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.

Abstract 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 $EQS$ .

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.

Abstract 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 computer

We aim at investigating the solvability/insolvability of nondeterministic logarithmic-space (NL) decision, search, and optimization problems parameterized by size parameters using simultaneously polynomial time and sub-linear space on multi-tape deterministic Turing machines. We are particularly focused on a special NL-complete problem, 2SAT - the 2CNF Boolean formula satisfiability problem-parameterized by the number of Boolean variables. It is shown that 2SAT with n variables and m clauses can be solved simultaneously polynomial time and (n/2^{c sqrt{log(n)}}) polylog(m+n) space for an absolute constant c>0. This fact inspires us to propose a new, practical working hypothesis, called the linear space hypothesis (LSH), which states that 2SAT_3-a restricted variant of 2SAT in which each variable of a given 2CNF formula appears as literals in at most 3 clauses-cannot be solved simultaneously in polynomial time using strictly "sub-linear" (i.e., n^{epsilon} polylog(n) for a certain constant epsilon in (0,1)) space. An immediate consequence of this working hypothesis is L neq NL. Moreover, we use our hypothesis as a plausible basis to lead to the insolvability of various NL search problems as well as the nonapproximability of NL optimization problems. For our investigation, since standard logarithmic-space reductions may no longer preserve polynomial-time sub-linear-space complexity, we need to introduce a new, practical notion of "short reduction." It turns out that overline{2SAT}_3 is complete for a restricted version of NL, called Syntactic NL or simply SNL, under such short reductions. This fact supports the legitimacy of our working hypothesis.

Given a symmetric $D\times D$ matrix $M$ over \0,1,*\, a list $M$-partition of a graph $G$ is a partition of the vertices of $G$ into $D$ parts which are associated with the rows of $M$. The part of each vertex is chosen from a given list in such a way that no edge of $G$ is mapped to a 0 in $M$ and no nonedge of $G$ is mapped to a 1 in $M$. Many important graph-theoretic structures can be represented as list $M$-partitions including graph colorings, split graphs, and homogeneous sets and pairs, which arise in the proofs of the weak and strong perfect graph conjectures. Thus, there has been quite a bit of work on determining for which matrices $M$ computations involving list $M$-partitions are tractable. This paper focuses on the problem of counting list $M$-partitions, given a graph $G$ and given a list for each vertex of $G$. We identify a certain set of “tractable” matrices $M$. We give an algorithm that counts list $M$-partitions in polynomial time for every (fixed) matrix $M$ in this set. The algorithm relies on data structures such as sparse-dense partitions and subcube decompositions to reduce each problem instance to a sequence of problem instances in which the lists have a certain useful structure that restricts access to portions of $M$ in which the interactions of 0s and 1s are controlled. We show how to solve the resulting restricted instances by converting them into particular counting constraint satisfaction problems (${\#CSP}$s), which we show how to solve using a constraint satisfaction technique known as arc-consistency. For every matrix $M$ for which our algorithm fails, we show that the problem of counting list $M$-partitions is ${\#P}$-complete. Furthermore, we give an explicit characterization of the dichotomy theorem: counting list $M$-partitions is tractable (in ${FP}$) if the matrix $M$ has a structure called a derectangularizing sequence. If $M$ has no derectangularizing sequence, we show that counting list $M$-partitions is ${\#P}$-hard. We show that the metaproblem of determining whether a given matrix has a derectangularizing sequence is ${NP}$-complete. Finally, we show that list $M$-partitions can be used to encode cardinality restrictions in $M$-partitions problems, and we use this to give a polynomial-time algorithm for counting homogeneous pairs in graphs.

Given a symmetric DxD matrix M over {0, 1, *}, a list M-partition of a graph G is a partition of the vertices of G into D parts which are associated with the rows of M. The part of each vertex is chosen from a given list in such a way that no edge of G is mapped to a 0 in M and no non-edge of G is mapped to a 1 in M. Many important graph-theoretic structures can be represented as list M-partitions including graph colourings, split graphs and homogeneous sets, which arise in the proofs of the weak and strong perfect graph conjectures. Thus, there has been quite a bit of work on determining for which matrices M computations involving list M-partitions are tractable. This paper focuses on the problem of counting list M-partitions, given a graph G and given lists for each vertex of G. We give an algorithm that solves this problem in polynomial time for every (fixed) matrix M for which the problem is tractable. The algorithm relies on data structures such as sparse-dense partitions and sub cube decompositions to reduce each problem instance to a sequence of problem instances in which the lists have a certain useful structure that restricts access to portions of M in which the interactions of 0s and 1s is controlled. We show how to solve the resulting restricted instances by converting them into particular counting constraint satisfaction problems (#CSPs) which we show how to solve using a constraint satisfaction technique known as "arc-consistency". For every matrix M for which our algorithm fails, we show that the problem of counting list M-partitions is #P-complete. Furthermore, we give an explicit characterisation of the dichotomy theorem - counting list M-partitions is tractable (in FP) if and only if the matrix M has a structure called a derectangularising sequence. Finally, we show that the meta-problem of determining whether a given matrix has a derectangularising sequence is NP-complete.

Hardcore functions have been used as a technical tool to construct secure cryptographic systems; however, little is known on their quantum counterpart, called quantum hardcore functions. With a new insight into fundamental properties of quantum hardcores, we present three new quantum hardcore functions for any (strong) quantum one-way function. We also give a “quantum” solution to Damgård's question [Advances in Cryptology, Lecture Notes in Comput. Sci. 403, Springer, Berlin, 1990, pp. 163–172] on a classical hardcore property of his pseudorandom generator by proving its quantum hardcore property. Our major technical tool is the new notion of quantum list-decoding of “classical” error-correcting codes (rather than “quantum” error-correcting codes), which is defined on the platform of computational complexity theory and computational cryptography (rather than information theory). In particular, we give a simple but powerful criterion that makes a polynomial-time computable classical block code (seen as a function) a quantum hardcore for all quantum one-way functions. On their own interest, we construct efficient quantum list-decoding algorithms for classical block codes whose associated quantum states (called codeword states) form a nearly phase-orthogonal basis.