On space-bounded synchronized alternating Turing machines

AbstractWe continue the study of the computational power of synchronized alternating Turing machines (SATM) introduced in (Hromkovič 1986, Slobodová 1987, 1988a, b) to allow communication via synchronization among processes of alternating Turing machines. We are interested in comparing the four main classes of space-bounded synchronized alternating Turing machines obtained by adding or removing off-line capability and nondeterminism (1SUTM(S(n)), SUTM(S(n)), 1SATM(S(n)), and SATM(S(n)) against one another and against other variants of alternating Turing machines. Denoting the class of languages accepted by machines in C by L(C), we show as our main results that L(1SUTM(S(n))) ⊂ L(SUTM(S(n))) ⊂ L(1SATM(S(n)))= L(SATM(S(n))) for all space-bounded functions S(n)ϵo(n), and L(1SUTM(S(n)))= L(SUTM(S(n))) ⊂ L(1SATM(S(n)))=L(SATM(S(n))) for S(n)) ⩾ n. Furthermore, we show that for log log(n) ⩽ S(n)ϵo(log(n)), L(1SUTM(S(n))) is incomparable to L[1] ATM(S(n))). L(UTM(S(n))), L(1MUTM(S(n))), and L(MUTM(S(n))), where MATMs are alternating Turing machines with modified acceptance proposed in (Inoue 1989); in contrast, we show that these relationships become proper inclusions when log(n) ⩽ S(n)ϵo(n).For deterministic synchronized alternating finite automata with at most k processes (1DSA(k)FA and DSA(k)FA) we establish a tight hierarchy on the number of processes for the one-way case, namely, L(1DSA(n)FA) ⊂ L(1DSA(n+1)FA) for all n > 0, and show that L(1DFA(2)) − ∪k=1∞L(DSA(k)FA) ≠ ∅, where DFA(k) denotes deterministic k-head finite automata. Finally we investigate closure properties under Boolean operations for some of these classes of languages

On space efficiency of algorithms working on structural decompositions of graphs

Dynamic programming on path and tree decompositions of graphs is a technique that is ubiquitous in the field of parameterized and exponential-time algorithms. However, one of its drawbacks is that the space usage is exponential in the decomposition's width. Following the work of Allender et al. [Theory of Computing, '14], we investigate whether this space complexity explosion is unavoidable. Using the idea of reparameterization of Cai and Juedes [J. Comput. Syst. Sci., '03], we prove that the question is closely related to a conjecture that the Longest Common Subsequence problem parameterized by the number of input strings does not admit an algorithm that simultaneously uses XP time and FPT space. Moreover, we complete the complexity landscape sketched for pathwidth and treewidth by Allender et al. by considering the parameter tree-depth. We prove that computations on tree-depth decompositions correspond to a model of non-deterministic machines that work in polynomial time and logarithmic space, with access to an auxiliary stack of maximum height equal to the decomposition's depth. Together with the results of Allender et al., this describes a hierarchy of complexity classes for polynomial-time non-deterministic machines with different restrictions on the access to working space, which mirrors the classic relations between treewidth, pathwidth, and tree-depth.Comment: An extended abstract appeared in the proceedings of STACS'16. The new version is augmented with a space-efficient algorithm for Dominating Set using the Chinese remainder theore

Nesting Depth of Operators in Graph Database Queries: Expressiveness Vs. Evaluation Complexity

Designing query languages for graph structured data is an active field of research, where expressiveness and efficient algorithms for query evaluation are conflicting goals. To better handle dynamically changing data, recent work has been done on designing query languages that can compare values stored in the graph database, without hard coding the values in the query. The main idea is to allow variables in the query and bind the variables to values when evaluating the query. For query languages that bind variables only once, query evaluation is usually NP-complete. There are query languages that allow binding inside the scope of Kleene star operators, which can themselves be in the scope of bindings and so on. Uncontrolled nesting of binding and iteration within one another results in query evaluation being PSPACE-complete. We define a way to syntactically control the nesting depth of iterated bindings, and study how this affects expressiveness and efficiency of query evaluation. The result is an infinite, syntactically defined hierarchy of expressions. We prove that the corresponding language hierarchy is strict. Given an expression in the hierarchy, we prove that it is undecidable to check if there is a language equivalent expression at lower levels. We prove that evaluating a query based on an expression at level i can be done in $\Sigma_i$ in the polynomial time hierarchy. Satisfiability of quantified Boolean formulas can be reduced to query evaluation; we study the relationship between alternations in Boolean quantifiers and the depth of nesting of iterated bindings.Comment: Improvements from ICALP 2016 review comment

Conjunctive query inseparability in OWL2QL is ExpTime-hard

We settle an open question on the complexity of the following problem: given two OWL2QL TBoxes and a signature, decide whether these TBoxes return the same answers to any conjunctive query over any data formulated in the given signature. It has been known that the complexity of this problem is between PSpace and ExpTime. Here we show that the problem is ExpTime-complete and, in fact, deciding whether two OWL2QL knowledge bases (each with its own data) give the same answers to any conjunctive query is ExpTime-hard