Formula Size-Depth Tradeoffs for Iterated Sub-Permutation Matrix Multiplication

We study the formula complexity of Iterated Sub-Permutation Matrix Multiplication, the logspace-complete problem of computing the product of $k$ $n$-by-$n$ Boolean matrices with at most a single $1$ in each row and column. For all $d \le \log k$, this problem is solvable by $n^{O(dk^{1/d})}$ size monotone formulas of two distinct types: (unbounded fan-in) $AC^0$ formulas of depth $d+1$ and (semi-unbounded fan-in) $SAC^0$ formulas of $\bigwedge$-depth $d$ and $\bigwedge$-fan-in $k^{1/d}$. The results of this paper give matching $n^{\Omega(dk^{1/d})}$ lower bounds for monotone $AC^0$ and $SAC^0$ formulas for all $k \le \log\log n$, as well as slightly weaker $n^{\Omega(dk^{1/2d})}$ lower bounds for non-monotone $AC^0$ and $SAC^0$ formulas. These size-depth tradeoffs converge at $d = \log k$ to tight $n^{\Omega(\log k)}$ lower bounds for both unbounded-depth monotone formulas [Ros15] and bounded-depth non-monotone formulas [Ros18]. Our non-monotone lower bounds extend to the more restricted Iterated Permutation Matrix Multiplication problem, improving the previous $n^{k^{1/\exp(O(d))}}$ tradeoff for this problem [BIP98].


INTRODUCTION
Motivated by the major open question of NC 1 vs. L (polynomial-size formulas vs. branching programs), we study the formula complexity of Iterated Sub-Permutation Matrix Multiplication (sub-pmm n,k ), the monotone logspace-complete problem of computing the product of k n-by-n sub-permutation matrices.Our main results are asymptotically tight (resp.nearly tight) size-depth tradeo s for monotone (resp.non-monotone) AC 0 and SAC 0 formulas solving this problem.Along the way, we establish new reductions from formulas solving sub-pmm n,k to union trees computing the graph Path k and develop a combinatorial framework for analyzing the latter objects.
The rest of this introduction is structured as follows: §1.1 introduces iterated matrix multiplication problems imm n,k , bmm n,k , pmm n,k and sub-pmm n,k .§1.2 discusses the formula complexity of these problems in relation to the NC 1 vs. L question.§1.3 reviews some upper bounds on the AC 0 and SAC 0 formula size of sub-pmm n,k .§1.4 states our main results.§1.5 discusses prior related work.§1.6 outlines the rest of the paper.

Iterated matrix multiplication problems
Iterated Matrix Multiplication (imm n,k ) is the task, given n × n matrices M (1) , . . ., M (k ) (over some speci ed eld), of computing the 1, 1-entry of the matrix product M (1) • • • M (k ) .In algebraic complexity theory, this problem is represented by a formal polynomial in kn 2 variables corresponding to matrix entries (2) Iterated Boolean Matrix Multiplication (bmm n,k ) is the related problem for Boolean matrices with entries in the ring {0, 1} with addition ∨ and multiplication ∧.This problem is represented by a monotone Boolean function {0, 1} kn 2 → {0, 1} described by the formula 1,a 1 ∧ M (2) An important special case is when the input M (1) , . . ., M (k ) is restricted to permutation matrices, or more generally sub-permutation matrices with at most a single 1 in each row and column.(For such matrices, note that bmm n,k coincides with the polynomial imm n,k as a function of over any eld.)We refer to these restricted promise problems as Iterated (Sub-)Permutation Matrix Multiplication, denoted pmm n,k and sub-pmm n,k .Note an important di erence in computational complexity: whereas bmm n,k and imm n,k are complete for nondeterministic logspace N L and its algebraic analogue V BP (polynomial-size algebraic branching programs), both pmm n,k and sub-pmm n,k are complete for deterministic logspace L [8].
While problems pmm n,k and sub-pmm n,k are equivalent under randomized AC 0 reductions, in this paper we focus on sub-pmm n,k for a simple reason: like bmm n,k , it is naturally represented by a monotone Boolean function {0, 1} kn 2 → {0, 1}, and there is interest in understanding its complexity in both monotone and nonmonotone models of computation.In contrast, pmm n,k is a "slice function" where all relevant inputs (i.e., k-tuples of permutation matrices) have equal Hamming weight kn; for such functions, it is known that that monotone and non-monotone complexity coincide up to a polynomial factor [6].
1.2 Formula complexity and the question of NC 1 vs. L Formulas are rooted trees whose leaves ("inputs") are labeled by constants (0 or 1) or literals (X i or X i ) and whose non-leaves ("gates") are labeled or (or ∧ / ∨ in the setting of binary "DeMorgan" formulas).Understanding the formula complexity of iterated matrix multiplication problems is a major question at the frontier of lower bounds in Circuit Complexity.Recall that NC 1 is (by one of its equivalent de nitions) the complexity class of languages decidable by polynomial-size formulas.At present, no lower bound better than n 3−o (1) is known on the minimum formula size of any explicit sequence of Boolean functions (describing a language in P, or even N P) [24].Historically, the rst lower bounds against weak circuit classes have usually targeted functions of barely higher complexity.Consider the classical theorems parity AC 0 [1,9] and maj AC 0 [⊕] [17,23].Even the best n 3−o (1) formula lower bounds [11,24] are for a function (Andreev's function) with O (n 3 ) size formulas.This makes sense insofar as lower bounds are dual to upper bounds.From this perspective, problems bmm n,k and sub-pmm n,k are ideal targets for a super-polynomial lower bound.The smallest known formulas for these problems have size n O (log k ) ; so too the smallest known arithmetic formulas for imm n,k .These upper bounds are believed to be asymptotically optimal.Note that a matching n Ω(log k ) lower bound on the formula size of sub-pmm n,k (respectively: bmm n,k , imm n,k ) for any super-constant k (n) would separate NC 1 from L (respectively: Motivated by this goal, previous work of the author [18,20] took aim at this lower bound question in two restricted classes of formulas: (+) (unbounded-depth) monotone formulas, (−) (non-monotone) bounded-depth formulas.The results of [18,20] are lower bounds for an average-case version of bmm n,k on random matrices with i.i.d.Bernoulli( 1 n ) entries, which straightforwardly extend to the following lower bounds for sub-pmm n,k and pmm n,k .Theorem 1.1 ( [18,20]).For all k ≤ log log n, size n Ω(log k ) is required by both 1) solving pmm n,k .These lower bounds come "close" to showing NC 1 L, which would follow from either extending (+) from sub-pmm n,k to pmm n,k , or extending (−) from depth log n (log log n) O (1) to log n.

Upper bounds
Lower bounds (+) and (−) are simultaneously shown to be tight by formulas implementing the standard divide-and-conquer (a.k.a.recursive doubling) technique of Savitch [22].3).For sub-permutation matrices M (1) , . . ., M (5) ∈ {0, 1} n×n and indices a 0 , a 5 ∈ [n], the a 0 , a 5 -entry of the product M (1) • • • M (5) is computed by the following monotone Σ 2 and Π 2 (a.k.a.DNF and CNF) formulas: Notice that formulas (I) Σ 2d (and their algebraic counterparts with and replacing and ) implement a standard divide-andconquer algorithm for problems bmm n,k and imm n,k .This algorithm recursively splits a k-ary matrix product M (1) for sub-pmm n,k , which utilize a similar divide-and-conquer, generalize the well-known Σ d+1 and Π d +1 formulas for parity k , which are essentially formulas (II) in the case n = 2.
1.4 Our tradeo s for AC 0 and SAC 0 formulas Lower bounds (+) and (−) of Theorem 1.1 were proved using a technique, known as the Pathset Framework, which reduces formulas computing sub-pmm n,k to simpler combinatorial objects called union trees (binary trees with ∪ gates that compute the path graph Path k out of its individual edges).This reduction provides a means of "lifting" lower bounds for union trees to lower bounds for formulas computing sub-pmm n,k .
Perhaps surprisingly, in the reductions of papers [18,20] which prove (+) and (−), there is no obvious relationship between the depth of formulas solving sub-pmm n,k and the the depth of the corresponding union trees.For this reason, the lower bound technique of [18,20], which extends to formulas of depth well beyond O (log k ), does not (in any obvious way) yield lower bounds stronger than n Ω(log k ) for formulas of depths 2, 3, 4, . . .below O (log k ).
The results of this paper expand the Pathset Framework in order to extend lower bounds (+) and (−) of Theorem 1.1 to sizedepth tradeo s for both unbounded fan-in AC 0 formulas and "semiunbounded fan-in" SAC 0 formulas.We obtain tradeo s for monotone formulas which match the upper bounds of Proposition 1.3.We are able to extend these tradeo s to non-monotone formulas, at the expense of a slightly weaker lower bound and smaller range of k.
Theorem 1.6.We have the following size-depth tradeo s for AC 0 and SAC 0 formulas solving sub-pmm n,k .For all k ≤ log log n and d ≤ log k, size n Ω(dk 1/d ) is required by both (I) + monotone SAC 0 formulas of -depth d and -fan-in n 1/k , (II) + monotone AC 0 formulas of depth d + 1.
Additionally, for all k ≤ log * n and d ≤ log k, size n Ω(dk 1/2d ) is required by both (I) − SAC 0 formulas of -depth d and -fan-in n 1/k , (II) − AC 0 formulas of depth d + 1.
Monotone tradeo s (I) + and (II) + are tight up to the big-Ω constant in light of Proposition 1.3.(Notice that the maximum -fan-in n 1/k in lower bounds (I) ± exceeds the -fan-in k 1/d of formulas (I) Σ 2d .)Furthermore, tradeo s (II) ± converge at d = log k to the tight n Ω(log k ) lower bounds (+) and (−) of Theorem 1.1.(The range k ≤ log * n of non-monotone tradeo s (I) − and (II) − can conservatively be improved to k ≤ 1 2 log log log log n.)As an immediate corollary, lower bounds n Ω(dk 1/d ) and n Ω(dk 1/2d ) of Theorem 1.6 directly imply n Ω(k 1/d ) and n Ω(k 1/2d ) lower bounds for the corresponding SAC 0 and AC 0 circuits.Additionally, tradeo s (I) − and (II) − extend to slightly weaker n Ω(dk 1/(d +O (1)) ) and n Ω(dk 1/(2d +O (1)) ) lower bounds for non-monotone SAC 0 and AC 0 formulas solving the more restricted problem pmm n,k , as a corollary of the randomized AC 0 reduction from sub-pmm n,k to pmm n,k (see [26]).

Related work
Previous lower bounds.Prior to the results of this paper, the strongest size-depth tradeo for pmm n,k (or sub-pmm n,k ) was n k exp(−O (d )) , proved by Beame, Impagliazzo and Pitassi [4] using a special purpose switching lemma.This tradeo , which is non-trivial to depth O (log log k ), improved previous tradeo s of [2,5] which were only non-trivial to depth O (log * k ).
With respect to the signi cantly more general bmm n,k problem, Chen, Oliveira, Servedio and Tan [7] established a nearly tight tradeo n Ω((1/d )k 1/d ) lower bound for depth-d AC 0 circuits.This lower bound is non-trivial to depth O ( log k log log k ), and for depths d = o( log k log log k ) it is quantitatively stronger than the n Ω(k 1/(2d −1) ) lower bound implied by Theorem 1.6 for depth-d AC 0 circuits.However, the result of [7] is fundamentally a size-depth tradeo for Sipser functions (read-once formulas with a carefully chosen fan-in sequence), combined with a reduction from the NC 1 -complete sipser problem to bmm.(In contrast to the present paper and [4], this result neither extends to sub-pmm n,k , nor sheds light on the questions of NC 1 vs. L or NC 1 vs. N L.) Di erent regimes of parameters n and k.Our focus in this paper is on the formula complexity of sub-pmm n,k in the regime where k is an arbitrarily slow-growing but super-constant function of n.We have not attempted to optimize the ranges of k ≤ log log n and k ≤ log * n in Theorem 1.6.We remark that the circuit lower bound of Beame et al [4] and Chen et al [7] extend to larger ranges k ≤ log n and k ≤ n 1/5 , respectively.
In the opposite parameter regime where n = 2 and k is unbounded, all three problems bmm n,k , sub-pmm n,k and pmm n,k are equivalent to parity k under AC 0 reductions.Here tight 2 Ω(k 1/d ) and 2 Ω(dk 1/d ) lower bounds for depth d + 1 AC 0 circuits and formulas computing parity k were shown by Håstad [10] and the author [19] respectively.
The complexity jumps at n = 5, where all three problems are complete for NC 1 by Barrington's Theorem [3].The complexity of bmm n,k (resp.sub-pmm n,k and pmm n,k ) then jumps at k = n to being complete for N L (resp.L). (See the papers of Mereghetti and Palano [16] and Wang [27] concerning TC 0 and ACC 0 circuits for these problems.)Algebraic circuit classes.The Iterated Matrix Multiplication polynomial (imm n,k ) is computable by set-multilinear formulas of productdepth d and size n O (dk 1/d ) by construction (I) of Proposition 1.3 with , replacing , .A recent breakthrough of Limaye, Srinivasan and Tavenas [15] showed that arithmetic formulas of productdepth d computing imm n,k require size n k exp(−O (d )) .This tradeo , quantitatively similar to [4], is the rst super-polynomial lower bound for constant-depth arithmetic formulas.A related paper of Tavenas, Limaye, and Srinivasan [25] gives a di erent (log n) Ω(dk 1/d ) lower bound for set-multilinear formulas.In the further restricted class of set-multilinear formulas with the few parse trees property, Legarde, Limaye and Srinivasan [14] established a tight n Ω(dk 1/d ) tradeo .
He and Rossman [12] studied the complexity of pmm n,k in the hybrid Boolean-algebraic model of AC 0 formula that are invariant under a group action of Sym(n) k −1 .This is the class of AC 0 formulas on kn 2 variables (encoding permutation matrices M (1) , . . ., M (k ) ) which are syntactically invariant under a certain action of Sym(n) k −1 , where the ith symmetric group Sym(n) acts by permuting both the columns of M (i ) and rows of M (i+1) .For instance, construction (II) of Proposition 1.3 produces Sym(n) k −1 -invariant AC 0 formulas of depth d + 1 and size n dk 1/d + O (1) which solve pmm k,n for all k ≤ n and d ≤ log k.The paper [12] proves a sharply matching n dk 1/d − O (1) lower bound in the Sym(n) k −1 -invariant setting.This lower bound even extends to Sym(n) k −1 -invariant TC 0 formulas.
Smallest known formulas for sub-pmm n,k .Construction (II) of Proposition 1.3 in the limit d = log 2 k produces AC 0 formulas for sub-pmm n,k of depth O (log k ) and size n log 2 k + O (1) .These formulas are moreover Sym(n) k −1 -invariant, monotone, and uniform.However, these formulas are not the smallest known!For all k ≤ n, Rossman [20] showed that there exist (non-Sym(n . This was subsequently improved to n where φ = 1+ √ 5 2 by Kush and Rossman [13], who have conjectured (with evidence from the Pathset Framework) that 1  3 log φ k ≈ 0.49 log 2 k is optimal in this upper bound.

Outline of the paper
The paper is organized as follows: • In §2 we state de nitions pertaining to AC 0 and SAC 0 formulas, subgraphs of Path k , and union trees.• In §3 we state the combinatorial main results at the heart of this paper: two tradeo s for union trees, labeled (I) and (II), which correspond to tradeo s (I) ± and (II) ± of Theorem 1.6.• In §4 we introduce the Pathset Framework of papers [18,20] and prove two tradeo s for pathset complexity, which follow from our tradeo s for union trees.• Our combinatorial main theorems, tradeo s (I) and (II) for union trees (Theorem 3.6), are proved in §5 and §6 respectively.• we prove Theorem 1.6 Our size-depth tradeo s for AC 0 and SAC 0 formulas are proved in §7 by reduction to the Pathset Framework and our tradeo s for pathset complexity (Theorem 4.11).
Due the page limit, this extended abstract contains §1-4 only.

Graphs and trees
Graphs in this paper are simple graphs without isolated vertices.That is, a graph G consists of sets V (G) and The empty graph is denoted by the empty set symbol ∅.With the exception of the in nite path graph Path Z , all other graphs considered in this paper are nite.
Trees in this paper are nite rooted trees.Binary trees are trees in which each non-leaf node has exactly two children, designed "left" and "right".For binary trees T 1 and T 2 , we denote by T 1 ∪ T 2 the binary tree consisting of a root with T 1 and T 2 as its left and right subtrees.
2.2 AC 0 and SAC 0 formulas An AC 0 formula on variables X 1 , . . ., X N is a rooted tree whose leaves (called "inputs") are labeled by a constant 0 or 1 or literals X i or X i , and whose non-leaves (called "gates") are labeled by or .An AC 0 formula is monotone if no input is labeled by a negative literal X i .Every [monotone] AC 0 formula computes a [monotone] Boolean function {0, 1} N → {0, 1} in the usual way.
We measure the size of an AC 0 formula by the number of leaves that are labeled by literals.Depth (resp.-depth) is the maximum number of gates (resp.-gates) on a root-to-leaf-branch.Fan-in (resp.-fan-in) is the maximum number of children of any gate (resp.-gate).
Depth 0 AC 0 formulas (i.e., literals) are also known as Σ 0 formulas and Π 0 formulas.For d ≥ 1, a Σ d formula (resp.Π d formula) is either a Π d −1 formula (resp.Σ d −1 formula) or a depth d AC 0 formula whose output gate is labeled (resp.).Note that Σ 2d+1 formulas are precisely the class of AC 0 formulas with -depth d.SAC 0 formulas (where "S" stands for "semi-unbounded fan-in") are AC 0 formulas with bounded -fan-in.This usually refers tofan-in 2 (i.e., formulas with unbounded fan-in gates and fan-in 2 ∧ gates).However, when speaking of problems bmm n,k , sub-pmm n,k and pmm n,k , we allow -fan-in O (k 1/d ) in the context of upper bounds, and at most n 1/k in the context of lower bounds.Note that k 1/d ≪ n 1/k in the regime of parameters that we consider (i.e., k ≤ log log n and d ≤ log k).

Subgraphs of paths
The graphs that we will mainly consider are disjoint unions of paths.It is convenient to regard these as nite subgraphs of the in nite path graph Path Z with vertex set Z and edge set {{i − 1, i} : i ∈ Z}.
For integers s < t, let Path s,t denote subgraph of Path Z with vertex set {i ∈ Z : s ≤ i ≤ t } and edge set {{i − 1, i} ∈ Z : s < i ≤ t }.For positive integers k, we write Path k for Path 0,k .
Note that every nite subgraph of Path Z is is a union of paths The next three de nitions introduce notation that will be heavily used throughout the paper.

De nition 2.1 (Parameters
De nition 2.2 (The operation G ⊖ F ).For graphs G, F ⊂ Path Z , we denote by G ⊖ F the subgraph of G comprised of the connected components of G that are vertex-disjoint from F .

De nition 2.3. For a sequence of nite graphs
For F ⊂ Path Z , we also introduce a convenient "conditional #» ∆ " notation: The value of #» ∆ (G 1 , . . ., G m ) is sensitive to the order of G 1 , . . ., G m , as the following example shows.
For later reference, the following lemma records some basic properties of #» ∆ ( • | • ), which follow directly from de nitions.
decomposes via the following "chain rule":

Union trees
We de ne the notion of union trees with respect to arbitrary nite graphs G.However, our focus will soon narrow to union trees for subgraphs of Path k .
De nition 2.6 (Union tree).For any nite graph G, a G-union tree is a binary tree T together a labeling of nodes of T by subgraphs of G such that • each leaf is labeled by a single-edge subgraph of G or the empty graph, • each non-leaf is labeled by the union of the graphs labeling its children, and • the root is labeled by G.
Note that the labeling function is induced by its value on leaves of T , given by a surjective partial function from {leaves of T } to E(G).
We consider non-standard notions of "depth" and "size" for union trees.Our combinatorial main theorem gives tradeo s between these parameters, which we de ne next.
De nition 2.7 (OperationsT 1 ∪T 2 , ⟨⟨T 1 , . . .,T m ⟩⟩ and T 1 , . . .,T m ).If T 1 is a G 1 -union tree and T 2 is a G 2 -union tree, then we denote by T 1 ∪ T 2 the G 1 ∪ G 2 -union tree consisting of a root with left and right subtrees T 1 and T 2 with the obvious induced labeling of nodes.
∪-depth is the standard notion of "depth" for binary trees, that is, the maximum length (= number of parent-to-child descents) of a root-to-leaf branch.⟨⟨⟩⟩-depth is what might be called "leftdepth", that is, the maximum number of left descents on a rootto-leaf-branch.-depth is perhaps a new notion, which arises in connection to our tradeo s for unbounded fan-in AC 0 formulas.
In order to de ne our "size" measure for union trees, we rst introduce the notion of branch coverings.De nition 2.9 (Branch coverings).Let T be a G-union tree.We associate each root-to-leaf branch (b 1 , . . ., b ℓ ) in T (where b 1 is the root and b ℓ a leaf) with the sequence of graphs B 1 , . . ., B ℓ where • B j labels the sibling of b j+1 (i.e., the opposite-side child of b j ) for each j ∈ {1, . . ., ℓ − 1}, and We refer to sets {B 1 , . . ., B ℓ } arising from root-to-leaf branches as T -branch coverings of G.
One standard way of measuring the "size" of a binary tree is by the number of leaves (= number of root-to-leaf branches).For union trees T with graph G ⊂ Path k , we consider a completely di erent "size" measure de ned as the maximum of a certain complexity measure over all T -branch coverings {B 1 , . . ., B ℓ } of G.This complexity measure is itself the maximum #» ∆ -value among all orderings B π (1) , . . ., B π (ℓ) .This rather complicated "size" measure, denoted by Ψ, arises naturally in the context of pathset complexity (see §4 and in particular Corollary 4.7 and Lemma 4.10, which explain the roles of ⃗ ∆ and Ψ).
De nition 2.10 (Ψ-size of union trees).Let T be a G-union tree where G ⊂ Path Z .We de ne the Ψ-size of T by

TRADEOFFS FOR UNION TREES
In this section we state the combinatorial main theorem of this paper, Theorem 3.6, which gives tight tradeo s between the Ψsize and the ⟨⟨⟩⟩-and -depth of Path k -union trees.We state two auxiliary results, Lemmas 3.4 and 3.5, which concern coverings of Path k .Proofs of these results are postponed to §5 and §6 (in the full paper).We also state and prove a key numerical inequality, Lemma 3.9, which will be used later on in the inductive proofs of our tradeo s for union trees.

Shift permutations
Given a sequence G 1 , . . ., G m ⊂ Path Z , we will be interested in the question of maximizing #» ∆ (G π (1) , . . ., G π (m) ) over permutations π .We will also be studying a variant of this question over a restricted class of permutations, which we call "shift permutations".(The author isWe are unaware if there is any standard terminology for this class of permutations.) (For clarity of notation, we will consistently write σ for shift permutations and π for general permutations.)There are exactly 2 m−1 shift permutations, which we will index via a bijection Note that π is not a shift permutation.However, we can achieve the same value #» ∆ (E σ (1) , . . ., E σ (m) ) = 13 via the shift permutation σ = σ {1,3,5, ...,25} .That is, we have Generalizing from 25 to any odd k, the shift permutation σ {1,3,5, ...,k } produces a #» ∆ -value of k +1 2 .Example 3.3.Suppose that instead of E 1 , . . ., E 25 in the usual order, we are given the sequence Note that the #» ∆ -value of this sequence equals 5. What is the best #» ∆ -value we can obtain by applying a shift permutation to this sequence of graphs?It turns out that the answer is 7.Among many possibilities, this is achieved by the shift permutation σ {15,25} , which produces the sequence Generalizing this example from 25 to any odd square k, we get an ordering E 1 , E √ k+1 , E 2 √ k +1 , . . ., E k where applying the best shift permutation increases the

Combinatorial results of this paper
We now state our combinatorial main results, which we prove in §5 and §6 (of the full paper).Our scheme of numbering lower bounds (I) and (II) corresponds to upper bounds (I) and (II) of Proposition 1.3 (as well as lower bounds (I) and (II) in the tradeo s for pathset complexity and AC 0 formulas).The following auxiliary lemmas, which are interesting in their own right, concern coverings of Path k .
(II) There exists a shi permutation σ such that As the name suggests, Lemma 3.4 is used to prove Lemma 3.5, which is in term used to prove the main result of this paper: Theorem 3.6 (Main Theorem).Let T be any Path k -union tree.
Note that inequalities (I) and (II) of Theorem 3.6 are respectively Ω(dk 1/d ) and Ω(dk 1/2d ) up to some d = O (log k ), yet both inequalities become trivial for some slightly larger d = Ω(log k ).However, this doesn't bother us since a di erent bound Φ(T ) ≥ 0.35 log k is known by a result of [13] for di erent parameter Φ(T ) satisfying Φ(T ) ≥ Ψ(T ).As we describe later in Corollary 4.12, we end up with tradeo s n Ω(d (k 1/d −1)) and n Ω(d (k 1/2d −1)) for pathset complexity, which are tight for all d.In particular, these tradeo s both converge to n Ω(log k ) as d → ∞.

Matching upper bounds
The following Examples 3.7 and 3.8 show that the lower bounds of Theorem 3.6 are tight up to a constant factor.

Numerical inequality used in the induction
Both inequalities of Theorem 3.6 are proved by induction on the parameter d.The following numerical inequality plays a key role in the induction step.Assume that Z > 0, since otherwise the inequality is trivially valid.Toward a contradiction, assume that for all j ∈ {1, . . ., m}, we have .
It follows that We now get a contradiction Z < Z as follows: In the induction step of Theorem 3.6, Lemma 3.9 gets invokes once and twice, respectively, in bounds (I) and (II).This fact accounts for the di erent values 1/d vs. 1/2d in the exponent of k in these two bounds.
We remark that in the simplest case m = 1 of Lemma 3.9, one gets a stronger bound (without the constant 1/e) by elementary calculus.We record this as a separate lemma, since we will use it later on (as one of the two applications of Lemma 3.9 in the proof of Theorem 3.6(II)).
Lemma 3.10 (The m = 1 case of Lemma 3.9).For all real numbers x, y ≥ 0 and d > 1, we have We remark that Lemma 3.10 is a common inequality that shows up in formula size-depth tradeo s including [14,19].We suspect that the more general inequality given by Lemma 3.9 might nd applications beyond the present paper to additional formula tradeo s which involve the function dk 1/d .

TRADEOFFS FOR PATHSETS
In this section, we review the Pathset Framework of papers [13,18,20,21] and prove new tradeo s for pathset complexity (Theorem 4.11) that follow from our tradeo s for union trees (Theorem 3.6).

Relations and joins
Throughout this section, we x arbitrary positive integers k and n.We additionally x an arbitrary parameter ñ ≤ n.In our application, we will set ñ := n (k −1)/k .

De nition 4.1 (G-relations).
Note that the join behaves as direct product A × B when V (G) ∩ V (H ) = ∅, and as intersection A ∩ B when V (G) = V (H ).

De nition 4.3 (Density
For a graph F ⊆ Path k and I := V (F ) ∩V (G), the maximum density of A conditioned on F is de ned by The following lemma and corollary relate the density of a join to the maximum conditional densities of the constituent relations.The proof is again a simple exercise in relational algebra.
Furthermore, since the density of the join does not depend on the ordering of relations A 1 , . . ., A m , we have The lifting technique of papers [18,20] focuses on a special class of G-relations that are subject to a family of density constraints in terms of ∆(•|•).

De nition 4.6 (G-pathsets). For a graph
The set of G-pathsets is denoted by P G .(Note that P G is a proper subset of R G when G is nonempty.) The next result Corollary 4.5 applied to pathsets) bounds the density of a join of pathsets in terms of the operation #» ∆ (•).

Pathset complexity
For a graph G ⊆ Path k , we de ne a family of complexity measures µ T : P G → Z ≥0 for each G-union tree T .
De nition 4.8 (Pathset complexity).Let G ⊆ Path k , let A ∈ P G , and let T be a G-union tree.The T -pathset complexity of A, denoted by χ T (A), is de ned inductively as follows: Here i ranges over an arbitrary nite index set.
The following lower bound on pathset complexity was rst proved in [20], then again in [13] with an improved big-Ω constant.
The lower bounds of Corollary 4.12 are asymptotically tight for every xed d and k, and both bounds converge to n Ω(log k ) as d → ∞ (since d (k 1/d − 1) → ln k).

Proposition 1 .Proposition 1 . 3 .
2. sub-pmm n,k is solvable by monotone formulas of depth O (log k ) and size n O (log k ) for all n and k.This raises the question: what about formulas of very small depths 2, 3, 4, . . .below O (log k )?Here the known upper bounds exhibit a typical size-depth tradeo .In fact, there are two distinct tradeo s given by unbounded fan-in AC 0 formulas and semiunbounded fan-in SAC 0 formulas.We label these two settings by (I) and (II), respectively.(Note: Classes of depth-d formulas Σ d and Π d are formally de ned in §2.2.) Whenever k 1/d is an integer, sub-pmm n,k is solvable by both (I) monotone Σ 2d formulas with -fan-in k 1/d , (II) monotone Σ d +1 and Π d+1 formulas, of size at most kn dk 1/d .Note that Proposition 1.3 implies the n O (log k ) upper bound of Proposition 1.2 when d = log k.We remark that the circuit versions of (I) and (II) have smaller size k • n k 1/d , which shrinks to merely polynomial O (kn 2 ) at d = log k.Formulas (I) Σ 2d , (II) Σ d +1 and (II) Π d +1 of Proposition 1.3 are best understood by looking at concrete examples in cases d = 1, 2, where we consider k = 5, 25 for concreteness sake.Example 1.4 (The case d = 1 and k = 5 of Proposition 1.
For a graph G ⊆ Path k , we refer to sets A ⊆ [n] V (G ) as G-relations.We denote the set of all G-relations by R G .(That is, R G is the set of "V (G)-ary" relations on [n].)The join operation ▷◁ combines a G-relation and an H -relation into a G ∪ H -relation.De nition 4.2 (Join).For graphs G, H ⊆ Path k and relations A ∈ R G and B ∈ R H , the join of A and B is the relation A ▷◁ B ∈ R G∪H de ned by

Lemma 4 . 4 (
Chain rule for density of a join).For all graphs F , G, H ⊆ Path k and relations A ∈ R G and B ∈ R H , we have µ