Benjamin Rossman

Homomorphism preservation theorems

The homomorphism preservation theorem (h.p.t.), a result in classical model theory, states that a first-order formula is preserved under homomorphisms on all structures (finite and infinite) if and only if it is equivalent to an existential-positive formula. Answering a long-standing question in finite model theory, we prove that the h.p.t. remains valid when restricted to finite structures (unlike many other classical preservation theorems, including the Łoś--Tarski theorem and Lyndons positivity theorem). Applications of this result extend to constraint satisfaction problems and to database theory via a correspondence between existential-positive formulas and unions of conjunctive queries. A further result of this article strengthens the classical h.p.t.: we show that a first-order formula is preserved under homomorphisms on all structures if and only if it is equivalent to an existential-positive formula of equal quantifier-rank.

Existential positive types and preservation under homomorphisms

We prove the finite homomorphism preservation theorem: a first-order formula is preserved under homomorphisms on finite structures iff it is equivalent in the finite to an existential positive formula. We also strengthen the classical homomorphism preservation theorem by showing that a formula is preserved under homomorphisms on all structures iff it is equivalent to an existential positive formula of the same quantifier rank. Our method involves analysis of existential positive types and a new notion of existential positive saturation.

An optimal decomposition algorithm for tree edit distance

The edit distance between two ordered rooted trees with vertex labels is the minimum cost of transforming one tree into the other by a sequence of elementary operations consisting of deleting and relabeling existing nodes, as well as inserting new nodes. In this paper, we present a worst-case O(n3)-time algorithm for this problem, improving the previous best O(n3 log n)-time algorithm [7]. Our result requires a novel adaptive strategy for deciding how a dynamic program divides into subproblems, together with a deeper understanding of the previous algorithms for the problem. We prove the optimality of our algorithm among the family of decomposition strategy algorithms--which also includes the previous fastest algorithms--by tightening the known lower bound of Ω(n2 log2 n) [4] to O(n3), matching our algorithms running time. Furthermore, we obtain matching upper and lower bounds of Θ(nm2(1+log n/m)) when the two trees have sizes m and n where m < n.

On the constant-depth complexity of k-clique

We prove a lower bound of ω(n^k/4) on the size of constant-depth circuits solving the k-clique problem on n-vertex graphs (for every constant k). This improves a lower bound of ω(n^k/89d²) due to Beame where d is the circuit depth. Our lower bound has the advantage that it does not depend on the constant d in the exponent of n, thus breaking the mold of the traditional size-depth tradeoff.n Our k-clique lower bound derives from a stronger result of independent interest. Suppose f_n :0,1^n/2 → {0,1} is a sequence of functions computed by constant-depth circuits of size O(n^t). Let G be an Erdos-Renyi random graph with vertex set {1,...,n} and independent edge probabilities n^-α where α ≤ 1/2t-1. Let A be a uniform random k-element subset of {1,...,n} (where k is any constant independent of n) and let K_A denote the clique supported on A. We prove that f_n(G) = f_n(G ∪ K_A) asymptotically almost surely.n These results resolve a long-standing open question in finite model theory (going back at least to Immerman in 1982). The <i>m-variable fragment of first-order logic</i>, denoted by FO^m, consists of the first-order sentences which involve at most m variables. Our results imply that the <i>bounded variable hierarchy</i> FO¹ ⊂ FO² ⊂ ... ⊂ FO^m ⊂ ... is strict in terms of expressive power on finite ordered graphs. It was previously unknown that FO³ is less expressive than full first-order logic on finite ordered graphs.

Interactive Small-Step Algorithms II: Abstract State Machines and the Characterization Theorem

In earlier work, the Abstract State Machine Thesis -- that arbitrarynalgorithms are behaviorally equivalent to abstract state machines -- wasnestablished for several classes of algorithms, including ordinary, interactive,nsmall-step algorithms. This was accomplished on the basis of axiomatizations ofnthese classes of algorithms. In Part I (Interactive Small-Step Algorithms I:nAxiomatization), the axiomatization was extended to cover interactivensmall-step algorithms that are not necessarily ordinary. This means that thenalgorithms (1) can complete a step without necessarily waiting for replies tonall queries from that step and (2) can use not only the environments repliesnbut also the order in which the replies were received. In order to prove thenthesis for algorithms of this generality, we extend here the definition ofnabstract state machines to incorporate explicit attention to the relativentiming of replies and to the possible absence of replies. We prove thencharacterization theorem for extended abstract state machines with respect tongeneral algorithms as axiomatized in Part I.

Choiceless polynomial time, counting and the Cai–Fürer–Immerman graphs

Abstract We consider Choiceless Polynomial Time ( C PT ), a language introduced by Blass, Gurevich and Shelah, and show that it can express a query originally constructed by Cai, Furer and Immerman to separate fixed-point logic with counting ( IFP + C ) from P . This settles a question posed by Blass et al. The program we present uses sets of unbounded finite rank: we demonstrate that this is necessary by showing that the query cannot be computed by any program that has a constant bound on the rank of sets used, even in C PT(Card) , an extension of C PT with counting.

Exponential Lower Bounds for Monotone Span Programs

Monotone span programs are a linear-algebraic model of computation which were introduced by Karchmer and Wigderson in 1993 [1]. They are known to be equivalent to linear secret sharing schemes, and have various applications in complexity theory and cryptography. Lower bounds for monotone span programs have been difficult to obtain because they use non-monotone operations to compute monotone functions, in fact, the best known lower bounds are quasipolynomial for a function in (nonmonotone) P [2]. A fundamental open problem is to prove exponential lower bounds on monotone span program size for any explicit function. We resolve this open problem by giving exponential lower bounds on monotone span program size for a function in monotone P. This also implies the first exponential lower bounds for linear secret sharing schemes. Our result is obtained by proving exponential lower bounds using Razborovs rank method [3], a measure that is strong enough to prove lower bounds for many monotone models. As corollaries we obtain new proofs of exponential lower bounds for monotone formula size, monotone switching network size, and the first lower bounds for monotone comparator circuit size for a function in monotone P. We also obtain new polynomial degree lower bounds for Nullstellensatz refutations using an interpolation theorem of Pudlak and Sgall [4]. Finally, we obtain quasipolynomial lower bounds on the rank measure for the st-connectivity function, implying tight bounds for st-connectivity in all of the computational models mentioned above.

An Average-Case Depth Hierarchy Theorem for Boolean Circuits

We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of AND, OR, and NOT gates. Our hierarchy theorem says that for every d ≥ 2, there is an explicit n-variable Boolean function f, computed by a linear-size depth-d formula, which is such that any depth-(d - 1) circuit that agrees with f on (1/2 + on(1)) fraction of all inputs must have size exp(nΩ(1/d)). This answers an open question posed by Hastad in his Ph.D. thesis [Has86b]. Our average-case depth hierarchy theorem implies that the polynomial hierarchy is infinite relative to a random oracle with probability 1, confirming a conjecture of Hastad [Has86a], Cai [Cai86], and Babai [Bab87]. We also use our result to show that there is no “approximate converse” to the results of Linial, Mansour, Nisan [LMN93] and Boppana [Bop97] on the total influence of constant-depth circuits, thus answering a question posed by Kalai [Kal12] and Hatami [Hat14]. A key ingredient in our proof is a notion of random projections which generalize random restrictions.

The Monotone Complexity of

It is widely suspected that ErdH{o}s-Renyi random graphs are a source of hard instances for clique problems. Giving further evidence for this belief, we prove the first average-case hardness result for the

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