87 research outputs found
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Register complexity and determinisation of max-plus automata
We survey some results about the sequentiality problem for max-plus automata and its generalisation, the register complexity problem for cost register automata. We compare classes of functions computed by maxplus automata and by cost register automata with respect to the notion of ambiguity. The two models are introduced gently, so the novice reader is welcome
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 variables and clauses can be
solved simultaneously polynomial time and space for an absolute constant . This fact inspires us to
propose a new, practical working hypothesis, called the linear space hypothesis
(LSH), which states that 2SAT---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., for a certain constant
) space on all instances . An immediate consequence of
this working hypothesis is . 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, is
complete for a syntactically restricted version of NL, called Syntactic
NL, 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
Projection Theorems Using Effective Dimension
In this paper we use the theory of computing to study fractal dimensions of projections in Euclidean spaces. A fundamental result in fractal geometry is Marstrand\u27s projection theorem, which shows that for every analytic set E, for almost every line L, the Hausdorff dimension of the orthogonal projection of E onto L is maximal.
We use Kolmogorov complexity to give two new results on the Hausdorff and packing dimensions of orthogonal projections onto lines. The first shows that the conclusion of Marstrand\u27s theorem holds whenever the Hausdorff and packing dimensions agree on the set E, even if E is not analytic. Our second result gives a lower bound on the packing dimension of projections of arbitrary sets. Finally, we give a new proof of Marstrand\u27s theorem using the theory of computing
Results on the Dimension Spectra of Planar Lines
In this paper we investigate the (effective) dimension spectra of lines in the Euclidean plane. The dimension spectrum of a line L_{a,b}, sp(L), with slope a and intercept b is the set of all effective dimensions of the points (x, ax + b) on L. It has been recently shown that, for every a and b with effective dimension less than 1, the dimension spectrum of L_{a,b} contains an interval. Our first main theorem shows that this holds for every line. Moreover, when the effective dimension of a and b is at least 1, sp(L) contains a unit interval.
Our second main theorem gives lower bounds on the dimension spectra of lines. In particular, we show that for every alpha in [0,1], with the exception of a set of Hausdorff dimension at most alpha, the effective dimension of (x, ax + b) is at least alpha + dim(a,b)/2. As a consequence of this theorem, using a recent characterization of Hausdorff dimension using effective dimension, we give a new proof of a result by Molter and Rela on the Hausdorff dimension of Furstenberg sets
A Robust Class of Linear Recurrence Sequences
We introduce a subclass of linear recurrence sequences which we call poly-rational sequences because they are denoted by rational expressions closed under sum and product. We show that this class is robust by giving several characterisations: polynomially ambiguous weighted automata, copyless cost-register automata, rational formal series, and linear recurrence sequences whose eigenvalues are roots of rational numbers
Algebraic Model Checking for Discrete Linear Dynamical Systems
Model checking infinite-state systems is one of the central challenges in automated verification. In this survey we focus on an important and fundamental subclass of infinite-state systems, namely discrete linear dynamical systems. While such systems are ubiquitous in mathematics, physics, engineering, etc., in the present context our motivation stems from their relevance to the formal analysis and verification of program loops, weighted automata, hybrid systems, and control systems, amongst many others. Our main object of study is the problem of model checking temporal properties on the infinite orbit of a linear dynamical system, and our principal contribution is to show that for a rich class of properties this problem can be reduced to certain classical decision problems on linear recurrence sequences, notably the Skolem Problem. This leads us to discuss recent advances on the latter and to highlight the prospects for further progress on charting the algorithmic landscape of linear recurrence sequences and linear dynamical systems
Algebraic model checking for discrete linear dynamical systems
Model checking infinite-state systems is one of the central challenges in automated verification. In this survey we focus on an important and fundamental subclass of infinite-state systems, namely discrete linear dynamical systems. While such systems are ubiquitous in mathematics, physics, engineering, etc., in the present context our motivation stems from their relevance to the formal analysis and verification of program loops, weighted automata, hybrid systems, and control systems, amongst many others. Our main object of study is the problem of model checking temporal properties on the infinite orbit of a linear dynamical system, and our principal contribution is to show that for a rich class of properties this problem can be reduced to certain classical decision problems on linear recurrence sequences, notably the Skolem Problem. This leads us to discuss recent advances on the latter and to highlight the prospects for further progress on charting the algorithmic landscape of linear recurrence sequences and linear dynamical systems
Existential Definability over the Subword Ordering
We study first-order logic (FO) over the structure consisting of finite words over some alphabet A, together with the (non-contiguous) subword ordering. In terms of decidability of quantifier alternation fragments, this logic is well-understood: If every word is available as a constant, then even the ?? (i.e., existential) fragment is undecidable, already for binary alphabets A.
However, up to now, little is known about the expressiveness of the quantifier alternation fragments: For example, the undecidability proof for the existential fragment relies on Diophantine equations and only shows that recursively enumerable languages over a singleton alphabet (and some auxiliary predicates) are definable.
We show that if |A| ? 3, then a relation is definable in the existential fragment over A with constants if and only if it is recursively enumerable. This implies characterizations for all fragments ?_i: If |A| ? 3, then a relation is definable in ?_i if and only if it belongs to the i-th level of the arithmetical hierarchy. In addition, our result yields an analogous complete description of the ?_i-fragments for i ? 2 of the pure logic, where the words of A^* are not available as constants
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