arXiv subjects

Tomoyuki Yamakami

Publications and source records attributed to Tomoyuki Yamakami.

At least 19 recordsLinked to original sources

arXiv2026

How Can Size and Ceiling Bounds Affect the Complexity of Nonuniform Automata Families?

In the past literature, families of two-way finite automata and pushdown automata having limited state complexity (i.e., the total number of inner states) and stack-state complexity (i.e., the total number of inner states multiplied by the total number of strings "pushable" to a stack), have been studied in direct connection to (mainstream) space-bounded complexity classes equipped with Karp-Lipton style advice of limited size when all inputs given to the automata have bounded length. Here, we acknowledge two major factors -- size and ceiling -- of such families, which have a significant impact on the complexity of finite and pushdown automata families, where the "size" refers to (stack-)state complexity and the "ceiling" refers to an input's length bound. In this line of study, we further explore those effects caused by different sizes and ceilings.

cs.FL↗

arXiv2025

Unambiguous and Co-Nondeterministic Computations of Finite Automata and Pushdown Automata Families and the Effects of Multiple Counters

Nonuniform families of polynomial-size finite automata and pushdown automata respectively have strong connections to nonuniform-NL and nonuniform-LOGCFL. We examine the behaviors of unambiguous and co-nondeterministic computations produced by such families of automata operating multiple counters, where a counter is a stack using only a single non-bottom symbol. As immediate consequences, we obtain various collapses of the complexity classes of families of promise problems solvable by finite and pushdown automata families when all valid instances are limited to either polynomially long strings or unary strings. A key technical ingredient of our proofs is an inductive counting of reachable vertices of each computation graph of finite and pushdown automata that operate multiple counters simultaneously.

cs.CC↗

arXiv2025

Machine Learning by Adiabatic Evolutionary Quantum System

A computational model of adiabatic evolutionary quantum system (or AEQS, pronounced "eeh-ks") was introduced in [Yamakami,2022] as a sort of quantum annealing and its underlying input-driven Hamiltonians are generated quantum-algorithmically by various forms of quantum automata families (including 1qqaf's). We study an efficient way to accomplish certain machine learning tasks by training these AEQSs quantumly. When AEQSs are controlled by 1qqaf's, it suffices in essence to find an optimal 1qqaf that approximately solves a target relational problem. For this purpose, we develop a basic idea of approximately utilizing well-known quantum algorithms for quantum counting, quantum amplitude estimation, and quantum approximation. We then provide a rough estimation of the efficiency of our quantum learning algorithms for AEQSs.

quant-ph↗

arXiv2025

The No Endmarker Theorem for One-Way Probabilistic Pushdown Automata

In various models of one-way pushdown automata, the explicit use of two designated endmarkers on a read-once input tape has proven to be extremely useful for making a conscious, final decision on the acceptance/rejection of each input word immediately after reading the right endmarker. With no endmarkers, by contrast, a machine must constantly stay in either accepting or rejecting states at any moment since it never notices the end of the input word. This situation, however, helps us analyze the behavior of the machine whose tape head makes the consecutive moves on all prefixes of a given extremely long input word. Since those two machine formulations have their own advantages, it is natural to ask whether the endmarkers are truly necessary to correctly recognize languages. In the deterministic and nondeterministic models, it is well-known that the endmarkers are removable without changing the acceptance criteria of each input word. This paper proves that, for a more general model of one-way probabilistic pushdown automata, the endmarkers are always removable. This is proven by employing probabilistic transformations from an "endmarker" machine to an equivalent "no-endmarker" machine at the cost of double exponential stack-state complexity without compromising its error probability. By setting this error probability appropriately, our proof also provides an alternative proof to both the deterministic and the nondeterministic models as well.

cs.FL↗

arXiv2025

Logical Expressibility of Syntactic NL for Complementarity, Monotonicity, and Maximization

Syntactic NL or succinctly SNL was first introduced in 2017, analogously to SNP, as a ``syntactically''-defined natural subclass of NL (nondeterministic logarithmic-space complexity class) using a restricted form of logical sentences, starting with second-order ``functional'' existential quantifiers followed by first-order universal quantifiers, in close connection to the so-called linear space hypothesis. We further explore various properties of this complexity class SNL to achieve the better understandings of logical expressibility in NL. For instance, SNL does not enjoy the dichotomy theorem unless L=NL. To express the ``complementary'' problems of SNL problems logically, we introduce $μ$SNL, which is an extension of SNL by allowing the use of $μ$-terms. As natural variants of SNL, we further study the computational complexity of monotone and optimization versions of SNL, respectively called MonoSNL and MAXSNL. We further consider maximization problems that are logarithmic-space approximable with only constant approximation ratios. We then introduce a natural subclass of MAXSNL, called MAX$τ$SNL, which enjoys such limited approximability.

cs.CC↗

arXiv2025

Power of Counting by Nonuniform Families of Polynomial-Size Finite Automata

Lately, there have been intensive studies on strengths and limitations of nonuniform families of promise decision problems solvable by various types of polynomial-size finite automata families, where ``polynomial-size'' refers to the polynomially-bounded state complexity of a finite automata family. In this line of study, we further expand the scope of these studies to families of partial counting and gap functions, defined in terms of nonuniform families of polynomial-size nondeterministic finite automata, and their relevant families of promise decision problems. Counting functions have an ability of counting the number of accepting computation paths produced by nondeterministic finite automata. With no unproven hardness assumption, we show numerous separations and collapses of complexity classes of those partial counting and gap function families and their induced promise decision problem families. We also investigate their relationships to pushdown automata families of polynomial stack-state complexity.

cs.CC↗

arXiv2025

Intersection and Union Hierarchies of Deterministic Context-Free Languages and Pumping Lemmas

We study the computational complexity of finite intersections and finite unions of deterministic context-free (dcf) languages. Earlier, Wotschke [J. Comput. System Sci. 16 (1978) 456--461] demonstrated that intersections of $(d+1)$ dcf languages are in general more powerful than intersections of $d$ dcf languages for any positive integer $d$ based on the separation result of the intersection hierarchy of Liu and Weiner [Math. Systems Theory 7 (1973) 185--192]. The argument of Liu and Weiner, however, works only on bounded languages of particular forms, and therefore Wotschke's result is not directly extendable to other non-bounded languages. To deal with a wide range of languages for the non-membership to the intersection hierarchy, we circumvent the specialization of their proof technics and devise a new and practical technical tool: two pumping lemmas for finite unions of dcf languages. Since the family of dcf languages is closed under complementation and also under intersection with regular languages, these pumping lemmas help us establish the non-membership relation of languages formed by finite intersections of target languages. We also concern ourselves with a relationship to deterministic limited automata of Hibbard [Inf. Control 11 (1967) 196--238] in this regard.

cs.FL↗

arXiv2025

Quantum First-Order Logics That Capture Logarithmic-Time/Space Quantum Computability

We introduce a quantum analogue of classical first-order logic (FO) and develop a theory of quantum first-order logic as a basis of the productive discussions on the power of logical expressiveness toward quantum computing. The purpose of this work is to logically express "quantum computation" by introducing specially-featured quantum connectives and quantum quantifiers that quantify fixed-dimensional quantum states. Our approach is founded on the recently introduced recursion-theoretical schematic definitions of time-bounded quantum functions, which map finite-dimensional Hilbert spaces to themselves. The quantum first-order logic (QFO) in this work therefore looks quite different from the well-known old concept of quantum logic based on lattice theory. We demonstrate that quantum first-order logics possess an ability of expressing bounded-error quantum logarithmic-time computability by the use of new "functional" quantum variables. In contrast, an extra inclusion of quantum transitive closure operator helps us characterize quantum logarithmic-space computability. The same computability can be achieved by the use of different "functional" quantum variables.

quant-ph↗

arXiv2024

Nondeterministic Auxiliary Depth-Bounded Storage Automata and Semi-Unbounded Fan-in Cascading Circuits

We discuss a nondeterministic variant of the recently introduced machine model of deterministic auxiliary depth-$k$ storage automata (or aux-$k$-sda's) by Yamakami. It was proven that all languages recognized by polynomial-time logarithmic-space aux-$k$-sda's are located between $\mathrm{LOGDCFL}$ and $\mathrm{SC}^k$ (the $k$th level of Steve's class SC). We further propose a new and simple computational model of semi-unbounded fan-in Boolean circuits composed partly of cascading blocks, in which the first few AND gates of unbounded fan-out (called AND$_{(ω)}$ gates) at each layer from the left (where all gates at each layer are indexed from left to right) are linked in a "cascading" manner to their right neighbors though specific AND and OR gates. We use this new circuit model to characterize a nondeterministic variant of the aux-$2k$-sda's (called aux-$2k$-sna's) that run in polynomial time using logarithmic work space. By relaxing the requirement for cascading circuits, we also demonstrate how such cascading circuit families characterize the complexity class $\mathrm{P}$. This yields an upper bound on the computational complexity of $\mathrm{LOG}k\mathrm{SNA}$ by $\mathrm{P}$.

cs.CC↗

arXiv2024

Elementary Quantum Recursion Schemes That Capture Quantum Polylogarithmic Time Computability of Quantum Functions

Quantum computing has been studied over the past four decades based on two computational models of quantum circuits and quantum Turing machines. To capture quantum polynomial-time computability, a new recursion-theoretic approach was taken lately by Yamakami [J. Symb. Logic 80, pp.~1546--1587, 2020] by way of recursion schematic definition, which constitutes six initial quantum functions and three construction schemes of composition, branching, and multi-qubit quantum recursion. By taking a similar approach, we look into quantum polylogarithmic-time computability and further explore the expressing power of elementary schemes designed for such quantum computation. In particular, we introduce an elementary form of the quantum recursion, called the fast quantum recursion, and formulate $EQS$ (elementary quantum schemes) of ``elementary'' quantum functions. This class $EQS$ captures exactly quantum polylogarithmic-time computability, which forms the complexity class BQPOLYLOGTIME. We also demonstrate the separation of BQPOLYLOGTIME from NLOGTIME and PPOLYLOGTIME. As a natural extension of $EQS$, we further consider an algorithmic procedural scheme that implements the well-known divide-and-conquer strategy. This divide-and-conquer scheme helps compute the parity function but the scheme cannot be realized within our system $EQS$.

cs.CC↗

arXiv2024

Complexity Classification of Complex-Weighted Counting Acyclic Constraint Satisfaction Problems

We study the computational complexity of counting constraint satisfaction problems (#CSPs) whose constraints assign complex numbers to Boolean inputs when the corresponding constraint hypergraphs are acyclic. These problems are called acyclic #CSPs or succinctly, #ACSPs. We wish to determine the computational complexity of all such #ACSPs when arbitrary unary constraints are freely available. Depending on whether we further allow or disallow the free use of the specific constraint XOR (binary disequality), we present two complexity classifications of the #ACSPs according to the types of constraints used for the problems. When XOR is freely available, we first obtain a complete dichotomy classification. On the contrary, when XOR is not available for free, we then obtain a trichotomy classification. To deal with an acyclic nature of constraints in those classifications, we develop a new technical tool called acyclic-T-constructibility or AT-constructibility, and we exploit it to analyze a complexity upper bound of each #ACSPs.

cs.CC↗

arXiv2023

When Input Integers are Given in the Unary Numeral Representation

Many NP-complete problems take integers as part of their input instances. These input integers are generally binarized, that is, provided in the form of the "binary" numeral representation, and the lengths of such binary forms are used as a basis unit to measure the computational complexity of the problems. In sharp contrast, the "unarization" (or the "unary" numeral representation) of numbers has been known to bring a remarkably different effect onto the computational complexity of the problems. When no computational-complexity difference is observed between binarization and unarization of instances, on the contrary, the problems are said to be strong NP-complete. This work attempts to spotlight an issue of how the unarization of instances affects the computational complexity of various combinatorial problems. We present numerous NP-complete (or even NP-hard) problems, which turn out to be easily solvable when input integers are represented in unary. We then discuss the computational complexities of such problems when taking unary-form integer inputs. We hope that a list of such problems signifies the structural differences between strong NP-completeness and non-strong NP-completeness.

cs.CC↗

arXiv2023

Unambiguity and Fewness for Nonuniform Families of Polynomial-Size Nondeterministic Finite Automata

Nonuniform families of polynomial-size finite automata, which are series of indexed finite automata having polynomially many inner states, are used in the past literature to solve nonuniform families of promise decision problems. Among such nonuniform families of finite automata, we focus our attention, in particular, on the variants of nondeterministic finite automata, which have at most "one" (unambiguous), "polynomially many" (few) accepting computation paths, or unambiguous/few computation paths leading to each fixed configuration. When such machines are limited to make only one-way head moves, we can prove with no unproven hardness assumptions that some of these variants are different in computational power from each other. As for two-way machines restricted to instances of polynomially-bounded length, families of two-way polynomial-size nondeterministic finite automata are equivalent in power to families of polynomial-size unambiguous finite automata.

cs.FL↗

arXiv2023

The 2CNF Boolean Formula Satisfiability Problem and the Linear Space Hypothesis

We aim at investigating the solvability/insolvability of nondeterministic logarithmic-space (NL) decision, search, and optimization problems parameterized by natural size parameters using simultaneously polynomial time and sub-linear space. We are particularly focused on $\mathrm{2SAT}_3$ -- a restricted variant of the 2CNF Boolean (propositional) formula satisfiability problem in which each variable of a given 2CNF formula appears at most 3 times in the form of literals -- parameterized by the total number $m_{vbl}(ϕ)$ of variables of each given Boolean formula $ϕ$. We propose a new, practical working hypothesis, called the linear space hypothesis (LSH), which asserts that $(\mathrm{2SAT}_3,m_{vbl})$ cannot be solved in polynomial time using only ``sub-linear'' space (i.e., $m_{vbl}(x)^{\varepsilon}\, polylog(|x|)$ space for a constant $\varepsilon\in[0,1)$) on all instances $x$. Immediate consequences of LSH include $\mathrm{L}l\neq\mathrm{NL}$, $\mathrm{LOGDCFL}\neq\mathrm{LOGCFL}$, and $\mathrm{SC}\neq \mathrm{NSC}$. For our investigation, we fully utilize a key notion of ``short reductions'', under which the class $\mathrm{PsubLIN}$ of all parameterized polynomial-time sub-linear-space solvable problems is indeed closed.

cs.CC↗

arXiv2022

Between SC and LOGDCFL: Families of Languages Accepted by Logarithmic-Space Deterministic Auxiliary Depth-k Storage Automata

The closure of deterministic context-free languages under logarithmic-space many-one reductions ($\mathrm{L}$-m-reductions), known as LOGDCFL, has been studied in depth from an aspect of parallel computability because it is nicely situated between $\mathrm{L}$ and $\mathrm{AC}^{1}\cap\mathrm{SC}^2$. By replacing a memory device from pushdown stacks with access-controlled storage tapes, we introduce a computational model of one-way deterministic depth-$k$ storage automata ($k$-sda's) whose tape cells are freely modified during the first $k$ accesses and then become blank forever. These $k$-sda's naturally induce the language family $k\mathrm{SDA}$. Similarly to $\mathrm{LOGDCFL}$, we study the closure $\mathrm{LOG}k\mathrm{SDA}$ of all languages in $k\mathrm{SDA}$ under $\mathrm{L}$-m-reductions. We demonstrate that $\mathrm{DCFL}\subseteq k\mathrm{SDA}\subseteq \mathrm{SC}^k$ by significantly extending Cook's early result (1979) of $\mathrm{DCFL}\subseteq \mathrm{SC}^2$. The entire hierarch of $\mathrm{LOG}k\mathrm{SDA}$ for all $k\geq1$ therefore lies between $\mathrm{LOGDCFL}$ and $\mathrm{SC}$. As an immediate consequence, we obtain the same simulation bounds for Hibbard's limited automata. We further characterize $\mathrm{LOG}k\mathrm{SDA}$ in terms of a new machine model, called logarithmic-space deterministic auxiliary depth-$k$ storage automata that run in polynomial time. These machines are as powerful as a polynomial-time two-way multi-head deterministic depth-$k$ storage automata. We also provide a ``generic'' $\mathrm{LOG}k\mathrm{SDA}$-complete language under $\mathrm{L}$-m-reductions by constructing a two-way universal simulator working for all $k$-sda's.

cs.FL↗

arXiv2022

Parameterized-NL Completeness of Combinatorial Problems by Short Logarithmic-Space Reductions and Immediate Consequences of the Linear Space Hypothesis

The concept of space-bounded computability has become significantly important in handling vast data sets on memory-limited computing devices. To replenish the existing short list of NL-complete problems whose instance sizes are dictated by log-space size parameters, we propose new additions obtained directly from natural parameterizations of three typical NP-complete problems -- the vertex cover problem, the exact cover by 3-sets problem, and the 3-dimensional matching problem. With appropriate restrictions imposed on their instances, the proposed decision problems parameterized by appropriate size parameters are proven to be equivalent in computational complexity to either the parameterized $3$-bounded 2CNF Boolean formula satisfiability problem or the parameterized degree-$3$ directed $s$-$t$ connectivity problem by ``short'' logarithmic-space reductions. Under the assumption of the linear space hypothesis, furthermore, none of the proposed problems can be solved in polynomial time if the memory usage is limited to sub-linear space.

cs.CC↗

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↗