Las Vegas versus determinism for one-way communication complexity, finite automata, and polynomial-time computations

1997 ◽  
pp. 117-128 ◽  
Author(s):  
Pavol Ďuriš ◽  
Juraj Hromkovič ◽  
José D. P. Rolim ◽  
Georg Schnitger
Keyword(s):  
Polynomial Time ◽  
Communication Complexity ◽  
Las Vegas ◽  
Finite Automata

scholarly journals A Pattern Logic for Automata with Outputs

2020 ◽  
Vol 31 (06) ◽  
pp. 711-748
Author(s):  
Emmanuel Filiot ◽  
Nicolas Mazzocchi ◽  
Jean-François Raskin
Keyword(s):  
Model Checking ◽  
Structural Properties ◽  
Polynomial Time ◽  
Finite Automata ◽  
Structural Patterns ◽  
Complexity Results

We introduce a logic to express structural properties of automata with string inputs and, possibly, outputs in some monoid. In this logic, the set of predicates talking about the output values is parametric. We then consider three particular automata models (finite automata, transducers and automata weighted by integers here called sum-automata) and instantiate the generic logic for each of them. We give tight complexity results for the three logics with respect to the model-checking problem, depending on whether the formula is fixed or not. We study the expressiveness of our logics by expressing classical structural patterns characterising for instance finite ambiguity and polynomial ambiguity in the case of finite automata, determinisability and finite-valuedness in the case of transducers and sum-automata. As a consequence of our complexity results, we directly obtain that these classical properties can be decided in polynomial time.

ORDINAL AUTOMATA AND CANTOR NORMAL FORM

2012 ◽  
Vol 23 (01) ◽  
pp. 87-98
Author(s):  
ZOLTÁN ÉSIK
Keyword(s):  
Normal Form ◽  
Polynomial Time ◽  
Regular Language ◽  
Finite Automata ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Order Type ◽  
Deterministic Finite Automaton ◽  
Lexicographic Ordering ◽  
Language L

It is known that an ordinal is the order type of the lexicographic ordering of a regular language if and only if it is less than ωω. We design a polynomial time algorithm that constructs, for each well-ordered regular language L with respect to the lexicographic ordering, given by a deterministic finite automaton, the Cantor Normal Form of its order type. It follows that there is a polynomial time algorithm to decide whether two deterministic finite automata accepting well-ordered regular languages accept isomorphic languages. We also give estimates on the state complexity of the smallest "ordinal automaton" representing an ordinal less than ωω, together with an algorithm that translates each such ordinal to an automaton.

Weak cardinality theorems

2005 ◽  
Vol 70 (3) ◽  
pp. 861-878
Author(s):  
Till Tantau
Keyword(s):  
Turing Machine ◽  
Polynomial Time ◽  
Computational Models ◽  
Finite Automata ◽  
Good Reason ◽  
Recursion Theory ◽  
Middle Ground ◽  
Logical Structures

AbstractKummer's Cardinality Theorem states that a language A must be recursive if a Turing machine can exclude for any n words , …, one of the n + 1 possibilities for the cardinality of {, …, }⋂ A. There was good reason to believe that this theorem is a peculiarity of recursion theory: neither the Cardinality Theorem nor weak forms of it hold for resource-bounded computational models like polynomial time. This belief may be flawed. In this paper it is shown that weak cardinality theorems hold for finite automata and also for other models. An explanation is proposed as to why recursion-theoretic and automata-theoretic weak cardinality theorems hold, but not corresponding 'middle-ground theorems': The recursion- and automata-theoretic weak cardinality theorems are instantiations of purely logical weak cardinality theorems. The logical theorems can be instantiated for logical structures characterizing recursive computations and finite automata computations. A corresponding structure characterizing polynomial time computations does not exist.

scholarly journals The Bit Probe Complexity Measure Revisited

1992 ◽  
Vol 21 (396) ◽  
Author(s):  
Peter Bro Miltersen
Keyword(s):  
Data Structure ◽  
Polynomial Time ◽  
Communication Complexity ◽  
Complexity Measure ◽  
Additive Constant ◽  
Static Structure ◽  
Worst Case ◽  
Static Data ◽  
On Line ◽  
Probe Complexity

The bit probe complexity of a static data structure problem within a given size bound was defined by Elias and Flower. It is the number of bits one needs to probe in the data structure for worst case data and query with an optimal encoding of the data within the space bound. We make some furtber investigations into the properties of the bit probe complexity measure. We determine the complexity of the full problem, which is the problem where every possible query is allowed, within an additive constant. We show a trade off-between structure size and the number of bit probes for all problems. We show that the complexity of almost every problem, even with small query sets, equals that of the full problem. We show how communication complexity can be used to give small, but occasionally tight lower bounds for natural functions. We define the class of access feasible static structure problems and conjecture that not every polynomial time computable problem is access feasible. We show a link to dynamic problems by showing that if polynomial time computable functions without feasible static structures exist, then there are problems in P which can not be reevaluated efficiently on-line.

scholarly journals Communication Complexity Method for Measuring Nondeterminism in Finite Automata

2002 ◽  
Vol 172 (2) ◽  
pp. 202-217 ◽  
Author(s):  
Juraj Hromkovič ◽  
Sebastian Seibert ◽  
Juhani Karhumäki ◽  
Hartmut Klauck ◽  
Georg Schnitger
Keyword(s):  
Communication Complexity ◽  
Finite Automata

scholarly journals Unary automatic graphs: an algorithmic perspective

2009 ◽  
Vol 19 (1) ◽  
pp. 133-152 ◽  
Author(s):  
BAKHADYR KHOUSSAINOV ◽  
JIAMOU LIU ◽  
MIA MINNES
Keyword(s):  
Polynomial Time ◽  
Finite Automata ◽  
Infinite Graphs ◽  
Input Graph ◽  
Finite Degree ◽  
Finite Graphs ◽  
Polynomial Time Algorithms ◽  
Infinite Component ◽  
Fixed Input ◽  
Finite Presentations

This paper studies infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced using such operations are of finite degree and automatic over the unary alphabet (that is, they can be described by finite automata over the unary alphabet). We investigate algorithmic properties of such unfolded graphs given their finite presentations. In particular, we ask whether a given node belongs to an infinite component, whether two given nodes in the graph are reachable from one another and whether the graph is connected. We give polynomial-time algorithms for each of these questions. For a fixed input graph, the algorithm for the first question is in constant time and the second question is decided using an automaton that recognises the reachability relation in a uniform way. Hence, we improve on previous work, in which non-elementary or non-uniform algorithms were found.

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