302,561 research outputs found
Complexity of Two-Dimensional Patterns
In dynamical systems such as cellular automata and iterated maps, it is often
useful to look at a language or set of symbol sequences produced by the system.
There are well-established classification schemes, such as the Chomsky
hierarchy, with which we can measure the complexity of these sets of sequences,
and thus the complexity of the systems which produce them.
In this paper, we look at the first few levels of a hierarchy of complexity
for two-or-more-dimensional patterns. We show that several definitions of
``regular language'' or ``local rule'' that are equivalent in d=1 lead to
distinct classes in d >= 2. We explore the closure properties and computational
complexity of these classes, including undecidability and L-, NL- and
NP-completeness results.
We apply these classes to cellular automata, in particular to their sets of
fixed and periodic points, finite-time images, and limit sets. We show that it
is undecidable whether a CA in d >= 2 has a periodic point of a given period,
and that certain ``local lattice languages'' are not finite-time images or
limit sets of any CA. We also show that the entropy of a d-dimensional CA's
finite-time image cannot decrease faster than t^{-d} unless it maps every
initial condition to a single homogeneous state.Comment: To appear in J. Stat. Phy
Effects of semantic and syntactic complexities and aspectual class on past tense production
This paper reports results from a series of experiments that investigated whether semantic and/or syntactic complexity influences young Dutch children’s production of past tense forms. The constructions used in the three experiments were (i) simple sentences (the Simple Sentence Experiment), (ii) complex sentences with CP complements (the Complement Clause Experiment) and (iii) complex sentences with relative clauses (the Relative Clause Experiment). The stimuli involved both atelic and telic predicates. The goal of this paper is to address the following questions.
Q1. Does semantic complexity regarding temporal anchoring influence the types of errors that children make in the experiments? For example, do children make certain types of errors when a past tense has to be anchored to the Utterance Time (UT), as compared to when it has to be anchored to the matrix topic time (TT)?
Q2. Do different syntactic positions influence children’s performance on past-tense production? Do children perform better in the Simple Sentence Experiment compared to complex sentences involving two finite clauses (the Complement Clause Experiment and the Relative Clause Experiment)? In complex sentence trials, do children perform differently when the CPs are complements vs. when the CPs are adjunct clauses? (Lebeaux 1990, 2000)
Q3. Do Dutch children make more errors with certain types of predicate (such as atelic predicates)? Alternatively, do children produce a certain type of error with a certain type of predicates (such as producing a perfect aspect with punctual predicates)? Bronckart and Sinclair (1973), for example, found that until the age of 6, French children showed a tendency to use passé composé with perfective events and simple present with imperfective events; we will investigate whether or not the equivalent of this is observed in Dutch
Graph classes and forbidden patterns on three vertices
This paper deals with graph classes characterization and recognition. A
popular way to characterize a graph class is to list a minimal set of forbidden
induced subgraphs. Unfortunately this strategy usually does not lead to an
efficient recognition algorithm. On the other hand, many graph classes can be
efficiently recognized by techniques based on some interesting orderings of the
nodes, such as the ones given by traversals.
We study specifically graph classes that have an ordering avoiding some
ordered structures. More precisely, we consider what we call patterns on three
nodes, and the recognition complexity of the associated classes. In this
domain, there are two key previous works. Damashke started the study of the
classes defined by forbidden patterns, a set that contains interval, chordal
and bipartite graphs among others. On the algorithmic side, Hell, Mohar and
Rafiey proved that any class defined by a set of forbidden patterns can be
recognized in polynomial time. We improve on these two works, by characterizing
systematically all the classes defined sets of forbidden patterns (on three
nodes), and proving that among the 23 different classes (up to complementation)
that we find, 21 can actually be recognized in linear time.
Beyond this result, we consider that this type of characterization is very
useful, leads to a rich structure of classes, and generates a lot of open
questions worth investigating.Comment: Third version version. 38 page
The Complexity of Infinite Computations In Models of Set Theory
We prove the following surprising result: there exist a 1-counter B\"uchi
automaton and a 2-tape B\"uchi automaton such that the \omega-language of the
first and the infinitary rational relation of the second in one model of ZFC
are \pi_2^0-sets, while in a different model of ZFC both are analytic but non
Borel sets.
This shows that the topological complexity of an \omega-language accepted by
a 1-counter B\"uchi automaton or of an infinitary rational relation accepted by
a 2-tape B\"uchi automaton is not determined by the axiomatic system ZFC.
We show that a similar result holds for the class of languages of infinite
pictures which are recognized by B\"uchi tiling systems.
We infer from the proof of the above results an improvement of the lower
bound of some decision problems recently studied by the author
Highly Undecidable Problems For Infinite Computations
We show that many classical decision problems about 1-counter
omega-languages, context free omega-languages, or infinitary rational
relations, are -complete, hence located at the second level of the
analytical hierarchy, and "highly undecidable". In particular, the universality
problem, the inclusion problem, the equivalence problem, the determinizability
problem, the complementability problem, and the unambiguity problem are all
-complete for context-free omega-languages or for infinitary rational
relations. Topological and arithmetical properties of 1-counter
omega-languages, context free omega-languages, or infinitary rational
relations, are also highly undecidable. These very surprising results provide
the first examples of highly undecidable problems about the behaviour of very
simple finite machines like 1-counter automata or 2-tape automata.Comment: to appear in RAIRO-Theoretical Informatics and Application
Hybrid tractability of soft constraint problems
The constraint satisfaction problem (CSP) is a central generic problem in
computer science and artificial intelligence: it provides a common framework
for many theoretical problems as well as for many real-life applications. Soft
constraint problems are a generalisation of the CSP which allow the user to
model optimisation problems. Considerable effort has been made in identifying
properties which ensure tractability in such problems. In this work, we
initiate the study of hybrid tractability of soft constraint problems; that is,
properties which guarantee tractability of the given soft constraint problem,
but which do not depend only on the underlying structure of the instance (such
as being tree-structured) or only on the types of soft constraints in the
instance (such as submodularity). We present several novel hybrid classes of
soft constraint problems, which include a machine scheduling problem,
constraint problems of arbitrary arities with no overlapping nogoods, and the
SoftAllDiff constraint with arbitrary unary soft constraints. An important tool
in our investigation will be the notion of forbidden substructures.Comment: A full version of a CP'10 paper, 26 page
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Unconditional Relationships within Zero Knowledge
Zero-knowledge protocols enable one party, called a prover, to "convince" another party, called a verifier, the validity of a mathematical statement such that the verifier "learns nothing" other than the fact that the proven statement is true. The different ways of formulating the terms "convince" and "learns nothing" gives rise to four classes of languages having zero-knowledge protocols, which are: statistical zero-knowledge proof systems, computational zero-knowledge proof systems, statistical zero-knowledge argument systems, and computational zero-knowledge argument systems.
We establish complexity-theoretic characterization of the classes of languages in NP having zero-knowledge argument systems. Using these characterizations, we show that for languages in NP:
-- Instance-dependent commitment schemes are necessary and sufficient for zero-knowledge protocols. Instance-dependent commitment schemes for a given language are commitment schemes that can depend on the instance of the language, and where the hiding and binding properties are required to hold only on the YES and NO instances of the language, respectively.
-- Computational zero knowledge and computational soundness (a property held by argument systems) are symmetric properties. Namely, we show that the class of languages in NP intersect co-NP having zero-knowledge arguments is closed under complement, and that a language in NP has a statistical zero-knowledge **argument** system if and only if its complement has a **computational** zero-knowledge proof system.
-- A method of transforming any zero-knowledge protocol that is secure only against an honest verifier that follows the prescribed protocol into one that is secure against malicious verifiers. In addition, our transformation gives us protocols with desirable properties like having public coins, being black-box simulatable, and having an efficient prover.
The novelty of our results above is that they are **unconditional**, meaning that they do not rely on any unproven complexity assumptions such as the existence of one-way functions. Moreover, in establishing our complexity-theoretic characterizations, we give the first construction of statistical zero-knowledge argument systems for NP based on any one-way function
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