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## Complexidade computacional e o problema P vs NP; Computational complexity and the P vs NP problem

Fonte: Biblioteca Digital da Unicamp
Publicador: Biblioteca Digital da Unicamp

Tipo: Dissertação de Mestrado
Formato: application/pdf

Publicado em 02/08/2010
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#Complexidade computacional#Diagonalização#Algoritmos#Computational complexity#Diagonalization#Algorithms

A teoria de complexidade computacional procura estabelecer limites para a eficiência dos algoritmos, investigando a dificuldade inerente dos problemas computacionais. O problema P vs NP é uma questão central em complexidade computacional. Informalmente, ele procura determinar se, para uma classe importante de problemas computacionais, a busca exaustiva por soluções é essencialmente a melhor alternativa algorítmica possível. Esta dissertação oferece tanto uma introdução clássica ao tema, quanto uma exposição a diversos teoremas mais avançados, resultados recentes e problemas em aberto. Em particular, o método da diagonalização é discutido em profundidade. Os principais resultados obtidos por diagonalização são os teoremas de hierarquia de tempo e de espaço (Hartmanis e Stearns [54, 104]). Apresentamos uma generalização desses resultados, obtendo como corolários os teoremas clássicos provados por Hartmanis e Stearns. Essa é a primeira vez que uma prova unificada desses resultados aparece na literatura; Computational complexity theory is the field of theoretical computer science that aims to establish limits on the efficiency of algorithms. The main open question in computational complexity is the P vs NP problem. Intuitively...

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## Existential Theorems in Computational Complexity Theory: Size and Robustness

Fonte: University of Rochester. Computer Science Department.
Publicador: University of Rochester. Computer Science Department.

Tipo: Thesis; Technical Report

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#p-selective sets#pseudo-random generator#Kolmogorov complexity#AC^0#NP optimization problems#descriptive complexity#approximation algorithms#computational complexity#abstract complexity#gap theorem#polynomial-degrees

Thesis (Ph. D.)--University of Rochester. Dept. of Computer Science, 1996. Simultaneously published in the Technical Report series.; How strong are the results in computational complexity that assert, under certain hypotheses, the existence of an object? Are there many such objects, or are there few? To what extent can we relax the hypotheses and still maintain the same conclusions? These are the types of questions that are studied in this thesis. More precisely, we investigate some of the central existential results in computational complexity from the point of view of size and robustness. Below is a sample of the results in the thesis. We show that for any effective enumeration of computational devices that cover the whole set of computable functions and for any complexity measure satisfying a single axiom, neither the set of speedable functions nor the set of functions that generate complexity gaps is small from a topological point of view. We show that, with probability one on the set of oracles, there is a set in NP^A that asymptotically splits in half any infinite set in P^A. This is the strongest currently known relativized separation of NP from P. We also show that most (in the resource-bounded measure sense) sets that are computable in exponential time do not have even very weak membership-related properties that are computable in polynomial time. We prove that in almost all relativized worlds...

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## Computational Complexity Analysis of Genetic Programming - Initial Results and Future Directions

Fonte: Springer; United States
Publicador: Springer; United States

Tipo: Parte de Livro

Publicado em //2011
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The computational complexity analysis of evolutionary algorithmsworking on binary strings has significantly increased the rigorous understanding on how these types of algorithm work. Similar results on the computational complexity of genetic programming would fill an important theoretic gap. They would significantly increase the theoretical understanding on how and why genetic programming algorithms work and indicate, in a rigorous manner, how design choices of algorithm components impact its success. We summarize initial computational complexity results for simple tree-based genetic programming and point out directions for future research.; Frank Neumann, Una-May O’Reilly and Markus Wagner; Genetic and Evolutionary Computation Series

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## Computational Complexity of the Minimum Cost Homomorphism Problem on Three-Element Domains

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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In this paper we study the computational complexity of the (extended) minimum
cost homomorphism problem (Min-Cost-Hom) as a function of a constraint
language, i.e. a set of constraint relations and cost functions that are
allowed to appear in instances. A wide range of natural combinatorial
optimisation problems can be expressed as Min-Cost-Homs and a classification of
their complexity would be highly desirable, both from a direct, applied point
of view as well as from a theoretical perspective.
Min-Cost-Hom can be understood either as a flexible optimisation version of
the constraint satisfaction problem (CSP) or a restriction of the
(general-valued) valued constraint satisfaction problem (VCSP). Other
optimisation versions of CSPs such as the minimum solution problem (Min-Sol)
and the minimum ones problem (Min-Ones) are special cases of Min-Cost-Hom.
The study of VCSPs has recently seen remarkable progress. A complete
classification for the complexity of finite-valued languages on arbitrary
finite domains has been obtained Thapper and Zivny [STOC'13]. However,
understanding the complexity of languages that are not finite-valued appears to
be more difficult. Min-Cost-Hom allows us to study problematic languages of
this type without having to deal with with the full generality of the VCSP. A
recent classification for the complexity of three-element Min-Sol...

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## Is Computational Complexity a Barrier to Manipulation?

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/07/2010
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#Computer Science - Artificial Intelligence#Computer Science - Computational Complexity#Computer Science - Computer Science and Game Theory#I.2.4

When agents are acting together, they may need a simple mechanism to decide
on joint actions. One possibility is to have the agents express their
preferences in the form of a ballot and use a voting rule to decide the winning
action(s). Unfortunately, agents may try to manipulate such an election by
misreporting their preferences. Fortunately, it has been shown that it is
NP-hard to compute how to manipulate a number of different voting rules.
However, NP-hardness only bounds the worst-case complexity. Recent theoretical
results suggest that manipulation may often be easy in practice. To address
this issue, I suggest studying empirically if computational complexity is in
practice a barrier to manipulation. The basic tool used in my investigations is
the identification of computational "phase transitions". Such an approach has
been fruitful in identifying hard instances of propositional satisfiability and
other NP-hard problems. I show that phase transition behaviour gives insight
into the hardness of manipulating voting rules, increasing concern that
computational complexity is indeed any sort of barrier. Finally, I look at the
problem of computing manipulation of other, related problems like stable
marriage and tournament problems.; Comment: To appear in Proceedings of 11th International Workshop on
Computational Logic in Multi-Agent Systems (CLIMA XI 2010)

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## Computational complexity of the quantum separability problem

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Ever since entanglement was identified as a computational and cryptographic
resource, researchers have sought efficient ways to tell whether a given
density matrix represents an unentangled, or separable, state. This paper gives
the first systematic and comprehensive treatment of this (bipartite) quantum
separability problem, focusing on its deterministic (as opposed to randomized)
computational complexity. First, I review the one-sided tests for separability,
paying particular attention to the semidefinite programming methods. Then, I
discuss various ways of formulating the quantum separability problem, from
exact to approximate formulations, the latter of which are the paper's main
focus. I then give a thorough treatment of the problem's relationship with the
complexity classes NP, NP-complete, and co-NP. I also discuss extensions of
Gurvits' NP-hardness result to strong NP-hardness of certain related problems.
A major open question is whether the NP-contained formulation (QSEP) of the
quantum separability problem is Karp-NP-complete; QSEP may be the first natural
example of a problem that is Turing-NP-complete but not Karp-NP-complete.
Finally, I survey all the proposed (deterministic) algorithms for the quantum
separability problem...

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## Physical portrayal of computational complexity

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/06/2009
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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 many natural processes have been
recognized to complete in non-polynomial time (NP). The irreversible process
with three or more degrees of freedom is found intractable because, in terms of
physics, flows of energy are inseparable from their driving forces. In
computational terms, when solving problems in the class NP, decisions will
affect subsequently available sets of decisions. The state space of a
non-deterministic finite automaton is evolving due to the computation itself
hence it cannot be efficiently contracted using a deterministic finite
automaton that will arrive at a solution in super-polynomial time. The solution
of the NP problem itself is verifiable in polynomial time (P) because the
corresponding state is stationary. Likewise the class P 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 class P set of states is inherently smaller than
the set of class NP. Since the computational time to contract a given set is
proportional to dissipation...

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## Computational Complexity in Electronic Structure

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/08/2012
Português

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In quantum chemistry, the price paid by all known efficient model chemistries
is either the truncation of the Hilbert space or uncontrolled approximations.
Theoretical computer science suggests that these restrictions are not mere
shortcomings of the algorithm designers and programmers but could stem from the
inherent difficulty of simulating quantum systems. Extensions of computer
science and information processing exploiting quantum mechanics has led to new
ways of understanding the ultimate limitations of computational power.
Interestingly, this perspective helps us understand widely used model
chemistries in a new light. In this article, the fundamentals of computational
complexity will be reviewed and motivated from the vantage point of chemistry.
Then recent results from the computational complexity literature regarding
common model chemistries including Hartree-Fock and density functional theory
are discussed.; Comment: 14 pages, 2 figures, 1 table. Comments welcome

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## Mathematical foundations of modern cryptography: computational complexity perspective

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/11/2002
Português

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#Computer Science - Cryptography and Security#Computer Science - Computational Complexity#68Qxx, 11xx

Theoretical computer science has found fertile ground in many areas of
mathematics. The approach has been to consider classical problems through the
prism of computational complexity, where the number of basic computational
steps taken to solve a problem is the crucial qualitative parameter. This new
approach has led to a sequence of advances, in setting and solving new
mathematical challenges as well as in harnessing discrete mathematics to the
task of solving real-world problems.
In this talk, I will survey the development of modern cryptography -- the
mathematics behind secret communications and protocols -- in this light. I will
describe the complexity theoretic foundations underlying the cryptographic
tasks of encryption, pseudo-randomness number generators and functions, zero
knowledge interactive proofs, and multi-party secure protocols. I will attempt
to highlight the paradigms and proof techniques which unify these foundations,
and which have made their way into the mainstream of complexity theory.

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## Statistical mechanics of classical and quantum computational complexity

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/09/2010
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#Condensed Matter - Statistical Mechanics#Computer Science - Computational Complexity#Quantum Physics

The quest for quantum computers is motivated by their potential for solving
problems that defy existing, classical, computers. The theory of computational
complexity, one of the crown jewels of computer science, provides a rigorous
framework for classifying the hardness of problems according to the
computational resources, most notably time, needed to solve them. Its extension
to quantum computers allows the relative power of quantum computers to be
analyzed. This framework identifies families of problems which are likely hard
for classical computers (``NP-complete'') and those which are likely hard for
quantum computers (``QMA-complete'') by indirect methods. That is, they
identify problems of comparable worst-case difficulty without directly
determining the individual hardness of any given instance. Statistical
mechanical methods can be used to complement this classification by directly
extracting information about particular families of instances---typically those
that involve optimization---by studying random ensembles of them. These pose
unusual and interesting (quantum) statistical mechanical questions and the
results shed light on the difficulty of problems for large classes of
algorithms as well as providing a window on the contrast between typical and
worst case complexity. In these lecture notes we present an introduction to
this set of ideas with older work on classical satisfiability and recent work
on quantum satisfiability as primary examples. We also touch on the connection
of computational hardness with the physical notion of glassiness.; Comment: 25 pages...

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## On the Computational Complexity of Defining Sets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/12/2006
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Suppose we have a family ${\cal F}$ of sets. For every $S \in {\cal F}$, a
set $D \subseteq S$ is a {\sf defining set} for $({\cal F},S)$ if $S$ is the
only element of $\cal{F}$ that contains $D$ as a subset. This concept has been
studied in numerous cases, such as vertex colorings, perfect matchings,
dominating sets, block designs, geodetics, orientations, and Latin squares.
In this paper, first, we propose the concept of a defining set of a logical
formula, and we prove that the computational complexity of such a problem is
$\Sigma_2$-complete.
We also show that the computational complexity of the following problem about
the defining set of vertex colorings of graphs is $\Sigma_2$-complete:
{\sc Instance:} A graph $G$ with a vertex coloring $c$ and an integer $k$.
{\sc Question:} If ${\cal C}(G)$ be the set of all $\chi(G)$-colorings of
$G$, then does $({\cal C}(G),c)$ have a defining set of size at most $k$?
Moreover, we study the computational complexity of some other variants of
this problem.

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## Why Philosophers Should Care About Computational Complexity

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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One might think that, once we know something is computable, how efficiently
it can be computed is a practical question with little further philosophical
importance. In this essay, I offer a detailed case that one would be wrong. In
particular, I argue that computational complexity theory---the field that
studies the resources (such as time, space, and randomness) needed to solve
computational problems---leads to new perspectives on the nature of
mathematical knowledge, the strong AI debate, computationalism, the problem of
logical omniscience, Hume's problem of induction, Goodman's grue riddle, the
foundations of quantum mechanics, economic rationality, closed timelike curves,
and several other topics of philosophical interest. I end by discussing aspects
of complexity theory itself that could benefit from philosophical analysis.; Comment: 58 pages, to appear in "Computability: G\"odel, Turing, Church, and
beyond," MIT Press, 2012. Some minor clarifications and corrections; new
references added

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## Computational Complexity of Interactive Behaviors

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/09/2012
Português

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#Computer Science - Computational Complexity#Computer Science - Logic in Computer Science#F.1.1#F.1.3

The theory of computational complexity focuses on functions and, hence,
studies programs whose interactive behavior is reduced to a simple
question/answer pattern. We propose a broader theory whose ultimate goal is
expressing and analyzing the intrinsic difficulty of fully general interactive
behaviors. To this extent, we use standard tools from concurrency theory,
including labelled transition systems (formalizing behaviors) and their
asynchronous extension (providing causality information). Behaviors are
implemented by means of a multiprocessor machine executing CCS-like processes.
The resulting theory is shown to be consistent with the classical definitions:
when we restrict to functional behaviors (i.e., question/answer patterns), we
recover several standard computational complexity classes.; Comment: 17 pages

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## Proceedings International Workshop on Developments in Implicit Computational complExity

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/05/2010
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#Computer Science - Logic in Computer Science#Computer Science - Computational Complexity#Computer Science - Programming Languages#F.3.2#F.3.3#F.4.1

This volume contains the proceedings of the International Workshop on
Developments in Implicit Computational complExity (DICE 2010), which took place
on March 27-28 2010 in Paphos, Cyprus, as a satellite event of the Joint
European Conference on Theory and Practice of Software, ETAPS 2010.
Implicit Computational Complexity aims at studying computational complexity
without referring to external measuring conditions or particular machine
models, but instead by considering restrictions on programming languages or
logical principles implying complexity properties. The aim of this workshop was
to bring together researchers working on implicit computational complexity,
from its logical and semantical aspects to those related to the static analysis
of programs, so as to foster their interaction and to give newcomers an
overview of the current trends in this area.

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## Computational complexity of solving polynomial differential equations over unbounded domains

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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In this paper we investigate the computational complexity of solving ordinary
differential equations (ODEs) $y^{\prime}=p(y)$ over \emph{unbounded time
domains}, where $p$ is a vector of polynomials. Contrarily to the bounded
(compact) time case, this problem has not been well-studied, apparently due to
the "intuition" that it can always be reduced to the bounded case by using
rescaling techniques. However, as we show in this paper, rescaling techniques
do not seem to provide meaningful insights on the complexity of this problem,
since the use of such techniques introduces a dependence on parameters which
are hard to compute.
We present algorithms which numerically solve these ODEs over unbounded time
domains. These algorithms have guaranteed precision, i.e. given some
arbitrarily large time $t$ and error bound $\varepsilon$ as input, they will
output a value $\tilde{y}$ which satisfies $\|y(t)-\tilde{y}\|\leq\varepsilon$.
We analyze the complexity of these algorithms and show that they compute
$\tilde{y}$ in time polynomial in several quantities including the time $t$,
the precision of the output $\varepsilon$ and the length of the curve $y$ from
$0$ to $t$, assuming it exists until time $t$. We consider both algebraic
complexity and bit complexity. As far as we know...

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## Towards Understanding the Predictability of Stock Markets from the Perspective of Computational Complexity

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Computer Science - Computational Engineering, Finance, and Science#Computer Science - Computational Complexity#F.1.3#F.2.2#G.2.1#G.2.3#G.3#J.1

This paper initiates a study into the century-old issue of market
predictability from the perspective of computational complexity. We develop a
simple agent-based model for a stock market where the agents are traders
equipped with simple trading strategies, and their trades together determine
the stock prices. Computer simulations show that a basic case of this model is
already capable of generating price graphs which are visually similar to the
recent price movements of high tech stocks. In the general model, we prove that
if there are a large number of traders but they employ a relatively small
number of strategies, then there is a polynomial-time algorithm for predicting
future price movements with high accuracy. On the other hand, if the number of
strategies is large, market prediction becomes complete in two new
computational complexity classes CPP and BCPP, which are between P^NP[O(log n)]
and PP. These computational completeness results open up a novel possibility
that the price graph of an actual stock could be sufficiently deterministic for
various prediction goals but appear random to all polynomial-time prediction
algorithms.; Comment: The conference version will appear in SODA 2001

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## On the computational complexity of a game of cops and robbers

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We study the computational complexity of a perfect-information two-player
game proposed by Aigner and Fromme. The game takes place on an undirected graph
where n simultaneously moving cops attempt to capture a single robber, all
moving at the same speed. The players are allowed to pick their starting
positions at the first move. The question of the computational complexity of
deciding this game was raised in the '90s by Goldstein and Reingold. We prove
that the game is hard for PSPACE.; Comment: 15 pages, 2 figures. Final accepted version, to be published in
Theoretical Computer Science

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## Computational Complexity and Phase Transitions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/05/2000
Português

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#Computer Science - Computational Complexity#Computer Science - Data Structures and Algorithms#F.2.2

Phase transitions in combinatorial problems have recently been shown to be
useful in locating "hard" instances of combinatorial problems. The connection
between computational complexity and the existence of phase transitions has
been addressed in Statistical Mechanics and Artificial Intelligence, but not
studied rigorously.
We take a step in this direction by investigating the existence of sharp
thresholds for the class of generalized satisfiability problems defined by
Schaefer. In the case when all constraints are clauses we give a complete
characterization of such problems that have a sharp threshold.
While NP-completeness does not imply (even in this restricted case) the
existence of a sharp threshold, it "almost implies" this, since clausal
generalized satisfiability problems that lack a sharp threshold are either
1. polynomial time solvable, or
2. predicted, with success probability lower bounded by some positive
constant by across all the probability range, by a single, trivial procedure.; Comment: A (slightly) revised version of the paper submitted to the 15th IEEE
Conference on Computational Complexity

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## Computational Complexity Analysis of Simple Genetic Programming On Two Problems Modeling Isolated Program Semantics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Computer Science - Neural and Evolutionary Computing#Computer Science - Computational Complexity#Computer Science - Data Structures and Algorithms

Analyzing the computational complexity of evolutionary algorithms for binary
search spaces has significantly increased their theoretical understanding. With
this paper, we start the computational complexity analysis of genetic
programming. We set up several simplified genetic programming algorithms and
analyze them on two separable model problems, ORDER and MAJORITY, each of which
captures an important facet of typical genetic programming problems. Both
analyses give first rigorous insights on aspects of genetic programming design,
highlighting in particular the impact of accepting or rejecting neutral moves
and the importance of a local mutation operator.; Comment: 26 pages

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## Proceedings Second Workshop on Developments in Implicit Computational Complexity

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/01/2012
Português

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#Computer Science - Logic in Computer Science#Computer Science - Computational Complexity#Computer Science - Programming Languages#F.3.2#F.3.3#F.4.1

This volume contains the proceedings of the Second International Workshop on
Developments in Implicit Computational complExity (DICE 2011), which took place
on April 2-3 2011 in Saarbruecken, Germany, as a satellite event of the Joint
European Conference on Theory and Practice of Software, ETAPS 2011. Implicit
Computational Complexity aims at studying computational complexity without
referring to external measuring conditions or particular machine models, but
instead by considering restrictions on programming languages or logical
principles implying complexity properties. The aim of this workshop was to
bring together researchers working on implicit computational complexity, from
its logical and semantics aspects to those related to the static analysis of
programs, so as to foster their interaction and to give newcomers an overview
of the current trends in this area.
The first DICE workshop was held in 2010 at ETAPS and published in EPTCS,
volume 23 (http://eptcs.org/content.cgi?DICE2010).; Comment: EPTCS 75, 2012

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