One-Way Reversible and Quantum Finite Automata with Advice

We examine the characteristic features of reversible and quantum computations in the presence of supplementary external information, known as advice. In particular, we present a simple, algebraic characterization of languages recognized by one-way reversible finite automata augmented with deterministic advice. With a further elaborate argument, we prove a similar but slightly weaker result for bounded-error one-way quantum finite automata with advice. Immediate applications of those properties lead to containments and separations among various language families when they are assisted by appropriately chosen advice. We further demonstrate the power and limitation of randomized advice and quantum advice when they are given to one-way quantum finite automata.Comment: A4, 10pt, 1 figure, 31 pages. This is a complete version of an extended abstract appeared in the Proceedings of the 6th International Conference on Language and Automata Theory and Applications (LATA 2012), March 5-9, 2012, A Coruna, Spain, Lecture Notes in Computer Science, Springer-Verlag, Vol.7183, pp.526-537, 201

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 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 at most 3 times in the form of literals---cannot be solved simultaneously in polynomial time using strictly "sub-linear" (i.e., $m(x)^{\varepsilon}\, polylog(|x|)$ for a certain constant $\varepsilon\in(0,1)$) space on all instances $x$. An immediate consequence of this working hypothesis is $\mathrm{L}\neq\mathrm{NL}$. 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, parameterized with the number of variables, $\overline{\mathrm{2SAT}_3}$ is complete for a syntactically restricted version of NL, called Syntactic NL$_{\omega}$, under such short reductions. This fact supports the legitimacy of our working hypothesis.Comment: (A4, 10pt, 25 pages) This current article extends and corrects its preliminary report in the Proc. of the 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017), August 21-25, 2017, Aalborg, Denmark, Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik 2017, vol. 83, pp. 62:1-62:14, 201

Approximation Complexity of Complex-Weighted Degree-Two Counting Constraint Satisfaction Problems

Constraint satisfaction problems have been studied in numerous fields with practical and theoretical interests. In recent years, major breakthroughs have been made in a study of counting constraint satisfaction problems (or #CSPs). In particular, a computational complexity classification of bounded-degree #CSPs has been discovered for all degrees except for two, where the "degree" of an input instance is the maximal number of times that each input variable appears in a given set of constraints. Despite the efforts of recent studies, however, a complexity classification of degree-2 #CSPs has eluded from our understandings. This paper challenges this open problem and gives its partial solution by applying two novel proof techniques--T_{2}-constructibility and parametrized symmetrization--which are specifically designed to handle "arbitrary" constraints under randomized approximation-preserving reductions. We partition entire constraints into four sets and we classify the approximation complexity of all degree-2 #CSPs whose constraints are drawn from two of the four sets into two categories: problems computable in polynomial-time or problems that are at least as hard as #SAT. Our proof exploits a close relationship between complex-weighted degree-2 #CSPs and Holant problems, which are a natural generalization of complex-weighted #CSPs.Comment: A4, 10pt, 23 pages. This is a complete version of the paper that appeared in the Proceedings of the 17th Annual International Computing and Combinatorics Conference (COCOON 2011), Lecture Notes in Computer Science, vol.6842, pp.122-133, Dallas, Texas, USA, August 14-16, 201