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-(.)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 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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 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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 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-(w)n(e)f(deal)h(with)g(in)g(La)n(v)-5 b(a)24 b(are)g(so-called)f -Fn(safety)515 2847 y(pr)l(op)l(erties)p Fq(.)45 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 b(matter)h(what)g(the)h(input)g(is?)35 b(T)-7 -b(o)21 b(answ)n(er)g(the)h(question,)h(w)n(e)f(use)g(the)g(La)n(v)-5 -b(a)21 b(op)r(eration)515 4295 y Fp(verify)p Fq(.)602 -4481 y Fp(Main>)41 b Fl(verify)h(prop_HalfAddOut)o(pu)o(tNe)o(ve)o(rBo) -o(th)o(Tru)o(e)602 4581 y Fp(Proving:)e(...)j(Valid.)515 -4767 y Fq(This)30 b(pro)r(cess)f(w)n(orks)f(in)j(the)f(follo)n(wing)g -(w)n(a)n(y)-7 b(.)43 b(Just)30 b(as)g(w)n(e)g(can)f(generate)g(VHDL)i -(from)515 4867 y(a)36 b(circuit)g(description,)i(w)n(e)e(can)g(also)f -(generate)g(a)h(logical)f(form)n(ula)h(represen)n(ting)f(the)515 -4966 y(circuit.)j(This)27 b(logical)g(form)n(ula)g(is)h(then)g(giv)n -(en)f(to)h(an)g(external)f(theorem)g(pro)n(v)n(er)f(whic)n(h)1905 -5255 y(18)p eop -%%Page: 19 20 -19 19 bop 515 523 a Fq(will)34 b(pro)n(v)n(e)f(\(or)h(dispro)n(v)n(e\)) -f(the)i(v)-5 b(alidit)n(y)35 b(of)f(the)h(form)n(ula.)56 -b(The)35 b(result)f(is)h(then)g(tak)n(en)515 623 y(bac)n(k)26 -b(in)n(to)i(La)n(v)-5 b(a.)515 762 y(Here)23 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 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b(General)46 b(Prop)t(erties)515 2042 -y Fq(General)22 b(prop)r(erties)g(are)g(prop)r(erties)g(that)h(are)f -(parametrized)g(b)n(y)h(one)f(or)g(more)g(circuits.)515 -2142 y(They)31 b(can)h(b)r(e)g(de\014ned)g(just)h(lik)n(e)e(connection) -h(patterns.)49 b(Here)32 b(is)f(a)h(general)e(prop)r(ert)n(y)515 -2241 y(that)e(p)r(oses)f(the)h(question)f(if)h(the)g(t)n(w)n(o)f(giv)n -(en)g(circuits)g(are)f(equiv)-5 b(alen)n(t.)602 2427 -y Fp(prop_Equivalent)37 b(circ1)42 b(circ2)f(a)i(=)h(ok)689 -2527 y(where)776 2627 y(out1)e(=)h(circ1)f(a)776 2726 -y(out2)g(=)h(circ2)f(a)776 2826 y(ok)130 b(=)43 b(out1)f(<==>)g(out2) -515 3012 y Fq(The)27 b(follo)n(wing)g(prop)r(ert)n(y)f(c)n(hec)n(ks)h -(if)h(a)f(giv)n(en)g(circuit)g(is)h(comm)n(utativ)n(e.)602 -3198 y Fp(prop_Commutative)37 b(circ)42 b(\(as,)g(bs\))g(=)h(ok)689 -3297 y(where)776 3397 y(out1)f(=)h(circ)f(\(as,)g(bs\))776 -3497 y(out2)g(=)h(circ)f(\(bs,)g(as\))776 3596 y(ok)130 -b(=)43 b(out1)f(<==>)g(out2)515 3782 y Fq(Of)30 b(course,)f(the)h 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-eop -%%Page: 21 22 -21 21 bop 515 523 a Fq(In)30 b(La)n(v)-5 b(a,)30 b(w)n(e)h(can)f(do)g -(this)h(b)n(y)f(v)n(erifying)f(not)i(just)g Fn(one)f -Fq(question,)h(but)g(a)f(whole)g(list)h(of)515 623 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) 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3844 y Fp(Proving:)j(pin)j -(1)g(...)f(Valid.)602 3944 y(Proving:)e(pin)j(2)g(...)f(Valid.)602 -4043 y(Proving:)e(pin)j(3)g(...)f(Valid.)602 4143 y(Proving:)e(pin)j(4) -g(...)f(Valid.)602 4242 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 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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 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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 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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 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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 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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 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-634 V 1861 637 173 4 v 2030 1272 56 4 v 2030 1161 V 2030 -1051 V 2030 940 V 2030 829 V 2030 718 V 2030 608 V 2030 -497 V 1624 1508 a Fq(Figure)27 b(8.2:)36 b Fp(map)42 -b(g)602 1776 y(Main>)f Fl(simulate)g(halveList)f([high,low,high,)o(low) -o(])602 1875 y Fp(\([high,low],[hig)o(h,)o(low)o(]\))515 -2078 y Fq(Then,)i(w)n(e)d(de\014ne)h(the)g(circuit)f -Fp(append)p Fq(,)h(whic)n(h)f(tak)n(es)g(a)g(pair)g(of)g(lists)g(of)h -(length)f Fj(m)515 2178 y Fq(and)29 b Fj(n)p Fq(,)g(and)g(joins)g(them) -g(together)f(\(or)h Fn(c)l(onc)l(atenates)g Fq(them\),)h(to)f(giv)n(e)f -(a)h(list)g(of)g(length)515 2277 y Fj(m)11 b Fq(+)g Fj(n)p -Fq(.)36 b(This)24 b(circuit)f(is)h(de\014ned)h(in)f(terms)g(of)g(Hask)n -(ell's)f(built-in)i(in\014x)f(list)g(concatenate)515 -2377 y(op)r(erator)i(\()p Fp(++)p Fq(\).)602 2563 y 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 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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 Fq(Circuits)27 -b(in)h(the)g(shap)r(e)f(of)g(trees,)g(lik)n(e)g(that)h(sho)n(wn)f(in)h -(\014gure)e(8.4,)h(can)g(b)r(e)h(used)g(to)f(sys-)515 -3901 y(tematically)g(apply)g(a)g(function)g(that)h(com)n(bines)f(data)f -(v)-5 b(alues)27 b(together)g(to)g(a)g(collection)515 -4000 y(of)e(data.)36 b(A)26 b(binary)f(tree)g(circuit)h(\014rst)f(com)n -(bines)g(eac)n(h)g(half)h(of)f(the)h(input)h(v)-5 b(alues,)25 -b(using)515 4100 y(t)n(w)n(o)31 b(smaller)g(trees)g(and)h(then)h(com)n -(bines)e(the)h(t)n(w)n(o)f(remaining)g(results.)50 b(One)31 -b(example)515 4200 y(of)c(suc)n(h)h(a)f(circuit)g(is)h(an)f(adder)g -(tree)g(that)h(adds)f(up)h(a)f(list)h(of)g(n)n(umhers.)515 -4339 y(The)f(outline)h(of)g(the)g(recursiv)n(e)e(de\014nition)i(of)f(a) -g(tree)h(connection)f(pattern)g(is:)602 4523 y Fp(tree)42 -b(circ)g([inp])f(=)i(...)g(inp)f(...)602 4623 y(tree)g(circ)g(inps)85 -b(=)43 b(...)g(tree)f(circ)g(...)g(tree)g(circ)g(...)g(circ)g(...)g -(inps)515 4807 y Fq(W)-7 b(e)26 b(call)f(the)g(parameter)f -Fp(circ)g Fq(the)i Fn(c)l(omp)l(onent)i(cir)l(cuit)p -Fq(.)36 b(The)26 b(\014rst)f(line)h(in)f(this)h(outline)515 -4907 y(de\014nes)c(what)g(should)g(b)r(e)h(done)f(when)g(w)n(e)g(get)g 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b(\022)1720 1303 y(@)1659 -1243 y(@)g(I)1659 1016 y(\000)1742 933 y(\000)1825 850 -y(\000)1864 811 y(\000)g(\022)2091 1303 y(\000)2152 1243 -y(\000)g(\022)2296 1303 y(@)2235 1243 y(@)g(I)2152 1016 -y(@)2069 933 y(@)1986 850 y(@)1947 811 y(@)g(I)p 1945 -584 V 1947 523 a(6)1350 1774 y Fq(Figure)27 b(8.4:)36 -b(A)28 b(tree)f(shap)r(ed)g(circuit)515 2036 y(using)h -Fp(circ)p Fq(.)38 b(Exactly)28 b(ho)n(w)g(these)h(de\014nitions)f -(should)h(lo)r(ok)f(dep)r(ends)h(partly)f(on)g(what)515 -2135 y(the)g(comp)r(onen)n(t)f Fp(circ)f Fq(lo)r(oks)h(lik)n(e,)g(and)g -(in)h(particular)f(on)g(its)h(t)n(yp)r(e.)515 2275 y(F)-7 -b(or)27 b(example,)h(if)h Fp(circ)d Fq(is)i(a)g(binary)f(function)i -(taking)e(a)h(pair)f(of)h(inputs)h(and)f(returning)515 -2374 y(a)c(single)g(output,)h(then)g(it)g(mak)n(es)f(sense)f(to)i(mak)n -(e)e(the)i(follo)n(wing)f(de\014nition)g(of)h(a)f(binary)515 -2474 y(tree)j(connection)g(pattern,)h Fp(binTree)p Fq(.)602 -2657 y Fp(binTree)41 b(circ)h([inp])f(=)i(inp)602 2756 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-(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 y(fact)f(equiv)-5 -b(alen)n(t:)43 b(the)31 b(same)g(net)n(w)n(ork)e(of)i(comp)r(onen)n(ts) -f(can)h(b)r(e)g(recursiv)n(ely)e(describ)r(ed)515 1959 -y(in)f(t)n(w)n(o)g(completely)g(di\013eren)n(t)g(w)n(a)n(ys.)37 -b(And)29 b(indeed)g(it)f(turns)h(out)f(that)g(there)g(are)g(man)n(y)515 -2059 y(more)f(w)n(a)n(ys)f(to)h(describ)r(e)g(the)h(same)f(net)n(w)n -(ork.)36 b(W)-7 b(e)28 b(will)g(study)f(some)g(of)h(them.)515 -2198 y(Butter\015y)c(circuits)g(are)g(used)g(for)g(example)g(to)g -(build)h(routing)f(net)n(w)n(orks)e(from)j(switc)n(hes,)515 -2298 y(and)38 b(in)h(building)f(e\016cien)n(t)h(sorting)e(circuits.)69 -b(P)n(erhaps)37 b(the)i(b)r(est)f(kno)n(wn)g(butter\015y-)515 -2397 y(lik)n(e)29 b(circuit)h(is)g(the)h(standard)e(Co)r(oley-T)-7 -b(uk)n(ey)28 b(algorithm)h([4)o(])h(for)g(computing)g(the)g(F)-7 -b(ast)515 2497 y(F)g(ourier)34 b(T)-7 b(ransform)35 b(\(FFT\).)h(W)-7 -b(e)36 b(will)g(not)g(consider)f(the)h(FFT)g(here.)60 -b(The)36 b(t)n(widdle-)515 2597 y(factors)31 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 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-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 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1611 858 -173 4 v 1611 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 497 V -111 w(@)2251 717 y(@)2224 800 y(A)2265 -883 y(A)2293 938 y(A)2224 910 y(B)2251 993 y(B)2279 1077 -y(B)2307 1160 y(B)g(B)p 2334 1272 56 4 v 2334 1161 V -2334 1051 V 2334 940 V 2334 829 V 2334 718 V 2334 608 -V 2334 497 V 2386 1132 173 4 v 2386 1298 4 167 v 2451 -1225 a Fj(g)p 2556 1298 V 2386 1301 173 4 v 2386 910 -V 2386 1077 4 167 v 2451 1003 a(g)p 2556 1077 V 2386 -1080 173 4 v 2386 689 V 2386 855 4 167 v 2451 782 a(g)p -2556 855 V 2386 858 173 4 v 2386 468 V 2386 634 4 167 -v 2451 561 a(g)p 2556 634 V 2386 637 173 4 v 2556 1272 -56 4 v 2556 1161 V 2556 1051 V 2556 940 V 2556 829 V -2556 718 V 2556 608 V 2556 497 V 1449 884 9 4 v 1449 -442 V 2215 884 V 2215 442 V 1474 884 17 4 v 1507 884 -V 1540 884 V 1573 884 V 1606 884 V 1640 884 V 1673 884 -V 1706 884 V 1739 884 V 1773 884 V 1806 884 V 1839 884 -V 1872 884 V 1905 884 V 1939 884 V 1972 884 V 2005 884 -V 2038 884 V 2071 884 V 2105 884 V 2138 884 V 2171 884 -V 1474 442 V 1507 442 V 1540 442 V 1573 442 V 1606 442 -V 1640 442 V 1673 442 V 1706 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 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2401 456 V 2434 456 V 2467 456 V 2500 456 V 2534 456 -V 2333 482 4 17 v 2333 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 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-(t)g(that)h(tak)n(es)e(a)h(pair)f(as)h(input)h(and)f(pro)r(duces)g(a)f -(pair)h(as)f(output,)515 623 y(then)34 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) 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b(circuit)g(that)g(w)n(e)f(call)h Fp(s2)p -Fq(,)h(then)f(w)n(e)f(\014nd)i(that)f(a)f(butter\015y)h(of)g(suc)n(h)g -(comp)r(onen)n(ts)515 3153 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 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y(v)n(eri\014cation,)h(30)515 2482 y(v)n(erify)681 -2582 y(com)n(binational,)g(18)681 2682 y(one-b)n(y-one,)g(21)681 -2781 y(options,)h(77)681 2881 y(sequen)n(tial,)g(29)515 -2980 y Fp(verify)p Fq(,)e(18,)i(77)515 3080 y Fp(verifyWith)p -Fq(,)c(34,)k(77)515 3180 y(VHDL)681 3279 y(generation,)f(9)515 -3462 y Fp(writeVhdl)p Fq(,)e(9,)j(77)515 3562 y Fp(writeVhdlInput)p -Fq(,)22 b(9,)27 b(77)515 3661 y Fp(writeVhdlInputOu)o(tp)o(ut)o -Fq(,)22 b(9,)27 b(77)515 3844 y Fp(xnor2)p Fq(,)e(73)515 -3944 y Fp(xor2)p Fq(,)h(73)515 4043 y Fp(xorl)p Fq(,)g(73)515 -4226 y Fp(zero)p Fq(,)g(26,)h(69)515 4325 y Fp(zeroList)p -Fq(,)d(25)515 4425 y Fp(zipp)p Fq(,)i(74,)h(80)1905 5255 -y(92)p eop -%%Trailer -end -userdict /end-hook known{end-hook}if -%%EOF diff --git a/INSTALL b/INSTALL --- a/INSTALL +++ b/INSTALL @@ -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. diff --git a/chalmers-lava2000.cabal b/chalmers-lava2000.cabal --- a/chalmers-lava2000.cabal +++ b/chalmers-lava2000.cabal @@ -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: category: Language, Hardware license: BSD3 license-file: LICENSE copyright: (c) 2008. Koen Claessen author: Koen Claessen maintainer: Emil Axelsson -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