286 research outputs found
Which Regular Languages can be Efficiently Indexed?
In the present work, we tackle the regular language indexing problem by first
studying the hierarchy of -sortable languages: regular languages accepted by
automata of width . We show that the hierarchy is strict and does not
collapse, and provide (exponential in ) upper and lower bounds relating the
minimum widths of equivalent NFAs and DFAs. Our bounds indicate the importance
of being able to index NFAs, as they enable indexing regular languages with
much faster and smaller indexes. Our second contribution solves precisely this
problem, optimally: we devise a polynomial-time algorithm that indexes any NFA
with the optimal value for its width, without explicitly computing
(NP-hard to find). In particular, this implies that we can index in polynomial
time the well-studied case (Wheeler NFAs). More in general, in polynomial
time we can build an index breaking the worst-case conditional lower bound of
, whenever the input NFA's width is .Comment: Extended versio
Complexity Theory, Game Theory, and Economics: The Barbados Lectures
This document collects the lecture notes from my mini-course "Complexity
Theory, Game Theory, and Economics," taught at the Bellairs Research Institute
of McGill University, Holetown, Barbados, February 19--23, 2017, as the 29th
McGill Invitational Workshop on Computational Complexity.
The goal of this mini-course is twofold: (i) to explain how complexity theory
has helped illuminate several barriers in economics and game theory; and (ii)
to illustrate how game-theoretic questions have led to new and interesting
complexity theory, including recent several breakthroughs. It consists of two
five-lecture sequences: the Solar Lectures, focusing on the communication and
computational complexity of computing equilibria; and the Lunar Lectures,
focusing on applications of complexity theory in game theory and economics. No
background in game theory is assumed.Comment: Revised v2 from December 2019 corrects some errors in and adds some
recent citations to v1 Revised v3 corrects a few typos in v
Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages
A language over an alphabet is suffix-convex if, for any words
, whenever and are in , then so is .
Suffix-convex languages include three special cases: left-ideal, suffix-closed,
and suffix-free languages. We examine complexity properties of these three
special classes of suffix-convex regular languages. In particular, we study the
quotient/state complexity of boolean operations, product (concatenation), star,
and reversal on these languages, as well as the size of their syntactic
semigroups, and the quotient complexity of their atoms.Comment: 20 pages, 11 figures, 1 table. arXiv admin note: text overlap with
arXiv:1605.0669
Languages convex with respect to binary relations, and their closure properties
A language is prefix-convex if it satisfies the condition that, if a word w and its prefix u are in the language, then so is every prefix of w that has u as a prefix. Prefix-convex languages include prefix-closed languages at one end of the spectrum, and prefix-free languages, which include prefix codes, at the other. In a similar way, we define suffix-, bifix-, factor-, and subword-convex languages and their closed and free counterparts. This provides a common framework for diverse languages such as codes, factorial languages and ideals. We examine the relationships among these languages. We generalize these notions to arbitrary binary relations on the set of all words over a given alphabet, and study the closure properties of such languages
Lukasiewicz mu-Calculus
We consider state-based systems modelled as coalgebras whose type incorporates branching, and show that by suitably adapting the definition of coalgebraic bisimulation, one obtains a general and uniform account of the linear-time behaviour of a state in such a coalgebra. By moving away from a boolean universe of truth values, our approach can measure the extent to which a state in a system with branching is able to exhibit a particular linear-time behaviour. This instantiates to measuring the probability of a specific behaviour occurring in a probabilistic system, or measuring the minimal cost of exhibiting a specific behaviour in the case of weighted computations
Most Complex Regular Right-Ideal Languages
A right ideal is a language L over an alphabet A that satisfies L = LA*. We
show that there exists a stream (sequence) (R_n : n \ge 3) of regular right
ideal languages, where R_n has n left quotients and is most complex under the
following measures of complexity: the state complexities of the left quotients,
the number of atoms (intersections of complemented and uncomplemented left
quotients), the state complexities of the atoms, the size of the syntactic
semigroup, the state complexities of the operations of reversal, star, and
product, and the state complexities of all binary boolean operations. In that
sense, this stream of right ideals is a universal witness.Comment: 19 pages, 4 figures, 1 tabl
Quotient Complexity of Ideal Languages
The final publication is available at Elsevier via http://dx.doi.org/10.1016/j.tcs.2012.10.055 © 2013. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/A language L over an alphabet ÎŁ is a right (left) ideal if it satisfies L=LÎŁâ (L=ÎŁâL). It is a two-sided ideal if L=ÎŁâLÎŁâ, and an all-sided ideal if L=ÎŁâL, the shuffle of ÎŁâ with L. Ideal languages are not only of interest from the theoretical point of view, but also have applications to pattern matching. We study the state complexity of common operations in the class of regular ideal languages, but prefer to use the equivalent term âquotient complexityâ, which is the number of distinct left quotients of a language. We find tight upper bounds on the complexity of each type of ideal language in terms of the complexity of an arbitrary generator and of the minimal generator, and also on the complexity of the minimal generator in terms of the complexity of the language. Moreover, tight upper bounds on the complexity of union, intersection, set difference, symmetric difference, concatenation, star, and reversal of ideal languages are derived.Natural Sciences and Engineering Research Council of Canada grant [OGP0000871]VEGA grant 2/0111/0
Residual Nominal Automata
Nominal automata are models for accepting languages over infinite alphabets.
In this paper we refine the hierarchy of nondeterministic nominal automata, by
developing the theory of residual nominal automata. In particular, we show that
they admit canonical minimal representatives, and that the universality problem
becomes decidable. We also study exact learning of these automata, and settle
questions that were left open about their learnability via observations
Towards a Uniform Theory of Effectful State Machines
Using recent developments in coalgebraic and monad-based semantics, we
present a uniform study of various notions of machines, e.g. finite state
machines, multi-stack machines, Turing machines, valence automata, and weighted
automata. They are instances of Jacobs' notion of a T-automaton, where T is a
monad. We show that the generic language semantics for T-automata correctly
instantiates the usual language semantics for a number of known classes of
machines/languages, including regular, context-free, recursively-enumerable and
various subclasses of context free languages (e.g. deterministic and real-time
ones). Moreover, our approach provides new generic techniques for studying the
expressivity power of various machine-based models.Comment: final version accepted by TOC
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