scholarly journals State Complexity of Neighbourhoods and Approximate Pattern Matching

2018 ◽  
Vol 29 (02) ◽  
pp. 315-329 ◽  
Author(s):  
Timothy Ng ◽  
David Rappaport ◽  
Kai Salomaa
Keyword(s):  
Lower Bound ◽  
Pattern Matching ◽  
Finite Automaton ◽  
The State ◽  
Worst Case ◽  
State Complexity ◽  
Approximate Pattern Matching ◽  
The Given ◽  
Nondeterministic Finite Automaton ◽  
Distance Formula

The neighbourhood of a language [Formula: see text] with respect to an additive distance consists of all strings that have distance at most the given radius from some string of [Formula: see text]. We show that the worst case deterministic state complexity of a radius [Formula: see text] neighbourhood of a language recognized by an [Formula: see text] state nondeterministic finite automaton [Formula: see text] is [Formula: see text]. In the case where [Formula: see text] is deterministic we get the same lower bound for the state complexity of the neighbourhood if we use an additive quasi-distance. The lower bound constructions use an alphabet of size linear in [Formula: see text]. We show that the worst case state complexity of the set of strings that contain a substring within distance [Formula: see text] from a string recognized by [Formula: see text] is [Formula: see text].

scholarly journals State Complexity of Suffix Distance

2019 ◽  
Vol 30 (06n07) ◽  
pp. 1197-1216
Author(s):  
Timothy Ng ◽  
David Rappaport ◽  
Kai Salomaa
Keyword(s):  
Lower Bound ◽  
Finite Automaton ◽  
Regular Language ◽  
Upper Bounds ◽  
The State ◽  
Deterministic Finite Automaton ◽  
Tight Bound ◽  
State Complexity

The neighbourhood of a regular language with respect to the prefix, suffix and subword distance is always regular and a tight bound for the state complexity of prefix distance neighbourhoods is known. We give upper bounds for the state complexity of the neighbourhood of radius [Formula: see text] of an [Formula: see text]-state deterministic finite automaton language with respect to the suffix distance and the subword distance, respectively. For restricted values of [Formula: see text] and [Formula: see text] we give a matching lower bound for the state complexity of suffix distance neighbourhoods.

A Model of Secure Functioning of Computer Systems

2021 ◽  
Vol 12 (3) ◽  
pp. 150-156
Author(s):  
A. V. Galatenko ◽  
◽  
V. A. Kuzovikhina ◽  
Keyword(s):  
Lower Bound ◽  
Computer System ◽  
Finite Automaton ◽  
Nonnegative Integer ◽  
The Other ◽  
Shannon Function ◽  
Security Breaches ◽  
System Functioning ◽  
The Cost ◽  
The Given

We propose an automata model of computer system security. A system is represented by a finite automaton with states partitioned into two subsets: "secure" and "insecure". System functioning is secure if the number of consecutive insecure states is not greater than some nonnegative integer k. This definition allows one to formally reflect responsiveness to security breaches. The number of all input sequences that preserve security for the given value of k is referred to as a k-secure language. We prove that if a language is k-secure for some natural and automaton V, then it is also k-secure for any 0 < k < k and some automaton V = V (k). Reduction of the value of k is performed at the cost of amplification of the number of states. On the other hand, for any non-negative integer k there exists a k-secure language that is not k"-secure for any natural k" > k. The problem of reconstruction of a k-secure language using a conditional experiment is split into two subcases. If the cardinality of an input alphabet is bound by some constant, then the order of Shannon function of experiment complexity is the same for al k; otherwise there emerges a lower bound of the order nk.

STATE COMPLEXITY OF ADDITIVE WEIGHTED FINITE AUTOMATA

2007 ◽  
Vol 18 (06) ◽  
pp. 1407-1416 ◽  
Author(s):  
KAI SALOMAA ◽  
PAUL SCHOFIELD
Keyword(s):  
Upper Bound ◽  
Finite Automaton ◽  
Regular Language ◽  
Finite Automata ◽  
The State ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Weighted Finite Automata

It is known that the neighborhood of a regular language with respect to an additive distance is regular. We introduce an additive weighted finite automaton model that provides a conceptually simple way to reprove this result. We consider the state complexity of converting additive weighted finite automata to deterministic finite automata. As our main result we establish a tight upper bound for the state complexity of the conversion.

SIMULATIONS OF UNARY ONE-WAY MULTI-HEAD FINITE AUTOMATA

2014 ◽  
Vol 25 (07) ◽  
pp. 877-896 ◽  
Author(s):  
MARTIN KUTRIB ◽  
ANDREAS MALCHER ◽  
MATTHIAS WENDLANDT
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Finite Automaton ◽  
Finite Automata ◽  
Upper And Lower Bounds ◽  
Unary Languages ◽  
Order Of Magnitude ◽  
Simulation Results ◽  
Special Case ◽  
Nondeterministic Finite Automaton

We investigate the descriptional complexity of deterministic one-way multi-head finite automata accepting unary languages. It is known that in this case the languages accepted are regular. Thus, we study the increase of the number of states when an n-state k-head finite automaton is simulated by a classical (one-head) deterministic or nondeterministic finite automaton. In the former case upper and lower bounds that are tight in the order of magnitude are shown. For the latter case we obtain an upper bound of O(n2k) and a lower bound of Ω(nk) states. We investigate also the costs for the conversion of one-head nondeterministic finite automata to deterministic k-head finite automata, that is, we trade nondeterminism for heads. In addition, we study how the conversion costs vary in the special case of finite and, in particular, of singleton unary lanuages. Finally, as an application of the simulation results, we show that decidability problems for unary deterministic k-head finite automata such as emptiness or equivalence are LOGSPACE-complete.

State Complexity of Insertion

2016 ◽  
Vol 27 (07) ◽  
pp. 863-878 ◽  
Author(s):  
Yo-Sub Han ◽  
Sang-Ki Ko ◽  
Timothy Ng ◽  
Kai Salomaa
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Regular Language ◽  
Finite Automata ◽  
The State ◽  
Tight Bound ◽  
State Complexity ◽  
Nondeterministic State

It is well known that the resulting language obtained by inserting a regular language to a regular language is regular. We study the nondeterministic and deterministic state complexity of the insertion operation. Given two incomplete DFAs of sizes m and n, we give an upper bound (m+2)·2mn−m−1·3m and find a lower bound for an asymp-totically tight bound. We also present the tight nondeterministic state complexity by a fooling set technique. The deterministic state complexity of insertion is 2Θ(mn) and the nondeterministic state complexity of insertion is precisely mn+2m, where m and n are the size of input finite automata. We also consider the state complexity of insertion in the case where the inserted language is bifix-free or non-returning.

Square on Deterministic, Alternating, and Boolean Finite Automata

2019 ◽  
Vol 30 (06n07) ◽  
pp. 1117-1134
Author(s):  
Galina Jirásková ◽  
Ivana Krajňáková
Keyword(s):  
Upper Bound ◽  
Finite Automaton ◽  
Finite Automata ◽  
Upper Bounds ◽  
The State ◽  
Final States ◽  
Deterministic Finite Automaton ◽  
State Complexity

We investigate the state complexity of the square operation on languages represented by deterministic, alternating, and Boolean automata. For each [Formula: see text] such that [Formula: see text], we describe a binary language accepted by an [Formula: see text]-state deterministic finite automaton with [Formula: see text] final states meeting the upper bound [Formula: see text] on the state complexity of its square. We show that in the case of [Formula: see text], the corresponding upper bound cannot be met. Using the binary deterministic witness for square with [Formula: see text] states where half of them are final, we get the tight upper bounds [Formula: see text] and [Formula: see text] on the complexity of the square operation on alternating and Boolean automata, respectively.

Operational State Complexity of Subtree-Free Regular Tree Languages

2016 ◽  
Vol 27 (06) ◽  
pp. 705-724
Author(s):  
Sang-Ki Ko ◽  
Hae-Sung Eom ◽  
Yo-Sub Han
Keyword(s):  
The State ◽  
Tree Automaton ◽  
Top Down ◽  
Tree Language ◽  
Worst Case ◽  
Bottom Up ◽  
State Complexity ◽  
Regular Tree ◽  
Tree Languages

We introduce subtree-free regular tree languages that are closely related to XML schemas and investigate the state complexity of basic operations on subtree-free regular tree languages. The state complexity of an operation for regular tree languages is the number of states that are sufficient and necessary in the worst-case for the minimal deterministic ranked tree automaton that accepts the tree language obtained from the operation. We establish the precise state complexity of (sequential, parallel) concatenation, (bottom-up, top-down) star, intersection and union for subtree-free regular tree languages.

scholarly journals Physical Portrayal of Computational Complexity

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Arto Annila
Keyword(s):  
Computational Complexity ◽  
Finite Automaton ◽  
Degrees Of Freedom ◽  
Irreversible Process ◽  
Driving Forces ◽  
The State ◽  
Computational Time ◽  
Natural Processes ◽  
Final Acceptance ◽  
Nondeterministic Finite Automaton

Computational complexity is examined using the principle of increasing entropy. To consider computation as a physical process from an initial instance to the final acceptance is motivated because information requires physical representations and because many natural processes complete in nondeterministic polynomial time (NP). The irreversible process with three or more degrees of freedom is found intractable when, in terms of physics, flows of energy are inseparable from their driving forces. In computational terms, when solving a problem in the class NP, decisions among alternatives will affect subsequently available sets of decisions. Thus the state space of a nondeterministic finite automaton is evolving due to the computation itself, hence it cannot be efficiently contracted using a deterministic finite automaton. Conversely when solving problems in the class P, the set of states does not depend on computational history, hence it can be efficiently contracted to the accepting state by a deterministic sequence of dissipative transformations. Thus it is concluded that the state set of class P is inherently smaller than the state set of class NP. Since the computational time needed to contract a given set is proportional to dissipation, the computational complexity class P is a proper (strict) subset of NP.

PATTERN MATCHING WITH SWAPS IN PRACTICE

2012 ◽  
Vol 23 (02) ◽  
pp. 323-342 ◽  
Author(s):  
MATTEO CAMPANELLI ◽  
DOMENICO CANTONE ◽  
SIMONE FARO ◽  
EMANUELE GIAQUINTA
Keyword(s):  
Pattern Matching ◽  
Time Complexity ◽  
Matching Problem ◽  
Worst Case ◽  
Word Size ◽  
Approximate Pattern Matching ◽  
Text Location

The Pattern Matching problem with Swaps consists in finding all occurrences of a pattern P in a text T, when disjoint local swaps in the pattern are allowed. In the Approximate Pattern Matching problem with Swaps one seeks, for every text location with a swapped match of P, the number of swaps necessary to obtain a match at the location. In this paper we devise two general algorithms for both Standard and Approximate Pattern Matching with Swaps, named CROSS-SAMPLING and BACKWARD-CROSS-SAMPLING, with a [Formula: see text] and [Formula: see text] worst-case time complexity, respectively. Then we provide efficient implementations of them, based on bit-parallelism, which achieve [Formula: see text] and [Formula: see text] worst-case time complexity, with patterns whose length is comparable to the word-size of the target machine. From an extensive comparison with some of the most effective algorithms for the swap matching problem, it turns out that our algorithms are very flexible and achieve very good results inpractice.

State Complexity of k-Union and k-Intersection for Prefix-Free Regular Languages

2015 ◽  
Vol 26 (02) ◽  
pp. 211-227 ◽  
Author(s):  
Hae-Sung Eom ◽  
Yo-Sub Han ◽  
Kai Salomaa
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Finite Automata ◽  
Constant Factor ◽  
The State ◽  
Regular Languages ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Binary Alphabet ◽  
Multiple Intersections

We investigate the state complexity of multiple unions and of multiple intersections for prefix-free regular languages. Prefix-free deterministic finite automata have their own unique structural properties that are crucial for obtaining state complexity upper bounds that are improved from those for general regular languages. We present a tight lower bound construction for k-union using an alphabet of size k + 1 and for k-intersection using a binary alphabet. We prove that the state complexity upper bound for k-union cannot be reached by languages over an alphabet with less than k symbols. We also give a lower bound construction for k-union using a binary alphabet that is within a constant factor of the upper bound.

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