Manindra Agrawal

Arithmetic Circuits: A Chasm at Depth Four

We show that proving exponential lower bounds on depth four arithmetic circuits imply exponential lower bounds for unrestricted depth arithmetic circuits. In other words, for exponential sized circuits additional depth beyond four does not help. We then show that a complete black-box derandomization of identity testing problem for depth four circuits with multiplication gates of small fanin implies a nearly complete derandomization of general identity testing.

Proving lower bounds via pseudo-random generators

In this paper, we formalize two stepwise approaches, based on pseudo-random generators, for proving P≠NP and its arithmetic analog: Permanent requires superpolynomial sized arithmetic circuits.

Primality and identity testing via Chinese remaindering

We give a simple and new randomized primality testing algorithm by reducing primality testing for number n to testing if a specific univariate identity over Zn holds.We also give new randomized algorithms for testing if a multivariate polynomial, over a finite field or over rationals, is identically zero. The first of these algorithms also works over Zn for any n. The running time of the algorithms is polynomial in the size of arithmetic circuit representing the input polynomial and the error parameter. These algorithms use fewer random bits and work for a larger class of polynomials than all the previously known methods, for example, the Schwartz--Zippel test [Schwartz 1980; Zippel 1979], Chen--Kao and Lewin--Vadhan tests [Chen and Kao 1997; Lewin and Vadhan 1998].

FST TCS 2002: Foundations of Software Technology and Theoretical Computer Science

Invited Papers.- Primality Testing with Gaussian Periods.- From Hilbert Spaces to Dilbert Spaces: Context Semantics Made Simple.- Encoding Generic Judgments.- Model Checking Algol-Like Languages Using Game Semantics.- Modeling Software: From Theory to Practice.- Contributed Papers.- Local Normal Forms for Logics over Traces.- On the Hardness of Constructing Minimal 2-Connected Spanning Subgraphs in Complete Graphs with Sharpened Triangle Inequality.- Communication Interference in Mobile Boxed Ambients.- The Seal Calculus Revisited: Contextual Equivalence and Bisimilarity.- Composing Strand Spaces.- Generalising Automaticity to Modal Properties of Finite Structures.- An Automata-Theoretic Approach to Constraint LTL.- Hardness Results for Multicast Cost Sharing.- How to Compose Presburger-Accelerations: Applications to Broadcast Protocols.- State Space Reductions for Alternating Buchi Automata Quotienting by Simulation Equivalences.- Algorithmic Combinatorics Based on Slicing Posets.- Pattern Matching for Arc-Annotated Sequences.- Knowledge over Dense Flows of Time (from a Hybrid Point of View).- The Complexity of the Inertia.- The Quantum Communication Complexity of the Pointer Chasing Problem: The Bit Version.- The Decidability of the First-Order Theory of the Knuth-Bendix Order in the Case of Unary Signatures.- Deciding the First Level of the ?-Calculus Alternation Hierarchy.- Dynamic Message Sequence Charts.- The Complexity of Compositions of Deterministic Tree Transducers.- On the Hardness of Approximating Minimum Monopoly Problems.- Hereditary History Preserving Bisimulation Is Decidable for Trace-Labelled Systems.- Lower Bounds for Embedding Graphs into Graphs of Smaller Characteristic.- Nearest Neighbors Search Using Point Location in Balls with Applications to Approximate Voronoi Decompositions.- Formal Languages and Algorithms for Similarity Based Retrieval from Sequence Databases.- Decomposition in Asynchronous Circuit Design.- Queue Layouts, Tree-Width, and Three-Dimensional Graph Drawing.

Reductions in Circuit Complexity

We show that all sets that are complete for NP under nonuniformAC0reductions are isomorphic under nonuniformAC0-computable isomorphisms. Furthermore, these sets remain NP-complete even under nonuniformNC0reductions. More generally, we show two theorems that hold for any complexity class C closed under (uniform)NC1-computable many?one reductions.Gap: The sets that are complete for C underAC0andNC0reducibility coincide.Isomorphism: The sets complete for C underAC0reductions are all isomorphic under isomorphisms computable and invertible byAC0circuits of depth three. Our Gap Theorem does not hold for strongly uniform reductions; we show that there are Dlogtime-uniformAC0-complete sets forNC1that are not Dlogtime-uniformNC0-complete.

Jacobian hits circuits: hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits

We present a single common tool to strictly subsume all known cases of polynomial time blackbox polynomial identity testing (PIT), that have been hitherto solved using diverse tools and techniques, over fields of zero or large characteristic. In particular, we show that polynomial time hitting-set generators for identity testing of the two seemingly different and well studied models - depth-3 circuits with bounded top fanin, and constant-depth constant-read multilinear formulas - can be constructed using one common algebraic-geometry theme: Jacobian captures algebraic independence. By exploiting the Jacobian, we design the first efficient hitting-set generators for broad generalizations of the above-mentioned models, namely: - depth-3 (Ω Π Ω) circuits with constant transcendence degree of the polynomials computed by the product gates (no bounded top fanin restriction), and - constant-depth constant-occur formulas (no multilinear restriction). Constant-occur of a variable, as we define it, is a much more general concept than constant-read. Also, earlier work on the latter model assumed that the formula is multilinear. Thus, our work goes further beyond the related results obtained by Saxena & Seshadhri (STOC 2011), Saraf & Volkovich (STOC 2011), Anderson et al. (CCC 2011), Beecken et al. (ICALP 2011) and Grenet et al. (FSTTCS 2011), and brings them under one unifying technique. In addition, using the same Jacobian based approach, we prove exponential lower bounds for the immanant (which includes permanent and determinant) on the same depth-3 and depth-4 models for which we give efficient PIT algorithms. Our results reinforce the intimate connection between identity testing and lower bounds by exhibiting a concrete mathematical tool - the Jacobian - that is equally effective in solving both the problems on certain interesting and previously well-investigated (but not well understood) models of computation.

Reducing the complexity of reductions

Abstract. We build on the recent progress regarding isomorphisms of complete sets that was reported in Agrawal et al. (1998). In that paper, it was shown that all sets that are complete under (non-uniform) AC0 reductions are isomorphic under isomorphisms computable and invertible via (non-uniform) depth-three AC0 circuits. One of the main tools in proving the isomorphism theorem in Agrawal et al. (1998) is a “Gap Theorem”, showing that all sets complete under AC0 reductions are in fact already complete under NC0 reductions. The following questions were left open in that paper:¶1. Does the “gap” between NC0 and AC0 extend further? In particular, is every set complete under polynomial-time reducibility already complete under NC0reductions?¶2. Does a uniform version of the isomorphism theorem hold?¶3. Is depth-three optimal, or are the complete sets isomorphic under isomorphisms computable by depth-two circuits?¶We answer all of these questions. In particular, we prove that the Berman—Hartmanis isomorphism conjecture is true for P-uniform AC0 reductions. More precisely, we show that for any class

Polynomial time truth-table reductions to p-selective sets

{\cal C}

The Boolean isomorphism problem

closed under uniform TC0-computable many-one reductions, the following three theorems hold:¶1. If

Pseudo-random generators and structure of complete degrees

{\cal C}