Dmitry Chistikov

The Taming of the Semi-Linear Set

Semi-linear sets, which are rational subsets of the monoid (Z^d,+), have numerous applications in theoretical computer science. Although semi-linear sets are usually given implicitly, by formulas in Presburger arithmetic or by other means, the effect of Boolean operations on semi-linear sets in terms of the size of description has primarily been studied for explicit representations. In this paper, we develop a framework suitable for implicitly presented semi-linear sets, in which the size of a semi-linear set is characterized by its norm—the maximal magnitude of a generator. We put together a toolbox of operations and decompositions for semi-linear sets which gives bounds in terms of the norm (as opposed to just the bit-size of the description), a unified presentation, and simplified proofs. This toolbox, in particular, provides exponentially better bounds for the complement and set-theoretic difference. We also obtain bounds on unambiguous decompositions and, as an application of the toolbox, settle the complexity of the equivalence problem for exponent-sensitive commutative grammars.

Unary Pushdown Automata and Straight-Line Programs

We consider decision problems for deterministic pushdown automata over the unary alphabet (udpda, for short). Udpda are a simple computation model that accept exactly the unary regular languages, but can be exponentially more succinct than finite-state automata. We complete the complexity landscape for udpda by showing that emptiness (and thus universality) is P-hard, equivalence and compressed membership problems are P-complete, and inclusion is coNP-complete. Our upper bounds are based on a translation theorem between udpda and straight-line programs over the binary alphabet (SLPs). We show that the characteristic sequence of any udpda can be represented as a pair of SLPs—one for the prefix, one for the lasso—that have size linear in the size of the udpda and can be computed in polynomial time. Hence, decision problems on udpda are reduced to decision problems on SLPs. Conversely, any SLP can be converted in logarithmic space into a udpda, and this forms the basis for our lower bound proofs. We show coNP-hardness of the ordered matching problem for SLPs, from which we derive coNP-hardness for inclusion. In addition, we complete the complexity landscape for unary nondeterministic pushdown automata by showing that the universality problem is Π2 P-hard, using a new class of integer expressions. Our techniques have applications beyond udpda. We show that our results imply Π2 P-completeness for a natural fragment of Presburger arithmetic and coNP lower bounds for compressed matching problems with one-character wildcards.

A uniformization theorem for nested word to word transductions

We study the class of relations implemented by nested word to word transducers (also known as visibly pushdown transducers). We show that any such relation can be uniformized by a functional relation from the same class, implemented by an unambiguous transducer. We give an exponential upper bound on the state complexity of the uniformization, improving a previous doubly exponential upper bound. Our construction generalizes a classical construction by Schutzenberger for the disambiguation of nondeterministic finite-state automata, using determinization and summarization constructions on nested word automata. Besides theoretical interest, our procedure can be the basis for synthesis procedures for nested word to word transductions.

Testing monotone read-once functions

A checking test for a monotone read-once function f depending essentially on all its n variables is a set of vectors M distinguishing f from all other monotone read-once functions of the same variables. We describe an inductive procedure for obtaining individual lower and upper bounds on the minimal number of vectors T(f) in a checking test for any function f. The task of deriving the exact value of T(f) is reduced to a combinatorial optimization problem related to graph connectivity. We show that for almost all functions f expressible by read-once conjunctive or disjunctive normal forms, T(f) ~n / ln n. For several classes of functions our results give the exact value of T(f).

Nonnegative Matrix Factorization Requires Irrationality

Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative

Synchronizing Automata over Nested Words

n \times m

Context-free commutative grammars with integer counters and resets

matrix

On rationality of nonnegative matrix factorization

M

The complexity of regular abstractions of one-counter languages

into a product of a nonnegative

On Restricted Nonnegative Matrix Factorization

n \times d