chalmers-lava2000 1.3 → 1.4.1

raw patch · 4 files changed

+15/−16550 lines, 4 filesbinary-addedPVP ok

version bump matches the API change (PVP)

API changes (from Hackage documentation)

Files

Doc/tutorial.pdf binary
Doc/tutorial.ps +0/−16541
INSTALL +10/−4
chalmers-lava2000.cabal +5/−5

+ Doc/tutorial.pdf view

binary file changed (absent → 464081 bytes)

− Doc/tutorial.ps

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b(.)41-b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h-(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(20)639 4402 y(4.4)84-b(Helping)28 b(V)-7 b(eri\014cation)80 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f-(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)-90 b(20)639 4541 y(4.5)84 b(Exercises)e(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h-(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)-f(.)h(.)f(.)h(.)f(.)h(.)90 b(22)515 4764 y Fr(5)76 b(Sequen)m(tial)32-b(Circuits)1850 b(23)639 4903 y Fq(5.1)84 b(The)28 b(Dela)n(y)f(Comp)r-(onen)n(t)43 b(.)e(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h-(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(23)1926 5255-y(1)p eop-%%Page: 2 3-2 2 bop 639 523 a Fq(5.2)84 b(Multiple)29 b(Dela)n(ys)34-b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)-f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(24)639-663 y(5.3)84 b(Coun)n(ters)24 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f-(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)-h(.)f(.)h(.)f(.)h(.)90 b(25)639 802 y(5.4)84 b(Sequen)n(tialization)55-b(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)-h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(25)639 941-y(5.5)84 b(V)-7 b(ariations)27 b(on)g Fp(rowSeq)f Fq(.)42-b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g-(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(26)639 1081 y(5.6)84-b(Exercises)e(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)-h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90-b(27)515 1303 y Fr(6)76 b(Sequen)m(tial)32 b(V)-8 b(eri\014cation)1702-b(29)639 1443 y Fq(6.1)84 b(Sequen)n(tial)27 b(Safet)n(y)h(Prop)r-(erties)33 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g-(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(29)639 1582 y(6.2)84-b(Sequen)n(tial)27 b(Logic)77 b(.)42 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b(63)639 4621 y(9.3)84 b(Exercises)e(.)41-b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h-(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90-b(67)1926 5255 y(2)p eop-%%Page: 3 4-3 3 bop 515 523 a Fr(10)28 b(T)m(yp)s(es)2394 b(68)639-663 y Fq(10.1)42 b(Signals)27 b(and)g(Circuits)73 b(.)42-b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g-(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(68)639 802 y(10.2)42-b(Connection)27 b(P)n(atterns)55 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h-(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90-b(69)639 941 y(10.3)42 b(Ov)n(erloading)d(.)j(.)f(.)h(.)g(.)f(.)h(.)f-(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)-h(.)f(.)h(.)f(.)h(.)90 b(69)515 1164 y Fr(A)52 b(Quic)m(k)33-b(Reference)e(Guide)1677 b(72)639 1303 y Fq(A.1)64 b(The)28-b Fp(lava)e Fq(command)79 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h-(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90-b(72)639 1443 y(A.2)64 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b(In)n(terpretations)j(.)42-b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h-(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(76)639 2419-y(A.9)64 b(Errors)56 b(.)41 b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h-(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)-f(.)h(.)f(.)h(.)90 b(77)515 2642 y Fr(B)56 b(Answ)m(ers)2295-b(79)639 2864 y(Bibliograph)m(y)2109 b(88)639 3087 y(Index)2409-b(90)1926 5255 y Fq(3)p eop-%%Page: 4 5-4 4 bop 515 1186 a Fo(Chapter)64 b(1)515 1641 y Fs(In)-6-b(tro)6 b(duction)515 2112 y Fq(La)n(v)-5 b(a)27 b(is)h(an)g(exp)r-(erimen)n(tal)f(to)r(ol)h(for)g(hardw)n(are)e(design)h(and)h(v)n-(eri\014cation.)37 b(Using)28 b(La)n(v)-5 b(a,)515 2212-y(one)36 b(can)f(describ)r(e)h(circuits)g(using)g(a)g(simple)g-(functional)h(hardw)n(are)d(description)i(lan-)515 2311-y(guage.)51 b(The)33 b(descriptions)f(are)g(short)h(and)f(sw)n(eet,)i-(and)f(do)g(not)g(su\013er)f(from)h(the)g(v)n(er-)515-2411 y(b)r(osit)n(y)22 b(of)g(more)f(standard)g(hardw)n(are)f-(description)i(languages)e(\(HDLs\))j(lik)n(e)f(VHDL)h(and)515-2511 y(V)-7 b(erilog.)34 b(On)22 b(the)g(other)f(hand,)j(w)n(e)d-(cannot)h(express)f(the)h(same)g(things)g(as)f(in)h(these)g(large,)515-2610 y(expressiv)n(e)32 b(\(and)i(complicated\))g(languages.)54-b(F)-7 b(or)34 b(example,)h(w)n(e)e(cannot)h(express)f(lo)n(w)515-2710 y(lev)n(el)22 b(details)g(ab)r(out)h(timing.)36-b(What)23 b(w)n(e)f(can)g(express)g(v)n(ery)g(nicely)-7-b(,)23 b(though,)h(is)e(the)i(w)n(a)n(ys)515 2809 y(in)29-b(whic)n(h)g(circuits)g(are)f(built)h(from)g(sub-circuits.)41-b(La)n(v)-5 b(a)27 b(facilitates)i(the)h(description)e(of)515-2909 y Fn(c)l(onne)l(ction)g(p)l(atterns)e Fq(so)f(that)i(they)f(are)f-(easily)h(reusable.)35 b(F)-7 b(or)26 b(some)f(kinds)h(of)g(circuits,)-515 3009 y(for)h(example)h(in)g(signal)f(pro)r(cessing,)f(this)i(is)g-(exactly)g(what)f(w)n(e)h(w)n(an)n(t)f(to)h(do.)37 b(La)n(v)-5-b(a)27 b(also)515 3108 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y(row)g(circ)g(\(carryIn,)e(a:as\))-i(=)h(\(b:bs,)e(carryOut\))689 2017 y(where)776 2117-y(\(b,)i(carry\))215 b(=)43 b(circ)f(\(carryIn,)e(a\))776-2216 y(\(bs,)i(carryOut\))e(=)j(row)g(circ)e(\(carry,)g(as\))515-2385 y Fq(Once)25 b(w)n(e)g(ha)n(v)n(e)g(made)g(this)h(de\014nition,)g-(w)n(e)g(do)f(not)h(need)f(to)h(use)f(recursion)f(an)n(ymore)g(to)515-2485 y(de\014ne)k(circuits)f(of)g(this)h(sp)r(eci\014c)g(pattern.)515-2624 y(Here)f(are)g(alternativ)n(e)f(de\014nitions)i(of)f-Fp(bitAdder)e Fq(and)i Fp(adder)p Fq(:)602 2793 y Fp(bitAdder')40-b(\(carry,)g(inps\))i(=)h(row)f(halfAdd)f(\(carry,)g(inps\))602-2892 y(adder')172 b(\(carry,)40 b(inps\))i(=)h(row)f(fullAdd)f-(\(carry,)g(inps\))515 3061 y Fq(Doing)33 b(so)g(not)h(only)f(mak)n(es)-g(the)h(circuit)f(descriptions)g(shorter.)54 b(F)-7 b(or)33-b(someb)r(o)r(dy)g(who)515 3160 y(kno)n(ws)21 b(ho)n(w)g(the)i(pattern)-f Fp(row)e Fq(is)i(de\014ned,)i(it)e(is)g(m)n(uc)n(h)g(easier)f(to)h-(understand)g(the)g(circuit)515 3260 y(de\014nition.)36-b(It)27 b(is)f(also)f(v)n(ery)f(con)n(v)n(enien)n(t)h(to)h(b)r(e)h-(able)e(to)h(mix)g(these)g(t)n(w)n(o)g(di\013eren)n(t)g(st)n(yles)515-3360 y(of)h(descriptions.)515 3499 y(Note)f(that)g(the)g(reason)e(wh)n-(y)h(w)n(e)h(c)n(hose)f(the)h(input)g(to)g Fp(fullAdd)d-Fq(to)j(ha)n(v)n(e)e(the)i(particular)515 3599 y(form)32-b(it)h(has)f(is)h(b)r(ecause)f(w)n(e)g(w)n(an)n(ted)g(it)h(to)f(b)r(e)h-(used)g(with)g Fp(row)p Fq(.)51 b(W)-7 b(e)33 b(will)g(alw)n(a)n(ys)d-(use)515 3698 y(that)i(con)n(v)n(en)n(tion,)f(so)h(that)g(it)g(is)g-(easier)e(to)i(use)g(circuits)f(with)i(prede\014ned)f(connection)515-3798 y(patterns.)515 3937 y(Ev)n(en)26 b(shorter)h(de\014nitions)g(of)h-(the)g(same)f(circuits)g(are:)602 4106 y Fp(bitAdder')40-b(=)j(row)f(halfAdd)602 4206 y(adder')172 b(=)43 b(row)f(fullAdd)515-4374 y Fq(Note)27 b(that)h(the)g Fn(typ)l(e)g Fq(of)g-Fp(adder')d Fq(is)i(sligh)n(tly)g(di\013eren)n(t)h(from)f-Fp(adder)p Fq(,)f(see)h(exercise)g(3.3.)515 4685 y Fm(3.3)135-b(Arithmetic)515 4907 y Fq(In)27 b(La)n(v)-5 b(a,)26-b(w)n(e)h(can)g(not)g(only)g(deal)g(with)h(lo)n(w-lev)n(el)e(wire)g(t)n-(yp)r(es)i(lik)n(e)e(bits,)i(and)f(gates)f(lik)n(e)515-5006 y Fp(and2)e Fq(and)h Fp(xor2)p Fq(,)g(but)h(also)f(with)h(more)f-(abstract)g(wire)g(t)n(yp)r(es)h(and)g(gates.)35 b(One)25-b(of)h(these)1905 5255 y(14)p eop-%%Page: 15 16-15 15 bop 515 523 a Fq(t)n(yp)r(es)27 b(is)h(in)n(tegers.)515-663 y(On)h(these)g(in)n(tegers,)g(w)n(e)g(ha)n(v)n(e)f(op)r(erations)g-(corresp)r(onding)f(to)i(abstract)f(gates)h(o)n(v)n(er)e(in-)515-762 y(tegers.)36 b(A)27 b(list)h(of)g(these)g(gates)e(can)h(b)r(e)h-(found)g(in)g(app)r(endix)g(A.)515 902 y(A)33 b(simple)g(circuit)g-(using)g(these)g(arithmetical)g(gates)f(is)h(called)g-Fp(numBreak)p Fq(.)50 b(It)33 b(tak)n(es)f(a)515 1001-y(n)n(um)n(b)r(er)e(as)h(input,)h(and)f(has)g(a)f(pair)g(of)h(a)g(bit)g-(and)g(a)g(n)n(um)n(b)r(er)f(as)h(output.)47 b(The)31-b(bit)h(in)515 1101 y(the)i(pair)g(corresp)r(onds)e(to)i(the)g(v)-5-b(alue)34 b(of)g(the)h(\014rst)f(binary)f(digit)h(of)g(the)h(n)n(um)n-(b)r(er;)i(the)515 1200 y(resulting)27 b(n)n(um)n(b)r(er)g(is)h(the)g-(input)g(n)n(um)n(b)r(er)f(divided)h(b)n(y)f(2.)602 1386-y Fp(numBreak)40 b(num)j(=)g(\(bit,)e(num'\))689 1486-y(where)776 1586 y(digit)h(=)h(imod)f(\(num,)f(2\))776-1685 y(bit)130 b(=)43 b(int2bit)e(digit)776 1785 y(num')86-b(=)43 b(idiv)f(\(num,)f(2\))515 1971 y Fq(The)24 b(circuit)f-Fp(int2bit)e Fq(con)n(v)n(erts)h(a)i(n)n(um)n(b)r(er)f(in)n(to)h(a)f-(bit,)i(b)n(y)f(transforming)e(a)h(0)h(in)n(to)f Fp(low)p-Fq(,)515 2071 y(and)k(an)n(y)g(other)g(n)n(um)n(b)r(er)g(in)n(to)h-Fp(high)p Fq(.)515 2210 y(W)-7 b(e)31 b(can)g(use)g(this)h-(arithmetical)f(circuit)g(to)g(build)h(a)f(circuit)g(that)h(con)n(v)n-(erts)d(a)i(n)n(um)n(b)r(er)515 2310 y(in)n(to)36 b(a)h(binary)f(n)n-(um)n(b)r(er,)j(that)e(is,)j(a)c(list)h(of)g(bits.)65-b(The)37 b(circuit)g(tak)n(es)f(a)g(parameter,)515 2409-y(corresp)r(onding)e(to)i(the)h(size)f(of)g(the)h(list)f(it)h(has)f(to)-g(pro)r(duce,)i(and)e(has)g(as)g(input)h(the)515 2509-y(n)n(um)n(b)r(er)27 b(that)h(needs)f(to)h(b)r(e)g(con)n(v)n(erted.)515-2648 y(The)d(con)n(v)n(erter)f Fp(int2bin)f Fq(con)n(v)n(erts)h(an)h-(in)n(teger)g(to)g(a)h(binary)f(n)n(um)n(b)r(er.)35 b(It)26-b(has)g(an)f(extra)515 2748 y Fn(p)l(ar)l(ameter)p Fq(,)32-b(whic)n(h)f(sp)r(eci\014es)f(the)i(n)n(um)n(b)r(er)e(of)h(bits)g(the)g-(binary)f(n)n(um)n(b)r(er)h(should)f(ha)n(v)n(e.)515-2848 y(Note)25 b(again)f(that)h(parameters)e(of)j(circuits)e(are)g-(di\013eren)n(t)i(from)e(inputs;)j Fp(int2bin)22 b Fq(is)j(not)515-2947 y(really)g(a)h(circuit,)h(but)g Fp(int2bin)40 b(16)26-b Fq(is.)37 b(W)-7 b(e)26 b(de\014ne)h(this)g(circuit)f(b)n(y)g-(recursion)f(o)n(v)n(er)g(the)515 3047 y(size)i(of)h(the)g(binary)e(n)n-(um)n(b)r(er.)602 3233 y Fp(int2bin)41 b(0)i(num)f(=)h([])602-3432 y(int2bin)e(n)i(num)f(=)h(\(bit:bits\))689 3532-y(where)776 3631 y(\(bit,num'\))d(=)j(numBreak)d(num)776-3731 y(bits)304 b(=)43 b(int2bin)e(\(n-1\))g(num')515-3917 y Fq(Other)27 b(arithmetical)g(gates)g(include)g-Fp(plus)p Fq(,)g Fp(times)p Fq(,)e(etc.)515 4056 y(Here)i(are)g(some)g-(example)g(sim)n(ulations)g(of)g(these)h(circuits:)602-4242 y Fp(Main>)41 b Fl(simulate)g(numBreak)f(7)602 4342-y Fp(\(high,3\))602 4541 y(Main>)h Fl(simulate)g(\(int2bin)f(3\))j(7)-602 4641 y Fp([high,)e(high,)h(high])1905 5255 y Fq(15)p-eop-%%Page: 16 17-16 16 bop 515 523 a Fm(3.4)135 b(Compilation)515 745-y Fq(As)38 b(so)r(on)f(as)g(w)n(e)g(start)g(writing)h(bigger)e-(circuits,)k(the)f(need)e(migh)n(t)h(arise)f(to)g Fn(c)l(ompile)515-844 y Fq(them.)48 b(So)31 b(far,)g(w)n(e)g(ha)n(v)n(e)f(only)h(used)g-(an)g(in)n(terpreter,)g(whic)n(h)g(can)g(b)r(e)h(a)e(little)i(bit)g-(slo)n(w)515 944 y(for)25 b(larger)f(pro)5 b(jects.)35-b(Compiling)25 b(can)h(b)r(e)g(done)f(b)n(y)h(running)f(the)h-Fp(lava)e Fq(command)i(with)515 1044 y(the)i(option)f-Fp(-c)p Fq(.)515 1183 y(But)32 b(\014rst,)g(w)n(e)f(m)n(ust)h(sp)r-(ecify)f(what)h(w)n(e)f(w)n(an)n(t)g(the)h(compiled)f(program)f(to)h-(do.)48 b(W)-7 b(e)32 b(do)515 1283 y(this)23 b(b)n(y)f(making)g(a)g-(de\014nition)h(for)f(the)h(function)h Fp(main)d Fq(in)i(the)g(\014le)g-(w)n(e)f(are)g(w)n(orking)f(with.)515 1382 y(F)-7 b(or)27-b(example,)g(if)h(w)n(e)f(w)n(an)n(t)g(to)h(sim)n(ulate)f(an)g(adder,)g-(w)n(e)g(migh)n(t)h(de\014ne:)602 1561 y Fp(main)42 b(=)h(print)f-(\(simulate)d(adder)j(\(high,\([high,low)o(],)o([lo)o(w,)o(hig)o(h])o-(\)\))o(\))515 1740 y Fq(Then,)28 b(w)n(e)f(run)g(the)h(La)n(v)-5-b(a)27 b(compiler,)g(b)n(y)g(in)n(v)n(oking:)602 1918-y Fp(\045)43 b Fl(lava)f(-c)h(First)602 2018 y Fp(--)g(Lava2000)d-(----------------)o(--)o(---)o(--)o(---)o(--)o(---)o(--)o(--)o(---)o-(--)o(---)o(--)o(--)o(---)o(-)602 2118 y(Compiling)g(First)h(...)602-2217 y(...)515 2396 y Fq(The)23 b(compiler)g(has)h(no)n(w)f(pro)r-(duced)g(an)g(executable)h(program,)e(called)h Fp(First)p-Fq(,)g(whic)n(h)g(w)n(e)515 2495 y(can)k(run)g(b)n(y)h(sa)n(ying:)602-2674 y Fp(\045)43 b Fl(First)602 2774 y Fp(\([high,low],high)o(\))515-2952 y Fq(F)-7 b(or)27 b(more)g(con)n(trol)f(o)n(v)n(er)g(the)i-(compilation)f(pro)r(cess,)f(see)h(app)r(endix)h(A.1.)515-3265 y Fm(3.5)135 b(Exercises)575 3478 y Fq(3.1)40 b(De\014ne)27-b(a)f(bit)h(subtractor,)e(called)h Fp(bitSubber)p Fq(,)d(whic)n(h)k-(tak)n(es)e(a)h(bit)h(and)f(a)g(binary)722 3578 y(n)n(um)n(b)r(er)i(as)-f(input,)h(and)f(subtracts)g(the)h(bit)g(from)g(the)g(binary)f(n)n(um)n-(b)r(er.)575 3740 y(3.2)40 b(De\014ne)29 b(a)f(binary)g(adder,)g-(called)g Fp(adder2)p Fq(,)e(whic)n(h)i(do)r(es)g(not)h(tak)n(e)e(in)i-(a)f(carry)e(bit,)722 3840 y(and)i(thro)n(ws)e(a)n(w)n(a)n(y)g(the)i-(resulting)f(carry)-7 b(.)575 4002 y(3.3)40 b(What)e(is)g(the)g-(di\013erence)g(b)r(et)n(w)n(een)g Fp(adder)d Fq(and)j-Fp(adder')p Fq(?)65 b(Hin)n(t:)58 b(lo)r(ok)37 b(at)h(the)722-4102 y(t)n(yp)r(es)28 b(of)f(the)h(inputs.)575 4264 y(3.4)40-b(De\014ne)28 b(a)g(circuit)f Fp(bin2int)p Fq(,)e(whic)n(h)j(con)n(v)n-(erts)d(a)j(bit)g(v)n(ector)e(in)n(to)h(an)h(in)n(teger.)575-4427 y(3.5)40 b(De\014ne)29 b(the)g(circuit)f Fp(zipp)p-Fq(,)f(whic)n(h)i(tak)n(es)e(a)h(pair)g(of)g(list)h(as)f(inputs)h(and)f-(pro)r(duces)722 4526 y(a)f(list)h(of)g(pairs,)f(one)g(b)n(y)g(one)g-(group)r(ed)g(together.)809 4717 y Fp(Main>)42 b Fl(simulate)e(zipp)i-(\([low,high,low],[)o(hi)o(gh)o(,hi)o(gh)o(,lo)o(w])o(\))809-4816 y Fp([\(low,high\),\(high)o(,hi)o(gh)o(\),\()o(lo)o(w,)o(low)o-(\)])722 5006 y Fq(Also)28 b(de\014ne)f(the)h(circuit)g-Fp(unzipp)p Fq(,)d(whic)n(h)j(is)f(the)h(in)n(v)n(erse)e(of)i-Fp(zipp)p Fq(.)1905 5255 y(16)p eop-%%Page: 17 18-17 17 bop 575 523 a Fq(3.6)40 b(De\014ne)29 b(the)f(circuit)g-Fp(pair)p Fq(,)f(whic)n(h)h(tak)n(es)f(a)h(list)g(as)f(input)i(and)f-(pro)r(duces)f(a)h(list)g(of)722 623 y(pairs,)f(with)h(the)g(neigh)n(b)-r(ours)e(group)r(ed)h(together.)809 822 y Fp(Main>)42-b Fl(simulate)e(pair)i(\([low,high,low,hi)o(gh)o(,h)o(igh)o(,l)o(ow])o-(\))809 922 y Fp([\(low,high\),\(low,)o(hig)o(h\))o(,\(h)o(ig)o(h,)o-(low)o(\)])722 1121 y Fq(Also)28 b(de\014ne)f(the)h(circuit)g-Fp(unpair)p Fq(,)d(whic)n(h)j(is)f(the)h(in)n(v)n(erse)e(of)i-Fp(pair)p Fq(.)575 1287 y(3.7)40 b(De\014ne)i(a)f(connection)g(pattern)-g(called)g Fp(par)f Fq(whic)n(h)i(turns)f(t)n(w)n(o)f(circuits,)45-b(eac)n(h)722 1386 y(taking)33 b(in)h(one)g(input)g(and)g(ha)n(ving)e-(one)h(output,)j(in)n(to)e(one)f(circuit)g(taking)h(in)f(a)722-1486 y(pair)27 b(of)h(inputs)g(and)f(ha)n(ving)g(a)g(pair)g(of)g-(outputs.)575 1652 y(3.8)40 b(De\014ne,)31 b(using)e(recursion,)f(a)h-(binary)g(m)n(ultiplier.)42 b(What)30 b(is)f(the)h(recursiv)n(e)e-(struc-)722 1752 y(ture?)575 1918 y(3.9)40 b(Lo)r(oking)20-b(at)h(the)h(de\014nition)g(of)f Fp(row)p Fq(,)g(de\014ne)h(a)f-(connection)g(pattern)g(called)f Fp(column)722 2017 y-Fq(whic)n(h)32 b(carries)e(the)j(righ)n(t)e(part)g(of)h(the)g(input)h-(and)f(the)g(left)h(part)e(of)h(the)h(output)722 2117-y(through.)722 2250 y(\(*\))28 b(Can)f(y)n(ou)g(de\014ne)h-Fp(column)d Fq(in)j(terms)f(of)h Fp(row)p Fq(?)533 2416-y(3.10)40 b(De\014ne)33 b(a)f(connection)g(pattern)g(called)g-Fp(grid)p Fq(,)g(whic)n(h)h(puts)f(a)g(n)n(um)n(b)r(er)g(of)h(copies)-722 2516 y(of)g(circuits)g(in)g(a)f(matrix.)53 b(The)33-b(left)g(parts)g(of)g(the)g(inputs)g(are)f(carried)g(through)722-2615 y(from)22 b(left)h(to)f(righ)n(t,)h(and)f(the)g(righ)n(t)g(parts)f-(of)h(the)h(inputs)g(and)f(outputs)g(are)f(carried)722-2715 y(through)27 b(from)g(top)h(to)f(b)r(ottom.)722-2848 y(Hin)n(t:)38 b(think)28 b(of)f(a)h(grid)e(as)h(a)h(ro)n(w)e(of)h-(columns)h(\(or)f(a)g(column)g(of)h(ro)n(ws\).)533 3014-y(3.11)40 b(Can)31 b(y)n(ou)f(think)i(of)f(a)g(useful)g(circuit)g(that)-h(mak)n(es)e(use)h(of)g(the)g Fp(grid)f Fq(connection)722-3113 y(pattern?)533 3279 y(3.12)40 b(Lo)r(oking)22 b(at)h(the)g-(recursiv)n(e)e(de\014nition)i(of)g(an)f(adder,)h(de\014ne)g(a)f-(simple)h(subtractor.)722 3379 y(It)30 b(will)g(only)f(ha)n(v)n(e)f(to)-i(subtract)f(smaller)f(n)n(um)n(b)r(ers)h(from)g(bigger)f(n)n(um)n(b)r-(ers.)42 b(Can)722 3479 y(y)n(ou)27 b(use)h(an)n(y)f(of)h(the)h-(connection)e(patterns)h(describ)r(ed)f(in)i(this)f(c)n(hapter)f(to)h-(mak)n(e)722 3578 y(a)f(non-recursiv)n(e)f(description?)533-3744 y(3.13)40 b(De\014ne)35 b(a)f(sw)n(app)r(er,)h(a)f(circuit)g(that)-h(tak)n(es)e(in)i(t)n(w)n(o)e(inputs:)51 b(an)34 b(activ)-5-b(ate)34 b(signal)722 3844 y(and)d(a)g(pair)g(of)g(signals,)g(and)h-(the)f(output)h(is)f(a)g(pair)g(of)g(signals.)47 b(If)32-b(the)f(activ)-5 b(ate)722 3944 y(signal)32 b(is)g(high,)h(the)g(order)-d(of)j(the)f(input)h(pair)f(is)g(sw)n(app)r(ed,)h(otherwise)e(is)h(sta)-n(ys)722 4043 y(the)c(same.)809 4242 y Fp(swapper)41-b(\(swap,)g(\(a,)i(b\)\))f(=)h(\(x,)f(y\))897 4342 y(where)f(...)533-4541 y Fq(3.14)f(De\014ne)29 b(a)f(comparator,)e(a)i(circuit)g(that)h-(tak)n(es)e(in)i(t)n(w)n(o)e(binary)h(n)n(um)n(b)r(ers)f(of)i(equal)722-4641 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b(These)29-b(are)g(prop)r(erties)g(whic)n(h)g(can)h(b)r(e)g(de\014ned)g(in)g(suc)n-(h)g(a)f(w)n(a)n(y)g(that)h(they)515 2947 y(state)d(that)h(some)f-(condition)g(is)h Fn(always)h Fq(true)f(\(or,)f(equiv)-5-b(alen)n(tly)e(,)27 b(nev)n(er)g(false\).)515 3087 y(Here)f(is)g(an)f-(example;)i(a)f(prop)r(ert)n(y)f(that)h(c)n(hec)n(ks)f(that)h(the)h-(outputs)f(of)h(a)e(half)i(adder)e(are)515 3186 y(nev)n(er)h(b)r(oth)i-(true.)602 3372 y Fp(prop_HalfAddOutp)o(ut)o(Nev)o(er)o(Bot)o(hT)o(ru)o-(e)38 b(\(a,)k(b\))h(=)g(ok)689 3472 y(where)776 3571-y(\(sum,)f(carry\))f(=)i(halfAdd)e(\(a,)h(b\))776 3671-y(ok)479 b(=)43 b(nand2)f(\(sum,)f(carry\))515 3857 y-Fq(Note)28 b(that)h(this)g(prop)r(ert)n(y)e(lo)r(oks)h(prett)n(y)g(m)n-(uc)n(h)h(lik)n(e)f(a)g(normal)g(circuit)g(de\014nition,)h(and)515-3957 y(in)f(fact)f(it)h(is.)515 4096 y(The)e(actual)g(v)n-(eri\014cation)e(question)i(is:)36 b(do)r(es)26 b(this)h(prop)r(ert)n-(y)e(circuit)h(alw)n(a)n(ys)e(yield)i(true,)515 4196-y(no)21 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b(is)h(another)f(example;)-h(w)n(e)g(form)n(ulate)f(that)h(a)f(full)h(adder)f(do)r(es)h(not)f-(care)g(ab)r(out)h(the)515 862 y(order)29 b(of)i(the)g(argumen)n(ts,)f-(it)h(will)g(alw)n(a)n(ys)e(pro)r(duce)h(the)h(same)g(result.)46-b(This)30 b(prop)r(ert)n(y)515 961 y(is)d(in)h(general)e(called)i-Fn(c)l(ommutativity)p Fq(.)602 1135 y Fp(prop_FullAddComm)o(ut)o(ati)o-(ve)37 b(\(c,)42 b(\(a,)h(b\)\))f(=)h(ok)689 1234 y(where)776-1334 y(out1)f(=)h(fullAdd)e(\(c,)h(\(a,)h(b\)\))776 1434-y(out2)f(=)h(fullAdd)e(\(c,)h(\(b,)h(a\)\))776 1533 y(ok)130-b(=)43 b(out1)f(<==>)g(out2)515 1707 y Fq(Note)31 b(that,)i(since)e(w)n-(e)f(are)h(not)g(in)n(terested)g(in)g(the)h(exact)f(shap)r(e)g(of)g-(the)h(output)f(of)h(the)515 1806 y(t)n(w)n(o)26 b(full)i(adders,)e(w)n-(e)h(can)f(just)i(giv)n(e)e(a)h(name)g(to)g(the)g(whole)g(output,)h(in)-f(this)g(case)g Fp(out1)515 1906 y Fq(and)d Fp(out2)p-Fq(.)34 b(Another)24 b(thing)g(to)g(notice)g(is)g(that)h(w)n(e)f(use)g-(the)g(general)f(equalit)n(y)h Fp(<==>)p Fq(.)34 b(W)-7-b(e)515 2006 y(can)27 b(also)f(use)i(the)g(circuit)f-Fp(equal)f Fq(for)h(that.)515 2317 y Fm(4.2)135 b(Quan)l(ti\014cation)-515 2539 y Fq(The)25 b(comm)n(utativit)n(y)g(prop)r(ert)n(y)g(is)g(not)-h(only)f(true)h(for)f(full)h(adders,)f(but)h(also)f(in)g(general)515-2639 y(for)i(binary)g(adders.)36 b(Here)27 b(is)g(ho)n(w)g(w)n(e)g-(state)h(that)g(prop)r(ert)n(y:)602 2812 y Fp(prop_AdderCommut)o(at)o-(ive)37 b(\(as,)42 b(bs\))g(=)h(ok)689 2912 y(where)776-3011 y(out1)f(=)h(adder2)e(\(as,)h(bs\))776 3111 y(out2)g(=)h(adder2)e-(\(bs,)h(as\))776 3211 y(ok)130 b(=)43 b(out1)f(<==>)g(out2)515-3384 y Fq(Note)27 b(that)g(w)n(e)f(use)h(the)g(adder)f-Fp(adder2)e Fq(w)n(e)i(de\014ned)i(in)f(exercise)e(3.2)h(\(the)h(answ)n-(er)f(is)g(on)515 3484 y(page)g(80\).)515 3623 y(The)34-b(problem)f(is)h(that)h(this)f(prop)r(ert)n(y)f(holds)h(for)f-Fn(al)t(l)i Fq(circuit)f(sizes,)h(but)g(w)n(e)f(can)f(only)515-3723 y(v)n(erify)f(it)i(for)f(sp)r(eci\014c)h(sizes!)54-b(This)33 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y(questions.)52-b(F)-7 b(or)33 b(example,)h(to)f(v)n(erify)f(that)i(t)n(w)n(o)e-(circuits)h(ha)n(v)n(e)e(the)j(same)e(output,)j(w)n(e)515-722 y(can)e(pro)n(v)n(e)e(this)j(pin-b)n(y-pin)f(instead)g(of)g(at)g-(one)g(go.)53 b(T)-7 b(o)33 b(do)g(this,)i(w)n(e)e(ha)n(v)n(e)f(to)h-(de\014ne)515 822 y Fn(one-by-one)d(e)l(quality)p Fq(.)602-1008 y Fp(equalOneByOne)38 b(\([],)129 b([]\))h(=)43-b([])602 1207 y(equalOneByOne)38 b(\([],)129 b(y:ys\))42-b(=)h([low])602 1406 y(equalOneByOne)38 b(\(x:xs,)j([]\))130-b(=)43 b([low])602 1606 y(equalOneByOne)38 b(\(x:xs,)j(y:ys\))h(=)h(eq)-g(:)g(eqs)689 1705 y(where)776 1805 y(eq)87 b(=)43 b(equal)e(\(x,)i-(y\))776 1905 y(eqs)g(=)g(equalOneByOne)38 b(\(xs,)k(ys\))515-2090 y Fq(Note)27 b(that)h(the)h(comparison)d(fails)h(if)h(the)g(lists)-g(are)f(not)g(of)h(the)g(same)f(length.)38 b(T)-7 b(o)27-b(c)n(hec)n(k)515 2190 y(that)h(a)f(binary)g(adder)f(is)i(comm)n-(utativ)n(e,)f(w)n(e)g(can)g(no)n(w)g(instead)h(sa)n(y:)602-2376 y 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y(Proving:)e(pin)j(5)g(...)f(Valid.)602-4342 y(--)602 4442 y(Result:)f(Valid.)515 4628 y Fq(Note)27-b(that)h(in)g(the)g(pro)r(of)f(for)g Fp(pin)43 b(2)p-Fq(,)27 b(w)n(e)g(can)g(assume)g(that)h Fp(pin)42 b(1)28-b Fq(holds,)f(etc.)1905 5255 y(21)p eop-%%Page: 22 23-22 22 bop 515 523 a Fm(4.5)135 b(Exercises)575 738 y-Fq(4.1)40 b(T)-7 b(ak)n(e)33 b(a)h(lo)r(ok)f(at)h(the)g(t)n(w)n(o)f-(bit)i(sorter)e(y)n(ou)g(de\014ned)h(in)g(exercise)f(2.2.)55-b(T)-7 b(o)34 b(v)n(erify)722 838 y(that)28 b(it)g(is)g(correct,)e(t)n-(w)n(o)h(prop)r(erties)f(need)i(to)g(b)r(e)g(true:)816-1004 y Fr({)41 b Fq(The)g(left)g(part)f(of)h(the)g(output)g(is)g-(smaller)f(than)h(the)g(righ)n(t)f(part)g(of)h(the)905-1103 y(output,)816 1236 y Fr({)g Fq(The)34 b(output)g(of)g(the)g-(circuit)f(con)n(tains)g(the)h(same)f(bits)h(as)g(the)g(input)g(\(but)-905 1336 y(p)r(ossibly)27 b(in)h(a)f(di\013eren)n(t)h(order\).)722-1502 y(State)g(these)g(t)n(w)n(o)f(prop)r(erties)f(separately)-7-b(,)26 b(and)i(v)n(erify)f(them)h(using)f Fp(verify)p-Fq(.)575 1668 y(4.2)40 b(Some)21 b(prop)r(erties)g(are)f(so)h(easy)f-(to)h(v)n(erify)g(that)g(w)n(e)g(can)g(actually)g(do)g(it)h(b)n(y)f-(sim)n(ulat-)722 1768 y(ing)i(them)g(for)f(all)g(inputs)h(\(using)g-Fp(domain)p Fq(\).)33 b(There)22 b(are)g(a)g(few)h(of)f(these)h(prop)r-(erties)722 1867 y(in)i(this)f(c)n(hapter.)35 b(V)-7-b(erify)24 b(them)h(b)n(y)e(testing)h(them)h(for)f(all)f(inputs.)37-b(Can)23 b(y)n(ou)h(think)722 1967 y(of)k(other)f(suc)n(h)g(easy-to-v)n-(erify)e(prop)r(erties?)575 2133 y(4.3)40 b(Chec)n(k)32-b(that)h(the)g(v)-5 b(arious)31 b(adders)g(in)i(the)g(previous)e(c)n-(hapter)g(are)h(all)g(comm)n(uta-)722 2232 y(tiv)n(e,)i(for)d(sizes)h-(up)h(to)f(16)f(bits.)51 b(What)33 b(happ)r(ens)f(if)h(y)n(ou)e(try)h-(to)h(pro)n(v)n(e)d(that)j(the)722 2332 y(subtractor)26-b(is)i(comm)n(utativ)n(e?)575 2498 y(4.4)40 b(Chec)n(k)29-b(that)g(the)h(subtractor)d(y)n(ou)i(de\014ned)g(in)h(the)f(previous)f-(c)n(hapter)g(is)h(really)f(a)722 2598 y(subtractor.)35-b(Ho)n(w)24 b(do)h(y)n(ou)f(form)n(ulate)h(y)n(our)f(prop)r(ert)n(y;)g-(what)h(is)g(the)h("de\014nition")722 2697 y(of)37 b(subtraction?)62-b(Mak)n(e)36 b(sure)g(y)n(ou)f(do)i(not)f(mess)g(up)h(the)g(sizes)f(of)-g(the)h(binary)722 2797 y(n)n(um)n(b)r(ers.)575 2963-y(4.5)j(De\014ne)35 b(a)f(general)e(prop)r(ert)n(y)h(that)i(states)e-(that)i(a)f(giv)n(en)f(circuit)h(is)g(asso)r(ciativ)n(e.)722-3063 y(An)29 b(op)r(erator)e Fi(\016)h Fq(is)h(asso)r(ciativ)n(e,)e(if)-i(for)f(ev)n(ery)f Fj(x;)14 b(y)s(;)g(z)32 b Fq(it)d(holds)f(that)h(\()-p Fj(x)20 b Fi(\016)f Fj(y)s Fq(\))g Fi(\016)f Fj(z)28-b Fq(=)722 3162 y Fj(x)19 b Fi(\016)f Fq(\()p Fj(y)k-Fi(\016)c Fj(z)t Fq(\).)36 b(Are)27 b(all)h(the)g(adders)e(asso)r-(ciativ)n(e?)575 3328 y(4.6)40 b(V)-7 b(erify)37 b(that)g(the)g-(carry-sa)n(v)n(e)c(adder)j(y)n(ou)f(de\014ned)i(in)g(the)g(previous)e-(c)n(hapter)h(is)722 3428 y(equiv)-5 b(alen)n(t)27 b(to)f(a)h(binary)f-(adder.)36 b(Be)26 b(careful)g(in)h(ho)n(w)g(y)n(ou)f(form)n(ulate)g(y)-n(our)f(prop-)722 3528 y(ert)n(y)-7 b(,)27 b(since)h(the)g(inputs)g(do)-f(not)h(ha)n(v)n(e)e(the)i(same)f(shap)r(e.)575 3694-y(4.7)40 b(Pro)n(v)n(e)27 b(that,)i(for)g(an)f(adder)h(and)f-(subtractor)g(of)h(y)n(our)e(c)n(hoice,)i(it)g(holds)f(that)i-Fj(x)19 b Fq(+)722 3793 y(\()p Fj(y)c Fi(\000)d Fj(z)t-Fq(\))22 b(=)h(\()p Fj(x)12 b Fq(+)g Fj(y)s Fq(\))g Fi(\000)g-Fj(z)t Fq(.)35 b(What)25 b(extra)e(condition)i(should)f(hold)g(for)g-Fj(y)j Fq(and)d Fj(z)t Fq(?)36 b(Ho)n(w)722 3893 y(do)28-b(y)n(ou)e(express)h(that?)575 4059 y(4.8)40 b(\(Hask)n(ell\))29-b(Ho)n(w)f(w)n(ould)h(y)n(ou)f(pro)r(ceed)g(if)h(y)n(ou)g(w)n(an)n(t)f-(to)h(v)n(erify)f(a)g(prop)r(ert)n(y)g(for)g(all)722-4159 y(sizes)f(in)h(b)r(et)n(w)n(een,)g(sa)n(y)-7 b(,)26-b(1)i(and)f(n?)1905 5255 y(22)p eop-%%Page: 23 24-23 23 bop 515 1186 a Fo(Chapter)64 b(5)515 1641 y Fs(Sequen)-6-b(tial)76 b(Circuits)515 2112 y Fq(In)34 b(this)h(c)n(hapter)e(w)n(e)h-(describ)r(e)g(ho)n(w)g(to)g(deal)g(with)g(sequen)n(tial)g(circuits)g-(in)h(La)n(v)-5 b(a.)55 b(Se-)515 2212 y(quen)n(tial)22-b(circuits)g(in)h(La)n(v)-5 b(a)22 b(are)g(sync)n(hronous)e(circuits,)k-(whic)n(h)e(means)g(that)h(there)g(is)f(one)515 2311-y(global)k(clo)r(c)n(k)h(a\013ecting)g(all)h(dela)n(y)f(comp)r(onen)n-(ts)g(in)g(the)h(circuit.)515 2626 y Fm(5.1)135 b(The)45-b(Dela)l(y)h(Comp)t(onen)l(t)515 2847 y Fq(A)37 b(new)h(comp)r(onen)n-(t)f(in)g(sequen)n(tial)g(circuits)g(is)g(the)h Fn(delay)h-Fq(comp)r(onen)n(t.)65 b(It)38 b(is)f(a)g(cir-)515 2947-y(cuit)29 b(with)h(one)e(parameter)g(\(the)h(initial)g(output)h(of)f-(the)g(dela)n(y\))g(and)f(one)h(input,)h(whic)n(h)515-3047 y(b)r(ecomes)d(its)h(output)g(in)g(the)g(next)g(clo)r(c)n(k)e-(cycle.)515 3186 y(Here)40 b(is)g(an)g(example)g(of)g(a)g(simple)g-(circuit)g(called)g Fp(edge)p Fq(,)i(that)f(c)n(hec)n(ks)e(if)i(its)f-(input)515 3286 y(has)31 b(c)n(hanged)h(with)h(resp)r(ect)f(to)g(its)g-(previous)f(input.)52 b(It)33 b(uses)f(a)g(dela)n(y)f(comp)r(onen)n(t)h-(to)515 3385 y(remem)n(b)r(er)27 b(the)h(previous)e(input.)602-3571 y Fp(edge)42 b(inp)g(=)h(change)689 3671 y(where)776-3771 y(inp')129 b(=)44 b(delay)d(low)h(inp)776 3870 y(change)f(=)j-(xor2)d(\(inp,)h(inp'\))515 4056 y Fq(W)-7 b(e)33 b(can)g(sim)n(ulate)g-(a)g(sequen)n(tial)g(circuit)g(b)n(y)g(using)g(the)h(op)r(eration)e-Fp(simulateSeq)p Fq(.)50 b(It)515 4156 y(needs)37 b(a)h(circuit)g(and)f-(a)h(list)g(of)f(inputs.)68 b(The)38 b(list)g(of)g(inputs)h(is)e(in)n-(terpreted)h(as)f(the)515 4255 y(di\013eren)n(t)27 b(inputs)i(at)e(eac)-n(h)g(clo)r(c)n(k)g(tic)n(k.)602 4441 y Fp(Main>)41 b-Fl(simulateSeq)e(edge)j([high,)f(low,)h(low,)g(high])602-4541 y Fp([high,)f(high,)h(low,)f(high])515 4727 y Fq(Here)34-b(is)h(another)e(sequen)n(tial)h(circuit,)i(whic)n(h)f(is)g(called)f-Fp(toggle)p Fq(.)55 b(It)35 b(has)f(an)h(in)n(ternal)515-4827 y(state,)29 b(whic)n(h)g(it)g(outputs,)h(and)f(it)g(tak)n(es)f-(one)h(input.)42 b(If)30 b(the)f(input)h(is)f(high,)g(it)g(c)n(hanges)-515 4926 y(the)f(state.)36 b(If)28 b(not,)g(it)g(sta)n(ys)f(the)h-(same.)1905 5255 y(23)p eop-%%Page: 24 25-24 24 bop 602 523 a Fp(toggle)41 b(change)g(=)i(out)689-623 y(where)776 722 y(out')f(=)h(delay)f(low)g(out)776-822 y(out)86 b(=)43 b(xor2)f(\(change,)f(out'\))515 1005-y Fq(As)31 b(w)n(e)g(can)f(see,)i(the)f(de\014nition)h(of)f-Fp(out')e Fq(is)i(dep)r(enden)n(t)h(on)f Fp(out)p Fq(,)f(whose)h-(de\014nition)g(is)515 1105 y(dep)r(enden)n(t)d(on)g-Fp(out')p Fq(.)36 b(Th)n(us,)28 b(there)f(is)h(a)f(lo)r(op)h(in)g(the)g-(circuit.)38 b(Lo)r(ops)27 b(are)g(not)g(allo)n(w)n(ed)515-1204 y(in)i(com)n(binational)f(circuits,)i(since)f(the)g(meaning)g(of)g-(suc)n(h)g(circuits)g(is)g(unclear.)41 b(But)30 b(in)515-1304 y(sequen)n(tial)d(circuits,)g(they)h(are)e(essen)n(tial)h(to)h-(implemen)n(t)g(an)n(y)f(in)n(teresting)f(b)r(eha)n(vior.)515-1443 y(Sim)n(ulating)h Fp(toggle)e Fq(giv)n(es:)602 1626-y Fp(Main>)41 b Fl(simulateSeq)e(toggle)j([high,)f(low,)h(low,)g(high])-602 1726 y Fp([high,)f(high,)h(high,)f(low])515 2040-y Fm(5.2)135 b(Multiple)46 b(Dela)l(ys)515 2262 y Fq(W)-7-b(e)22 b(ha)n(v)n(e)g(seen)g(ho)n(w)f(w)n(e)h(can)g(dela)n(y)g(a)g-(signal)f Fn(one)i Fq(time)g(instan)n(t,)g(so)f(that)h(w)n(e)f(can)g-(refer)f(to)515 2361 y(the)26 b(signal's)e(previous)h(v)-5-b(alue.)36 b(Sometimes,)26 b(w)n(e)g(w)n(an)n(t)f(to)g(dela)n(y)g(a)g-(signal)g(m)n(ultiple)i(time)515 2461 y(instances.)38-b(W)-7 b(e)29 b(can)f(do)g(this)h(b)n(y)f(de\014ning)g(a)g-(parametrized)f(circuit,)i(called)f Fp(delayN)p Fq(.)e(It)515-2560 y(has)e(t)n(w)n(o)f(parameters,)h Fj(n)p Fq(,)h(the)g(n)n(um)n(b)r-(er)f(of)h(dela)n(ys)e(to)i(use,)g(and)f Fp(init)p Fq(,)g(the)h-(initial)f(v)-5 b(alues)515 2660 y(of)27 b(these)h(dela)n(ys.)515-2799 y(W)-7 b(e)28 b(use)f(recursion)f(o)n(v)n(er)g Fj(n)i-Fq(to)f(de\014ne)h(this)g(circuit.)602 2983 y Fp(delayN)41-b(0)i(init)f(inp)g(=)i(inp)602 3182 y(delayN)d(n)i(init)f(inp)g(=)i-(out)689 3281 y(where)776 3381 y(out)86 b(=)43 b(delay)f(init)g(rest)-776 3481 y(rest)g(=)h(delayN)e(\(n-1\))h(init)g(inp)515-3664 y Fq(A)26 b(useful)h(sequen)n(tial)e(circuit)h(that)g(w)n(e)g(can)-g(implemen)n(t)g(using)g Fp(delayN)p Fq(,)e(is)i(called)g-Fp(puls)p Fq(.)515 3763 y(It)h(has)g(no)g(inputs,)g(one)g(output,)h-(and)f(one)g(parameter)e Fj(n)p Fq(.)37 b(Its)27 b(output)h(is)f-(normally)f(lo)n(w,)515 3863 y(except)h(on)h(the)g Fj(n)p-Fq(-th,)f(2)p Fj(n)p Fq(-th,)g(3)p Fj(n)p Fq(-th,)h(...)37-b(clo)r(c)n(k)27 b(tic)n(k,)g(where)g(it)h(outputs)g(high.)515-4002 y(W)-7 b(e)20 b(implemen)n(t)h(the)f(circuit)g(b)n(y)g(creating)f-Fj(n)s Fi(\000)s Fq(1)g(dela)n(y)g(comp)r(onen)n(ts)h(in)g(a)g(ro)n(w,)-g(initialized)515 4102 y(b)n(y)27 b Fp(low)p Fq(,)g(ended)g(with)h(one)-g(dela)n(y)e(comp)r(onen)n(t)i(initialized)g(b)n(y)f-Fp(high)p Fq(.)602 4285 y Fp(puls)42 b(n)h(\(\))g(=)g(out)689-4385 y(where)776 4484 y(out)86 b(=)43 b(delayN)e(\(n-1\))h(low)g(last)-776 4584 y(last)g(=)h(delay)f(high)g(out)515 4767 y Fq(Note)24-b(that)g(w)n(e)g(need)g(to)g(use)g(a)f(lo)r(op)h(bac)n(k)f(here.)35-b(This)24 b(implemen)n(tation)g(is)g(not)g(optimal,)515-4867 y(in)k(the)g(sense)f(that)h(it)g(uses)f(to)r(o)g(man)n(y)g(dela)n-(y)g(comp)r(onen)n(ts;)g(see)g(exercise)g(5.6.)515 5006-y(Sim)n(ulating)g Fp(puls)42 b(3)27 b Fq(giv)n(es:)1905-5255 y(24)p eop-%%Page: 25 26-25 25 bop 602 523 a Fp(Main>)41 b Fl(simulateSeq)e(\(puls)j(3\))h-([\(\),)f(\(\),)g(\(\),)g(\(\),)h(\(\),)f(\(\),)g(\(\)])602-623 y Fp([low,)f(low,)h(high,)g(low,)g(low,)g(high,)f(low])515-937 y Fm(5.3)135 b(Coun)l(ters)515 1159 y Fq(An)24 b-Fp(n)p Fq(-bit)g(coun)n(ter)g(is)g(a)g(circuit)g(that)g(outputs)h(an)e-Fp(n)p Fq(-bit)h(binary)g(n)n(um)n(b)r(er)g(at)g(ev)n(ery)f(clo)r(c)n-(k)515 1258 y(tic)n(k,)i(starting)f(with)h(0,)g(and)g(increasing)e(it)i-(b)n(y)g(1)f(ev)n(ery)f(clo)r(c)n(k)h(tic)n(k.)36 b(W)-7-b(e)25 b(implemen)n(t)h(this)515 1358 y(b)n(y)k(k)n(eeping)g(an)h(in)n-(ternal)f(state,)h(whic)n(h)g(is)g(a)f(binary)g(n)n(um)n(b)r(er.)46-b(The)31 b(circuit)g(tak)n(es)e(one)515 1458 y(parameter,)d(whic)n(h)h-(indicates)h(the)g(n)n(um)n(b)r(er)f(of)h(bits)g(to)f(use,)h(and)f(has)-g(no)g(inputs.)602 1644 y Fp(counter)41 b(n)i(\(\))g(=)g(number')689-1743 y(where)776 1843 y(number')520 b(=)44 b(delay)d(\(zeroList)f(n\))j-(number)776 1942 y(\(number,)e(carryOut\))e(=)44 b(bitAdder)c(\(high,)h-(number'\))515 2128 y Fq(W)-7 b(e)37 b(use)g(the)g(function)h-Fp(zeroList)p Fq(,)e(whic)n(h)h(creates)f(a)h(list)g(of)g-Fj(n)g Fq(zeros,)h(denoting)f(the)515 2228 y(initial)30-b(v)-5 b(alue.)46 b(Note)30 b(that)h(the)g(dela)n(y)f(comp)r(onen)n(t)g-(not)h(only)f(w)n(orks)e(for)i(bits,)i(but)f(also)515-2328 y(for)c(example)g(for)g(pairs)g(of)g(bits)h(and)g(lists)f(\(as)g-(in)h(this)g(case\).)515 2467 y(Sim)n(ulating)f Fp(counter)e-Fq(giv)n(es:)602 2653 y Fp(Main>)41 b Fl(simulateSeq)e(\(counter)i(3\))-h([\(\),)g(\(\),)h(\(\)])602 2753 y Fp([[low,)e(low,)h(low],)f([high,)h-(low,)f(low],)h([low,)g(high,)f(low]])515 2939 y Fq(A)22-b(v)-5 b(arian)n(t)21 b(on)h(this)g(circuit)g(is)f(the)i-Fn(up-c)l(ounter)p Fq(,)f(whic)n(h)g(tak)n(es)f(an)g(input,)j(whic)n(h)-e(indicates)515 3038 y(if)k(the)h(n)n(um)n(b)r(er)f(should)g(increase)f-(or)g(not.)37 b(In)26 b(this)h(case,)f(w)n(e)f(w)n(an)n(t)h(the)h-(desired)e(increase)515 3138 y(to)i(tak)n(e)g(e\013ect)h(immediately)-7-b(,)28 b(so)f(w)n(e)g(output)h(the)g(n)n(um)n(b)r(er)g-Fn(b)l(efor)l(e)g Fq(w)n(e)g(dela)n(y)e(it.)602 3324-y Fp(counterUp)40 b(n)j(up)g(=)g(number)689 3424 y(where)776-3523 y(number')520 b(=)44 b(delay)d(\(zeroList)f(n\))j(number)776-3623 y(\(number,)e(carryOut\))e(=)44 b(bitAdder)c(\(up,)i(number'\))515-3809 y Fq(Sim)n(ulating)27 b Fp(counterUp)d Fq(giv)n(es:)602-3995 y Fp(Main>)41 b Fl(simulateSeq)e(\(counterUp)h(3\))j([high,)e-(low,)h(high])602 4094 y Fp([[high,)f(low,)h(low],)f([high,)g(low,)h-(low],)g([low,)f(high,)h(low]])515 4409 y Fm(5.4)135-b(Sequen)l(tialization)515 4630 y Fq(In)27 b(c)n(hapter)g(3,)g(w)n(e)f-(ha)n(v)n(e)g(seen)i(a)e(com)n(binational)g(binary)h(adder.)36-b(As)27 b(an)g(input,)h(it)g(tak)n(es)515 4730 y(t)n(w)n(o)g-Fj(n)p Fq(-bit)h(binary)g(n)n(um)n(b)r(ers,)g(and)g(adds)g(them)g(up.)-42 b(F)-7 b(or)29 b(large)e Fj(n)p Fq(,)j(this)f(circuit)h(can)e(get)-515 4830 y(quite)h(large,)f(whic)n(h)h(means)g(it)g(tak)n(es)f(more)g-(circuit)h(area)f(and)h(consumes)f(more)g(p)r(o)n(w)n(er,)515-4929 y(and)f(will)h(need)g(a)f(lo)n(w)n(er)f(clo)r(c)n(k)h(frequency)g-(to)g(w)n(ork)g(prop)r(erly)-7 b(.)1905 5255 y(25)p eop-%%Page: 26 27-26 26 bop 515 523 a Fq(W)-7 b(e)37 b(can)g(mak)n(e)f(use)h(of)g(the)g-(regularit)n(y)e(in)i(the)h(circuit)f(to)f(mak)n(e)h(a)f(small)h(v)n-(ersion)e(of)515 623 y(the)27 b(circuit)g(that)h(ho)n(w)n(ev)n(er)d-(needs)i(sev)n(eral)e(clo)r(c)n(k)i(cycles)f(to)i(compute)f(the)g-(result.)37 b(If)27 b(w)n(e)515 722 y(apply)d(this)g(tec)n(hnique)h(on)-f(the)g(binary)g(adder,)g(w)n(e)g(obtain)g(a)g Fn(se)l(quential)j-(adder)p Fq(.)37 b(It)25 b(tak)n(es)515 822 y(one)32-b(new)h(digit)g(of)g(b)r(oth)g(binary)f(n)n(um)n(b)r(ers)g(at)h(eac)n-(h)f(clo)r(c)n(k)g(cycle.)52 b(This)33 b(is)f(sometimes)515-922 y(called)27 b Fn(bit)j(serial)p Fq(.)515 1061 y(W)-7-b(e)35 b(can)f(implemen)n(t)h(this)g(b)n(y)g(storing)e(the)i(carry)e-(as)h(an)g(in)n(ternal)g(state,)j(so)d(that)h(the)515-1161 y(curren)n(t)26 b(carry-in)g(of)i(the)g(circuit)f(is)h(the)g-(previous)e(carry-out.)602 1347 y Fp(adderSeq)40 b(\(a,b\))i(=)h(sum)-689 1446 y(where)776 1546 y(carryIn)346 b(=)43 b(delay)f(low)g-(carryOut)776 1645 y(\(sum,carryOut\))c(=)43 b(fullAdd)e(\(carryIn,)f-(\(a,b\)\))515 1831 y Fq(Sim)n(ulating)27 b Fp(adderSeq)e-Fq(giv)n(es:)602 2017 y Fp(Main>)41 b Fl(simulateSeq)e(adderSeq)i-([\(high,low\),)d(\(high,high\),)h(\(low,high\)])602-2117 y Fp([high,)i(low,)h(low])515 2303 y Fq(Because)29-b(w)n(e)i(\014nd)g(that)g(man)n(y)f(sequen)n(tial)g(circuits)g(ha)n(v)n-(e)f(this)i(structure,)g(w)n(e)g(de\014ne)f(a)515 2403-y Fn(se)l(quential)35 b Fq(connection)f(pattern,)i(called)e-Fp(rowSeq)e Fq(whic)n(h)j(w)n(orks)e(m)n(uc)n(h)h(lik)n(e)g-Fp(row)p Fq(,)h(but)515 2502 y(instead)27 b(in)n(terprets)g(the)h(ro)n-(w)e Fn(over)31 b(time)p Fq(.)602 2688 y Fp(rowSeq)41-b(circ)h(inp)g(=)h(out)689 2788 y(where)776 2888 y(carryIn)390-b(=)43 b(delay)e(zero)h(carryOut)776 2987 y(\(out,)g(carryOut\))e(=)j-(circ)f(\(carryIn,)e(inp\))515 3173 y Fq(W)-7 b(orth)33-b(noting)g(is)h(that)g(w)n(e)f(mak)n(e)f(use)i(of)f(the)h(generic)f-Fp(delay)e Fq(and)i Fp(zero)f Fq(comp)r(onen)n(t)515-3273 y(here.)k(The)28 b(structure)f(is)g(exactly)g(the)h(same)f(as)g-(in)h(the)g(sequen)n(tial)f(adder.)515 3412 y(Recalling)f(the)h-(de\014nition)f(of)h(a)f(binary)g(adder)g(in)g(terms)h(of)f-Fp(row)p Fq(,)g(w)n(e)g(can)g(rep)r(eat)g(it)h(and)515-3512 y(implemen)n(t)h(a)f(sequen)n(tial)g(adder)g(in)h(terms)f(of)h-Fp(rowSeq)p Fq(:)602 3698 y Fp(adder')172 b(=)43 b(row)173-b(fullAdd)84 b(--)43 b Fl(combinational)602 3797 y Fp(adderSeq')d(=)j-(rowSeq)e(fullAdd)84 b(--)43 b Fl(sequential)515 3983-y Fq(In)30 b(this)g(w)n(a)n(y)-7 b(,)29 b(using)g(a)g(connection)h-(pattern)f(to)h(de\014ne)g(a)f(com)n(binational)f(circuit)i(helps)515-4083 y(us)d(to)h(de\014ne)g(the)g(sequen)n(tial)e(v)n(ersion)h(of)g-(the)h(circuit.)515 4397 y Fm(5.5)135 b(V)-11 b(ariations)46-b(on)f Fh(rowSeq)515 4619 y Fq(The)35 b(sequen)n(tial)f(ro)n(w)f-(connection)h(pattern)h(is)f(sometimes)h(useful,)i(but)e(certainly)f-(not)515 4719 y(alw)n(a)n(ys.)g(If)26 b(w)n(e)f(use)g(it)h(to)f-(implemen)n(t)h(a)f(sequen)n(tial)g(adder,)g(as)g(w)n(e)g(did,)h(w)n(e)-f(can)g(also)f(use)515 4818 y(it)32 b(to)h(add)f(up)g(\\in\014nitely)h-(big")e(binary)h(n)n(um)n(b)r(ers.)50 b(The)33 b(addition)f(nev)n(er)f-(ends,)j(so)d(w)n(e)515 4918 y(can)c(nev)n(er)g(start)g(o)n(v)n(er)e-(adding)j(t)n(w)n(o)e(new)i(n)n(um)n(b)r(ers.)1905 5255-y(26)p eop-%%Page: 27 28-27 27 bop 515 523 a Fq(Therefore,)24 b(it)i(is)f(handy)g(to)g(ha)n(v)n-(e)g(a)f(connection)h(pattern,)h(called)f Fp(rowSeqReset)p-Fq(,)c(whic)n(h)515 623 y(tak)n(es)27 b(one)h(extra)f(input)i-Fp(reset)p Fq(.)37 b(When)29 b Fp(reset)d Fq(is)i(high,)g(the)h(in)n-(ternal)e(carry)g(state)h(will)515 722 y(b)r(e)g(reset)f(to)g-Fp(zero)p Fq(.)602 908 y Fp(rowSeqReset)39 b(circ)j(\(reset,inp\))d(=)k-(out)689 1008 y(where)776 1108 y(carryIn)390 b(=)43 b(delay)e(zero)h-(carry)776 1207 y(carry)478 b(=)43 b(mux)f(\(reset,)f(\(carryOut,)e-(zero\)\))776 1307 y(\(out,)j(carryOut\))e(=)j(circ)f(\(carryIn,)e-(inp\))515 1493 y Fq(W)-7 b(e)25 b(use)g(the)g(standard)f(m)n-(ultiplexer)h(comp)r(onen)n(t)f Fp(mux)g Fq(here,)h(whic)n(h)g(c)n(ho)r-(oses)e(the)i(left)h(or)515 1592 y(righ)n(t)31 b(comp)r(onen)n(t)h(of)h-(an)f(input)h(pair,)g(dep)r(ending)f(on)g(if)h(the)g(\014rst)f-(incoming)g(signal)f(is)515 1692 y(lo)n(w)c(or)f(high,)i(resp)r(ectiv)n-(ely)-7 b(.)515 1831 y(No)n(w)27 b(w)n(e)g(can)g(de\014ne)h(a)f-(resettable)g(sequen)n(tial)g(adder)g Fp(adderSeqReset)22-b Fq(as)27 b(follo)n(ws:)602 2017 y Fp(adderSeqReset)38-b(=)43 b(rowSeqReset)c(fullAdd)515 2203 y Fq(V)-7 b(ery)31-b(often,)i(it)e(is)h(the)g(case)e(that)i(the)g(in)n(ternal)e(carry)g-(state)h(has)g(to)h(b)r(e)g(reset)e(p)r(erio)r(di-)515-2303 y(cally)-7 b(,)30 b(that)h(is,)g(on)f(ev)n(ery)f-Fj(n)p Fq(-th,)h(2)p Fj(n)p Fq(-th,)h(...)45 b(clo)r(c)n(k)29-b(tic)n(k.)45 b(Therefore,)30 b(w)n(e)g(create)f(a)h(third)515-2403 y(sequen)n(tial)f(ro)n(w)g(v)-5 b(ariation,)30 b(whic)n(h)g(tak)n-(es)f(a)h(parameter)f Fj(n)p Fq(,)i(whic)n(h)f(indicates)g(the)h(reset)-515 2502 y(p)r(erio)r(d.)602 2688 y Fp(rowSeqPeriod)39-b(n)k(circ)f(inp)g(=)h(out)689 2788 y(where)776 2888-y(reset)f(=)h(puls)f(n)h(\(\))776 2987 y(out)130 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b(design)f(should)h(use)f(a)h(minimal)g(n)n(um)n(b)r-(er)f(of)722 1951 y(dela)n(y)f(comp)r(onen)n(ts)g(\(ho)n(w)g(man)n(y)g-(is)h(that?\).)575 2117 y(5.7)40 b(De\014ne)29 b(an)f(up-do)n(wn)g-(coun)n(ter.)38 b(The)28 b(coun)n(ter)f(gets)h(a)g(pair)f(of)i(inputs.)-39 b(If)29 b(the)f(left)722 2217 y(input)f(is)g(high,)f(it)h(coun)n(ts)-f(up.)36 b(Otherwise,)26 b(if)h(the)g(righ)n(t)e(input)i(is)f(high,)h-(it)g(coun)n(ts)722 2316 y(do)n(wn.)37 b(Otherwise,)26-b(the)i(state)g(sta)n(ys)e(the)i(same.)575 2482 y(5.8)40-b(De\014ne)32 b(a)e(0-to-9)g(coun)n(ter.)46 b(The)31-b(coun)n(ter)f(has)g(no)h(inputs,)h(and)f(a)g(4-bit)f(n)n(um)n(b)r(er)-722 2582 y(as)d(output.)37 b(Initially)-7 b(,)28 b(the)g(output)g-(starts)e(at)h(0,)g(and)h(incremen)n(ts)e(at)i(ev)n(ery)e(clo)r(c)n(k)-722 2682 y(tic)n(k,)i(but)g(after)f(the)h(output)g(9,)g(it)g(returns)f-(to)g(0.)722 2814 y(Connect)h(the)g(displa)n(y)f(from)g(exercise)f(2.6)-h(to)g(y)n(our)g(coun)n(ter.)575 2980 y(5.9)40 b(De\014ne)26-b(a)e Fn(synchr)l(onizer)p Fq(,)j(whic)n(h)e(has)f(t)n(w)n(o)h(inputs)g-Fp(go1)f Fq(and)h Fp(go2)p Fq(,)f(and)h(one)g(output)722-3080 y Fp(go)p Fq(.)35 b(The)23 b(output)h(only)e(b)r(ecomes)h(true)g-(when)g(b)r(oth)g Fp(go1)f Fq(and)h Fp(go2)f Fq(ha)n(v)n(e)g(b)r(een)h-(high)722 3180 y(in)28 b(the)g(past)f(or)g(are)g(high)g(no)n(w)g(since)-h(the)g(last)f Fp(go)p Fq(.)36 b(.)722 3313 y(Here)28-b(is)f(an)g(example)h(sim)n(ulation:)809 3512 y Fp(Main>)42-b Fl(simulateSeq)d(synchronize)1420 3611 y([\(low,high\),\(hi)o(gh,)o-(lo)o(w\),)o(\(h)o(ig)o(h,h)o(ig)o(h\),)1725 3711 y-(\(high,low\),\(low,)o(lo)o(w\))o(,\(l)o(ow)o(,hi)o(gh)o(\)])809-3811 y Fp([low,)j(high,)f(high,)h(low,)g(low,)g(high])533-4010 y Fq(5.10)e(De\014ne)f(a)e(circuit)h(called)f Fp(outputList)p-Fq(,)f(whic)n(h)i(has)f(one)h(parameter,)h(a)e(list)h(of)722-4110 y(v)-5 b(alues,)36 b(no)e(inputs,)i(and)f(one)e(output.)58-b(The)34 b(circuit)g(outputs)h(the)g(elemen)n(ts)f(in)722-4209 y(the)f(parameter)d(list)j(one)e(b)n(y)h(one)g(at)g(ev)n(ery)f-(clo)r(c)n(k)g(tic)n(k)h(rep)r(eatedly)-7 b(.)50 b(Here)31-b(is)h(an)722 4309 y(example)27 b(sim)n(ulation:)809-4508 y Fp(Main>)42 b Fl(simulateSeq)d(\(outputList)g([low,)j(low,)f-(high]\))2291 4608 y([\(\),\(\),\(\),\(\),\(\),\()o(\)])809-4707 y Fp([low,)h(low,)g(high,)f(low,)h(low,)g(high])1905-5255 y Fq(28)p eop-%%Page: 29 30-29 29 bop 515 1186 a Fo(Chapter)64 b(6)515 1641 y Fs(Sequen)-6-b(tial)76 b(V)-19 b(eri\014cation)515 2112 y Fq(In)29-b(this)h(c)n(hapter)e(w)n(e)h(describ)r(e)g(ho)n(w)f(w)n(e)h(can)g(v)n-(erify)f(prop)r(erties)h(of)g(sequen)n(tial)f(circuits.)515-2212 y(W)-7 b(e)28 b(restrict)f(ourselv)n(es)e(to)j(sequen)n(tial)f-(safet)n(y)g(prop)r(erties.)515 2525 y Fm(6.1)135 b(Sequen)l(tial)46-b(Safet)l(y)g(Prop)t(erties)515 2747 y Fq(Let)37 b(us)h(tak)n(e)f(a)g-(lo)r(ok)f(at)i(ho)n(w)f(to)g(de\014ne)h(prop)r(erties)e(ab)r(out)i-(sequen)n(tial)f(circuits.)66 b(In)515 2846 y(principle,)24-b(w)n(e)e(can)h(use)f(the)h(same)g(tec)n(hniques)f(as)h(w)n(e)f(did)h-(with)h(com)n(binational)d(circuits.)515 2946 y(Let)27-b(us)h(tak)n(e)f(a)g(lo)r(ok)g(at)g(some)g(examples.)515-3086 y(Here)g(is)g(ho)n(w)g(w)n(e)h(can)f(compare)f(the)i(t)n(w)n(o)f-(sequen)n(tial)g(adders)f(from)i(section)f(5.4.)602 3267-y Fp(prop_SameAdderSe)o(q)37 b(inp)43 b(=)g(ok)689 3366-y(where)776 3466 y(out1)f(=)h(adderSeq)e(inp)776 3566-y(out2)h(=)h(adderSeq')d(inp)776 3665 y(ok)130 b(=)43-b(out1)f(<==>)g(out2)515 3847 y Fq(Here)27 b(is)g(another)g(example;)g-(the)h(comp)r(osition)e(of)i Fp(edge)e Fq(and)h Fp(toggle)e-Fq(from)i(section)g(5.1)515 3946 y(giv)n(es)e(the)i(iden)n(tit)n(y)g-(circuit.)36 b(This)26 b(means)h(that)f(the)h(input)h(is)e(the)h(same)f-(as)g(the)g(output.)602 4128 y Fp(prop_ToggleEdgeI)o(de)o(nti)o(ty)37-b(inp)42 b(=)h(ok)689 4227 y(where)776 4327 y(mid)g(=)g(toggle)e(inp)-776 4426 y(out)i(=)g(edge)f(mid)776 4526 y(ok)87 b(=)43-b(out)f(<==>)g(inp)515 4707 y Fq(The)20 b(prop)r(erties)f(w)n(e)h(can)g-(describ)r(e)f(in)i(this)f(w)n(a)n(y)f(are)g(called)h-Fn(se)l(quential)j(safety)h(pr)l(op)l(erties)p Fq(.)515-4807 y(Recall)j(that)i(safet)n(y)e(prop)r(erties)g(are)g(prop)r(erties)-g(whic)n(h)h(can)f(b)r(e)i(describ)r(ed)e(as)h(a)f(circuit)515-4907 y(with)21 b(one)f(output,)i(whic)n(h)f(should)f(alw)n(a)n(ys)f(b)r-(e)i(true)f(\(or)g(nev)n(er)g(b)r(e)h(false\))f(for)g(the)h(prop)r(ert)-n(y)515 5006 y(to)27 b(hold.)1905 5255 y(29)p eop-%%Page: 30 31-30 30 bop 515 523 a Fq(Examples)25 b(of)h(prop)r(erties)f(whic)n(h)h-(are)f(not)h(safet)n(y)f(prop)r(erties)g(are)g(for)h(example)f-Fn(liveness)515 623 y Fq(prop)r(erties.)43 b(These)29-b(can)h(assert)f(that)h(a)g(certain)f(condition)h(m)n(ust)g(hold)g(at)g-(some)f(p)r(oin)n(t)515 722 y(in)f(the)g(future,)g(for)f(example.)515-1037 y Fm(6.2)135 b(Sequen)l(tial)46 b(Logic)515 1258-y Fq(Apart)22 b(from)h(the)g(tec)n(hniques)g(w)n(e)f(used)h(to)g-(de\014ne)g(com)n(binational)e(prop)r(erties,)i(there)g(are)515-1358 y(also)j(sp)r(ecial)i(tec)n(hniques)f(w)n(e)g(can)g(apply)h(to)f-(de\014ne)h(sequen)n(tial)f(prop)r(erties.)639 1581 y-Fi(\017)41 b Fq(When)28 b(w)n(e)e(w)n(an)n(t)h(to)g(refer)f(to)h(v)-5-b(alues)26 b(of)h(signals)f(at)h(di\013eren)n(t)g(time)h(instances,)e-(w)n(e)722 1680 y(can)34 b(use)f(a)g Fp(delay)f Fq(to)i(get)f(access)g-(to)g(previous)g(v)-5 b(alues.)55 b(But)34 b(b)r(e)g(careful)f(ab)r-(out)722 1780 y(what)28 b(initial)g(v)-5 b(alue)27 b(y)n(ou)g(c)n(ho)r-(ose)f(for)h(this)h(use)g(of)f(dela)n(y)-7 b(.)639 1946-y Fi(\017)41 b Fq(When)23 b(w)n(e)f(w)n(an)n(t)g(a)g(certain)g(prop)r-(ert)n(y)f(only)h(to)g(b)r(e)h(true)f(when)h(a)f(certain)f(condition)-722 2045 y(holds,)40 b(whic)n(h)e(do)r(es)f(not)h(necessarily)e(hold)i-(all)f(the)h(time,)j(w)n(e)c(can)h(use)f(logical)722-2145 y(implication.)j(Implication)29 b(is)f(implemen)n(ted)i(b)n(y)e-(the)h(La)n(v)-5 b(a)27 b(gate)h Fp(impl)p Fq(,)g(and)g(also)722-2245 y(b)n(y)g(the)g(binary)e(op)r(erator)g Fp(==>)p-Fq(.)515 2467 y(Here)32 b(is)h(an)g(example.)53 b(Supp)r(ose)33-b(w)n(e)g(w)n(an)n(t)f(to)h(de\014ne)h(the)f(follo)n(wing)f(prop)r(ert)-n(y)g(ab)r(out)515 2567 y(the)i Fp(toggle)e Fq(circuit:)49-b("if)35 b(the)f(input)h(is)f(high,)h(then)g(the)f(curren)n(t)f(output)-i(is)f(di\013eren)n(t)515 2666 y(from)27 b(the)h(previous)e(output".)-515 2806 y(The)h(w)n(a)n(y)g(w)n(e)g(de\014ne)h(this)g(in)g(La)n(v)-5-b(a)26 b(is:)602 2992 y Fp(prop_ToggleToggl)o(es)o(Whe)o(nH)o(igh)37-b(inp)42 b(=)h(ok)689 3092 y(where)776 3191 y(out)173-b(=)44 b(toggle)d(inp)776 3291 y(out')129 b(=)44 b(delay)d(low)h(out)-776 3390 y(change)f(=)j(xor2)d(\(out,)h(out'\))776 3490-y(ok)217 b(=)44 b(inp)e(==>)g(change)515 3676 y Fq(First,)21-b(w)n(e)d(compute)i(the)f(output)h Fp(out)e Fq(from)h-Fp(toggle)p Fq(.)32 b(Then,)21 b(w)n(e)e(use)g(a)g(dela)n(y)f(comp)r-(onen)n(t)515 3776 y(to)32 b(get)g(access)e(to)i(the)h(previous)e-(output)h Fp(out')p Fq(.)49 b(W)-7 b(e)33 b(de\014ne)f(the)g(situation)-g Fp(change)e Fq(in)515 3875 y(whic)n(h)j(these)g(outputs)h(di\013er.)-54 b(And)34 b(then)g(w)n(e)e(sa)n(y:)48 b("if)33 b(the)g(input)i(is)e-Fp(high)p Fq(,)g(then)h(the)515 3975 y(outputs)28 b(di\013er".)515-4289 y Fm(6.3)135 b(V)-11 b(eri\014cation)515 4511 y-Fq(After)33 b(de\014ning)f(these)h(prop)r(erties,)f(w)n(e)h(w)n(ould)f-(lik)n(e)g(to)g(formally)g(v)n(erify)f(them.)52 b(V)-7-b(eri\014-)515 4611 y(cation)28 b(of)h(a)f(sequen)n(tial)g(prop)r(ert)n-(y)g(means)h(that)g(w)n(e)f(ha)n(v)n(e)g(to)h(pro)n(v)n(e)e(that)i(the)-g(prop)r(ert)n(y)515 4710 y(holds)36 b(at)g(all)g(times.)63-b(In)37 b(La)n(v)-5 b(a,)37 b(w)n(e)f(do)h(this)f(b)n(y)g-Fn(induction)h Fq(o)n(v)n(er)e(time.)63 b(It)37 b(w)n(orks)e(as)515-4810 y(follo)n(ws.)1905 5255 y(30)p eop-%%Page: 31 32-31 31 bop 515 523 a Fq(Firstly)-7 b(,)23 b(w)n(e)f(ha)n(v)n(e)f(to)i-(do)f(the)g Fn(b)l(ase)k(c)l(ase)p Fq(:)35 b(pro)n(ving)20-b(that)j(the)g(prop)r(ert)n(y)e(holds)h(for)g(the)h Fn(\014rst)515-623 y Fq(time)28 b(instance.)38 b(Since)28 b(lo)r(oking)f(at)h(just)g-(one)g(time)g(instance)g(do)r(es)f(not)h(in)n(v)n(olv)n(e)f(time)h(at)-515 722 y(all,)f(w)n(e)g(can)h(use)f(the)h(same)f(tec)n(hniques)g(as)g-(w)n(e)g(did)h(in)g(the)g(com)n(binational)f(case.)515-862 y(Then,)f(w)n(e)g(do)g(the)g(inductiv)n(e)h Fn(step)p-Fq(.)36 b(W)-7 b(e)27 b(w)n(an)n(t)e(to)h(pro)n(v)n(e)f(that)h-Fn(if)h Fq(the)f(prop)r(ert)n(y)f(holds)h(at)515 961-y(time)h Fj(t)p Fq(,)f(it)h(also)f(holds)g(at)g(time)h-Fj(t)16 b Fq(+)f(1.)37 b(W)-7 b(e)26 b(do)g(this)h(as)f(follo)n(ws:)35-b(w)n(e)26 b(create)g(an)g Fn(arbitr)l(ary)515 1061 y-Fq(time)i(instance)f(b)n(y)g(\014lling)g(the)h(states)f(of)g(the)g-(circuits)g(with)h(fresh)f(v)-5 b(ariables.)36 b(Then,)27-b(w)n(e)515 1161 y(run)32 b(the)h(circuit)f(once)g(on)h(that)f(state,)i-(obtaining)e(an)g(output)h(and)g(new)f(state)g(v)-5 b(alues.)515-1260 y(Then)34 b(w)n(e)f(assert)f(that)i(the)g(output)g(is)g(true,)h-(and)e(run)h(the)g(circuit)f(on)h(the)g(new)f(state)515-1360 y(v)-5 b(alues.)36 b(Finally)-7 b(,)28 b(w)n(e)f(need)h(to)f(pro)n-(v)n(e)f(that)i(the)g(new)g(output)g(is)f(true.)515 1499-y(After)22 b(pro)n(ving)d(the)j(base)f(case)g(and)g(the)h(inductiv)n(e)-f(step,)i(w)n(e)e(ha)n(v)n(e)g(pro)n(v)n(ed)e(our)i(prop)r(ert)n(y)-7-b(.)515 1599 y(Here)27 b(is)g(what)h(happ)r(ens)g(in)g(La)n(v)-5-b(a:)602 1785 y Fp(Main>)41 b Fl(verify)h(prop_ToggleEdge)o(Id)o(ent)o-(it)o(y)602 1885 y Fp(Proving:)e(base)i(1)h(...)g(Valid.)602-1984 y(Proving:)d(step)i(1)h(...)g(Valid.)602 2084 y(--)602-2183 y(Result:)e(Valid.)602 2383 y(Lava>)g Fl(verify)h(prop_ToggleTogg)-o(le)o(sWh)o(en)o(Hig)o(h)602 2482 y Fp(Proving:)e(base)i(1)h(...)g-(Valid.)602 2582 y(Proving:)d(step)i(1)h(...)g(Valid.)602-2682 y(--)602 2781 y(Result:)e(Valid.)515 2967 y Fq(W)-7-b(e)28 b(giv)n(e)e(a)i(more)e(detailed)i(explanation)f(of)g(induction)h-(in)g(the)g(next)g(section.)515 3282 y Fm(6.4)135 b(Induction)515-3503 y Fq(T)-7 b(o)36 b(p)r(erform)f(induction)i(on)f(a)g(La)n(v)-5-b(a)35 b(prop)r(ert)n(y)-7 b(,)38 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gr-/Times-Italic ff 270.00 scf sf-1050 3900 m-gs 1 -1 sc (2) col0 sh gr-$F2psEnd-rs-%%EndDocument- @endspecial 994 3114 a(Figure)g(7.7:)36 b(A)28 b(time)g(transformed)e-(sequen)n(tial)h(circuit)h Fj(F)12 b Fq(.)776 3379 y-Fp(sum1)86 b(=)43 b(adderSeqPeriod)38 b(n)43 b(ab)776-3479 y(sum1')f(=)h(delayN)e(\(n-1\))h(low)g(sum1)776-3579 y(sum2)86 b(=)43 b(adderSlowedDown)37 b(n)43 b(ab)776-3678 y(ok)174 b(=)43 b(sum1')e(<==>)h(sum2)515 3864 y-Fq(Unfortunately)-7 b(,)43 b(this)d(w)n(a)n(y)f(of)h(sp)r(ecifying)g-(the)h(prop)r(ert)n(y)e(in)n(tro)r(duces)g(a)h(lot)g(of)g(extra)515-3964 y(logic,)34 b(and)f(moreo)n(v)n(er,)f(extra)g(state.)54-b(This)33 b(mak)n(es)g(the)g(v)n(eri\014cation)f(of)h(these)h(kind)f-(of)515 4063 y(prop)r(erties)d(v)n(ery)g(hard.)47 b(In)31-b(particular,)g(the)h(induction)f(metho)r(ds)g(need)h(an)f(extremely)-515 4163 y(high)24 b(induction)h(depth.)36 b(In)25 b(the)g(next)g-(section,)f(w)n(e)g(will)h(see)f(a)g(simpler)g(and)h(more)e(direct)515-4263 y(metho)r(d)28 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Fp(append)41-b(\(a,b\))h(=)h(a)g(++)g(b)515 2749 y Fq(Lastly)-7 b(,)27-b(w)n(e)g(de\014ne)h Fp(parl)p Fq(:)602 2935 y Fp(parl)42-b(circ1)f(circ2)h(=)689 3034 y(halveList)e(->-)i(\(circ1)f(-|-)i-(circ2\))e(->-)h(append)602 3237 y(Main>)f Fl(simulate)g(\(parl)g-(reverse)g(id\))h([1..16])602 3337 y Fp([8,7,6,5,4,3,2,1)o(,9)o(,10)o-(,1)o(1,1)o(2,)o(13)o(,14)o(,1)o(5,1)o(6])515 3539 y-Fq(Sometimes,)26 b(w)n(e)f(w)n(an)n(t)g(to)g(p)r(erform)g(an)h(op)r-(eration)e(of)i(eac)n(h)e(elemen)n(t)i(of)g(a)f(list)h(of)f(signals)515-3639 y(or)31 b(bus.)50 b(F)-7 b(or)31 b(this)h(w)n(e)g(use)g(the)g-(connection)g(pattern)f Fp(map)p Fq(,)h(whic)n(h)g(y)n(ou)f(will)i(ha)n-(v)n(e)d(seen)515 3738 y(if)36 b(y)n(ou)f(ha)n(v)n(e)f(used)i(a)f-(functional)h(programming)d(language.)60 b(F)-7 b(or)35-b(example,)i Fp(map)42 b(inv)515 3838 y Fq(in)n(v)n(erts)26-b(eac)n(h)h(of)h(a)f(list)h(of)f(bits.)602 4024 y Fp(Main>)41-b Fl(simulate)g(\(map)h(inv\))g([high,)f(low,)h(high,)f(low])602-4124 y Fp([low,high,low,hi)o(gh)o(])515 4310 y Fq(Buses)d(need)g(not)h-(con)n(tain)e(only)h(lists)h(of)f(bits.)70 b(They)38-b(can)g(b)r(e)h(more)f(structured,)j(so)515 4409 y(that)29-b(our)f(circuit)h(descriptions)f(can)h(matc)n(h)g(the)g-Fn(lo)l(gic)l(al)i Fq(structure)d(of)h(the)h(circuit.)41-b(F)-7 b(or)515 4509 y(example,)27 b(the)h(circuit)f-Fp(map)43 b(fullAdd)25 b Fq(mak)n(es)h(p)r(erfect)i(sense.)602-4695 y Fp(Main>)41 b Fl(simulate)g(\(map)h(fullAdd\))951-4794 y([\(low,\(high,low)o(\)\),)o(\(h)o(ig)o(h,\()o(hi)o(gh,)o(hi)o-(gh\))o(\),)o(\(l)o(ow,)o(\(h)o(igh)o(,h)o(ig)o(h\)\))o(])602-4894 y Fp([\(high,low\),\(hig)o(h,)o(hig)o(h\))o(,\(l)o(ow)o(,h)o(igh)o-(\)])1905 5255 y Fq(47)p eop-%%Page: 48 49-48 48 bop 1382 1383 1130 4 v 1759 1100 753 4 v 2135 818-377 4 v 1382 1100 95 4 v 1665 1100 V 1473 1005 195 4-v 1473 1193 4 189 v 1546 1119 a Fj(f)p 1665 1193 V 1473-1196 195 4 v 1382 818 95 4 v 1665 818 V 1473 722 195-4 v 1473 910 4 189 v 1546 837 a(f)p 1665 910 V 1473 913-195 4 v 1382 536 95 4 v 1665 536 V 1473 440 195 4 v 1473-628 4 189 v 1546 555 a(f)p 1665 628 V 1473 631 195 4-v 1759 818 95 4 v 2041 818 V 1849 722 195 4 v 1849 910-4 189 v 1922 837 a(f)p 2041 910 V 1849 913 195 4 v 1759-536 95 4 v 2041 536 V 1849 440 195 4 v 1849 628 4 189-v 1922 555 a(f)p 2041 628 V 1849 631 195 4 v 2135 536-95 4 v 2417 536 V 2226 440 195 4 v 2226 628 4 189 v 335-w(f)p 2417 628 V 2226 631 195 4 v 1624 1658 a Fq(Figure)27-b(8.3:)36 b Fp(tri)42 b(f)515 1921 y Fq(Figure)22 b(8.2)g(sho)n(ws)g(a)-h Fp(map)e Fq(in)j(the)f(case)f(where)g(the)i(input)g(is)e(a)h(4-list)f-(\(of)i(pairs)d(or)i(2-lists\).)515 2060 y(Strangely)34-b(enough,)j(the)f(connection)g(pattern)f(that)h(places)f(zero)f(copies)-h(of)h(a)f(circuit)515 2160 y(on)29 b(the)i(\014rst)e(signal)g(in)i(a)e-(bus,)i(one)e(cop)n(y)g(on)h(the)g(next,)h(t)n(w)n(o)e(on)h(the)g-(next,)h(and)f(so)f(on,)515 2260 y(is)i(one)f(that)i(arises)e(often)h-(in)g(hardw)n(are)e(design.)48 b(It)31 b(is)g(a)g(sort)f(of)h(mixture)g-(of)g Fp(map)f Fq(and)515 2359 y Fp(composeN)p Fq(.)g(W)-7-b(e)33 b(call)g(it)g Fp(tri)f Fq(for)h Fn(triangle)p-Fq(.)54 b(Y)-7 b(ou)33 b(should)g(understand)g(wh)n(y)g(when)g(y)n(ou)-515 2459 y(lo)r(ok)25 b(at)g(the)h(diagram)e(in)i(\014gure)f(8.3.)35-b(W)-7 b(e)26 b(lea)n(v)n(e)f(the)h(de\014nition)g(of)f-Fp(tri)f Fq(as)h(exercise)g(8.3.)515 2598 y(An)j(example)f(of)h(the)g-(use)f(of)h(triangle)e(is)602 2783 y Fp(Main>)41 b Fl(simulate)g(\(tri)-h(inc\))g(\(replicate)d(10)k(0\))602 2882 y Fp([0,1,2,3,4,5,6,7)o(,8)o-(,9])515 3066 y Fq(The)24 b(connection)g(patterns)g(that)g(w)n(e)g(ha)n-(v)n(e)f(seen)h(in)h(this)f(section)g(are)f(all)h(useful)h(in)g(man)n-(y)515 3166 y(di\013eren)n(t)38 b(kinds)g(of)g(circuits.)68-b(No)n(w)37 b(let)i(us)f(consider)e(ho)n(w)i(to)g(describ)r(e)f(tree)h-(shap)r(ed)515 3266 y(circuits.)515 3580 y Fm(8.2)135-b(T)-11 b(ree)45 b(shap)t(ed)f(circuits)515 3801 y 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bop 602 523 a Fp(Main>)41 b Fl(simulate)g(\(wrapAddTree)d(8\))43-b([3,4,5,6,10,9,8,)o(7])602 623 y Fp(52)515 825 y Fq(Bew)n(are,)34-b(this)g(adder)f(tree)h(w)n(orks)e(only)i(for)g(input)g(lists)g(whose)g-(length)g(is)g(a)f(p)r(o)n(w)n(er)g(of)515 925 y(t)n(w)n(o.)j(Exercise)-26 b(8.5)h(asks)f(y)n(ou)h(to)g(de\014ne)h(an)g(adder)e(tree)i(that)g-(w)n(orks)d(for)j(an)n(y)e(size.)515 1239 y Fm(8.3)135-b(Describing)46 b(Butter\015y)e(Circuits)515 1461 y Fq(Butter\015y)28-b(circuits)g(are)f(circuits)h(with)h(a)f(particular)f(recursiv)n(e)g-(structure.)38 b(Figures)28 b(8.6)515 1561 y(and)33 b(8.7)f(sho)n(w)g-(t)n(w)n(o)g(suc)n(h)h(circuits)f(and)h(also)f(indicate)h(their)g-(recursiv)n(e)e(structures)h(b)n(y)515 1660 y(sho)n(wing,)d(b)n(y)h-(means)f(of)h(dotted)g(b)r(o)n(xes,)g(where)f(to)h(\014nd)g-(sub-circuits)f(that)i(themselv)n(es)515 1760 y(ha)n(v)n(e)h(the)h-(same)g(recursiv)n(e)e(structure.)53 b(It)34 b(turns)f(out)g(that)g-(these)g(t)n(w)n(o)g(circuits)g(are)f(in)515 1859 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b(complicate)g(matters)h-(a)g(bit.)51 b(The)32 b(circuit)g(is)g(not)g(quite)g(as)g(uniform)g(as)-f(those)515 2696 y(that)c(w)n(e)f(consider.)36 b(Ho)n(w)n(ev)n(er,)25-b(the)i(in)n(terested)f(reader)g(is)g(referred)g(to)h([3)o(],)g(whic)n-(h)g(sho)n(ws)515 2796 y(ho)n(w)h(to)g(describ)r(e)g(and)g(compare)g(v)--5 b(arious)27 b(FFT)i(circuits)f(in)h(an)f(older)g(v)n(ersion)f(of)h-(La)n(v)-5 b(a.)515 2896 y(F)e(or)27 b(more)g(details)g(ab)r(out)g(ho)n-(w)g(the)h(v)n(eri\014cation)f(is)g(actually)g(done,)g(see)h([2)o(].)-515 3035 y(In)h(this)g(section,)f(w)n(e)g(\014rst)h(in)n(tro)r(duce)f-(t)n(w)n(o)g(new)h(connection)f(patterns,)g(and)h(then)g(sho)n(w)515-3135 y(that)41 b(butter\015y)g(circuits)g(can)f(b)r(e)i(made)f(with)g-(just)h(these)f(t)n(w)n(o)f(patterns)g(and)h(serial)515-3234 y(comp)r(osition.)515 3374 y(The)21 b(\014rst)f(of)h(these)g-(patterns)f(w)n(e)h(call)f Fp(two)p Fq(.)34 b(The)20-b(circuit)h Fp(two)43 b(circ)19 b Fq(con)n(tains)h(t)n(w)n(o)g(copies)-515 3473 y(of)31 b Fp(circ)p Fq(.)46 b(The)31 b(\014rst)g(of)g(these)g-(op)r(erates)f(on)h(the)h(\014rst)f(half)g(of)g(the)g(input)h(list,)h-(and)e(the)515 3573 y(second)26 b(on)g(the)h(second)e(half.)37-b(Eac)n(h)25 b(cop)n(y)h(of)g Fp(circ)f Fq(should)h(ha)n(v)n(e)g(a)g-(list)g(as)g(output,)h(and)515 3673 y(the)i(t)n(w)n(o)g(resulting)g-(lists)g(are)f(app)r(ended.)43 b(This)29 b(pattern)g(is)g(easily)g-(de\014ned)h(in)f(terms)g(of)515 3772 y Fp(parl)p Fq(,)d(whic)n(h)h(w)n-(as)g(in)n(tro)r(duced)g(earlier)f(in)i(this)g(c)n(hapter.)602-3958 y Fp(two)42 b(circ)g(=)h(parl)f(circ)g(circ)602-4161 y(Main>)f Fl(simulate)g(\(two)h(reverse\))e([1..16])602-4260 y Fp([8,7,6,5,4,3,2,1)o(,1)o(6,1)o(5,)o(14,)o(13)o(,1)o(2,1)o(1,)o-(10,)o(9])602 4460 y(Main>)h Fl(simulate)g(\(two)h(\(two)g(reverse\)\))-d([1..16])602 4559 y Fp([4,3,2,1,8,7,6,5)o(,1)o(2,1)o(1,)o(10,)o(9,)o-(16)o(,15)o(,1)o(4,1)o(3])515 4762 y Fq(Related)27 b(to)g-Fp(two)p Fq(,)f(w)n(e)g(also)g(in)n(tro)r(duce)h(the)g(pattern)g-Fp(ilv)p Fq(,)f(for)g Fn(interle)l(ave)p Fq(.)38 b(Whereas)26-b Fp(two)515 4862 y(f)21 b Fq(applies)g Fp(f)f Fq(to)i(the)f(top)g(and)-g(b)r(ottom)h(halv)n(es)e(of)h(a)g(list,)i Fp(ilv)42-b(f)21 b Fq(applies)g Fp(f)g Fq(to)g(the)h(o)r(dd)f(and)515-4961 y(ev)n(en)h(elemen)n(ts.)35 b(W)-7 b(e)23 b(de\014ne)h(it)f(in)g-(terms)f(of)h(the)g(wiring)g(pattern)f Fn(ri\017e)p Fq(,)j(whic)n(h)e-(p)r(erforms)1905 5255 y(50)p eop-%%Page: 51 52-51 51 bop 1200 1272 56 4 v 1200 1161 V 1200 1051 V 1200-940 V 1200 829 V 1200 718 V 1200 608 V 1200 497 V 1255-1272 111 4 v 1255 1132 a Fa(@)1283 1160 y(@)1255 910-y(A)1297 993 y(A)1324 1049 y(A)1255 689 y(B)1283 772-y(B)1310 855 y(B)1338 938 y(B)-28 b(B)p 1255 497 V 1255-717 a(\000)1283 689 y(\000)1255 938 y(\001)1297 855 y(\001)1324-800 y(\001)1255 1160 y(\002)1283 1077 y(\002)1310 994-y(\002)1338 910 y(\002)g(\002)p 1366 1272 56 4 v 1366-1161 V 1366 1051 V 1366 940 V 1366 829 V 1366 718 V 1366-608 V 1366 497 V 1418 910 173 4 v 1418 1298 4 388 v 1479-1125 a Fj(f)p 1587 1298 V 1418 1301 173 4 v 1418 468-V 1418 855 4 388 v 1479 682 a(f)p 1587 855 V 1418 858-173 4 v 1587 1272 56 4 v 1587 1161 V 1587 1051 V 1587-940 V 1587 829 V 1587 718 V 1587 608 V 1587 497 V 1642-1272 111 4 v 1642 1160 a Fa(\000)1670 1132 y(\000)1642-1049 y(\001)1684 966 y(\001)1712 911 y(\001)1642 938-y(\002)1670 855 y(\002)1698 772 y(\002)1725 689 y(\002)g(\002)p-1642 497 V -111 w(@)1670 717 y(@)1642 800 y(A)1684 883-y(A)1712 938 y(A)1642 910 y(B)1670 993 y(B)1698 1077-y(B)1725 1160 y(B)g(B)p 1753 1272 56 4 v 1753 1161 V-1753 1051 V 1753 940 V 1753 829 V 1753 718 V 1753 608-V 1753 497 V 2085 1272 V 2085 1161 V 2085 1051 V 2085-940 V 2085 829 V 2085 718 V 2085 608 V 2085 497 V 2141-1272 111 4 v 388 w(\000)2168 1132 y(\000)-110 b(@)2168-1160 y(@)p 2141 940 V 2141 829 V 2141 717 a(\000)2168-689 y(\000)g(@)2168 717 y(@)p 2141 497 V 2251 1272 56-4 v 2251 1161 V 2251 1051 V 2251 940 V 2251 829 V 2251-718 V 2251 608 V 2251 497 V 2303 1132 173 4 v 2303 1298-4 167 v 2368 1225 a Fj(g)p 2473 1298 V 2303 1301 173-4 v 2303 910 V 2303 1077 4 167 v 2368 1003 a(g)p 2473-1077 V 2303 1080 173 4 v 2303 689 V 2303 855 4 167 v-2368 782 a(g)p 2473 855 V 2303 858 173 4 v 2303 468 V-2303 634 4 167 v 2368 561 a(g)p 2473 634 V 2303 637 173-4 v 2473 1272 56 4 v 2473 1161 V 2473 1051 V 2473 940-V 2473 829 V 2473 718 V 2473 608 V 2473 497 V 2528 1272-111 4 v 2528 1160 a Fa(\000)2556 1132 y(\000)f(@)2556-1160 y(@)p 2528 940 V 2528 829 V 2528 717 a(\000)2556-689 y(\000)g(@)2556 717 y(@)p 2528 497 V 2639 1272 56-4 v 2639 1161 V 2639 1051 V 2639 940 V 2639 829 V 2639-718 V 2639 608 V 2639 497 V 1290 1508 a Fq(Figure)27-b(8.5:)36 b Fp(ilv)42 b(f)27 b Fq(and)h Fp(two)42 b(\(ilv)g(g\))515-1769 y Fq(the)22 b(p)r(erfect)g(sh)n(u\017e)g(on)g(a)f(list.)35-b(Think)23 b(of)e(taking)h(a)f(pac)n(k)g(of)h(cards,)g(halving)f(it,)j-(and)e(then)515 1869 y(in)n(terlea)n(ving)31 b(the)i(t)n(w)n(o)f(half)h-(pac)n(ks.)51 b(If)33 b(y)n(ou)f(no)n(w)g Fn(unri\017e)h-Fq(the)g(pac)n(k,)g(y)n(ou)f(rev)n(erse)f(the)515 1968-y(pro)r(cess,)h(returning)g(the)g(pac)n(k)g(to)g(its)h(original)d-(condition.)51 b(\(This)33 b(is)f(somewhat)g(more)515-2068 y(di\016cult)c(to)g(accomplish)e(with)j(aplom)n(b)d(at)i(the)g(p)r-(ok)n(er)e(table.\))602 2247 y Fp(Main>)41 b Fl(simulate)g(riffle)g-([1..16])602 2347 y Fp([1,9,2,10,3,11,4)o(,1)o(2,5)o(,1)o(3,6)o(,1)o-(4,)o(7,1)o(5,)o(8,1)o(6])602 2546 y(Main>)g Fl(simulate)g(\(riffle)f-(->-)j(unriffle\))d([1..16])602 2646 y Fp([1,2,3,4,5,6,7,8)o(,9)o(,10)o-(,1)o(1,1)o(2,)o(13)o(,14)o(,1)o(5,1)o(6])602 2845 y(Main>)h-Fl(simulate)g(unriffle)f([1..16])602 2945 y Fp([1,3,5,7,9,11,13)o(,1)o-(5,2)o(,4)o(,6,)o(8,)o(10)o(,12)o(,1)o(4,1)o(6])515 3124-y Fq(Note)29 b(that)h(unri\017ing)g(the)g(sequence)f(from)g(1)h(to)f-Fj(n)h Fq(divides)f(in)n(to)h(its)f(o)r(dd)h(and)g(its)g(ev)n(en)515-3223 y(elemen)n(ts.)36 b(W)-7 b(e)28 b(use)g(this)g(fact)g(to)f-(de\014ne)h Fp(ilv)e Fq(in)i(terms)f(of)h Fp(two)p Fq(.)602-3403 y Fp(ilv)42 b(circ)g(=)h(unriffle)e(->-)h(two)g(circ)g(->-)h-(riffle)602 3597 y(Main>)e Fl(simulate)g(\(ilv)h(reverse\))e([1..16])-602 3696 y Fp([15,16,13,14,11,)o(12)o(,9,)o(10)o(,7,)o(8,)o(5,)o(6,3)o-(,4)o(,1,)o(2])602 3896 y(Main>)h Fl(simulate)g(\(ilv)h(\(ilv)g-(reverse\)\))d([1..16])602 3995 y Fp([13,14,15,16,9,1)o(0,)o(11,)o(12)o-(,5,)o(6,)o(7,)o(8,1)o(,2)o(,3,)o(4])515 4189 y Fq(Figure)32-b(8.5)g(sho)n(ws)f Fp(ilv)43 b(f)32 b Fq(and)h Fp(two)42-b(\(ilv)g(g\))p Fq(.)52 b(W)-7 b(e)33 b(lea)n(v)n(e)e(the)j-(de\014nition)f(of)f Fp(riffle)515 4289 y Fq(and)27 b-Fp(unriffle)e Fq(as)h(exercise)h(8.6.)515 4428 y(W)-7-b(e)38 b(ha)n(v)n(e)e(seen)i(from)f(our)g(examples)g(that)h(it)g(mak)n-(es)f(sense)g(to)h(apply)f Fp(two)g Fq(and)g Fp(ilv)515-4528 y Fq(rep)r(eatedly)-7 b(.)36 b(W)-7 b(e)27 b(will)g(do)f(this)h-(so)g(often)g(in)g(the)g(butter\015y)g(circuits,)f(that)h(it)g(is)g-(useful)g(to)515 4628 y(de\014ne)h(sp)r(ecial)f(functions.)602-4807 y Fp(twoN)42 b(0)h(circ)f(=)h(circ)602 4907 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468 V 1611 634 4 167 v 1677 561 a(g)p 1781-634 V 1611 637 173 4 v 1781 1272 56 4 v 1781 1161 V 1781-1051 V 1781 940 V 1781 829 V 1781 718 V 1781 608 V 1781-497 V 1836 1272 111 4 v 1836 1160 a Fa(\000)1864 1132-y(\000)f(@)1864 1160 y(@)p 1836 940 V 1836 829 V 1836-717 a(\000)1864 689 y(\000)g(@)1864 717 y(@)p 1836 497-V 1947 1272 56 4 v 1947 1161 V 1947 1051 V 1947 940 V-1947 829 V 1947 718 V 1947 608 V 1947 497 V 1999 1132-173 4 v 1999 1298 4 167 v 2064 1225 a Fj(g)p 2168 1298-V 1999 1301 173 4 v 1999 910 V 1999 1077 4 167 v 2064-1003 a(g)p 2168 1077 V 1999 1080 173 4 v 1999 689 V 1999-855 4 167 v 2064 782 a(g)p 2168 855 V 1999 858 173 4-v 1999 468 V 1999 634 4 167 v 2064 561 a(g)p 2168 634-V 1999 637 173 4 v 2168 1272 56 4 v 2168 1161 V 2168-1051 V 2168 940 V 2168 829 V 2168 718 V 2168 608 V 2168-497 V 2224 1272 111 4 v 2224 1160 a Fa(\000)2251 1132-y(\000)2224 1049 y(\001)2265 966 y(\001)2293 911 y(\001)2224-938 y(\002)2251 855 y(\002)2279 772 y(\002)2307 689 y(\002)-28-b(\002)p 2224 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442 V 1739 442 V 1773 442-V 1806 442 V 1839 442 V 1872 442 V 1905 442 V 1939 442-V 1972 442 V 2005 442 V 2038 442 V 2071 442 V 2105 442-V 2138 442 V 2171 442 V 1447 883 4 9 v 2222 883 V 1447-448 V 2222 448 V 1447 493 4 17 v 1447 526 V 1447 559-V 1447 592 V 1447 625 V 1447 659 V 1447 692 V 1447 725-V 1447 758 V 1447 791 V 1447 825 V 1447 858 V 2222 493-V 2222 526 V 2222 559 V 2222 592 V 2222 625 V 2222 659-V 2222 692 V 2222 725 V 2222 758 V 2222 791 V 2222 825-V 2222 858 V 1587 649 17 4 v 1620 649 V 1654 649 V 1687-649 V 1720 649 V 1753 649 V 1786 649 V 1587 456 V 1620-456 V 1654 456 V 1687 456 V 1720 456 V 1753 456 V 1786-456 V 1585 482 4 17 v 1585 515 V 1585 548 V 1585 581-V 1585 614 V 1585 648 V 1807 482 V 1807 515 V 1807 548-V 1807 581 V 1807 614 V 1807 648 V 1559 1508 a Fq(Figure)26-b(8.6:)37 b Fp(bfly)k(3)j(g)602 1774 y(ilvN)e(0)h(circ)f(=)h(circ)602-1874 y(ilvN)f(n)h(circ)f(=)h(ilv)f(\(ilvN)g(\(n-1\))f(circ\))515-2058 y Fq(Clearly)-7 b(,)31 b(there)f(are)g(similarities)h(b)r(et)n(w)n-(een)g(these)g(t)n(w)n(o)f(de\014nitions.)47 b(W)-7 b(e)31-b(migh)n(t)g(just)h(as)515 2157 y(w)n(ell)27 b(ha)n(v)n(e)f(de\014ned)i-(a)g(function)g(that)g(tak)n(es)e(a)i Fn(c)l(onne)l(ction)h(p)l(attern)-f Fq(as)f(input.)602 2342 y Fp(iter)42 b(0)h(comb)f(circ)g(=)h(circ)602-2441 y(iter)f(n)h(comb)f(circ)g(=)h(comb)f(\(iter)f(\(n-1\))h(comb)g-(circ\))515 2625 y Fq(No)n(w,)23 b(w)n(e)g(can)g(use)f-Fp(iter)42 b(n)i(two)e(f)23 b Fq(instead)f(of)h Fp(twoN)42-b(n)i(f)22 b Fq(and)h Fp(iter)42 b(n)h(ilv)g(f)22 b Fq(instead)515-2725 y(of)27 b Fp(ilvN)42 b(n)h(f)p Fq(.)515 2865 y(No)n(w)31-b(w)n(e)g(are)g(in)h(a)f(p)r(osition)h(to)f(de\014ne)h(a)f(connection)h-(pattern)f(for)g(butter\015y)h(circuits,)515 2964 y(that)27-b(is)g(circuits,)f(lik)n(e)h(those)g(sho)n(wn)f(in)h(\014gures)f(8.6)g-(and)h(8.7,)f(that)h(ha)n(v)n(e)f(a)g(v)n(ery)g(partic-)515-3064 y(ular)31 b(recursiv)n(e)f(structure.)50 b(Because)30-b(the)j(circuits)e(are)g(recursiv)n(e,)h(the)g(corresp)r(onding)515-3163 y(connection)27 b(pattern)g(is)h(de\014ned)g(using)f(recursion.)-602 3348 y Fp(bfly)42 b(0)h(circ)f(=)h(id)602 3447 y(bfly)f(n)h(circ)f-(=)h(ilv)f(\(bfly)g(\(n-1\))f(circ\))h(->-)g(twoN)g(\(n-1\))g(circ)515-3631 y Fq(The)31 b(smallest)g(butter\015y)h(is)f(just)h(the)g(iden)n-(tit)n(y)-7 b(.)49 b(A)32 b(butter\015y)g(of)f(size)g-Fj(n)p Fq(,)i(for)d Fj(n)i Fq(greater)515 3731 y(than)k(zero,)i-(consists)e(of)h(t)n(w)n(o)f(in)n(terlea)n(v)n(ed)e(butter\015ies)j(of)-g(size)f Fj(n)24 b Fi(\000)g Fq(1,)39 b(the)e(output)g(of)515-3831 y(whic)n(h)30 b(is)f(fed)i(in)n(to)f(a)f(stac)n(k)g(of)h-Fp(circ)e Fq(comp)r(onen)n(ts,)j(whic)n(h)e(is)h(made)g(using)g-Fp(twoN)p Fq(.)e(This)515 3930 y(connection)f(pattern)g(is)h(sho)n(wn)f-(in)g(\014gure)g(8.6,)g(whic)n(h)h(sho)n(ws)e Fp(bfly)42-b(3)h(g)p Fq(.)515 4070 y(The)34 b(larger)e(dashed)i(b)r(o)n(x)f(sho)n-(ws)g(one)h(instance)g(of)g Fp(bfly)42 b(2)h(g)p Fq(,)35-b(and)f(there)g(is)g(another)515 4169 y(instance)f(just)i(b)r(elo)n(w)e-(it.)56 b(These)33 b(t)n(w)n(o)g(smaller)g(butter\015ies)h(are)e(in)n-(terlea)n(v)n(ed,)i(so)f(there)515 4269 y(is)j(actually)f(an)h-Fn(unri\017e)g Fq(to)g(their)g(left)g(and)g(a)g Fn(ri\017e)g-Fq(to)g(their)g(righ)n(t.)62 b(\(Mak)n(e)35 b(sure)g(to)515-4369 y(\014nd)e(these)h(wiring)e(patterns,)j(and)e(lo)r(ok)f(again)h-(at)g(the)g(de\014nition)h(of)f Fp(ilv)p Fq(.\))53 b(The)34-b(t)n(w)n(o)515 4468 y(in)n(terlea)n(v)n(ed)24 b(butter\015ies)i(feed)g-(their)f(outputs)h(in)n(to)g(four)f Fp(g)g Fq(comp)r(onen)n(ts,)h(one)f-(ab)r(o)n(v)n(e)g(the)515 4568 y(other,)41 b(that)e(is)g-Fp(twoN)j(2)h(g)p Fq(.)71 b(And)39 b(if)h(y)n(ou)e(lo)r(ok)g(inside)h-(the)g Fp(bfly)j(2)i(g)38 b Fq(in)h(the)h(outer)515 4667-y(dashed)27 b(b)r(o)n(x,)g(y)n(ou)g(will)h(\014nd)g(that)g(it)g(again)e-(has)h(the)h(same)f(recursiv)n(e)f(structure.)515 4807-y(Strangely)34 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515 V 2333 548 V 2333 581 V 2333-614 V 2333 648 V 2554 482 V 2554 515 V 2554 548 V 2554-581 V 2554 614 V 2554 648 V 1537 1508 a Fq(Figure)27-b(8.7:)36 b Fp(bfly1)41 b(3)i(g)602 1776 y(bfly1)e(0)j(circ)e(=)h(id)-602 1875 y(bfly1)e(n)j(circ)e(=)h(ilvN)f(\(n-1\))f(circ)h(->-)g(two)h-(\(bfly1)e(\(n-1\))g(circ\))515 2061 y Fq(This)35 b(time,)j(w)n(e)d-(start)g(with)h(a)g(rep)r(eatedly)f(in)n(terlea)n(v)n(ed)f(stac)n(k)g-(of)i(basic)f(comp)r(onen)n(ts,)515 2161 y(whose)e(outputs)i(are)e(fed)-i(in)n(to)f(t)n(w)n(o)f(smaller)h(butter\015ies,)i(whic)n(h)e(are)f-(com)n(bined)h(using)515 2261 y Fp(two)p Fq(.)h(Figure)27-b(8.7)g(sho)n(ws)g(this)h(recursiv)n(e)d(decomp)r(osition.)515-2400 y(It)g(turns)f(out)g(that)h Fp(ilv)43 b(\(bfly)e(n)i(circ\))23-b Fq(is)h(the)h(same)f(as)g Fp(bfly)42 b(n)h(\(ilv)f(circ\))p-Fq(.)34 b(\(See)515 2500 y(the)22 b(question)g(b)r(elo)n(w)g(ab)r(out)g-Fp(two)43 b(ilv)f(g)22 b Fq(if)g(y)n(ou)g(w)n(an)n(t)f(to)i(\014gure)e-(out)h(wh)n(y)-7 b(.\))35 b(This)23 b(means)515 2599-y(that)29 b(w)n(e)f(can)g(de\014ne)h(the)g(butter\015y)g(net)n(w)n(ork)-e(using)h(a)h(single)f(recursiv)n(e)f(call,)h(but)i(with)515-2699 y(a)d(larger)f(comp)r(onen)n(t:)602 2885 y Fp(bfly2)41-b(0)j(circ)e(=)h(id)602 2985 y(bfly2)e(n)j(circ)e(=)h(ilvN)f(\(n-1\))f-(circ)h(->-)g(bfly2)g(\(n-1\))f(\(two)h(circ\))602 3184-y(bfly3)f(0)j(circ)e(=)h(id)602 3283 y(bfly3)e(n)j(circ)e(=)h(bfly3)e-(\(n-1\))h(\(ilv)g(circ\))f(->-)i(twoN)e(\(n-1\))h(circ)515-3469 y Fq(The)d(surprising)f(thing)h(is)g(that)g(all)g(of)g(these)g-(connection)g(patterns)f(giv)n(e)g(equiv)-5 b(alen)n(t)515-3569 y(circuits)27 b(\(for)g(the)h(same)f(size)h(and)f(comp)r(onen)n-(t\).)515 3709 y(The)h(original)f(butter\015y)h(de\014n)n(tions)g(\()p-Fp(bfly)f Fq(and)h Fp(bfly1)p Fq(\))e(can)i(also)f(b)r(e)h(expressed)f-(using)515 3808 y(a)f(tree-lik)n(e)g(com)n(binator.)35-b(T)-7 b(ak)n(e)27 b(a)f(lo)r(ok)g(at)h(the)h(connection)e(pattern)h-Fp(listTree)p Fq(,)d(whic)n(h)515 3908 y(is)33 b(a)h(v)n(ersion)e(of)i-Fp(binTree)d 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b(w)n(e)f(can)h(describ)r(e)f(a)-g(stac)n(k)g(of)h(suc)n(h)f(comp)r(onen)n(ts)h(b)n(y)f(using)g-(pairing,)i(unpairing)515 722 y(and)d(map)h(as)e(follo)n(ws)h(\(see)g-(exercise)g(3.6)f(and)i(the)g(answ)n(er)e(on)h(page)g(80)f(for)h-Fp(pair)f Fq(and)515 822 y Fp(unpair)p Fq(\).)602 996-y Fp(pmap)42 b(circ)g(=)h(pair)f(->-)g(map)h(circ)e(->-)i(unpair)602-1183 y(Main>)e Fl(simulate)g(\(pmap)g(swap\))h([1..16])602-1282 y Fp([2,1,4,3,6,5,8,7)o(,1)o(0,9)o(,1)o(2,1)o(1,)o(14)o(,13)o(,1)o-(6,1)o(5])515 1470 y Fq(Then,)d(for)d(inputs)h(of)g(length)f(2)1573-1439 y Fg(n)1618 1470 y Fq(,)j Fp(ilvTree)i(\(pmap)g(circ\))35-b Fq(is)h(the)h(same)g(as)e Fp(bfly)42 b(n)515 1569 y(circl)p-Fq(,)22 b(where)h Fp(circl)e Fq(is)i(the)h(same)f(as)g-Fp(circ)e Fq(except)j(that)g(it)f(relates)g(a)g(2-list)f(to)i(a)f-(2-list.)515 1709 y(So)32 b(what)g(kinds)h(of)f(circuits)g(can)g(w)n(e)-h(build)g(with)g(these)f(remark)-5 b(ably)31 b(recursiv)n(e)g(struc-)-515 1808 y(tures?)63 b(W)-7 b(ell,)39 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y(p)r(erforms)27 b(the)h(p)r(erfect)g(sh)n-(u\017e!)602 3327 y Fp(s2)43 b([a,b,c,d])d(=)j([a,c,b,d])602-3514 y(Main>)e Fl(simulate)g(\(bfly)g(3)i(s2\))g([1..16])602-3614 y Fp([1,9,2,10,3,11,4)o(,1)o(2,5)o(,1)o(3,6)o(,1)o(4,)o(7,1)o(5,)o-(8,1)o(6])515 3801 y Fq(But)31 b(all)g(of)g(these)g(examples)g(w)n(ere)-f(just)h(wiring)g(functions.)48 b(What)31 b(happ)r(ens)g(when)g(w)n(e)-515 3901 y(add)c(some)g(functionalit)n(y)h(to)f(the)h(comp)r(onen)n(t?)-515 4212 y Fm(8.4)135 b(Batc)l(her's)46 b(Bitonic)f(Merger)515-4434 y Fq(One)21 b(of)h(the)g(b)r(est)h(kno)n(wn)e(uses)g(of)h(the)g-(butter\015y)g(net)n(w)n(ork)f(is)g(in)i(the)f(building)g(of)g(mergers)-515 4534 y(and)i(sorters)f(based)g(on)h(a)g(t)n(w)n(o-input)g(t)n(w)n-(o-output)g(comparator.)33 b(Let)25 b(us)f(start)g(with)h(t)n(w)n(o)515-4633 y(abstract)36 b(comparators)e(that)j(w)n(ork)e(on)i(in)n(teger)f-(inputs.)65 b(One)37 b(sorts)e(in)n(to)i(ascending)515-4733 y(order,)26 b(and)h(the)h(other)f(in)n(to)h(descending.)602-4907 y Fp(compUp)128 b([x,y])42 b(=)h([imin)e(\(x,y\),)g(imax)h-(\(x,y\)])602 5006 y(compDown)e([x,y])i(=)h([imax)e(\(x,y\),)g(imin)h-(\(x,y\)])1905 5255 y Fq(54)p eop-%%Page: 55 56-55 55 bop 602 523 a Fp(Main>)41 b Fl(simulate)g(\(two)h(compUp\))e-([1,2,4,3])602 623 y Fp([1,2,3,4])602 822 y(Main>)h Fl(simulate)g-(\(ilv)h(compDown\))d([1,2,4,3])602 922 y Fp([4,3,1,2])515-1104 y Fq(It)29 b(turns)f(out)g(that)h Fp(bfly)42 b(n)h(compUp)26-b Fq(sorts)h(\(in)n(to)i(ascending)e(order\))h(a)g(list)h(whose)e-(\014rst)515 1203 y(half)g(is)h(ascending)f(and)g(second)g(half)h(is)f-(descending)g(or)g(vice-v)n(ersa.)35 b(W)-7 b(e)28 b(call)f(suc)n(h)g-(lists)515 1303 y Fn(inc-de)l(c)i Fq(and)g Fn(de)l(c-inc)h-Fq(lists.)41 b(\(The)29 b(merger)f(sorts)f(man)n(y)i(other)f(lists)h-(to)r(o,)g(the)h(so-called)515 1403 y Fn(bitonic)23 b-Fq(lists,)h(but)f(w)n(e)g(don't)g(need)f(to)h(w)n(orry)e(ab)r(out)h-(them.\))36 b(This)23 b(net)n(w)n(ork)e(is)i(kno)n(wn)f(as)515-1502 y Fn(Batcher's)32 b(bitonic)f(mer)l(ger)d Fq([1].)39-b(Also,)28 b Fp(bfly)41 b(n)j(compDown)25 b Fq(sorts)i-Fn(inc-de)l(c)h Fq(and)g Fn(de)l(c-inc)515 1602 y Fq(lists)f(in)n(to)h-(descending)f(order.)602 1772 y Fp(Main>)41 b Fl(simulate)g(\(bfly)g(3)-i(compUp\))e([1,3,5,7,8,6,4,2)o(])602 1871 y Fp([1,2,3,4,5,6,7,8)o(])-602 2071 y(Main>)g Fl(simulate)g(\(bfly)g(3)i(compDown\))d-([1,3,5,7,8,6,4,2])602 2170 y Fp([8,7,6,5,4,3,2,1)o(])515-2340 y Fq(Kno)n(wing)22 b(that)i(the)g(merger)e(sorts)g-Fn(inc-de)l(c)27 b(lists)c Fq(allo)n(ws)g(us)g(to)g(build)i(a)e-(recursiv)n(e)e(sorter.)515 2440 y(In)e(fact,)j(w)n(e)d(can)g-(parameterise)f(the)i(circuit)f(on)g(the)h(comparator)d(\(the)j-Fp(comp)e Fq(parameter\),)515 2539 y(and)28 b(de\014ne)h(b)r(oth)h(an)e-(up)h(and)g(a)f(do)n(wn)h(sorter)e(at)i(the)g(same)f(time.)41-b Fp(sorter)g(n)i(compUp)515 2639 y Fq(sorts)26 b(in)n(to)i(ascending)f-(order,)g(while)h Fp(sorter)41 b(n)i(compDown)25 b Fq(sorts)h(in)n(to)i-(descending)f(or-)515 2738 y(der.)602 2908 y 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bop 515 523 a Fq(T)-7 b(o)40 b(transform)g(a)g-Fp(While)p Fq(,)i(w)n(e)f(\014rst)f(transform)g(the)h(subprogram)e-Fp(prog)p Fq(.)75 b(Then,)44 b(w)n(e)515 623 y(in)n(tro)r(duce)c(an)h-(auxiliary)e(signal)h(called)g Fp(active)p Fq(,)i(whic)n(h)e(is)h(high)-f(exactly)h(when)f(w)n(e)515 722 y(should)31 b(consider)f(starting)g-Fp(prog)p Fq(,)g(that)h(is)g(when)h(the)f(whole)g(while)g(lo)r(op)g(is)-f(started)h(or)515 822 y(when)i Fp(prog)e Fq(has)i(\014nished.)53-b(W)-7 b(e)34 b(actually)e(start)h Fp(prog)e Fq(when)i(w)n(e)g(are)f-(activ)n(e,)i(and)f(the)515 922 y(condition)24 b(is)f(true.)36-b(W)-7 b(e)24 b(\014nish)h(the)f(while)g(lo)r(op)g(when)g(w)n(e)g(are)f-(activ)n(e)g(but)i(the)f(condition)515 1021 y(is)j(false.)602-1206 y Fp(circuit)41 b(\(While)g(cond)h(prog\))f(start)h(=)h(\(emits,)e-(finish\))689 1306 y(where)776 1405 y(\(emits,)g(finish'\))f(=)j-(circuit)e(prog)h(start')776 1605 y(active)f(=)j(or2)e(\(start,)f-(finish'\))776 1704 y(start')g(=)j(and2)d(\(active,)g(cond\))776-1804 y(finish)g(=)j(and2)d(\(active,)g(inv)h(cond\))515-1989 y Fq(T)-7 b(ransforming)25 b(sequen)n(tial)i(comp)r(osition)f-(just)i(connects)f(the)g Fp(finish)e Fq(of)i(the)h(\014rst)f(with)515-2089 y(the)h Fp(start)d Fq(of)j(the)g(second,)f(and)g(collects)g(the)h-(emitted)h(messages.)602 2274 y Fp(circuit)41 b(\(prog1)g(:>>)h-(prog2\))f(start)h(=)h(\(emits,)e(finish\))689 2373 y(where)776-2473 y(\(emits1,)g(finish1\))f(=)j(circuit)e(prog1)g(start)776-2572 y(\(emits2,)g(finish\))84 b(=)43 b(circuit)e(prog2)g(finish1)776-2772 y(emits)h(=)h(emits1)e(++)i(emits2)515 2957 y Fq(And)25-b(lastly)-7 b(,)26 b(transforming)d(parallel)h(comp)r(osition)g(starts)-g(b)r(oth)i(circuits)e(when)i(started,)515 3056 y(collects)j(the)h-(emitted)g(messages,)f(and)g Fn(synchr)l(onizes)i Fq(the)f(\014nish)g-(signals)e(for)h(\014nishing.)515 3156 y(W)-7 b(e)28-b(use)g(the)h Fp(synchronize)23 b Fq(circuit,)29 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5255 y Fq(66)p eop-%%Page: 67 68-67 67 bop 776 623 a Fp(signals)41 b(=)863 722 y([)j(orl)e([)h(sig)1125-822 y(|)g(\(out',sig\))d(<-)i(emits)1125 922 y(,)h(out)f(==)h(out')1125-1021 y(])863 1121 y(|)h(out)e(<-)h(outputs)863 1220 y(])515-1406 y Fq(W)-7 b(e)32 b(\014rst)f(create)g(a)h(top-lev)n(el)e-Fp(start)g Fq(signal,)i(whic)n(h)g(is)f(to)h(b)r(e)g(high)g(on)f(the)h-(\014rst)g(clo)r(c)n(k)515 1506 y(tic)n(k,)e(and)f(then)h(lo)n(w)f-(forev)n(er,)f(then)i(\014lter)f(out)h(the)g(signals)e(w)n(e)h(are)g-(in)n(terested)g(in)g(from)515 1606 y(the)i(resulting)g(circuit.)48-b(Note)32 b(that)f(w)n(e)g(ha)n(v)n(e)g(to)g(tak)n(e)g(the)g(or)g(for)g-(these)g(signals,)g(since)515 1705 y(there)c(migh)n(t)h(b)r(e)g(sev)n-(eral)d(parts)i(of)h(the)g(P)n(ace)e(program)g(emitting)i(the)g(same)f-(signal.)515 1845 y(Here)g(is)g(ho)n(w)g(w)n(e)h(can)f(create)g(a)g-(toggle)f(circuit)i(from)f(the)h(giv)n(en)f(P)n(ace)f(program:)602-2031 y Fp(toggle')41 b(change)g(=)i(out)689 2130 y(where)776-2230 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b(rey)g(,)38-b(G)n(\023)-39 b(erard)33 b(Berry)-7 b(,)37 b(P)n(atrice)d(Bertin,)j(F)--7 b(ran\030)-37 b(cois)35 b(Bourdoncle,)h(and)685 3065-y(Jean)f(V)-7 b(uillemin.)62 b(Jazz.)d(Av)-5 b(ailable)35-b(from)h Fp(http://www.cma.)o(en)o(smp)o(.f)o(r/j)o(az)o(z)p-Fq(,)685 3164 y(1998.)556 3330 y([6])41 b(Nicolas)28-b(Halb)n(w)n(ac)n(hs,)e(P)n(aul)h(Caspi,)h(P)n(ascal)e(Ra)n(ymond,)i-(and)g(Daniel)g(Pilaud.)38 b(The)685 3430 y(sync)n(hronous)19-b(data\015o)n(w)h(programming)f(language)g(LUSTRE.)25-b Fn(Pr)l(o)l(c.)g(IEEE)p Fq(,)d(79\(9\),)685 3530 y(1991.)556-3696 y([7])41 b(Stev)n(en)34 b(Johnson.)56 b Fn(Synthesis)36-b(of)g(Digital)h(Designs)f(fr)l(om)g(R)l(e)l(cursion)f(Equations)p-Fq(.)685 3795 y(The)28 b(A)n(CM)g(Distinguished)g(Dissertation)f-(Series,)g(The)g(MIT)h(Press,)e(1984.)556 3961 y([8])41-b(M.)66 b(P)-7 b(.)65 b(Jones.)149 b(The)65 b(Hugs)g(distribution.)150-b(Curren)n(tly)64 b(a)n(v)-5 b(ailable)64 b(from)685-4061 y Fp(http://haskell.or)o(g/h)o(ug)o(s)-6 b 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b(J)f(McCann)n(y)-7 b(,)35 b(J)e(McWhirter,)j(and)d(E)h(Sw)n-(artzlander,)685 722 y(editors,)28 b Fn(Systolic)k(A)n(rr)l(ay)f(Pr)l-(o)l(c)l(essors)p Fq(,)e(pages)f(589)f({)h(598.)f(Pren)n(tice-Hall)g-(In)n(terna-)685 822 y(tional,)h(1989.)515 988 y([11])40-b(John)24 b(Matthews)h(and)f(John)g(Launc)n(h)n(bury)-7-b(.)30 b(Elemen)n(tary)23 b(microarc)n(hitecture)f(alge-)685-1088 y(bra.)29 b(In)23 b Fn(Int.)i(Conf.)h(on)g(Computer)g(A)n(ide)l(d)-g(V)-6 b(eri\014c)l(ation)p Fq(.)23 b(Springer)f(V)-7-b(erlag,)23 b(LNCS,)685 1187 y(1999.)515 1353 y([12])40-b(John)20 b(O'Donnell.)25 b(F)-7 b(rom)20 b(transistors)e(to)j-(computer)f(arc)n(hitecture:)32 b(T)-7 b(eac)n(hing)19-b(func-)685 1453 y(tional)i(circuit)f(sp)r(eci\014cation)h(in)g(Hydra.)-k(In)c Fn(F)-6 b(unctional)23 b(Pr)l(o)l(gr)l(amming)i(L)l(anguages)685-1553 y(in)30 b(Educ)l(ation)p Fq(.)f(Springer)d(V)-7-b(erlag,)27 b(LNCS,)h(1996.)515 1719 y([13])40 b(Mary)27-b(Sheeran.)36 b Fj(\026)p Fq(FP)-7 b(,)27 b(an)g(algebraic)f(VLSI)i-(design)f(language.)35 b(PhD)27 b(thesis,)h(Pro-)685-1818 y(gramming)f(Researc)n(h)f(Group,)h(Oxford)f(Univ)n(ersit)n(y)-7-b(,)27 b(1983.)515 1984 y([14])40 b(Mary)26 b(Sheeran.)34-b Fj(\026)p Fq(FP,)26 b(a)g(language)e(for)i(VLSI)h(design.)34-b(In)26 b Fn(A)n(CM)j(Symp.)g(on)g(LISP)685 2084 y(and)i(F)-6-b(unctional)29 b(Pr)l(o)l(gr)l(amming)p Fq(,)g(1984.)515-2250 y([15])40 b(Mary)c(Sheeran.)64 b(Designing)36 b(regular)f(arra)n-(y)g(arc)n(hitectures)g(using)h(higher)g(order)685 2350-y(functions.)i(In)28 b Fn(Int.)i(Conf.)h(on)f(F)-6 b(unctional)30-b(Pr)l(o)l(gr)l(amming)h(L)l(anguages)f(and)h(Com-)685-2449 y(puter)f(A)n(r)l(chite)l(ctur)l(e,)f(LNCS)h(201)p-Fq(.)f(Springer)d(V)-7 b(erlag,)27 b(1985.)515 2615 y([16])40-b(Mary)26 b(Sheeran.)35 b(Describing)27 b(and)f(reasoning)f(ab)r(out)i-(circuits)g(using)f(relations.)35 b(In)685 2715 y Fn(Workshop)d(on)e(F)--6 b(oundations)30 b(of)g(VLSI)g(Design)p Fq(.)e(Klu)n(w)n(er,)e(1990.)-515 2881 y([17])40 b(Mary)29 b(Sheeran.)43 b(Puzzling)29-b(p)r(erm)n(utations.)43 b(In)30 b Fn(Pr)l(o)l(c.)j(Glasgow)g(F)-6-b(unctional)32 b(Pr)l(o-)685 2980 y(gr)l(amming)f(Workshop)p-Fq(,)e(1996.)1905 5255 y(89)p eop-%%Page: 90 91-90 90 bop 515 999 a Fs(Index)515 1414 y Fp(++)p Fq(,)27-b(47)515 1513 y Fp(*)p Fq(,)g(73)515 1613 y Fp(+)p Fq(,)g(73)515-1713 y Fp(->-)p Fq(,)f(45,)h(74)515 1812 y Fp(-|-)p Fq(,)f(46,)h(74)515-1912 y Fp(-)p Fq(,)g(73)515 2011 y Fp(/)p Fq(,)g(73)515-2111 y Fp(<==>)p Fq(,)f(19)515 2211 y Fp(<=>)p Fq(,)g(72)515-2310 y Fp(<#>)p Fq(,)g(72)515 2410 y Fp(<&>)p Fq(,)g(72)515-2510 y Fp(<|>)p Fq(,)g(72)515 2609 y Fp(==>)p Fq(,)g(30,)h(72)515-2709 y Fp(>>==)p Fq(,)f(73)515 2808 y Fp(\045\045)p Fq(,)h(73)515-2991 y(adder)681 3091 y(binary)-7 b(,)27 b(12)681 3190-y(bit,)h(12)681 3290 y(full,)g(8)681 3390 y(half,)g(6)681-3489 y(sequen)n(tial,)f(26,)f(27)681 3589 y(tree,)h(49)515-3688 y Fp(adder)p Fq(,)e(12,)i(76)515 3788 y Fp(adder2)p-Fq(,)e(80)515 3888 y Fp(adderSeq)p Fq(,)f(26)515 3987-y 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eop-%%Trailer-end-userdict /end-hook known{end-hook}if-%%EOF

INSTALL view

@@ -1,5 +1,11 @@-Install the package by running:+First install the [Haskell Platform](http://www.haskell.org/platform). -  runghc Setup configure-  runghc Setup build-  runghc Setup install+Then install the package by running++    cabal install++from within this directory. Alternatively, run++    cabal install chalmers-lava2000++to install the latest version on Hackage.

chalmers-lava2000.cabal view

@@ -1,17 +1,17 @@ name:                chalmers-lava2000-version:             1.3-synopsis:            Hardware description library-description:         Hardware description library+version:             1.4.1+synopsis:            Hardware description EDSL+description:         For more info, see the tutorial: <http://projects.haskell.org/chalmers-lava2000/Doc/tutorial.pdf> category:            Language, Hardware license:             BSD3 license-file:        LICENSE copyright:           (c) 2008. Koen Claessen <koen@chalmers.se> author:              Koen Claessen <koen@chalmers.se> maintainer:          Emil Axelsson <emax@chalmers.se>-homepage:            http://projects.haskell.org/chalmers-lava2000/+homepage:            http://projects.haskell.org/chalmers-lava2000/Doc/tutorial.pdf cabal-version:       >= 1.6 build-type:          Simple-data-files:          README, INSTALL, Doc/tutorial.ps, Vhdl/lava.vhd, Scripts/satzoo.wrapper, Scripts/smv.wrapper+data-files:          README, INSTALL, Doc/tutorial.pdf, Vhdl/lava.vhd, Scripts/satzoo.wrapper, Scripts/smv.wrapper  source-repository head   type:     darcs