takes)h(three)g(v)o(alues)262 581 y(red,)h(green,)h(and)g(blue)
e(is)h(a)h(symbolic)e(v)o(ariable.)15 b(Symbolic)8 b(v)o(ariables)h
(are)h(declared)g(as)g(follo)o(ws.)262 672 y Fh(.mv)24
b(<variable-name-list>)f(<number-of-values>)g(<value-list>)324
764 y Fg(\017)d Ff(variable-name-list)11 b Fk(and)i Ff(number)o
(-of-values)f Fk(are)i(the)e(same)j(as)e(for)g(the)g(declaration)f(of)
365 814 y(enumerati)o(v)o(e)f(v)o(ariables.)324 897 y
Fg(\017)20 b Ff(value-list)10 b Fk(is)h(a)h(white-space)g(separated)g
(list)e(of)h(strings)f(\(terminated)h(by)g(the)g(end)g(of)g(the)365
946 y(line\))c(gi)o(ving)f(the)h(list)g(of)g(v)o(alues)h(the)f(v)o
(ariable)h(can)g(take.)14 b(The)8 b(number)g(of)f(v)o(alues)h(declared)
365 996 y(and)j(the)f(range)g(size)h(should)e(match.)324
1079 y Fg(\017)20 b Fk(Example:)15 b Fh(.mv)25 b(x,y)g(3)f(red)h(green)
g(blue)p Fk(.)324 1171 y(If)9 b(a)h(v)o(ariable)f(is)g(not)g(de\256ned)
g(using)g Fh(.mv)g Fk(in)g(a)h(model,)g(then)f(the)g(v)o(ariable)g(is)g
(assumed)i(to)d(be)262 1220 y(a)i(Boolean)g(v)o(ariable.)324
1270 y(T)m(wo)g(v)o(ariables)g(are)h(said)f(to)g(ha)o(v)o(e)h(the)f
(same)i(type)e(if)313 1353 y(1.)21 b(the)10 b(v)o(ariables)h(are)g
(enumerati)o(v)o(e)g(v)o(ariables)f(with)f(the)h(same)i(domain)e(size,)
h(or)313 1436 y(2.)21 b(the)15 b(v)o(ariables)g(are)h(symbolic)e(v)o
(ariables)h(with)f(the)h(same)i(domain)e(size)g(and)g(the)g(same)365
1486 y(symbolic)10 b(v)o(alues)g(de\256ned)h(in)f(the)g(same)h(order)f
(in)g(the)g Fh(.mv)g Fk(construct.)262 1569 y(Consider)f(the)h(follo)o
(wing)e(e)o(xample.)262 1652 y Fh(.mv)24 b(x)h(2)262
1702 y(.mv)f(y)h(2)g(red)f(blue)262 1785 y Fe(x)7 b Fk(and)g
Fe(y)j Fk(are)e(not)e(of)h(the)h(same)g(type)f(because)i
Fe(x)e Fk(is)h(an)f(enumerati)o(v)o(e)h(v)o(ariable)g(and)f
Fe(y)i Fk(is)e(a)h(symbolic)262 1835 y(v)o(ariable)i(although)e(both)h
(are)i(two-v)o(alued)e(v)o(ariables.)262 1918 y Fh(.mv)24
b(x)h(2)g(red)f(blue)262 1968 y(.mv)g(y)h(2)g(blue)f(red)262
2051 y Fe(x)13 b Fk(and)h Fe(y)h Fk(in)e(the)h(abo)o(v)o(e)h(e)o
(xample)f(are)h(not)e(of)g(the)h(same)h(type)e(because)i(symbolic)e(v)o
(alues)h(are)262 2100 y(de\256ned)c(in)g(dif)o(ferent)f(orders.)262
2219 y Fi(1.3)50 b(T)-5 b(ables)262 2297 y Fk(A)14 b(table)h(is)f(an)h
(abstract)g(representation)f(of)g(a)h(physical)f(gate.)28
b(A)15 b(table)f(is)h(dri)o(v)o(en)f(by)g(inputs)262
2347 y(and)g(generates)h(outputs)e(follo)o(wing)f(its)h(functionality)m
(.)25 b(Although)12 b(a)j(real)g(gate)f(generates)h(an)262
2397 y(output)d(deterministically)h(depending)h(on)g(what)h(inputs)e
(are)i(supplied,)g(tables)g(in)f(BLIF-MV)262 2447 y(can)h(represent)h
(non-deterministic)d(beha)o(viors)i(as)h(well.)29 b(The)16
b(functionality)c(of)j(the)g(table)g(is)967 2574 y(3)p
eop
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4 3 bop 262 307 a Fk(described)13 b(as)h(a)f(symbolic)g(relation,)g
(i.e.)25 b(the)13 b(table)g(enumerates)h(symbolically)e(all)h(the)g(v)o
(alid)262 357 y(combination)h(of)h(v)o(alues)h(among)g(the)f(inputs)g
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407 y(handle)e(multi-output)e(tables,)k(unlike)d(BLIF)m(,)i(where)g(e)o
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457 y(without)9 b(input)h(represents)i(a)g(constant)f(generator)n(.)18
b(If)12 b(the)f(table)g(allo)o(ws)g(more)h(than)f(one)h(v)o(alue)262
506 y(for)h(its)h(output,)f(then)h(the)g(table)g(is)f(a)i
Ff(nondeterministic)d Fk(constant)i(generator)n(,)h(which)f(we)h(call)
262 556 y Ff(pseudo)9 b(input)p Fk(.)14 b(T)m(ables)d(are)g(declared)g
(in)f(the)g(follo)o(wing)e(way)m(.)262 646 y Fh(.table)24
b(<in-1>)g(<in-2>)h(...)f(<in-n>)h(->)f(<out-1>)h(<out-2>...)e(<out-m>)
262 696 y(<relation>)262 745 y(...)262 795 y(<relation>)324
885 y Fg(\017)d Ff(in-1)p Fk(,)p Fe(:)7 b(:)g(:)e Fk(,)p
Ff(in-n)14 b Fk(are)j(strings)d(gi)o(ving)f(the)i(names)h(of)f(the)g
(inputs)f(to)h(the)g(table)g(being)g(de-)365 935 y(\256ned.)h(The)10
b(v)o(ariables)g(ha)o(v)o(e)h(to)e(be)i(de\256ned)f(using)f(the)h
Fh(.mv)f Fk(construct)h(before)g(the)g(table.)365 984
y(Otherwise,)h(the)o(y)f(are)h(assumed)g(to)f(be)h(Boolean)e(v)o
(ariables.)324 1067 y Fg(\017)20 b Ff(out-1)p Fk(,)p
Fe(:)7 b(:)g(:)e Fk(,)p Ff(out-m)14 b Fk(are)i(strings)e(gi)o(ving)f
(the)h(names)j(of)d(the)h(outputs)e(to)i(the)g(table)f(being)365
1116 y(de\256ned.)28 b(The)15 b(v)o(ariables)g(ha)o(v)o(e)g(to)f(be)h
(de\256ned)g(using)e(the)h Fh(.mv)h Fk(construct)e(before)i(the)365
1166 y(table.)g(Otherwise,)9 b(the)o(y)f(are)h(assumed)g(to)f(be)h
(Boolean)f(v)o(ariables.)14 b(An)o(y)8 b(table)h(must)f(ha)o(v)o(e)365
1216 y(at)j(least)f(one)g(output.)324 1298 y Fg(\017)20
b Fk(If)10 b(a)h(table)f(has)h(a)g(single)e(output,)g
Fh(->)h Fk(is)g(optional.)324 1388 y(A)h Ff(r)n(elation)f
Fk(is)g(a)i(white-space)f(separated)h(non-null)d(list)g(of)i
Fe(n)f Fd(+)f Fe(m)j Fk(strings,)e(gi)o(ving)f(a)j(v)o(alid)262
1438 y(combination)c(of)i(v)o(alues)g(among)h(inputs)d(and)j(outputs.)i
(The)e Fe(i)p Fk(-th)f(string)e(in)i(a)h(relation)e(speci\256es)262
1488 y(a)k(set)g(of)f(v)o(alues)h(for)f(the)g Fe(i)p
Fk(-th)h(v)o(ariable)f(in)g(the)g(input/output)d(declaration)j(of)h
Fh(.table)p Fk(.)22 b(Each)262 1537 y(relation)6 b(denotes)i(the)f
(Cartesian)h(product)f(of)g(all)h(the)f(sets)h(of)g(v)o(alues.)14
b(The)9 b(input-outpu)o(t)c(relation)262 1587 y(of)12
b(a)h(table)f(is)g(de\256ned)h(as)g(the)f(union)g(of)g(all)g(the)g
(relations.)21 b(A)13 b(set)f(of)g(v)o(alues)h(can)g(be)g(declared)262
1637 y(recursi)o(v)o(ely)d(in)f(the)h(follo)o(wing)e(form.)313
1726 y(1.)21 b(a)11 b(v)o(alue)f Fe(v)q Fk(,)i(or)313
1809 y(2.)21 b Fg(\000)p Fk(,)11 b(which)f(is)g(the)g(uni)o(v)o(erse,)h
(or)313 1891 y(3.)21 b(a)11 b(range)g Fg(f)p Fe(v)538
1897 y Fc(1)563 1891 y Fg(\000)f Fe(v)625 1897 y Fc(2)642
1891 y Fg(g)p Fk(,)g(or)313 1973 y(4.)21 b(a)11 b(list)e
Fd(\()p Fe(S)496 1979 y Fc(1)513 1973 y Fe(;)e(S)557
1979 y Fc(2)574 1973 y Fe(;)g(:)g(:)g(:)e(;)i(S)692 1979
y Fb(l)704 1973 y Fd(\))p Fk(,)k(where)g Fe(S)878 1979
y Fb(i)903 1973 y Fk(\()p Fe(i)g Fd(=)h Fk(1)p Fe(;)7
b(:)g(:)g(:)e(;)i(l)q Fk(\))j(is)g(a)g(set)h(of)f(v)o(alues,)h(or)313
2056 y(5.)21 b(!)p Fe(S)r Fk(,)12 b(which)e(is)g(a)g(complement)h(of)f
(a)h(set)f(of)g(v)o(alues)h Fe(S)r Fk(.)324 2145 y(Let)e
Fe(x)g Fk(be)g(an)g(enumerati)o(v)o(e)h(v)o(ariable)e(which)h(takes)f
(4)h(v)o(alues.)15 b(The)9 b(follo)o(wing)e(are)i(e)o(xamples)262
2195 y(of)g(a)i(set)g(of)f(v)o(alues)g(for)g Fe(x)p Fk(.)324
2285 y Fg(\017)20 b Fk(1)324 2367 y Fg(\017)g(\000)324
2449 y(\017)g(f)p Fk(2)9 b Fg(\000)h Fk(3)p Fg(g)967
2574 y Fk(4)p eop
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5 4 bop 324 307 a Fg(\017)20 b Fd(\()p Fk(0)p Fe(;)7
b Fg(f)p Fk(2)h Fg(\000)i Fk(3)p Fg(g)p Fd(\))324 390
y Fg(\017)20 b Fk(!)p Fg(f)p Fk(2)9 b Fg(\000)g Fk(3)p
Fg(g)324 482 y Fk(If)g(a)i(v)o(ariable)e(is)h(a)g(symbolic)f(v)o
(ariable,)h(the)g(range)g(construct)f(in)g(the)g(abo)o(v)o(e)i(cannot)f
(be)g(used)262 531 y(since)g Fg(f)p Fk(red-green)p Fg(g)p
Fk(,)g(for)g(e)o(xample,)i(does)f(not)e(make)i(sense.)324
581 y(Let)f(us)h(consider)f(the)g(follo)o(wing)e(e)o(xample.)262
672 y Fh(.mv)24 b(x,y)h(4)262 722 y(.table)f(x)h(->)f(y)262
772 y(!2)g({1-3})262 822 y(-)g(0)262 872 y(2)g(\(0,3\))324
963 y Fk(The)13 b(relation)e(speci\256ed)i(in)f(this)f(table)h(is:)18
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Fg(\002)h Fd(\()p Fk(1)p Fe(;)d Fk(2)p Fe(;)g Fk(3)p
Fd(\)])i Fg([)h Fd([\()p Fk(0)p Fe(;)d Fk(1)p Fe(;)g
Fk(2)p Fe(;)g Fk(3)p Fd(\))g Fg(\002)k Fd(\()p Fk(0)p
Fd(\)])f Fg([)262 1013 y Fd([\()p Fk(2)p Fd(\))e Fg(\002)i
Fd(\()p Fk(0)p Fe(;)d Fk(3)p Fd(\)])p Fk(.)262 1123 y
Fa(1.3.1)41 b(=)10 b(Construct)262 1201 y Fk(One)j(can)h(also)f(use)h
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b(Assume)14 b(that)e(in)h(the)g(column)262 1251 y(corresponding)8
b(to)i(v)o(ariable)g Fe(y)q Fk(,)i(we)e(ha)o(v)o(e)i
Fd(=)g Fe(x)e Fk(as)h(in)e(the)h(follo)o(wing)e(e)o(xample.)262
1342 y Fh(.table)24 b(x)h(->)f(y)262 1392 y(-)g(=x)324
1483 y Fk(The)11 b(interpretation)d(of)h(this)h(construct)f(is)h(that)g
(the)g(v)o(alue)g(of)g Fe(y)i Fk(should)d(be)h(equal)g(to)g
Fe(x)p Fk(.)15 b(This)262 1533 y(enables)10 b(us)h(to)e(describe)i(a)g
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1616 y Fh(.mv)24 b(select)g(2)262 1666 y(.mv)g(data0,data1,output)f
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1766 y(0)f(-)h(-)g(=data0)262 1816 y(1)f(-)h(-)g(=data1)262
1899 y Fk(Note)11 b(that)h(two)f(v)o(ariables)h(related)g(with)f(the)h
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b(An)o(y)262 1948 y(v)o(ariable)10 b(referred)g(to)g(with)f(the)h(=)h
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2059 y Fa(1.3.2)41 b(Default)10 b(Output)262 2137 y Fk(It)e(is)h
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(for)h(the)g(input)f(patterns)h(not)f(speci\256ed)262
2187 y(in)g(a)h(gi)o(v)o(en)g(relation.)14 b(The)9 b
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b(In)8 b(the)h(follo)o(wing)262 2236 y(e)o(xample,)i(no)e(relation)g
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Fe(x)p Fk(1)g(or)h Fe(x)p Fk(2)f(is)h(0.)k(Since)d(we)f(ha)o(v)o(e)262
2286 y(a)f(default)f(statement)h(in)f(the)h(table,)g(output)e(00)h(is)h
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2336 y(Therefore,)14 b(the)f(relation)f(of)h(this)f(table)g(is:)20
b Fd([\()p Fk(1)p Fd(\))10 b Fg(\002)h Fd(\()p Fk(1)p
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Fk(1)p Fd(\)])f Fg([)h Fd([\()p Fk(0)p Fd(\))f Fg(\002)h
Fd(\()p Fk(0)p Fd(\))g Fg(\002)g Fd(\()p Fk(0)p Fd(\))f
Fg(\002)262 2386 y Fd(\()p Fk(0)p Fd(\)])f Fg([)g Fd([\()p
Fk(0)p Fd(\))g Fg(\002)h Fd(\()p Fk(1)p Fd(\))g Fg(\002)g
Fd(\()p Fk(0)p Fd(\))g Fg(\002)g Fd(\()p Fk(0)p Fd(\)])f
Fg([)g Fd([\()p Fk(1)p Fd(\))h Fg(\002)g Fd(\()p Fk(0)p
Fd(\))f Fg(\002)h Fd(\()p Fk(0)p Fd(\))g Fg(\002)g Fd(\()p
Fk(0)p Fd(\)])p Fk(.)18 b(Each)12 b(table)f(can)h(ha)o(v)o(e)g(at)f
(most)262 2436 y(one)f Fh(.default)f Fk(declaration.)967
2574 y(5)p eop
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6 5 bop 262 307 a Fh(.mv)24 b(x1,x2,y1,y2)g(2)262 357
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b(Ev)o(ery)12 b(latch)f(has)h(to)e(be)262 1143 y(initialized)g
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(than)g(one)h(initial)d(v)o(alue,)j(in)f(which)262 1193
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b(A)262 1243 y(latch)14 b(can)g(be)h(seen)g(as)g(a)g(multi-v)o(alued)d
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1392 y(A)g(latch)g(is)g(declared)h(as)g(follo)o(ws.)262
1483 y Fh(.latch)24 b(<latch-input>)g(<latch-output>)324
1575 y Ff(latc)o(h)11 b(input)f Fk(and)h Ff(latc)o(h)g(output)f
Fk(are)i(strings,)f(gi)o(ving)f(the)h(name)i(of)e(the)g(latch)g(input)f
(and)i(the)262 1625 y(latch)h(output.)24 b(The)14 b(two)e(v)o(ariables)
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b(Notice)12 b(that)f(we)h(use)262 1774 y Fh(.reset)c
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b(If)9 b(a)h(latch)f(is)g(reset)g(to)g(a)h(constant)e(v)o(alue,)i(then)
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1459 1824 13 2 v 14 w(output)f Fk(whose)262 1874 y(reset)h(state)h(is)f
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%%Page: 7 7
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%%Page: 8 8
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%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF