arXiv subjects

Tomoyuki Yamakami

Publications and source records attributed to Tomoyuki Yamakami.

At least 37 records · Page 2Linked to original sources

arXiv2021

Behavioral Strengths and Weaknesses of Various Models of Limited Automata

We examine the behaviors of various models of $k$-limited automata, which naturally extend Hibbard's [Inf. Control, vol. 11, pp. 196--238, 1967] scan limited automata, each of which is a single-tape linear-bounded automaton satisfying the $k$-limitedness requirement that the content of each tape cell should be modified only during the first $k$ visits of a tape head. One central computation model is a probabilistic $k$-limited automaton (abbreviated as a $k$-lpa), which accepts an input exactly when its accepting states are reachable from its initial state with probability more than 1/2 within expected polynomial time. We also study the behaviors of one-sided-error and bounded-error variants of such $k$-lpa's as well as the deterministic, nondeterministic, and unambiguous models of $k$-limited automata, which can be viewed as natural restrictions of $k$-lpa's. We discuss fundamental properties of these machine models and obtain inclusions and separations among language families induced by them. In due course, we study special features -- the blank skipping property and the closure under reversal -- which are keys to the robustness of $k$-lpa's.

cs.FL↗

arXiv2021

Parameterizations of Logarithmic-Space Reductions, Stack-State Complexity of Nonuniform Families of Pushdown Automata, and a Road to the LOGCFL$\subseteq$LOGDCFL/poly Question

The complexity class LOGCFL (resp., LOGDCFL) consists of all languages that are many-one reducible to context-free (resp., deterministic context-free) languages using logarithmic space. These complexity classes have been studied over five decades in connection to parallel computation since they are located between Nick's classes $\mathrm{NC}^1$ and $\mathrm{NC}^2$. In contrast, the state complexity of nonuniform finite-automaton families was first discussed in the 1970s and it has been extensively explored lately for various finite-automata families. We extend this old subject to the stack-state complexity (i.e., the total number of inner states plus simultaneously pushable stack symbol series) of nonuniform families of various pushdown automata. We introduce reasonable "parameterizations" of LOGCFL and LOGDCFL and apply them as a technical tool to establish a close connection between the LOGCFL$\subseteq$LOGDCFL/poly question and the polynomial stack-state complexity of nonuniform families of two-way pushdown automata. We also discuss the precise computational complexity of polynomial-size one-way pushdown automata.

cs.FL↗

arXiv2021

Nonuniform Families of Polynomial-Size Quantum Finite Automata and Quantum Logarithmic-Space Computation with Polynomial-Size Advice

The state complexity of a finite(-state) automaton intuitively measures the size of the description of the automaton. Sakoda and Sipser [STOC 1972, pp. 275--286] were concerned with nonuniform families of finite automata and they discussed the behaviors of the nonuniform complexity classes defined by such families of finite automata having polynomial-size state complexity. In a similar fashion, we introduce nonuniform state complexity classes using nonuniform families of quantum finite automata empowered by the flexible use of garbage tapes. We first present general inclusion and separation relationships among nonuniform state complexity classes of various one-way finite automata, including deterministic, nondeterministic, probabilistic, and quantum finite automata having polynomially many inner states. For two-way quantum finite automata equipped with flexible garbage tapes, we show a close relationship between the nonuniform state complexity of the family of such polynomial-size quantum finite automata and the parameterized complexity class induced by logarithmic-space quantum computation assisted by polynomial-size advice. We further establish a direct connection between space-bounded quantum computation with quantum advice and quantum finite automata whose transitions are dictated by superpositions of transition tables.

cs.FL↗

arXiv2020

Synchronizing Deterministic Push-Down Automata Can Be Really Hard

The question if a deterministic finite automaton admits a software reset in the form of a so-called synchronizing word can be answered in polynomial time. In this paper, we extend this algorithmic question to deterministic automata beyond finite automata. We prove that the question of synchronizability becomes undecidable even when looking at deterministic one-counter automata. This is also true for another classical mild extension of regularity, namely that of deterministic one-turn push-down automata. However, when we combine both restrictions, we arrive at scenarios with a PSPACE-complete (and hence decidable) synchronizability problem. Likewise, we arrive at a decidable synchronizability problem for (partially) blind deterministic counter automata. There are several interpretations of what synchronizability should mean for deterministic push-down automata. This is depending on the role of the stack: should it be empty on synchronization, should it be always the same or is it arbitrary? For the automata classes studied in this paper, the complexity or decidability status of the synchronizability problem is mostly independent of this technicality, but we also discuss one class of automata where this makes a difference.

cs.FL↗

arXiv2020

How Does Adiabatic Quantum Computation Fit into Quantum Automata Theory?

Quantum computation has emerged as a powerful computational medium of our time, having demonstrated the remarkable efficiency in factoring a positive integer and searching databases faster than any currently known classical computing algorithm. Adiabatic evolution of quantum systems have been studied as a potential means that physically realizes quantum computation. Up to now, all the research on adiabatic quantum systems has dealt with polynomial time-bounded computation and little attention has been paid to, for instance, adiabatic quantum systems consuming only constant memory space. Such quantum systems can be modeled in a form similar to quantum finite automata. This exposition dares to ask a bold question of how to make adiabatic quantum computation fit into the rapidly progressing framework of quantum automata theory. As our answer to this eminent but profound question, we first lay out a fundamental platform to carry out adiabatic evolutionary quantum systems (AEQSs) with limited computational resources (in size, energy, spectral gap, etc.) and then establish how to construct such AEQSs by operating suitable families of quantum finite automata. We further explore fundamental structural properties of decision problems (as well as promise problems) solved quickly by the appropriately constructed AEQSs.

quant-ph↗

arXiv2020

One-Way Topological Automata and the Tantalizing Effects of Their Topological Features

We cast new light on the existing models of one-way deterministic topological automata by introducing a fresh but general, convenient model, in which, as each input symbol is read, an interior system of an automaton, known as a configuration, continues to evolve in a topological space by applying continuous transition operators one by one. The acceptance and rejection of a given input are determined by observing the interior system after the input is completely processed. Such automata naturally generalize one-way finite automata of various types, including deterministic, probabilistic, quantum, and pushdown automata. We examine the strengths and weaknesses of the power of this new automata model when recognizing formal languages. We investigate tantalizing effects of various topological features of our topological automata by analyzing their behaviors when different kinds of topological spaces and continuous maps, which are used respectively as configuration spaces and transition operators, are provided to the automata. Finally, we present goals and directions of future studies on the topological features of topological automata.

cs.FL↗

arXiv2020

A Schematic Definition of Quantum Polynomial Time Computability

In the past four decades, the notion of quantum polynomial-time computability has been mathematically modeled by quantum Turing machines as well as quantum circuits. This paper seeks the third model, which is a quantum analogue of the schematic (inductive or constructive) definition of (primitive) recursive functions. For quantum functions mapping finite-dimensional Hilbert spaces to themselves, we present such a schematic definition, composed of a small set of initial quantum functions and a few construction rules that dictate how to build a new quantum function from the existing ones. We prove that our schematic definition precisely characterizes all functions that can be computable with high success probabilities on well-formed quantum Turing machines in polynomial time, or equivalently uniform families of polynomial-size quantum circuits. Our new, schematic definition is quite simple and intuitive and, more importantly, it avoids the cumbersome introduction of the well-formedness condition imposed on a quantum Turing machine model as well as of the uniformity condition necessary for a quantum circuit model. Our new approach can further open a door to the descriptional complexity of quantum functions, to the theory of higher-type quantum functionals, to the development of new first-order theories for quantum computing, and to the designing of programming languages for real-life quantum computers.

cs.CC↗

arXiv2020

Not All Multi-Valued Partial CFL Functions Are Refined by Single-Valued Functions

Multi-valued partial CFL functions are functions computed along accepting computation paths by one-way nondeterministic pushdown automata, equipped with write-only output tapes, which are allowed to reject an input, in comparison with single-valued partial CFL functions. We give an answer to a fundamental question, raised by Konstantinidis, Santean, and Yu [Act. Inform. 43 (2007) 395-417], of whether all such multi-valued partial CFL functions can be refined by single-valued partial CFL functions. We negatively solve this open question by presenting a special multi-valued partial CFL function as an example function and by proving that no refinement of this particular function becomes a single-valued partial CFL function. This contrasts an early result of Kobayashi [Inform. Control 15 (1969) 95-109] that multi-valued partial NFA functions are always refined by single-valued NFA functions, where NFA functions are computed by one-way nondeterministic finite automata with output tapes. Our example function turns out to be unambiguously 2-valued, and thus we obtain a stronger separation result, in which no refinement of unambiguously 2-valued partial CFL functions can be single-valued. For the proof of this fact, we first introduce a new concept of colored automata having no output tapes but having "colors," which can simulate pushdown automata equipped with constant-space output tapes. We then conduct an extensive combinatorial analysis on the behaviors of transition records of stack contents (called stack histories) of these colored automata.

cs.FL↗

arXiv2019

State Complexity Characterizations of Parameterized Degree-Bounded Graph Connectivity, Sub-Linear Space Computation, and the Linear Space Hypothesis

The linear space hypothesis is a practical working hypothesis, which originally states the insolvability of a restricted 2CNF Boolean formula satisfiability problem parameterized by the number of Boolean variables. From this hypothesis, it naturally follows that the degree-3 directed graph connectivity problem (3DSTCON) parameterized by the number of vertices in a given graph cannot belong to PsubLIN, composed of all parameterized decision problems computable by polynomial-time, sub-linear-space deterministic Turing machines. This hypothesis immediately implies L$\neq$NL and it was used as a solid foundation to obtain new lower bounds on the computational complexity of various NL search and NL optimization problems. The state complexity of transformation refers to the cost of converting one type of finite automata to another type, where the cost is measured in terms of the increase of the number of inner states of the converted automata from that of the original automata. We relate the linear space hypothesis to the state complexity of transforming restricted 2-way nondeterministic finite automata to computationally equivalent 2-way alternating finite automata having narrow computation graphs. For this purpose, we present state complexity characterizations of 3DSTCON and PsubLIN. We further characterize a nonuniform version of the linear space hypothesis in terms of the state complexity of transformation.

cs.CC↗

arXiv2019

Supportive Oracles for Parameterized Polynomial-Time Sub-Linear-Space Computations in Relation to L, NL, and P

We focus our attention onto polynomial-time sub-linear-space computation for decision problems, which are parameterized by size parameters $m(x)$, where the informal term "sub linear" means a function of the form $m(x)^{\varepsilon}\cdot polylog(|x|)$ on input instances $x$ for a certain absolute constant $\varepsilon\in(0,1)$ and a certain polylogarithmic function $polylog(n)$. The parameterized complexity class PsubLIN consists of all parameterized decision problems solvable simultaneously in polynomial time using sub-linear space. This complexity class is associated with the linear space hypothesis. There is no known inclusion relationships between PsubLIN and para-NL (nondeterministic log-space class), where the prefix "para-" indicates the natural parameterization of a given complexity class. Toward circumstantial evidences for the inclusions and separations of the associated complexity classes, we seek their relativizations. However, the standard relativization of Turing machines is known to violate the relationships of L$\subseteq$NL=co-NL$\subseteq$DSPACE[O($\log^2{n}$)]$\cap$P. We instead consider special oracles, called NL-supportive oracles, which guarantee these relationships in the corresponding relativized worlds. This paper vigorously constructs such NL-supportive oracles that generate relativized worlds where, for example, para-L$\neq$para-NL$\nsubseteq$PsubLIN and para-L$\neq$para-NL$\subseteq$PsubLIN.

cs.CC↗

arXiv2016

Quantum List Decoding of Classical Block Codes of Polynomially Small Rate from Quantumly Corrupted Codewords

Given a classical error-correcting block code, the task of quantum list decoding is to produce from any quantumly corrupted codeword a short list containing all messages whose codewords exhibit high "presence" in the quantumly corrupted codeword. Efficient quantum list decoders have been used to prove a quantum hardcore property of classical codes. However, the code rates of all known families of efficiently quantum list-decodable codes are, unfortunately, too small for other practical applications. To improve those known code rates, we prove that a specific code family of polynomially small code rate over a fixed code alphabet, obtained by concatenating generalized Reed-Solomon codes as outer codes with Hadamard codes as inner codes, has an efficient quantum list-decoding algorithm if its codewords have relatively high codeword presence in a given quantumly corrupted codeword. As an immediate application, we use the quantum list decodability of this code family to solve a certain form of quantum search problems in polynomial time. When the codeword presence becomes smaller, in contrast, we show that the quantum list decodability of generalized Reed-Solomon codes with high confidence is closely related to the efficient solvability of the following two problems: the noisy polynomial interpolation problem and the bounded distance vector problem. Moreover, assuming that NP is not included in BQP, we also prove that no efficient quantum list decoder exists for the generalized Reed-Solomon codes.

quant-ph↗

arXiv2016

Complexity Bounds of Constant-Space Quantum Computation

We realize constant-space quantum computation by measure-many two-way quantum finite automata and evaluate their language recognition power by analyzing patterns of their exotic behaviors and by exploring their structural properties. In particular, we show that, when the automata halt "in finite steps" along all computation paths, they must terminate in worst-case liner time. In the bounded-error probability case, the acceptance of the automata depends only on the computation paths that terminate within exponentially many steps even if not all computation paths may terminate. We also present a classical simulation of those automata on two-way multi-head probabilistic finite automata with cut points. Moreover, we discuss how the recognition power of the automata varies as the automata's acceptance criteria change to error free, one-sided error, bounded error, and unbounded error by comparing the complexity of their computational powers. We further note that, with the use of arbitrary complex transition amplitudes, two-way unbounded-error quantum finite automata and two-way bounded-error 2-head quantum finite automata can recognize certain non-recursive languages, whereas two-way error-free quantum finite automata recognize only recursive languages.

cs.FL↗

arXiv2016

Uniform-Circuit and Logarithmic-Space Approximations of Refined Combinatorial Optimization Problems

A significant progress has been made in the past three decades over the study of combinatorial NP optimization problems and their associated optimization and approximate classes, such as NPO, PO, APX (or APXP), and PTAS. Unfortunately, a collection of problems that are simply placed inside the P-solvable optimization class PO never have been studiously analyzed regarding their exact computational complexity. To improve this situation, the existing framework based on polynomial-time computability needs to be expanded and further refined for an insightful analysis of various approximation algorithms targeting optimization problems within PO. In particular, we deal with those problems characterized in terms of logarithmic-space computations and uniform-circuit computations. We are focused on nondeterministic logarithmic-space (NL) optimization problems or NPO problems. Our study covers a wide range of optimization and approximation classes, dubbed as, NLO, LO, APXL, and LSAS as well as new classes NC1O, APXNC1, NC1AS, and AC0O, which are founded on uniform families of Boolean circuits. Although many NL decision problems can be naturally converted into NL optimization (NLO) problems, few NLO problems have been studied vigorously. We thus provide a number of new NLO problems falling into those low-complexity classes. With the help of NC1 or AC0 approximation-preserving reductions, we also identify the most difficult problems (known as complete problems) inside those classes. Finally, we demonstrate a number of collapses and separations among those refined optimization and approximation classes with or without unproven complexity-theoretical assumptions.

cs.CC↗

arXiv2015

The World of Combinatorial Fuzzy Problems and the Efficiency of Fuzzy Approximation Algorithms

We re-examine a practical aspect of combinatorial fuzzy problems of various types, including search, counting, optimization, and decision problems. We are focused only on those fuzzy problems that take series of fuzzy input objects and produce fuzzy values. To solve such problems efficiently, we design fast fuzzy algorithms, which are modeled by polynomial-time deterministic fuzzy Turing machines equipped with read-only auxiliary tapes and write-only output tapes and also modeled by polynomial-size fuzzy circuits composed of fuzzy gates. We also introduce fuzzy proof verification systems to model the fuzzification of nondeterminism. Those models help us identify four complexity classes: Fuzzy-FPA of fuzzy functions, Fuzzy-PA and Fuzzy-NPA of fuzzy decision problems, and Fuzzy-NPAO of fuzzy optimization problems. Based on a relative approximation scheme targeting fuzzy membership degree, we formulate two notions of "reducibility" in order to compare the computational complexity of two fuzzy problems. These reducibility notions make it possible to locate the most difficult fuzzy problems in Fuzzy-NPA and in Fuzzy-NPAO.

cs.AI↗

arXiv2015

Pseudorandom Generators Against Advised Context-Free Languages

Pseudorandomness has played a central role in modern cryptography, finding theoretical and practical applications to various fields of computer science. A function that generates pseudorandom strings from shorter but truly random seeds is known as a pseudorandom generator. Our generators are designed to fool languages (or equivalently, Boolean-valued functions). In particular, our generator fools advised context-free languages, namely, context-free languages assisted by external information known as advice, and moreover our generator is made almost one-to-one, stretching $n$-bit seeds to $n+1$ bits. We explicitly construct such a pseudorandom generator, which is computed by a deterministic Turing machine using logarithmic space and also belongs to CFLMV(2)/n---a functional extension of the 2-conjunctive closure of CFL with the help of appropriate deterministic advice. In contrast, we show that there is no almost one-to-one pseudorandom generator against context-free languages if we demand that it should be computed by a nondeterministic pushdown automaton equipped with a write-only output tape. Our generator naturally extends known pseudorandom generators against advised regular languages. Our proof of the CFL/n-pseudorandomness of the generator is quite elementary, and in particular, one part of the proof utilizes a special feature of the behaviors of nondeterministic pushdown automata, called a swapping property, which is interesting in its own right, generalizing the swapping lemma for context-free languages.

cs.FL↗

arXiv2015

Structural Complexity of Multi-Valued Partial Functions Computed by Nondeterministic Pushdown Automata

This paper continues a systematic and comprehensive study on the structural properties of CFL functions, which are in general multi-valued partial functions computed by one-way one-head nondeterministic pushdown automata equipped with write-only output tapes (or pushdown transducers), where CFL refers to a relevance to context-free languages. The CFL functions tend to behave quite differently from their corresponding context-free languages. We extensively discuss containments, separations, and refinements among various classes of functions obtained from the CFL functions by applying Boolean operations, functional composition, many-one relativization, and Turing relativization. In particular, Turing relativization helps construct a hierarchy over the class of CFL functions. We also analyze the computational complexity of optimization functions, which are to find optimal values of CFL functions, and discuss their relationships to the associated languages.

cs.FL↗

arXiv2015

Oracle Pushdown Automata, Nondeterministic Reducibilities, and the CFL Hierarchy over the Family of Context-Free Languages

To expand a fundamental theory of context-free languages, we equip nondeterministic one-way pushdown automata with additional oracle mechanisms, which naturally induce various nondeterministic reducibilities among formal languages. As a natural restriction of NP-reducibility, we introduce a notion of many-one CFL reducibility and conduct a ground work to formulate a coherent framework for further expositions. Two more powerful reducibilities--bounded truth-table and Turing CFL-reducibilities--are also discussed in comparison. The Turing CFL-reducibility, in particular, helps us introduce an exquisite hierarchy, called the CFL hierarchy, built over the family CFL of context-free languages. For each level of this hierarchy, its basic structural properties are proven and three alternative characterizations are presented. The second level is not included in NC(2) unless NP= NC(2). The first and second levels of the hierarchy are different. The rest of the hierarchy (more strongly, the Boolean hierarchy built over each level of the hierarchy) is also infinite unless the polynomial hierarchy over NP collapses. This follows from a characterization of the Boolean hierarchy over the k-th level of the polynomial hierarchy in terms of the Boolean hierarchy over the k+1st level of the CFL hierarchy using log-space many-one reductions. Similarly, the complexity class Theta(k) is related to the closure of the k-th level of the CFL hierarchy under log-space truth-table reductions. We also argue that the CFL hierarchy coincides with a hierarchy over CFL built by application of many-one CFL-reductions. We show that BPCFL--a bounded-error probabilistic version of CFL--is not included in CFL even in the presence of advice. Employing a known circuit lower bound and a switching lemma, we exhibit a relativized world where BPCFL is not located within the second level of the CFL hierarchy.

cs.FL↗

arXiv2015

Counting list matrix partitions of graphs

Given a symmetric D*D matrix M over {0,1,*}, a list M-partition of a graph G is a partition of G's vertices into D parts associated with the rows of M. The part of each vertex is chosen from a given list so that no edge of G maps to a 0 in M and no non-edge of G maps to a 1 in M. Many important graph-theoretic structures can be represented as list M-partitions, such as graph colourings, split graphs and homogeneous sets and pairs, which arise in the proofs of the weak and strong perfect graph conjectures. There has been quite a bit of work on determining for which matrices M computations involving list M-partitions are tractable. We focus on counting list M-partitions, given a graph G and a list for each vertex of G. We identify a set of "tractable" matrices and give an algorithm that counts list M-partitions in polynomial time for every (fixed) matrix M in this set. The algorithm uses data structures such as sparse-dense partitions and subcube decompositions to reduce each instance to a sequence of instances in which the lists restrict access to portions of M in which the interaction of 0s and 1s is controlled. We solve the resulting restricted instances by converting them into counting constraint satisfaction problems (#CSPs) which we solve using arc-consistency. For every matrix M for which our algorithm fails, we show that counting list M-partitions is #P-complete. Further, we give an explicit characterisation of the dichotomy theorem: counting list M-partitions is in FP if M has a structure called a derectangularising sequence; otherwise, counting list M-partitions is #P-hard. We show that the meta-problem of determining whether a given matrix has a derectangularising sequence is NP-complete. Finally, we show that lists can be used to encode cardinality restrictions in M-partitions problems and use this to give a polynomial-time algorithm for counting homogeneous pairs in graphs.

cs.CC↗