Limited Nondeterminism Research Articles

State complexity of finite partial languages

Partial word finite automata are deterministic finite automata that may have state transitions on a special symbol ⋄ which represents an unknown symbol or a hole in the word. Together with a subset of the input alphabet that gives the symbols which may be substituted for the ⋄, a partial word finite automaton (⋄-DFA) represents a regular language. However, this substitution implies a certain form of limited nondeterminism in the computations when the ⋄-transitions are replaced by ordinary transitions. In this paper we consider the state complexity of partial word finite automata accepting finite languages. We study the state complexity of the NFA to ⋄-DFA conversion for finite languages as well as the state complexity of the ⋄-DFA to DFA conversion for finite languages. Then we consider the operational state complexity with respect to complementation, union, reversal, and concatenation of finite languages. It turns out that the upper and lower bounds for all these operations are exponential. Moreover, we establish a state complexity hierarchy on the number of productive ⋄-transitions that may appear in ⋄-DFAs accepting finite languages. The levels of the hierarchy are separated by quadratic state costs.

One-Time Nondeterministic Computations

We introduce the concept of one-time nondeterminism as a new kind of limited nondeterminism for finite state machines and pushdown automata. Roughly speaking, one-time nondeterminism means that at the outset the computation is nondeterministic, but whenever it performs a guess, this guess is fixed for the rest of the computation. We characterize the computational power of one-time nondeterministic finite automata (OTNFAs) and one-time nondeterministic pushdown devices. Moreover, we study the descriptional complexity of these machines. For instance, we show that for an [Formula: see text]-state OTNFA with a sole nondeterministic state, that is nondeterministic for only one input symbol, [Formula: see text] states are sufficient and necessary in the worst case for an equivalent deterministic finite automaton. In case of pushdown automata, the conversion of a nondeterministic to a one-time nondeterministic as well as the conversion of a one-time nondeterministic to a deterministic one turn out to be non-recursive, that is, the trade-offs in size cannot be bounded by any recursive function.

Minimal partial languages and automata

Partial words are sequences of characters from an alphabet in which some positions may be marked with a “hole” symbol, ⋄. We can create a ⋄-substitution mapping this symbol to a subset of the alphabet, so that applying such a substitution to a partial word results in a set of total words (ones without holes). This setup allows us to compress regular languages into smaller partial languages. Deterministic finite automata for such partial languages, referred to as ⋄-DFAs, employ a limited non-determinism that can allow them to have lower state complexity than the minimal DFAs for the corresponding total languages. Our paper focuses on algorithms for the construction of minimal partial languages, associated with some ⋄-substitution, as well as approximation algorithms for the construction of minimal ⋄-DFAs.

Operational state complexity of unary NFAs with finite nondeterminism

We study the state complexity of language operations for unary NFAs with limited nondeterminism. We consider the Boolean operations, concatenation, and Kleene star. We give upper bounds for the state complexity of these language operations and lower bounds that are fairly close to the upper bounds. Our constructions rely on the fact that minimal unary NFAs with limited nondeterminism can be found in Chrobak normal form for most measures of nondeterminism. The measures of nondeterminism which are considered here with the above property are tree width, advice, and trace.

A Multivariate Complexity Analysis of Lobbying in Multiple Referenda

Assume that each of n voters may or may not approve each of m issues. If an agent (the lobby) may influence up to k voters, then the central question of the NP-hard Lobbying problem is whether the lobby can choose the voters to be influenced so that as a result each issue gets a majority of approvals. This problem can be modeled as a simple matrix modification problem: Can one replace k rows of a binary n x m-matrix by k all-1 rows such that each column in the resulting matrix has a majority of 1s? Significantly extending on previous work that showed parameterized intractability (W[2]-completeness) with respect to the number k of modified rows, we study how natural parameters such as n, m, k, or the "maximum number of 1s missing for any column to have a majority of 1s" (referred to as "gap value g") govern the computational complexity of Lobbying. Among other results, we prove that Lobbying is fixed-parameter tractable for parameter m and provide a greedy logarithmic-factor approximation algorithm which solves Lobbying even optimally if m < 5. We also show empirically that this greedy algorithm performs well on general instances. As a further key result, we prove that Lobbying is LOGSNP-complete for constant values g>0, thus providing a first natural complete problem from voting for this complexity class of limited nondeterminism.

IN MEMORIAM CHANDRA KINTALA

With this contribution we would like to commemorate Chandra M. R. Kintala, who passed away in November 2009. We will give short overviews of his CV and his contributions to the field of theoretical and applied computer science and, given the opportunity, will attempt to present his influence on areas like limited nondeterminism and resources as well as software reliability in an exemplary fashion. Finally, we will briefly touch on some research topics which hopefully will be addressed in the not-so-distant future.

Remembering Chandra Kintala

With this contribution we would like to remember Chandra M. R. Kintala who passed away in November 2009. We will give short overviews of his CV and his contributions to the field of theoretical and applied computer science and, given the opportunity, will attempt to present the current state of limited nondeterminism and limited resources for machines. Finally, we will briefly touch on some research topics which hopefully will be addressed in the not so distant future.

Complete open-state testing of limitedly nondeterministic systems

An approach to the problem of complete testing is proposed. Testing is interpreted as the check of an implementation's conformance to the given requirements described by a specification. The completeness means that a test suite finds all the possible implementation errors. In practice, testing must end in a finite amount of time. In the general case, the requirements of completeness and finiteness contradict each other. However, finite complete test suites can be constructed for certain classes of implementations and specifications provided that there are specific test capabilities. Test algorithms are proposed for finite specifications and finite implementations with limited nondeterminism for the case of open-state testing. The complexity of those algorithms is estimated.

Computational aspects of monotone dualization: A brief survey

Dualization of a monotone Boolean function represented by a conjunctive normal form (CNF) is a problem which, in different disguise, is ubiquitous in many areas including Computer Science, Artificial Intelligence, and Game Theory to mention some of them. It is also one of the few problems whose precise tractability status (in terms of polynomial-time solvability) is still unknown, and now open for more than 25 years. In this paper, we briefly survey computational results for this problem, where we focus on the famous paper by Fredman and Khachiyan [On the complexity of dualization of monotone disjunctive normal forms, J. Algorithms 21 (1996) 618–628], which showed that the problem is solvable in quasi-polynomial time (and thus most likely not co-NP-hard), as well as on follow-up works. We consider computational aspects including limited nondeterminism, probabilistic computation, parallel and learning-based algorithms, and implementations and experimental results from the literature. The paper closes with open issues for further research.

New Results on Monotone Dualization and Generating Hypergraph Transversals

We consider the problem of dualizing a monotone CNF (equivalently, computing all minimal transversals of a hypergraph) whose associated decision problem is a prominent open problem in NP-completeness. We present a number of new polynomial time, respectively, output-polynomial time results for significant cases, which largely advance the tractability frontier and improve on previous results. Furthermore, we show that duality of two monotone CNFs can be disproved with limited nondeterminism. More precisely, this is feasible in polynomial time with O(log2n /\log log n) suitably guessed bits. This result sheds new light on the complexity of this important problem.

Downward Separation Fails Catastrophically for Limited Nondeterminism Classes

The $\beta$ hierarchy consists of classes $\beta_k={\rm NP}[logkn]\subseteq {\rm NP}$. Unlike collapses in the polynomial hierarchy and the Boolean hierarchy, collapses in the $\beta$ hierarchy do not seem to translate up, nor does closure under complement seem to cause the hierarchy to collapse. For any consistent set of collapses and separations of levels of the hierarchy that respects ${\rm P} = \beta_1\subseteq \beta_2\subseteq \cdots \subseteq {\rm NP}$, we can construct an oracle relative to which those collapses and separations hold; at the same time we can make distinct levels of the hierarchy closed under computation or not, as we wish. To give two relatively tame examples: for any $k \geq 1$, we construct an oracle relative to which \[ {\rm P} = \beta_{k} \neq \beta_{k+1} \neq \beta_{k+2} \neq \cdots \] and another oracle relative to which \[ {\rm P} = \beta_{k} \neq \beta_{k+1} = {\rm PSPACE}. \] We also construct an oracle relative to which $\beta_{2k} = \beta_{2k+1} \neq \beta_{2k+2}$ for all k.

On finite automata with limited nondeterminism

We develop a new algorithm for determining if a given nondeterministic finite automaton is limited in nondeterminism. From this, we show that the number of nondeterministic moves of a finite automaton, if limited, is bounded by \(2^{n} - 2\) where \(n\) is the number of states. If the finite automaton is over a one-letter alphabet, using Gohon's result the number of nondeterministic moves, if limited, is less than \(n^{2}\). In both cases, we present families of finite automata demonstrating that the upper bounds obtained are almost tight. We also show that the limitedness problem of the number of nondeterministic moves of finite automata is PSPACE-hard. Since the problem is already known to be in PSPACE, it is therefore PSPACE-complete.

On the Amount of Nondeterminism and the Power of Verifying

The relationship between nondeterminism and other computational resources is investigated based on the "guess-then-check" model GC. Systematic techniques are developed to construct natural complete languages for the classes defined by this model. This improves a number of previous results in the study of limited nondeterminism. Connections of the model GC to computational optimization problems are exhibited.

Some intuitions behind realizability semantics for constructive logic: Tableaux and Läuchli countermodels

We use formal semantic analysis based on new constructions to study abstract realizability, introduced by Läuchli in 1970, and expose its algebraic content. We claim realizability so conceived generates semantics-based intuitive confidence that the Heyting Calculus is an appropriate system of deduction for constructive reasoning. Well-known semantic formalisms have been defined by Kripke and Beth, but these have no formal concepts corresponding to constructions, and shed little intuitive light on the meanings of formulae. In particular, the completeness proofs for these semantics do not generate confidence in the sufficiency of the Heyting Calculus, since we have no reason to believe that every intuitively constructive truth is valid in the formal semantics. Läuchli has proved completeness for a realizability semantics with formal concepts analogous to constructions. We argue in some detail that, in spite of a certain inherent inexactness of the analogy, every intuitively constructive truth is valid in Läuchli semantics, and therefore the Heyting Calculus is powerful enough to prove all constructive truths. Our argument is based on the postulate that a uniformly constructible object must be communicable in spite of imprecision in our language, and that the permutations in Läuchli's semantics represent conceivable imprecision in a language, while allowing a certain amount of freedom in choosing the particular structure of the language. We give a detailed generalization of Läuchli's proof of completeness for the propositional part of the Heyting Calculus, in order to make explicit constructive and algebraic content. In our treatment, we establish several new results about Läuchli models. We show how to extend the sconing and gluing constructions familiar from Kripke and Frame semantics and Topos theory, to Läuchli models, and use them to give an algebraic approach to countermodel construction. In particular, the Läuchli arguments are given without the restriction to the integers, Z, as a group of permutations, which makes much of the coding scheme used in Läuchli's original paper transparent. We also make use of a new propositions-as-types syntax for the Heyting calculus, with limited nondeterminism, in which validity of formulae can be decided without loop-detection.

On the Structure of Parameterized Problems in NP

Fixed-parameter intractability of optimization problems in NP is studied based on computational models with limited nondeterminism. Strong evidence is shown that many NP optimization problems are not fixed-parameter tractable and that the fixed-parameter intractability hierarchy (the W -hierarchy) does not collapse.

How hard are n 2 -hard problems?

Many of the " n 2 -hard" problems described by Gajentaan and Overmars can be solved using limited nondeterminism or other sharply-bounded quantifiers. Thus we suggest that problems are not among the hardest problems solvable in quadratic time

Nondeterminism within $P^ * $

Classes of machines using very limited amounts of nondeterminism are studied. The P=? NP question is related to questions about classes lying within P. Complete sets for these classes are given.