Kaspars Balodis

University of Latvia | LU · Faculty of Computing

In the claw detection problem we are given two functions $f:D\rightarrow R$ and $g:D\rightarrow R$ ($|D|=n$, $|R|=k$), and we have to determine if there is exist $x,y\in D$ such that $f(x)=g(y)$. We show that the quantum query complexity of this problem is between $\Omega\left(n^{1/2}k^{1/6}\right)$ and $O\left(n^{1/2+\varepsilon}k^{1/4}\right)$ wh...

We study the quantum query complexity of two problems. First, we consider the problem of determining if a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most $k$. We call this the $Dyck_{k,n}$ problem. We prove a lower bound of $\Omega(c^k \sqrt{n})$, showing that the complexity of this problem increases expone...

We show quantum lower bounds for two problems. First, we consider the problem of determining if a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most $k$. It has been known that, for any $k$, $\tilde{O}(\sqrt{n})$ queries suffice, with a $\tilde{O}$ term depending on $k$. We prove a lower bound of $\Omega(c^k \...

In this paper we study quantum algorithms for NP-complete problems whose best classical algorithm is an exponential time application of dynamic programming. We introduce the path in the hypercube problem that models many of these dynamic programming algorithms. In this problem we are asked whether there is a path from $0^n$ to $1^n$ in a given subg...

In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total Boolean function is given by the function f on n = 2k bits defined by a complete binary tree of NAND gates of depth k, which achieves R0(f) = O(D(f)0.7537…). We show that this is false by giv...

In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total boolean function is given by the function f on n=2k bits defined by a complete binary tree of NAND gates of depth k, which achieves R0(f) = O(D(f)0.7537…). We show this is false by giving an example of a t...

We show that an improvement to the best known quantum lower bound for GRAPH-COLLISION problem implies an improvement to the best known lower bound for TRIANGLE problem in the quantum query complexity model. In GRAPH-COLLISION we are given free access to a graph $(V,E)$ and access to a function $f:V\rightarrow \{0,1\}$ as a black box. We are asked t...

In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total boolean function is given by the function $f$ on $n=2^k$ bits defined by a complete binary tree of NAND gates of depth $k$, which achieves $R_0(f) = O(D(f)^{0.7537\ldots})$. We show this is false by giving...

B.A.Trakhtenbrot proved that in frequency computability (introduced by G. Rose) it is crucially important whether the frequency exceeds 1 2 . If it does then only recursive sets are frequency-computable. If the frequency does not exceed 1 2 then a continuum of sets is frequencycomputable. Similar results for �nite automata were proved by E.B. Kinbe...

This paper explores the language classes that arise with respect to the head count of a finite ultrametric automaton. First we prove that in the one-way setting there is a language that can be recognized by a one-head ultrametric finite automaton and cannot be recognized by any k-head non-deterministic finite automaton. Then we prove that in the tw...

We investigate the state complexity of probabilistic and ultrametric finite automata for the problem of counting, i.e. recognizing the one-word unary language \(C_n=\left\{ 1^n \right\} \). We also review the known results for other types of automata. For one-way probabilistic automata, we construct a minimal \(3\)-state automaton for counting to \...

We explore an alternative definition of query algorithms which proposes the use of p-adic numbers or their ordered tuples as amplitudes. The reader is introduced to the definition of p-adic numbers and their main properties and operations. Afterwards the reader is reminded of the notions of deterministic and randomized query algorithms, which are t...

We study the query complexity of Weak Parity: the problem of computing the parity of an n-bit input string, where one only has to succeed on a 1/2+eps fraction of input strings, but must do so with high probability on those inputs where one does succeed. It is well-known that n randomized queries and n/2 quantum queries are needed to compute parity...

We show a family of languages that can be recognized by a family of linear-size alternating one-way finite automata with one alternation but cannot be recognized by any family of polynomial-size bounded-error two-way probabilistic finite automata with the expected runtime bounded by a polynomial. In terms of finite automata complexity theory this m...

We present three new quantum algorithms in the quantum query model for \textsc{graph-collision} problem: \begin{itemize} \item an algorithm based on tree decomposition that uses $O\left(\sqrt{n}t^{\sfrac{1}{6}}\right)$ queries where $t$ is the treewidth of the graph; \item an algorithm constructed on a span program that improves a result by Gavinsk...

We investigate properties of an identification type of recursive functions, called co-learning. The inductive process refutes all possible programs but one, and, by definition, this program is demanded to be correct. This type of identification was introduced in [6]. M. Kummer in the paper [9] showed that this type characterizes computable numberin...

We consider representing of natural numbers by arithmetical expressions using ones, addition, multiplication and parentheses. The (integer) complexity of n -- denoted by ||n|| -- is defined as the number of ones in the shortest expressions representing n. We arrive here very soon at the problems that are easy to formulate, but (it seems) extremely...

We introduce a notion of ultrametric �nite automata using p-adic numbers to describe random branching of the process of computation. These automata have properties similar to the properties of probabilistic automata but the descriptional power of probabilistic automata and ultrametric automata can di�er very much.

String theory in physics and molecular biology use p-adic numbers as a tool to describe properties of microworld in a more adequate way. We consider a notion of automata and Turing machines using p-adic numbers to describe random branching of the process of computation.We prove that Turing machines of this type can have advantages in reversal compl...

Prediction of functions is one of processes considered in in-ductive inference. There is a "black box" with a given total function f in it. The result of the inductive inference machine F (< f (0), f (1), · · · , f (n) >) is expected to be f (n + 1). Deterministic and probabilistic prediction of functions has been widely studied. Frequency computat...

A transducer is a finite-state automaton with an input and an output. We compare possibilities of nondeterministic and probabilis-tic transducers, and prove several theorems which establish an infinite hierarchy of relations computed by these transducers. We consider only left-total relations (where for each input value there is exactly one al-lowe...

We initiate a study of random instances of nonlocal games. We show that quantum strategies are better than classical for almost any 2-player XOR game. More precisely, for large n, the entangled value of a random 2-player XOR game with n questions to every player is at least 1.21... times the classical value, for 1-o(1) fraction of all 2-player XOR...

Non-local games are studied in quantum information because they provide a simple way for proving the difference between the classical world and the quantum world. A non-local game is a cooperative game played by 2 or more players against a referee. The players cannot communicate but may share common random bits or a common quantum state. A referee...

This paper proposes statistical analysis methods for improvement of terminology entry compounding. Terminology entry compounding is a mechanism that identifies matching entries across multiple multilingual terminology collections. Bilingual or trilingual term entries are unified in compounded multilingual entry. We suggest that corpus analysis can...

Nonconstructive finite automata were first considered by R. Freivalds. We prove tight upper bound for amount of nonconstructivity that can be needed to recognize a language. We prove some theorems about saving amount of the nonconstructive help needed by encoding that information in automata. We also show that nonconstructive probabilistic automata...