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Fu(sp)-5 b(ac)g(e)p Fv(,)257 1304 y(transforming)31 b(a)i(previous)g (de\034nition)f(from)f([21].)44 b(The)34 b(space)g(of)e(a)h(resolution) e(refu-)257 1425 y(tation)e(is)i(the)f(n)m(um)m(b)s(er)h(of)f(clauses)h (that)g(ha)m(v)m(e)h(to)e(b)s(e)g(k)m(ept)i(sim)m(ultaneously)d(in)h (mem-)257 1545 y(ory)40 b(to)e(infer)h(a)g(con)m(tradiction.)62 b(As)39 b(in)g(the)g(de\034nition)f(of)h(space)h(in)e(a)h(T)-8 b(uring)39 b(ma-)257 1665 y(c)m(hine,)46 b(the)c(space)i(used)f(b)m(y)h (the)e(input)g(clauses)h(is)f(not)g(considered)h(in)f(the)h(mo)s(del) 257 1786 y(and)35 b(these)h(clauses)f(can)f(b)s(e)h(do)m(wnloaded)f(to) g(the)h(w)m(orking)g(memory)e(when)j(needed.)257 1906 y(Alekhno)m(vic)m(h,)48 b(Ben-Sasson,)g(Razb)s(oro)m(v)d(and)f (Wigderson)g([1])g(extended)i(the)f(de\034-)257 2027 y(nition)37 b(of)g(resolution)g(space)i(in)e(sev)m(eral)i(w)m(a)m(ys.) 61 b(They)40 b(de\034ned)f(a)f(space)h(notion)e(for)257 2147 y(stronger)23 b(pro)s(of)f(systems)i(lik)m(e)d(Prop)s(ositional)f 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e(the)i(resources)h(as)e(w)m(ell)g(as)g(purely)h(com)m(bi-)257 3240 y(natorial)j(c)m(haracterizations)i(of)g(the)h(concepts)h(of)d (resolution)g(width)i(and)f(tree-space)257 3360 y(in)g(terms)g(of)f (2-p)s(erson)h(games.)43 b(These)34 b(b)s(eautiful)d(c)m (haracterizations)g(stress)j(the)e(im-)257 3480 y(plicitness)k(of)f (these)i(measures)g(as)f(purely)g(com)m(binatorial)d(prop)s(erties)j (of)f(the)i(input)257 3601 y(form)m(ulas,)32 b(indep)s(enden)m(t)h(of)f (the)h(notion)f(of)g(resolution.)257 3990 y Fw(2)156 b(De\034nitions)257 4228 y Fv(W)-8 b(e)38 b(will)d(consider)j(form)m (ulas)d(o)m(v)m(er)k(a)e(set)h(of)f(prop)s(ositional)d(v)-5 b(ariables)36 b Ft(V)21 b Fv(.)58 b(A)37 b Fu(liter)-5 b(al)257 4348 y Fv(is)43 b(either)f(a)h(v)-5 b(ariable)40 b Ft(x)46 b Fs(2)f Ft(V)65 b Fv(or)42 b(its)g(negation)p 2134 4293 56 4 v 42 w Ft(x)p Fv(.)74 b(A)43 b Fu(clause)f Fv(is)g(a)h(disjunction)f(of)257 4468 y(literals)30 b(and)j(a)f(form)m (ula)e Ft(F)46 b Fv(in)32 b(conjunctiv)m(e)h(normal)d(form)h(\(CNF\))i (is)e(a)h(conjunction)257 4589 y(of)26 b(clauses.)42 b(F)-8 b(or)26 b(a)g(literal)d Ft(l)29 b Fv(and)e Ft(a)g Fs(2)h(f)p Fr(0)p Ft(;)17 b Fr(1)p Fs(g)p Fv(,)27 b Ft(F)2016 4604 y Fq(l)q Fp(=)p Fq(a)2161 4589 y Fv(represen)m(ts)i(the)d(form)m (ula)f(obtained)257 4709 y(from)34 b Ft(F)49 b Fv(b)m(y)36 b(giving)e(to)g Ft(l)k Fv(the)d(v)-5 b(alue)35 b Ft(a)g Fv(and)g(reducing)h(the)f(form)m(ula)e(in)i(the)g(in)m(tuitiv)m(e)257 4829 y(w)m(a)m(y)-8 b(.)404 4959 y(Resolution)47 b(is)h(a)g(refutation) g(pro)s(of)g(system)h(for)f(CNF)h(form)m(ulas.)90 b(The)50 b(only)257 5079 y(inference)33 b(rule)f(in)g(this)g(pro)s(of)g(system)i (is)e(the)h(resolution)e(rule:)1505 5302 y Ft(C)e Fs(_)23 b Ft(x)195 b(D)25 b Fs(_)j Fr(\026)-55 b Ft(x)p 1505 5347 688 4 v 1713 5438 a(C)29 b Fs(_)23 b Ft(D)2231 5370 y(:)p eop %%Page: 4 4 4 3 bop 257 573 a Fv(Cutting)28 b(v)-5 b(ariable)26 b Ft(x)j Fv(from)d(clauses)j Ft(C)20 b Fs(_)13 b Ft(x)29 b Fv(and)f Ft(D)16 b Fs(_)j Fr(\026)-55 b Ft(x)28 b Fv(one)h(gets)f (the)h Fu(r)-5 b(esolvent)27 b Fv(clause)257 693 y Ft(C)i Fs(_)23 b Ft(D)s Fv(.)43 b(A)33 b(resolution)e(refutation)h(of)g(a)g (CNF)h(form)m(ula)e Ft(F)46 b Fv(is)32 b(a)g(sequence)j(of)e(clauses) 257 814 y Ft(C)327 829 y Fp(1)367 814 y Ft(;)17 b(:)g(:)g(:)e(;)i(C)655 829 y Fq(s)723 814 y Fv(where)33 b(eac)m(h)f Ft(C)1292 829 y Fq(i)1351 814 y Fv(is)f(either)g(a)g(clause)h(from)e Ft(F)45 b Fv(or)31 b(is)g(inferred)g(from)f(earlier)257 934 y(clauses)39 b(b)m(y)g(the)f(resolution)f(rule,)i(and)f Ft(C)1853 949 y Fq(s)1928 934 y Fv(is)f(the)i(empt)m(y)f(clause.)60 b(W)-8 b(e)39 b(will)c(denote)257 1054 y(the)46 b(empt)m(y)f(clause)h (b)m(y)g Fo(2)p Fv(.)81 b(Resolution)44 b(is)g(sound)i(and)f(complete,) j(whic)m(h)d(means)257 1175 y(that)38 b(suc)m(h)h(a)e(refutation)g(for) g(a)h(form)m(ula)d(exists)k(if)d(and)i(only)f(if)g Ft(F)51 b Fv(is)37 b(unsatis\034able.)257 1295 y(A)45 b(resolution)f (refutation)g(can)h(b)s(e)g(seen)h(as)f(a)f(directed)i(acyclic)e (graph,)k(in)c(whic)m(h)257 1416 y(the)c(clauses)f(are)g(the)g(v)m (ertices,)j(and)c(if)g(t)m(w)m(o)i(clauses)f(are)g(resolv)m(ed)g(then)h (there)f(is)f(a)257 1536 y(directed)45 b(edge)g(going)d(from)h(eac)m(h) i(of)f(the)h(t)m(w)m(o)g(clauses)g(to)e(the)i(resolv)m(en)m(t.)80 b(If)44 b(the)257 1656 y(underlying)39 b(graph)f(in)g(a)h(refutation)e (happ)s(ens)j(to)f(b)s(e)g(a)f(tree,)j(w)m(e)f(talk)e(ab)s(out)g Fu(tr)-5 b(e)g(e-)257 1777 y(like)36 b Fv(resolution.)53 b(It)36 b(is)g(kno)m(wn)h(that)f(for)f(certain)h(form)m(ulas)f(general) g(resolution)g(can)257 1897 y(pro)s(duce)27 b(exp)s(onen)m(tially)d (shorter)i(refutations)f(than)h(tree-lik)m(e)f(resolution)f([9,)h(6].) 41 b(The)257 2017 y(reason)d(for)f(this)g(is)f(that,)j(con)m(trary)e (to)g(general)g(resolution,)g(in)g(a)g(tree-lik)m(e)f(pro)s(of)g(a)257 2138 y(clause)41 b(that)f(is)f(needed)j(more)d(than)h(once)h(in)f(the)g (refutation)f(m)m(ust)h(b)s(e)h(re-deriv)m(ed)257 2258 y(eac)m(h)34 b(time)d(from)g(the)i(initial)c(clauses.)404 2379 y(The)24 b Fu(size)f Fv(of)h(a)f(resolution)f(refutation)h(is)g (the)h(n)m(um)m(b)s(er)g(of)g(clauses)g(it)f(con)m(tains.)40 b(F)-8 b(or)257 2499 y(an)34 b(unsatis\034able)g(form)m(ula)e Ft(F)14 b Fv(,)34 b(size)p Fr(\()p Ft(F)14 b Fr(\))34 b Fv(denotes)h(the)g(minimal)30 b(size)k(of)g(a)f(resolution)257 2619 y(refutation)40 b(of)g Ft(F)14 b Fv(.)67 b(W)-8 b(e)41 b(denote)g(b)m(y)g(tree-size)p Fr(\()p Ft(F)14 b Fr(\))41 b Fv(the)g(minimal)36 b(size)41 b(of)f(a)g(tree-lik)m(e)257 2740 y(resolution)h(refutation)g(of)h Ft(F)14 b Fv(.)72 b(As)43 b(men)m(tioned)f(ab)s(o)m(v)m(e,)j(this)d(can)h(b)s(e)f (greater)g(than)257 2860 y(size)p Fr(\()p Ft(F)14 b Fr(\))p Fv(.)42 b(W)-8 b(e)26 b(de\034ne)i(no)m(w)e(resolution)f(width)h(and)h (space,)h(the)f(other)f(t)m(w)m(o)h(complexit)m(y)257 2980 y(measures)34 b(discussed)g(in)e(this)g(surv)m(ey)-8 b(.)257 3141 y Fn(De\034nition)40 b(1.)k Fu([7)o(])38 b(The)g(width)f(of)h(a)g(clause)f(is)h(the)g(numb)-5 b(er)37 b(of)h(liter)-5 b(als)38 b(app)-5 b(e)g(aring)257 3261 y(in)44 b(it.)71 b(F)-7 b(or)42 b(a)i(set)f(of)g(clauses)g Fs(C)50 b Fu(\()p Fs(C)g Fu(c)-5 b(an)43 b(b)-5 b(e)43 b(for)g(example)g(a)g(formula)g(in)g(CNF)g(or)257 3382 y(a)j(r)-5 b(esolution)46 b(r)-5 b(efutation\))46 b(the)g(width)g(of)g Fs(C)6 b Fu(,)49 b(denote)-5 b(d)45 b(by)h(initial-width)p Fr(\()p Fs(C)6 b Fr(\))p Fu(,)48 b(is)e(the)257 3502 y(maximal)34 b(width)g(of)h(a)f(clause)h(in)f(the)h(set)g Fs(C)6 b Fu(.)404 3623 y(The)37 b(width)h(ne)-5 b(e)g(de)g(d)38 b(for)g(the)g(r)-5 b(esolution)38 b(of)g(an)g(unsatis\034able)g(CNF)g (formula)g Ft(F)14 b Fu(,)257 3743 y(denote)-5 b(d)36 b(by)g(width)p Fr(\()p Ft(F)14 b Fr(\))p Fu(,)35 b(is)h(the)g(minimal)f (width)h(ne)-5 b(e)g(de)g(d)35 b(in)g(a)h(r)-5 b(esolution)36 b(of)g Ft(F)14 b Fu(,)36 b(that)257 3863 y(is,)f(the)g(minimum)e(of)i (initial-width)p Fr(\()p Ft(\031)t Fr(\))f Fu(over)g(al)5 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Fr(\))25 b(+)g(1)36 b Fv(p)s(ebbles.)58 b(Let)37 b(us)g(supp)s(ose)i(that)d Ft(d)2629 4898 y Fq(c)2664 4883 y Fr(\()p Ft(T)2759 4898 y Fp(1)2798 4883 y Fr(\))f Ft(<)h(d)3034 4898 y Fq(c)3068 4883 y Fr(\()p Ft(T)3163 4898 y Fp(2)3203 4883 y Fr(\))p Fv(,)i(then)257 5003 y Ft(d)308 5018 y Fq(c)343 5003 y Fr(\()p Ft(T)14 b Fr(\))32 b(=)h Ft(d)682 5018 y Fq(c)716 5003 y Fr(\()p Ft(T)811 5018 y Fp(2)851 5003 y Fr(\))i Fv(and)g(one)h(can)g(p)s(ebble) f(\034rst)h Ft(T)2055 5018 y Fp(2)2130 5003 y Fv(with)f Ft(d)2406 5018 y Fq(c)2441 5003 y Fr(\()p Ft(T)2536 5018 y Fp(2)2575 5003 y Fr(\))24 b(+)g(1)36 b Fv(p)s(ebbles,)g(lea)m(v)m(e)g (a)257 5124 y(p)s(ebble)30 b(in)e(the)i(ro)s(ot)e(of)h Ft(T)1211 5139 y Fp(2)1279 5124 y Fv(and)h(then)f(p)s(ebble)h Ft(T)2050 5139 y Fp(1)2118 5124 y Fv(with)f Ft(d)2388 5139 y Fq(c)2423 5124 y Fr(\()p Ft(T)2518 5139 y Fp(1)2557 5124 y Fr(\))15 b(+)g(1)29 b Fv(p)s(ebbles.)43 b(F)-8 b(or)29 b(this)p 257 5197 1296 4 v 370 5258 a Fi(2)433 5289 y Fh(A)e(tree)f Fe(T)748 5258 y Fd(0)797 5289 y Fh(is)h(em)n(b)r(edded)g(in)g Fe(T)37 b Fh(if)27 b Fe(T)38 b Fh(can)26 b(b)r(e)h(obtained)f(from)g Fe(T)2474 5258 y Fd(0)2523 5289 y Fh(b)n(y)g(adding)g(no)r(des)h(and)f(edges)257 5388 y(or)h(inserting)g(no)r(des)g(in)h(the)g(middle)g(of)g(edges)e(of) i Fe(T)12 b Fh(.)p eop %%Page: 10 10 10 9 bop 257 573 a Fv(second)41 b(part)d(of)h(the)g(p)s(ebbling)f(one)h (needs)i Ft(d)2002 588 y Fq(c)2036 573 y Fr(\()p Ft(T)2131 588 y Fp(1)2171 573 y Fr(\))26 b(+)g(2)39 b Fs(\024)g Ft(d)2592 588 y Fq(c)2626 573 y Fr(\()p Ft(T)2721 588 y Fp(2)2761 573 y Fr(\))26 b(+)h(1)p Fv(.)62 b(The)40 b(other)257 693 y(case)34 b(is)e(similar.)p 953 693 49 49 v 404 814 a(F)-8 b(rom)32 b(this)i(result)f(follo)m(ws)g(that)h(a)f (lo)m(w)m(er)i(b)s(ound)f(for)f(the)h(resolution)f(space)i(of)e(a)257 934 y(form)m(ula)28 b(implies)g(an)i(exp)s(onen)m(tially)f(larger)g (tree-lik)m(e)g(resolution)g(size)h(lo)m(w)m(er)g(b)s(ound.)257 1054 y(An)42 b(upp)s(er)g(b)s(ound)g(for)f(the)g(size)h(in)f(terms)g (of)g(the)h(space)h(for)e(the)g(case)i(of)e(general)257 1175 y(resolution)i(is)g(giv)m(en)h(in)f([17],)j(but)e(the)g(heigh)m(t) g(measure)g(is)f(also)g(considered)i(as)e(a)257 1295 y(parameter)30 b(in)f(the)i(relation.)40 b(The)31 b(heigh)m(t)f(of)g(a) g(resolution)f(refutation)g(is)g(the)i(length)257 1416 y(of)25 b(the)g(longest)f(path)h(from)e(an)i(initial)c(clause)k(to)f (the)h(empt)m(y)g(clause)g(in)f(the)h(refutation)257 1536 y(graph.)257 1725 y Fn(Theorem)37 b(10.)42 b Fu([17)o(])35 b(F)-7 b(or)34 b(an)h(unsatis\034able)e(formula)i Ft(F)48 b Fu(in)35 b(CNF)1122 1987 y(size)o Fr(\()p Ft(F)14 b Fr(\))27 b Fs(\024)1561 1846 y Fl(\022)1635 1919 y Fu(sp)-5 b(ac)g(e)o Fr(\()p Ft(F)14 b Fr(\))22 b(+)g Fu(height)o Fr(\()p Ft(F)14 b Fr(\))1897 2055 y Fu(sp)-5 b(ac)g(e)o Fr(\()p Ft(F)14 b Fr(\))2531 1846 y Fl(\023)2604 1987 y Ft(:)404 2242 y Fv(A)40 b(consequence)j(of)c(this)h(result)f(is)h (that)f(refutations)h(with)f(p)s(olynomial)d(heigh)m(t)257 2362 y(and)d(constan)m(t)g(space)h(m)m(ust)f(ha)m(v)m(e)g(p)s (olynomial)c(size.)257 2692 y Fw(5)156 b(T)-13 b(radeo\033s)257 2911 y Fv(Ben-Sasson)43 b(has)f(sho)m(wn)h(in)e([4])g(tradeo\033)h (results)g(for)f(the)h(size,)i(width)d(and)h(space)257 3032 y(complexit)m(y)33 b(measures)h(applied)e(to)h(certain)g(families) d(of)j(form)m(ulas.)44 b(He)33 b(pro)m(v)m(es)i(that)257 3152 y(for)24 b(these)i(form)m(ulas)d(there)i(are)g(optimal)c (resolution)i(refutations)h(with)g(resp)s(ect)i(to)e(one)257 3272 y(of)30 b(the)g(parameters,)g(but)g(the)g(optimization)c(of)j(one) h(complexit)m(y)e(measure)i(increases)257 3393 y(the)c(others.)42 b(A)25 b(cen)m(tral)h(part)f(in)g(the)g(pro)s(of)g(of)g(this)g(result)g (is)g(a)g(surprising)g(connection)257 3513 y(b)s(et)m(w)m(een)37 b(the)e(v)-5 b(ariable)32 b(space)k(needed)g(in)d(the)i(resolution)e (of)h(certain)g(form)m(ulas)f(and)257 3634 y(the)28 b(blac)m(k-white)f (p)s(ebbling)f(measure)h(of)g(an)g(underlying)g(graph.)41 b(This)27 b(connection)h(is)257 3754 y(stated)35 b(here)g(without)f (pro)s(of)f(in)g(Lemma)g(13.)48 b(The)35 b(blac)m(k-white)f(p)s (ebbling)f(measure)257 3874 y(of)27 b(a)f(circuit)g Ft(G)g Fv(in)m(tro)s(duced)h(in)f([13])h(in)m(tuitiv)m(ely)e(measures)i(the)g (space)h(needed)h(for)d(the)257 3995 y(sim)m(ulation)33 b(of)j(the)g(circuit)f(on)g(a)h(nondeterministic)e(T)-8 b(uring)35 b(mac)m(hine.)53 b(This)36 b(game)257 4115 y(is)i(more)g(in)m(v)m(olv)m(ed)h(than)g(the)g(standard)g(p)s(ebbling)e (game)h(and)g(includes)h(p)s(ebbles)g(of)257 4236 y(t)m(w)m(o)g(kinds:) 54 b(white)37 b(p)s(ebbles)h(sim)m(ulating)d(nondeterministic)h(steps)j (and)f(blac)m(k)f(ones)257 4356 y(for)28 b(the)g(sim)m(ulation)d(of)i (deterministic)f(steps.)43 b(W)-8 b(e)28 b(omit)e(here)j(the)f(details) e(ab)s(out)i(this)257 4476 y(game)k(and)h(refer)g(the)f(in)m(terested)i (reader)f(to)f([25].)404 4597 y(The)j(form)m(ulas)d(used)j(b)m(y)g (Ben-Sasson)g(to)f(pro)m(v)m(e)h(the)f(tradeo\033)g(b)s(eha)m(vior)g (are)g(the)257 4717 y(p)s(ebbling)29 b(con)m(tradictions)h(in)m(tro)s (duced)h(in)e([7].)43 b(They)32 b(express)g(the)f(principle)e(that)h (in)257 4837 y(a)41 b(directed)g(acyclic)f(graph,)i(p)s(ebbling)e(the)h (source)h(no)s(des)f(and)g(follo)m(wing)d(the)j(rule)257 4958 y(that)30 b(if)f(all)e(the)j(predecessors)j(of)c(a)h(no)s(de)g Ft(v)j Fv(con)m(tain)d(a)f(p)s(ebble)h(then)g Ft(v)k Fv(also)29 b(gets)h(one,)257 5078 y(implies)g(that)j(a)f(p)s(ebble)h (will)d(b)s(e)j(placed)f(on)g(the)h(sink.)257 5268 y Fn(De\034nition)45 b(11.)g Fu(L)-5 b(et)42 b Ft(G)d Fr(=)h(\()p Ft(V)5 b(;)17 b(E)6 b Fr(\))41 b Fu(b)-5 b(e)41 b(a)h(dir)-5 b(e)g(cte)g(d)40 b(acyclic)h(gr)-5 b(aph)41 b(in)g(which)f(every)257 5388 y(vertex)i(has)g(fan-in)g(2)g(or)g(0)g(with)g(a)h(unique)f(sink)g Ft(s)p Fu(.)67 b(W)-7 b(e)42 b(c)-5 b(al)5 b(l)42 b(a)g(gr)-5 b(aph)41 b(with)i(these)p eop %%Page: 11 11 11 10 bop 257 573 a Fu(pr)-5 b(op)g(erties)36 b(a)h(cir)-5 b(cuit-gr)g(aph.)49 b(L)-5 b(et)37 b Ft(k)e(>)30 b Fr(0)p Fu(.)50 b(W)-7 b(e)37 b(asso)-5 b(ciate)35 b Ft(k)40 b Fu(distinct)d(Bo)-5 b(ole)g(an)35 b(vari-)257 693 y(ables)47 b Ft(v)556 708 y Fp(1)596 693 y Ft(;)17 b(:)g(:)g(:)f(;)h(v)862 708 y Fq(k)952 693 y Fu(with)48 b(every)g(vertex)f Ft(v)56 b Fs(2)c Ft(V)22 b Fu(.)83 b Ft(P)14 b(eb)2324 657 y Fq(k)2324 718 y(G)2384 693 y Fu(,)51 b(the)c Ft(k)s Fu(-th)h(de)-5 b(gr)g(e)g(e)48 b(p)-5 b(ebbling)257 814 y(c)g(ontr)g(adiction)34 b(of)h Ft(G)p Fu(,)f(is)h(the)g(c)-5 b(onjunction)34 b(of:)306 1031 y Fn(Source)k(axioms:)1085 957 y Fl(W)1168 983 y Fq(k)1168 1060 y(i)p Fp(=1)1303 1031 y Ft(v)1350 1046 y Fq(i)1413 1031 y Fu(for)d(e)-5 b(ach)34 b(sour)-5 b(c)g(e)34 b Ft(v)t Fu(.)306 1243 y Fn(P)m(ebbling)e(axioms:)p 1182 1189 56 4 v 43 w Ft(u)1238 1258 y Fq(i)1282 1243 y Fs(_)p 1364 1189 51 4 v 16 w Ft(v)1414 1258 y Fq(j)1467 1243 y Fs(_)1549 1169 y Fl(W)1632 1195 y Fq(k)1632 1273 y(l)q Fp(=1)1764 1243 y Ft(w)1834 1258 y Fq(l)1892 1243 y Fu(for)g Ft(u)f Fu(and)g Ft(v)36 b Fu(the)c(two)g(pr)-5 b(e)g(de)g(c)g(essors)30 b(of)i Ft(w)s Fu(,)501 1364 y Ft(i;)17 b(j)34 b Fs(2)28 b(f)p Fr(1)17 b Ft(:)g(:)g(:)e(k)s Fs(g)p Fu(.)306 1565 y Fn(Sink)37 b(axioms:)p 973 1510 46 4 v 45 w Ft(s)1018 1580 y Fq(i)1082 1565 y Fu(for)d(the)h(sink)f Ft(s)h Fu(and)f Ft(i)28 b Fs(2)g(f)p Fr(1)17 b Ft(:)g(:)g(:)f(k)s Fs(g)p Fu(.)404 1782 y Fv(Ben-Sasson)25 b(sho)m(ws)g(that)f(for)g (certain)g(circuit-graphs)e Ft(G)i Fv(the)h(tree-lik)m(e)e(resolution) 257 1903 y(of)32 b(the)h(form)m(ulas)e Ft(P)14 b(eb)1095 1866 y Fp(1)1095 1928 y Fq(G)1187 1903 y Fv(presen)m(ts)35 b(the)e(follo)m(wing)d(tradeo\033:)257 2098 y Fn(Theorem)41 b(12.)i Fu([4])37 b(F)-7 b(or)37 b(in\034nitely)g(many)h(inte)-5 b(gers)37 b Ft(n)p Fu(,)h(ther)-5 b(e)38 b(exist)f(c)-5 b(ontr)g(adictions)257 2218 y Ft(F)320 2233 y Fq(n)402 2218 y Fu(of)35 b(size)f Ft(n)h Fu(such)g(that:)306 2414 y(1:)45 b Ft(F)494 2429 y Fq(n)576 2414 y Fu(has)34 b(tr)-5 b(e)g(e-like)34 b(r)-5 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y(2-p)s(erson)36 b(com)m(binatorial)c(games)k(pla)m(y)m(ed)g(on)g(the)g (form)m(ulas.)52 b(Besides)37 b(b)s(eing)e(useful)257 4907 y(for)29 b(pro)m(ving)g(new)g(width)g(or)g(space)h(b)s(ounds,)h (these)f(results)f(are)g(remark)-5 b(able)28 b(b)s(ecause)257 5027 y(they)h(sho)m(w)h(that)d(the)i(t)m(w)m(o)g(complexit)m(y)e (measures)i(only)f(dep)s(end)h(on)f(the)g(structure)i(of)257 5147 y(the)e(form)m(ulas)d(and)i(can)g(b)s(e)g(de\034ned)h(indep)s (enden)m(tly)f(of)g(the)g(notion)e(of)i(resolution.)40 b(In)257 5268 y(b)s(oth)31 b(cases)h(the)e(game)g(giving)f(the)i(c)m (haracterization)e(had)i(b)s(een)g(previously)g(de\034ned)257 5388 y(in)h(the)h(literature)e(for)h(other)h(purp)s(oses.)p eop %%Page: 13 13 13 12 bop 257 573 a Fc(6.1)131 b(A)44 b(Game)f(Characterization)j(of)d (Resolution)h(Width)257 769 y Fv(A)m(tserias)39 b(and)f(Dalmau)e([2])i (c)m(haracterized)h(resolution)d(width)i(in)f(terms)h(of)g(the)g(ex-) 257 889 y(isten)m(tial)43 b Ft(k)s Fv(-p)s(ebble)h(game.)78 b(This)44 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