Computing the maximum using (min,+) formulas

Transcription

1 Computig the maximum usig (mi,+) formulas Meea Mahaja 1, Prajakta Nimbhorkar 2, ad Auj Tawari 1 1 The Istitute of Mathematical Scieces, HBNI, Cheai, Idia. {meea,aujvt}@imsc.res.i 2 Cheai Mathematical Istitute, Idia. prajakta@cmi.ac.i Abstract We study computatio by formulas over (mi, +). We cosider the computatio of max{x 1,..., x } over N as a differece of (mi, +) formulas, ad show that size + log is sufficiet ad ecessary. Our proof also shows that ay (mi, +) formula computig the miimum of all sums of 1 out of variables must have log leaves; this too is tight. Our proofs use a complexity measure for (mi, +) fuctios based o miterm-like behaviour ad o the etropy of a associated graph ACM Subject Classificatio F.1.1 Models of Computatio; F.1.3 Complexity Measures ad Classes; F.2.3 Tradeoffs betwee Complexity Measures Keywords ad phrases formulas, circuits, lower bouds, tropical semirig Digital Object Idetifier /LIPIcs.MFCS Itroductio A (mi, +) formula is a formula (tree) i which the leaves are labeled by variables or costats. The iteral odes are gates labeled by either mi or +. A mi gate computes the miimum value amog its iputs while a + gate simply adds the values computed by its iputs. Such formulas ca compute ay fuctio expressible as the miimum over several liear polyomials with o-egative iteger coefficiets. I this work, we cosider the followig problem: Suppose we are give iput variables x 1, x 2,..., x ad we wat to fid a formula which computes the maximum value take by these variables, max(x 1, x 2,..., x ). If variables are restricted to take o-egative iteger values, t is easy to show that o (mi, +) formula ca compute max. Suppose ow we stregthe this model by allowig mius gates as well. Now we have a very small liear sized formula: max(x 1, x 2,..., x ) = 0 mi(0 x 1, 0 x 2,..., 0 x ). It is clear that mius gates add sigificat power to the model of (mi, +) formulas. But how may miuses do we actually eed? It turs out that oly oe mius gate, at the top, is sufficiet. Here is oe such formula: (Sum of all variables) - mi i (Sum of all variables except x i ). The secod expressio above ca be computed by a (mi, +) formula of size log usig recursio. So, we ca compute max usig mi, + ad oe mius gate at the top, at the cost of a slightly super-liear size. Ca we do ay better? We show that this simple differece formula is ideed the best we ca achieve for this model. The mai motivatio behid studyig this questio is the followig questio asked i [8]: Does there exist a aturally occurig fuctio f for which (mi, +) circuits are superpolyomially weaker tha (max, +) circuits? There are two possibilities: 1. Show that max ca be implemeted usig a small (mi, +) circuit. 2. Come up with a explicit fuctio f which has small (max, +) circuits but requires large (mi, +) circuits. Meea Mahaja ad Prajakta Nimbhorkar ad Auj Tawari; licesed uder Creative Commos Licese CC-BY 42d Iteratioal Symposium o Mathematical Foudatios of Computer Sciece (MFCS 2017). Editors: Kim G. Larse, Has L. Bodlaeder, ad Jea-Fracois Raski; Article No. 74; pp. 74:1 74:11 Leibiz Iteratioal Proceedigs i Iformatics Schloss Dagstuhl Leibiz-Zetrum für Iformatik, Dagstuhl Publishig, Germay

2 74:2 Computig max usig (mi,+) formulas Sice we show that o (mi, +) formula (or circuit) ca compute max, optio 1 is ruled out. I the weaker model of formulas istead of circuits, we show that ay differece of (mi, +) formulas computig max should have size at least log. This yields us a separatio betwee (max, +) formulas ad differece of (mi, +) formulas. Backgroud May dyamic programmig algorithms correspod to (mi, +) circuits over a appropriate semirig. Notable examples iclude the Bellma-Ford-Moore (BFM) algorithm for the sigle-source-shortest-path problem (SSSP) [2, 5, 14], the Floyd-Warshall (FW) algorithm for the All-Pairs-Shortest-Path (APSP) problem [4, 18], ad the Held-Karp (HK) algorithm for the Travellig Salesma Problem (TSP) [6]. All these algorithms are just recursively costructed (mi, +) circuits. For example, both the BFM ad the FW algorithms give O( 3 ) sized (mi, +) circuits while the HK algorithm gives a O( 2 2 ) sized (mi, +) circuit. Matchig lower bouds were proved for TSP i [7], for APSP i [8], ad for SSSP i [10]. So, provig tight lower bouds for circuits over (mi, +) ca help us uderstad the power ad limitatios of dyamic programmig. We refer the reader to [8, 9] for more results o (mi, +) circuit lower bouds. Note that algorithms for problems like computig the diameter of a graph are aturally expressed usig (mi, max, +) circuits. This makes the cost of covertig a max gate to a (mi, +) circuit or formula a iterestig measure. A related questio arises i the settig of coutig classes defied by arithmetic circuits ad formulas. Circuits over N, with specific resource bouds, cout acceptig computatio paths or proof-trees i a related resource-bouded Turig machie model defiig a class C. The coutig fuctio class is deoted #C. The differece of two such fuctios i a class #C is a fuctio i the class DiffC. O the other had, circuits with the same resource bouds, but over Z, or equivaletly, with subtractio gates, describe the fuctio class GapC. For most complexity classes C, a straightforward argumet shows that that DiffC ad GapC coicide. See [1] for further discussio o this. I this framework, we restrict attetio to computatio over N ad see that as a member of a Gap class over (mi, +), max has liear-size formulas, whereas as a member of a Diff class, it requires Ω( log ) size. Our results ad techiques: We ow formally state our results ad briefly commet o the techiques used to prove them. 1. For 2, o (mi, +) formula over N ca compute max(x 1, x 2,..., x ). (Theorem 10) The proof is simple: apply a carefully chose restrictio to the variables ad show that the (mi, +) formula does ot output the correct value of max o this restrictio. 2. max(x 1, x 2,..., x ) ca be computed by a differece of two (mi, +) formulas with total size + log. More geerally, the fuctio computig the sum of the topmost k values amogst the variables ca be computed by a differece of two (mi, +) formulas with total size + ( log ) mi{k, k}. (Theorem 11) Note that the sum of the topmost k values ca be computed by the followig formula: (Sum of all variables) - mi S (Sum of all variables except those i S). Here S rages over all possible subsets of {x 1, x 2,..., x } of cardiality k. Usig recursio, we obtai the claimed size boud. 3. Let F 1, F 2 be (mi, +) formulas over N such that F 1 F 2 = max(x 1, x 2,..., x ). The F 1 must have at least leaves ad F 2 at least log leaves. (Theorem 13)

3 M. Mahaja ad P. Nimbhorkar ad A. Tawari 74:3 A major igrediet i our proof is the defiitio of a measure for fuctios computable by costat-free (mi, +) formulas, ad relatig this measure to formula size. The measure ivolves terms aalogous to miterms of a mootoe Boolea fuctio, ad uses the etropy of a associated graph uder the uiform distributio o its vertices. I the settig of mootoe Boolea fuctios, this techique was used i i [15] to give formula size lower bouds. We adapt that techique to the (mi, +) settig. The same techique also yields the followig lower boud: Also, ay (mi, +) formula computig the miimum over the sums of 1 variables must have at least log leaves. This is tight. (Lemma 12 ad Corollary 18) 2 Prelimiaries 2.1 Notatio Let X deote the set of variables {x 1,..., x }. We use x to deote (x 1, x 2,..., x, 1). We use e i to deote the ( + 1)-dimesioal vector with a 1 i the ith coordiate ad zeroes elsewhere. For i [], we also use e i to deote a assigmet to the variables x 1, x 2,..., x where x i is set to 1 ad all other variables are set to 0. Defiitio 1. For 0 r, the -variate fuctios Sum, MiSum r ad MaxSum r are as defied below. Sum = i=1 x i { } MiSum r = mi x i S, S = r i S { } MaxSum r = max x i S, S = r i S Note that MiSum 0 ad MaxSum 0 are the costat fuctio 0, ad MiSum 1 ad MaxSum 1 are just the mi ad max respectively. Observatio 2. For 1 r <, MiSum = MaxSum = Sum = MiSum r +MaxSum r. 2.2 Formulas A (mi, +) formula is a directed tree. Each leaf of a formula has a label from X N; that is, it is labeled by a variable x i or a costat α N. Each iteral ode has exactly two childre ad is labeled by oe of the two operatios mi or +. The output ode of the formula computes a fuctio of the iput variables i the atural way. The iput odes of a formula are also referred to as gates. If all leaves of a formula are labeled from X, we say that the formula is costat-free. A (mi, +, ) formula is similarly defied; the operatio at a iteral ode may also be, i which case the childre are ordered ad the ode computes the differece of their values. We defie the size of a formula as the umber of leaves i the formula. For a formula F, we deote by L(F ) its size, the umber of leaves i it. For a fuctio f, we deote by L(f) the smallest size of a formula computig f. By L cf (f) we deote the smallest size of a costat-free formula computig f. M FC S

4 74:4 Computig max usig (mi,+) formulas 2.3 Graph Etropy The otio of the etropy of a graph or hypergraph, with respect to a probability distributio o its vertices, was first defied by Körer i [11]. I that ad subsequet works (e.g. [12, 13, 3, 15]), equivalet characterizatios of graph etropy were established ad are ofte used ow as the defiitio itself, see for istace [16, 17]. I this paper, we use graph etropy oly with respect to the uiform distributio, ad simply call it graph etropy. We use the followig defiitio, which is exactly the defiitio from [17] specialised to the uiform distributio. Defiitio 3. Let G be a graph with vertex set V (G) = {1,..., }. The vertex packig polytope V P (G) of the graph G is the covex hull of the characteristic vectors of idepedet sets of G. The etropy of G is defied as H(G) = mi a V P (G) i=1 1 log 1 a i. It ca easily be see that H(G) is a o-egative real umber, ad moreover, H(G) = 0 if ad oly if G has o edges. We list o-trivial properties of graph etropy that we use. Lemma 4 ([12, 13]). Let F = (V, E(F )) ad G = (V, E(G)) be two graphs o the same vertex set. The followig hold: 1. Mootoocity. If E(F ) E(G), the H(F ) H(G) 2. Subadditivity. Let Q be the graph with vertex set V ad edge set E(F ) E(G). The H(Q) H(F ) + H(G). Lemma 5 (see for istace [16, 17]). The followig hold: 1. Let K be the complete graph o vertices. The H(K ) = log. 2. Let G be a graph o vertices, whose edges iduce a bipartite graph o m (out of ) vertices. The H(G) m. 3 Trasformatios ad Easy bouds We cosider the computatio of max{x 1,..., x } over N usig (mi, +) formulas. To start with, we describe some properties of (mi, +) formulas that we use repeatedly. The first property, Propositio 7 below, is expressig the fuctio computed by a formula as a depth-2 polyomial where + plays the role of multiplicatio ad mi plays the role of additio. The ext properties, Propositio 8 ad 9 below, deal with removig redudat sub-expressios created by the costat zero or movig commo parts aside. Defiitio 6. Let F be a (mi, +) formula with leaves labeled from X N. For each gate v F, we costruct a set S v N +1 as described below. We costruct the sets iductively based o the depth of v. 1. Case 1. v is a leaf labeled α for some α N. The S v = {α e +1 }. (Recall, e i is the uit vector with 1 at the ith coordiate ad zero elsewhere). 2. Case 2: v is a leaf labeled x i for some i []. The S v = {e i }. 3. Case 3: v = mi{u, w}. The S v = S u S w. 4. Case 4: v = u + w. The S v = {ã + b ã S u, b S w } (coordiate-wise additio). Let r be the output gate of F. We deote by S(F ) the set S r so costructed.

5 M. Mahaja ad P. Nimbhorkar ad A. Tawari 74:5 It is straightforward to see that if F has o costats (so Case 1 is ever ivoked), the a +1 remais 0 throughout the costructio of the sets S v. Hece if F is costat-free, the for each ã S(F ), a +1 = 0. By costructio, the set S(F ) describes the fuctio computed by F. Thus we have the followig: Propositio 7. Let F be a formula with mi ad + gates, with leaves labeled by elemets of {x 1,..., x } N. For each gate v F, let f v deote the fuctio computed at v. The f v = mi{ ã x ã S v }. The followig propositio is a easy cosequece of the costructio i Defiitio 6. Propositio 8. Let F be a (mi, +) formula over N. Let G be the formula obtaied from F by replacig all costats by the costat 0. Let H be the costat-free formula obtaied from G by elimiatig 0s from G through repeated replacemets of 0 + A by A, mi{0, A} by 0. The 1. L(H) L(G) = L(F ), 2. S(G) = { b b +1 = 0, ã S(F ), i [], a i = b i }, ad 3. G ad H compute the same fuctio mi{ b x b S(G)}. (Note: It is ot the claim that S(G) = S(H). Ideed, this may ot be the case. eg. let F = x + mi{1, x + y}. The S(F ) = {101, 210}, S(G) = {100, 210}, S(H) = {100}, However, the fuctios computed are the same.) The ext propositio shows how to remove commo cotributors to S(F ) without icreasig the formula size. Propositio 9. Let F be a (mi, +) formula computig a fuctio f. If, for some i [], a i > 0 for every ã S(F ), the f x i ca be computed by a (mi, +) formula F of size at most size(f ). If a +1 > 0 for every ã S(F ), the f 1 ca be computed by a (mi, +) formula F of size at most size(f ). I both cases, S(F ) = { b ã S(F ), b = ã e i }. Proof. First cosider i []. Let X be the subset of odes i F defied as follows: X = {v F ã S v : a i > 0} Clearly, the output gate r of F belogs to X. By the costructio of the sets S v, wheever a mi ode v belogs to X, both its childre belog to X, ad wheever a + ode belogs to X, at least oe of its childre belogs to X. We pick a set T X as follows. Iclude r i T. For each mi ode i T, iclude both its childre i T. For each + ode i T, iclude i T oe child that belogs to X (if both childre are i X, choose ay oe arbitrarily). This gives a sub-formula of F where all leaves are labeled x i. Replace these occurreces of x i i F by 0 to get formula F. It is easy to see that S(F ) = {ã e i ã S}. Hece F computes f x i. For i = a +1, the same process as above yields a subformula where each leaf is labeled by a positive costat. Subtractig 1 from the costat at each leaf i T gives the formula computig f 1. It is ituitively clear that o (mi, +) formula ca compute max. A formal proof usig Propositio 7 appears below. Theorem 10. For 2, o (mi, +) formula over N ca compute max{x 1,..., x }. M FC S

6 74:6 Computig max usig (mi,+) formulas Proof. Suppose, to the cotrary, some formula C computes max. The its restrictio D to x 1 = X, x 2 = Y, x 3 = x 4 =... = x = 0, correctly computes max{x, Y }. Sice all leaves of D are labeled from {x 1, x 2 } N, the set S(D) is a set of triples. Let S N 3 be this set. For all X, Y N, max{x, Y } equals E(X, Y ) = mi{ax + BY + C (A, B, C) S}. Let K deote the maximum value take by C i ay triple i S. If for some B, C N, the triple (0, B, C) belogs to S, the E(K + 1, 0) C K < K + 1 = max{0, K + 1}. So for all (A, B, C) S, A 0, so A 1. Similarly, for all (A, B, C) S, B 1. Hece for all (A, B, C) S, A + B 2. Now E(1, 1) = mi{a + B + C (A, B, C) S} 2 > 1 = max{1, 1}. So E(X, Y ) does ot compute max(x, Y ) correctly. However, if we also allow the subtractio operatio at iteral odes, it is very easy to compute the maximum i liear size; max(x 1,..., x ) = mi{ x 1, x 2,..., x }. Here a is implemeted as 0 a, ad if we allow oly variables, ot costats, at leaves, we ca compute a as (x 1 x 1 ) a. Thus the subtractio operatio adds sigificat power. How much? Ca we compute the maximum with very few subtractio gates? It turs out that the max fuctio ca be computed as the differece of two (mi, +) formulas. Equivaletly, there is a (mi, +, ) formula with a sigle gate at the root, that computes the max fuctio. This formula is ot liear i size, but it is ot too big either; we show that it has size O( log ). A simple geeralisatio allows us to compute the sum of the largest k values. Theorem 11. For each 1, ad each 0 k, the fuctio MaxSum k ca be computed by a differece of two (mi, +) formulas with total size + ( log ) mi{k, k}. I particular, the fuctio max{x 1,..., x } ca be computed by a differece of two (mi, +) formulas with total size + log. Proof. Note that MaxSum k = Sum MiSum k. Lemma 12 below shows that MiSum k ca be computed by a formula of size ( log ) mi{k, k} for 0 k. Sice Sum ca be computed by a formula of size, the claimed upper boud for MaxSum k follows. Lemma 12. For all, k such that 1 ad 0 k <, the fuctios MiSum k, MiSum k ca be computed by a (mi, +) formula of size ( log ) k. Hece the fuctios MiSum k, MiSum k ca be computed by (mi, +) formulas of size ( log ) mi{k, k}. Proof. We prove the upper boud for MiSum k. The boud for MiSum k follows from a essetially idetical argumet. We prove this by iductio o k. Base Case: k = 0. For every 1, MiSum k = Sum ad ca be computed with size. Iductive Hypothesis: For all k < k, ad all > k, MiSum k ca computed i size ( log ) k. Iductive Step: We wat to prove the claim for k, where k 1, ad for all > k. We proceed by iductio o. Base Case: = k + 1. MiSum k = MiSum 1 is the miimum of the variables, ad ca be computed i size. Iductive Hypothesis: For all k < m <, MiSum m k m ca be computed i size m( log m ) k.

7 M. Mahaja ad P. Nimbhorkar ad A. Tawari 74:7 Iductive Step: Let m = /2, m = /2, Let X, X l, X r deote the sets of variables {x 1,..., x }, {x 1,..., x m }, {x m +1,..., x }. Note that X l = m, X r = m, m + m =. Let p deote log. Note that log m = log m = p 1. To compute MiSum k o X, we first compute, for various values of t, MiSum m t m o X r, ad add them up. We the take the miimum of o X l, MiSum m (k t) m these sums. Note that if m = t or m = k t, the that summad is simply 0 ad we oly compute the other summad. Now MiSum k { mi MiSum m t m (X) ca be computed as } (X l) + MiSum m (k t) m (X r) max{0, k m } t mi{m, k} For all the sub-expressios appearig i the above costructio, we ca use iductively costructed formulas. Usig the iductive hypotheses (both for t < k ad for t = k, m < ), we see that the umber of leaves i the resultig formula is give by = mi{m,k} t=max{0,k m } [ m (p 1) t + m (p 1) k t] k [ m (p 1) t + m (p 1) k t] t=0 [ k ] [ k ] m (p 1) t + m (p 1) t t=0 t=0 [ k ] = (m + m ) (p 1) t t=0 [(p 1) + 1] k = p k I the rest of this paper, our goal is to prove a matchig lower boud for the max fuctio. Note that the costructios i Theorem 11 ad Lemma 12 yield formulas that do ot use costats at ay leaves. Ituitively, it is clear that if a formula computes the maximum correctly for all atural umbers, the costats caot help. So the lower boud should hold eve i the presece of costats, ad ideed our lower boud does hold eve if costats are allowed. 4 The mai lower boud I this sectio, we prove the followig theorem: Theorem 13. Let F 1, F 2 be (mi, +) formulas over N such that F 1 F 2 = max(x 1,..., x ). The L(F 1 ), ad L(F 2 ) log. The proof proceeds as follows: we first trasform F 1 ad F 2 over a series of steps to formulas G 1 ad G 2 o larger tha F 1 ad F 2, such that G 1 G 2 equals F 1 F 2 ad hece still computes max, ad G 1 ad G 2 have some ice properties. These properties immediately imply that L(F 1 ) L(G 1 ). We further trasform G 2 to a costat-free formula H o larger tha G 2. We the defie a measure for fuctios computable by costat-free (mi, +) formulas, relate this measure to formula size, ad use the properties of G 2 ad H to show that the fuctio h computed by H has large measure ad large formula size. Trasformatio 1: For b {1, 2}, let S b deote the set S(F b ). For i [ + 1], let A i be the miimum value appearig i the ith coordiate i ay tuple i S 1 S 2. Let à deote M FC S

8 74:8 Computig max usig (mi,+) formulas the tuple (A 1,..., A, A +1 ). By repeatedly ivokig Propositio 9, we obtai formulas G b computig F b à x, with L(G b) L(F b ). For b {1, 2}, let T b deote the set S(G b ). We ow establish the followig properties of G 1 ad G 2. Lemma 14. Let F 1, F 2 be (mi, +) formulas such that F 1 F 2 computes max. Let G 1, G 2 be obtaied as described above. The 1. L(G 1 ) L(F 1 ), L(G 2 ) L(F 2 ), 2. max(x) = F 1 F 2 = G 1 G 2, 3. For every i [], for every ã T 1, a i > 0. Hece L(G 1 ). 4. For every i [], there exists ã T 2, a i = There exist ã T 1, b T 2, a +1 = b +1 = For every i, j [] with i j, for every ã T 2, a i + a j > 0. Proof. 1. This follows from propositio Obvious. 3. Suppose for some ã T 1 ad for some i [], a i = 0. Cosider the iput assigmet d where d i = 1 + a +1 ad d j = 0 for j [] \ {i}. The max{d 1,..., d } = 1 + a +1. However, ã d = a +1. Therefore o iput d, G 1 ( d) a +1. Sice G 2 0 o all assigmets, we get G 1 ( d) G 2 ( d) a +1 < max( d), cotradictig the assumptio that G 1 G 2 computes max. 4. This follows from the previous poit ad the choice of A i for each i. 5. From the choice of A +1, we kow that there is a ã i T 1 T 2 with a +1 = 0. Suppose there is such a tuple i exactly oe of the sets T 1, T 2. The exactly oe of G 1 ( 0), G 2 ( 0) equals 0, ad so G 1 G 2 does ot compute max( 0). 6. Suppose to the cotrary, some ã T 2 has a i = a j = 0. Cosider the iput assigmet d where d i = d j = 1 + a +1 ad d k = 0 for k [] \ {i, j}. The max{d 1,..., d } = 1 + a +1. Sice every x k figures i every tuple of T 1, G 1 ( d) d i + d j = 2a But G 2 ( d) a +1. Hece G 1 ( d) G 2 ( d) does ot compute max( d). We have already show above that L(F 1 ) L(G 1 ). Now the more tricky part: we eed to lower boud L(G 2 ). Trasformatio 2: Let H be the formula obtaied by simply replacig every costat i G 2 by 0. Let H be the costat-free formula obtaied from H by elimiatig the zeroes, repeatedly replacig 0 + A by A, mi{0, A} by 0. Let h be the fuctio computed by H. The, L cf (h) L(H) L(H ) = L(G 2 ) L(F 2 ). It thus suffices to show that L cf (h) log. To this ed, we defie a complexity measure µ, relate it to costat-free formula size, ad show that it is large for the fuctio h. Defiitio 15. For a -variate fuctio f computable by a costat-free (mi, +) formula, we defie (f) 1 = {i f(e i ) 1, f(0) = 0}. (f) 2 = {(i, j) f(e i + e j ) 1, f(e i ) = 0, f(e j ) = 0}. We defie G(f) to be the graph whose vertex set is [] ad edge set is (f) 2. The measure µ for fuctio f is defied as follows: µ = (f) 1 + H(G(f))

9 M. Mahaja ad P. Nimbhorkar ad A. Tawari 74:9 The followig lemma relates µ(f) with L(f). This relatio has bee used before, see for istace [15] for applicatios to mootoe Boolea circuits. Sice we have ot see a applicatio i the settig of (mi, +) formulas, we (re-)prove this i detail here; however, it is really the same proof. Lemma 16. Let f be a -variate fuctio computable by a costat-free (mi, +) formula. The L cf (f) µ(f). Proof. The proof is by iductio o the depth of a witessig formula F that computes f ad has L cf (F ) = L cf (f). Base case: F is a iput variable, say x i. The (f) 1 = {x i }, ad G(f) is the empty graph, so µ(f) = 1. Hece 1 = L cf (f) = µ(f). Iductive step: F is either F +F or mi{f, F } for some formulas F, F computig fuctios f, f respectively. Sice F is a optimal-size formula for f, F ad F are optimalsize formulas for f ad f as well. So L cf (f) = L(F ) = L(F )+L(F ) = L cf (f )+L cf (f ). Case a. F = F + F. The (f) 1 = (f ) 1 (f ) 1 ad G(f) G(f ) G(f ). Hece, µ(f) (f ) 1 (f ) 1 + H(G(f ) G(f )) (Lemma 4) (f ) 1 + (f ) 1 + H(G(f )) + H(G(f )) (Lemma 4) = µ(f ) + µ(f ) 1 L cf (f ) + 1 L cf (f ) (Iductio) = 1 L cf (f) (L cf (f) = L cf (f ) + L cf (f )) Case b. F = mi(f, F ). Let (f ) 1 = A ad (f ) 1 = B. The (f) 1 = A B ad G(f) G(f ) G(f ) G(A \ B, B \ A). Here, G(P, Q) deotes the bipartite graph G with parts P ad Q. Hece, µ(f) 1 ( A B ) + H(G(f ) G(f ) G(A \ B, B \ A)) (Lemma 4) 1 ( A B ) + H(G(f )) + H(G(f )) + H(G(A \ B, B \ A)) (Lemma 4) 1 ( A B ) + H(G(f )) + H(G(f )) + 1 ( A \ B + B \ A ) (Lemma 5) 1 ( A + B ) + H(G(f )) + H(G(f )) = µ(f ) + µ(f ) 1 L cf (f ) + 1 L cf (f ) (Iductio) = 1 L cf (f) (L cf (f) = L cf (f ) + L cf (f )) Hece, µ(f) 1 L cf (f). Usig this measure, we ca ow show the required lower boud. Lemma 17. For the fuctio h obtaied after Trasformatio 2, µ(h) log. Proof. Recall that we replaced costats i G 2 by 0 to get H, the elimiated the 0s to get costat-free H computig h. By Propositio 8, we kow that S(H ) = { b b +1 = 0, ã T 2, a i = b i i []} ad that h = mi{ x b b S(H )}. M FC S

10 74:10 Computig max usig (mi,+) formulas From item 4 i Lemma 14, it follows that (h) 1 =. (For every i, there is a b S(H ) with b i = 0. So h(e i ) e i b = 0.) Sice (h) 1 is empty, (i, j) G(h) exactly whe h(e i +e j ) 1. From item 6 i Lemma 14, it follows that every pair (i, j) is i G(h). Thus G(h) is the complete graph K. From Lemma 5 we coclude that µ(h) = log. Lemmas 16 ad 17 imply that L cf (h) log. Sice L cf (h) L(H) L(H ) = L(G 2 ) L(F 2 ), we coclude that L(F 2 ) log. This completes the proof of Theorem 13. A major igrediet i this proof is usig the measure µ. This yields lower bouds for costat-free formulas. For fuctios computable i a costat-free maer, it is hard to see how costats ca help. However, to trasfer a lower boud o L cf (f) to a lower boud o L(f), this idea of costats caot help eeds to be formalized. The trasformatios described before we defie µ do precisely this. For the MiSum 1 fuctio, applyig the measure techique immediately yields the lower boud L cf (MiSum 1 ) log. Trasferrig this lower boud to formulas with costats is a corollary of our mai result, ad with it we see that the upper boud from Lemma 12 is tight for MiSum 1. Corollary 18. Ay (mi, +) formula computig MiSum 1 must have size at least log. Proof. Let F be ay formula computig MiSum 1. Applyig Theorem 13 to F 1 = x x ad F 2 = F, we obtai L(F ) log. 5 Discussio Our results hold whe variables take values from N. I the stadard (mi, +) semi-rig, the value is also allowed, sice it serves as the idetity for the mi operatio. The proof of our mai result Theorem 13 does ot carry over to this settig. The mai stumblig block is the removal of the commo part of S(F ). However, if we allow as a value that a variable ca take, but ot as a costat appearig at a leaf, the the lower boud proof still seems to work. However, the upper boud o loger works; while takig a differece, what is? Apart from the may atural settigs where the tropical semirig (mi, +, N { }, 0, ) crops up, it is also iterestig because it ca simulate the Boolea semirig for mootoe computatio. The mappig is straightforward: 0, 1,, i the Boolea semirig are replaced by, 0, mi, + respectively i the tropical semirig. Provig lower bouds for (mi, +) formulas could be easier tha for mootoe Boolea formulas because the (mi, +) formula has to compute a fuctio correctly at all values, ot just at 0,. Hece it would be iterestig to exted our lower boud to this settig with as well. Our trasformatios crucially use the fact that there is a miimum elemet, 0. Thus, we do ot see how to exted these results to computatios over itegers. It appears that we will eed to iclude, ad sice we are curretly uable to hadle eve +, there is already a barrier. The lower boud method uses graph etropy which is always bouded above by log. Thus this method caot give a lower boud larger tha log. It would be iterestig to obtai a modified techique that ca show that all the upper bouds i Theorem 11 ad Lemma 12 are tight. It would also be iterestig to fid a direct combiatorial proof of our lower boud result, without usig graph etropy.

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