Pointers in Recursion: Exploring the Tropics

We translate the usual class of partial/primitive recursive functions to a pointer recursion framework, accessing actual input values via a pointer reading unit-cost function. These pointer recursive functions classes are proven equivalent to the usual partial/primitive recursive functions. Complexity-wise, this framework captures in a streamlined way most of the relevant sub-polynomial classes. Pointer recursion with the safe/normal tiering discipline of Bellantoni and Cook corresponds to polylogtime computation. We introduce a new, non-size increasing tiering discipline, called tropical tiering. Tropical tiering and pointer recursion, used with some of the most common recursion schemes, capture the classes logspace, logspace/polylogtime, ptime, and NC. Finally, in a fashion reminiscent of the safe recursive functions, tropical tiering is expressed directly in the syntax of the function algebras, yielding the tropical recursive function algebras

Quantum Information and the PCP Theorem

We show how to encode $2^n$ (classical) bits $a_1,...,a_{2^n}$ by a single quantum state $|\Psi>$ of size O(n) qubits, such that: for any constant $k$ and any $i_1,...,i_k \in \{1,...,2^n\}$, the values of the bits $a_{i_1},...,a_{i_k}$ can be retrieved from $|\Psi>$ by a one-round Arthur-Merlin interactive protocol of size polynomial in $n$. This shows how to go around Holevo-Nayak's Theorem, using Arthur-Merlin proofs. We use the new representation to prove the following results: 1) Interactive proofs with quantum advice: We show that the class $QIP/qpoly$ contains ALL languages. That is, for any language $L$ (even non-recursive), the membership $x \in L$ (for $x$ of length $n$) can be proved by a polynomial-size quantum interactive proof, where the verifier is a polynomial-size quantum circuit with working space initiated with some quantum state $|\Psi_{L,n} >$ (depending only on $L$ and $n$). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. 2) PCP with only one query: We show that the membership $x \in SAT$ (for $x$ of length $n$) can be proved by a logarithmic-size quantum state $|\Psi >$, together with a polynomial-size classical proof consisting of blocks of length $polylog(n)$ bits each, such that after measuring the state $|\Psi >$ the verifier only needs to read {\bf one} block of the classical proof. While the first result is a straight forward consequence of the new representation, the second requires an additional machinery of quantum low-degree-test that may be interesting in its own right.Comment: 30 page

A taxonomy of problems with fast parallel algorithms

The class NC consists of problems solvable very fast (in time polynomial in log n) in parallel with a feasible (polynomial) number of processors. Many natural problems in NC are known; in this paper an attempt is made to identify important subclasses of NC and give interesting examples in each subclass. The notion of NC1-reducibility is introduced and used throughout (problem R is NC1-reducible to problem S if R can be solved with uniform log-depth circuits using oracles for S). Problems complete with respect to this reducibility are given for many of the subclasses of NC. A general technique, the “parallel greedy algorithm,” is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in NC2 (solvable by uniform Boolean circuits of depth O(log2 n) and polynomial size). The class LOGCFL is given a new characterization in terms of circuit families. The class DET of problems reducible to integer determinants is defined and many examples given. A new problem complete for deterministic polynomial time is given, namely, finding the lexicographically first maximal clique in a graph. This paper is a revised version of S. A. Cook, (1983, in “Proceedings 1983 Intl. Found. Comut. Sci. Conf.,” Lecture Notes in Computer Science Vol. 158, pp. 78–93, Springer-Verlag, Berlin/New York)

On The Hardness of Approximate and Exact (Bichromatic) Maximum Inner Product

In this paper we study the (Bichromatic) Maximum Inner Product Problem (Max-IP), in which we are given sets A and B of vectors, and the goal is to find a in A and b in B maximizing inner product a * b. Max-IP is very basic and serves as the base problem in the recent breakthrough of [Abboud et al., FOCS 2017] on hardness of approximation for polynomial-time problems. It is also used (implicitly) in the argument for hardness of exact l_2-Furthest Pair (and other important problems in computational geometry) in poly-log-log dimensions in [Williams, SODA 2018]. We have three main results regarding this problem. - Characterization of Multiplicative Approximation. First, we study the best multiplicative approximation ratio for Boolean Max-IP in sub-quadratic time. We show that, for Max-IP with two sets of n vectors from {0,1}^{d}, there is an n^{2 - Omega(1)} time (d/log n)^{Omega(1)}-multiplicative-approximating algorithm, and we show this is conditionally optimal, as such a (d/log n)^{o(1)}-approximating algorithm would refute SETH. Similar characterization is also achieved for additive approximation for Max-IP. - 2^{O(log^* n)}-dimensional Hardness for Exact Max-IP Over The Integers. Second, we revisit the hardness of solving Max-IP exactly for vectors with integer entries. We show that, under SETH, for Max-IP with sets of n vectors from Z^{d} for some d = 2^{O(log^* n)}, every exact algorithm requires n^{2 - o(1)} time. With the reduction from [Williams, SODA 2018], it follows that l_2-Furthest Pair and Bichromatic l_2-Closest Pair in 2^{O(log^* n)} dimensions require n^{2 - o(1)} time. - Connection with NP * UPP Communication Protocols. Last, We establish a connection between conditional lower bounds for exact Max-IP with integer entries and NP * UPP communication protocols for Set-Disjointness, parallel to the connection between conditional lower bounds for approximating Max-IP and MA communication protocols for Set-Disjointness. The lower bound in our first result is a direct corollary of the new MA protocol for Set-Disjointness introduced in [Rubinstein, STOC 2018], and our algorithms utilize the polynomial method and simple random sampling. Our second result follows from a new dimensionality self reduction from the Orthogonal Vectors problem for n vectors from {0,1}^{d} to n vectors from Z^{l} where l = 2^{O(log^* d)}, dramatically improving the previous reduction in [Williams, SODA 2018]. The key technical ingredient is a recursive application of Chinese Remainder Theorem. As a side product, we obtain an MA communication protocol for Set-Disjointness with complexity O (sqrt{n log n log log n}), slightly improving the O (sqrt{n} log n) bound [Aaronson and Wigderson, TOCT 2009], and approaching the Omega(sqrt{n}) lower bound [Klauck, CCC 2003]. Moreover, we show that (under SETH) one can apply the O(sqrt{n}) BQP communication protocol for Set-Disjointness to prove near-optimal hardness for approximation to Max-IP with vectors in {-1,1}^d. This answers a question from [Abboud et al., FOCS 2017] in the affirmative

Algebra in Computational Complexity

At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things about these combinatorial objects is by establishing a connection to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some of the most prominent examples. The algebraic theme continues in some of the most exciting recent progress in computational complexity. There have been significant recent advances in algebraic circuit lower bounds, and the so-called "chasm at depth 4" suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model, and these are tied to central questions regarding the power of randomness in computation. Representation theory has emerged as an important tool in three separate lines of work: the "Geometric Complexity Theory" approach to P vs. NP and circuit lower bounds, the effort to resolve the complexity of matrix multiplication, and a framework for constructing locally testable codes. Coding theory has seen several algebraic innovations in recent years, including multiplicity codes, and new lower bounds. This seminar brought together researchers who are using a diverse array of algebraic methods in a variety of settings. It plays an important role in educating a diverse community about the latest new techniques, spurring further progress

Adventures in Monotone Complexity and TFNP

Separations: We introduce a monotone variant of Xor-Sat and show it has exponential monotone circuit complexity. Since Xor-Sat is in NC^2, this improves qualitatively on the monotone vs. non-monotone separation of Tardos (1988). We also show that monotone span programs over R can be exponentially more powerful than over finite fields. These results can be interpreted as separating subclasses of TFNP in communication complexity. Characterizations: We show that the communication (resp. query) analogue of PPA (subclass of TFNP) captures span programs over F_2 (resp. Nullstellensatz degree over F_2). Previously, it was known that communication FP captures formulas (Karchmer - Wigderson, 1988) and that communication PLS captures circuits (Razborov, 1995)

Descriptive Complexity of Deterministic Polylogarithmic Time and Space

We propose logical characterizations of problems solvable in deterministic polylogarithmic time (PolylogTime) and polylogarithmic space (PolylogSpace). We introduce a novel two-sorted logic that separates the elements of the input domain from the bit positions needed to address these elements. We prove that the inflationary and partial fixed point vartiants of this logic capture PolylogTime and PolylogSpace, respectively. In the course of proving that our logic indeed captures PolylogTime on finite ordered structures, we introduce a variant of random-access Turing machines that can access the relations and functions of a structure directly. We investigate whether an explicit predicate for the ordering of the domain is needed in our PolylogTime logic. Finally, we present the open problem of finding an exact characterization of order-invariant queries in PolylogTime.Comment: Submitted to the Journal of Computer and System Science