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

Logics for complexity classes

A new syntactic characterization of problems complete via Turing reductions is presented. General canonical forms are developed in order to define such problems. One of these forms allows us to define complete problems on ordered structures, and another form to define them on unordered non-Aristotelian structures. Using the canonical forms, logics are developed for complete problems in various complexity classes. Evidence is shown that there cannot be any complete problem on Aristotelian structures for several complexity classes. Our approach is extended beyond complete problems. Using a similar form, a logic is developed to capture the complexity class $NP\cap coNP$ which very likely contains no complete problem.Comment: This article has been accepted for publication in Logic Journal of IGPL Published by Oxford University Press; 23 pages, 2 figure

One-Tape Turing Machine Variants and Language Recognition

We present two restricted versions of one-tape Turing machines. Both characterize the class of context-free languages. In the first version, proposed by Hibbard in 1967 and called limited automata, each tape cell can be rewritten only in the first $d$ visits, for a fixed constant $d\geq 2$. Furthermore, for $d=2$ deterministic limited automata are equivalent to deterministic pushdown automata, namely they characterize deterministic context-free languages. Further restricting the possible operations, we consider strongly limited automata. These models still characterize context-free languages. However, the deterministic version is less powerful than the deterministic version of limited automata. In fact, there exist deterministic context-free languages that are not accepted by any deterministic strongly limited automaton.Comment: 20 pages. This article will appear in the Complexity Theory Column of the September 2015 issue of SIGACT New