The complexity of membership problems for circuits over sets of integers

AbstractWe investigate the complexity of membership problems for {∪,∩,-,+,×}-circuits computing sets of integers. These problems are a natural modification of the membership problems for circuits computing sets of natural numbers studied by McKenzie and Wagner [The complexity of membership problems for circuits over sets of natural numbers, Lecture Notes in Computer Science, Vol. 2607, 2003, pp. 571–582]. We show that there are several membership problems for which the complexity in the case of integers differs significantly from the case of the natural numbers: testing membership in the subset of integers produced at the output of a {∪,+,×}-circuit is NEXPTIME-complete, whereas it is PSPACE-complete for the natural numbers. As another result, evaluating {-,+}-circuits is shown to be P-complete for the integers and PSPACE-complete for the natural numbers. The latter result extends McKenzie and Wagner's work in nontrivial ways. Furthermore, evaluating {×}-circuits is shown to be NL∧⊕L-complete, and several other cases are resolved

On equations over sets of integers

Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition S+T=\makeset{m+n}{m \in S, \: n \in T} and with ultimately periodic constants is exactly the class of hyper-arithmetical sets. Equations using addition only can represent every hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can also be represented by equations over sets of natural numbers equipped with union, addition and subtraction S \dotminus T=\makeset{m-n}{m \in S, \: n \in T, \: m \geqslant n}. Testing whether a given system has a solution is $\Sigma^1_1$-complete for each model. These results, in particular, settle the expressive power of the most general types of language equations, as well as equations over subsets of free groups.Comment: 12 apges, 0 figure

The World of Combinatorial Fuzzy Problems and the Efficiency of Fuzzy Approximation Algorithms

We re-examine a practical aspect of combinatorial fuzzy problems of various types, including search, counting, optimization, and decision problems. We are focused only on those fuzzy problems that take series of fuzzy input objects and produce fuzzy values. To solve such problems efficiently, we design fast fuzzy algorithms, which are modeled by polynomial-time deterministic fuzzy Turing machines equipped with read-only auxiliary tapes and write-only output tapes and also modeled by polynomial-size fuzzy circuits composed of fuzzy gates. We also introduce fuzzy proof verification systems to model the fuzzification of nondeterminism. Those models help us identify four complexity classes: Fuzzy-FPA of fuzzy functions, Fuzzy-PA and Fuzzy-NPA of fuzzy decision problems, and Fuzzy-NPAO of fuzzy optimization problems. Based on a relative approximation scheme targeting fuzzy membership degree, we formulate two notions of "reducibility" in order to compare the computational complexity of two fuzzy problems. These reducibility notions make it possible to locate the most difficult fuzzy problems in Fuzzy-NPA and in Fuzzy-NPAO.Comment: A4, 10pt, 10 pages. This extended abstract already appeared in the Proceedings of the Joint 7th International Conference on Soft Computing and Intelligent Systems (SCIS 2014) and 15th International Symposium on Advanced Intelligent Systems (ISIS 2014), December 3-6, 2014, Institute of Electrical and Electronics Engineers (IEEE), pp. 29-35, 201

VPSPACE and a transfer theorem over the complex field

We extend the transfer theorem of [KP2007] to the complex field. That is, we investigate the links between the class VPSPACE of families of polynomials and the Blum-Shub-Smale model of computation over C. Roughly speaking, a family of polynomials is in VPSPACE if its coefficients can be computed in polynomial space. Our main result is that if (uniform, constant-free) VPSPACE families can be evaluated efficiently then the class PAR of decision problems that can be solved in parallel polynomial time over the complex field collapses to P. As a result, one must first be able to show that there are VPSPACE families which are hard to evaluate in order to separate P from NP over C, or even from PAR.Comment: 14 page

Complexity of Equivalence and Learning for Multiplicity Tree Automata

We consider the complexity of equivalence and learning for multiplicity tree automata, i.e., weighted tree automata over a field. We first show that the equivalence problem is logspace equivalent to polynomial identity testing, the complexity of which is a longstanding open problem. Secondly, we derive lower bounds on the number of queries needed to learn multiplicity tree automata in Angluin's exact learning model, over both arbitrary and fixed fields. Habrard and Oncina (2006) give an exact learning algorithm for multiplicity tree automata, in which the number of queries is proportional to the size of the target automaton and the size of a largest counterexample, represented as a tree, that is returned by the Teacher. However, the smallest tree-counterexample may be exponential in the size of the target automaton. Thus the above algorithm does not run in time polynomial in the size of the target automaton, and has query complexity exponential in the lower bound. Assuming a Teacher that returns minimal DAG representations of counterexamples, we give a new exact learning algorithm whose query complexity is quadratic in the target automaton size, almost matching the lower bound, and improving the best previously-known algorithm by an exponential factor

Boolean Operations, Joins, and the Extended Low Hierarchy

We prove that the join of two sets may actually fall into a lower level of the extended low hierarchy than either of the sets. In particular, there exist sets that are not in the second level of the extended low hierarchy, EL_2, yet their join is in EL_2. That is, in terms of extended lowness, the join operator can lower complexity. Since in a strong intuitive sense the join does not lower complexity, our result suggests that the extended low hierarchy is unnatural as a complexity measure. We also study the closure properties of EL_ and prove that EL_2 is not closed under certain Boolean operations. To this end, we establish the first known (and optimal) EL_2 lower bounds for certain notions generalizing Selman's P-selectivity, which may be regarded as an interesting result in its own right.Comment: 12 page

Finite-Degree Predicates and Two-Variable First-Order Logic

We consider two-variable first-order logic on finite words with a fixed number of quantifier alternations. We show that all languages with a neutral letter definable using the order and finite-degree predicates are also definable with the order predicate only. From this result we derive the separation of the alternation hierarchy of two-variable logic on this signature