195 research outputs found
Why Philosophers Should Care About Computational Complexity
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 adde
On the existence of strong proof complexity generators
Cook and Reckhow 1979 pointed out that NP is not closed under complementation
iff there is no propositional proof system that admits polynomial size proofs
of all tautologies. Theory of proof complexity generators aims at constructing
sets of tautologies hard for strong and possibly for all proof systems. We
focus at a conjecture from K.2004 in foundations of the theory that there is a
proof complexity generator hard for all proof systems. This can be equivalently
formulated (for p-time generators) without a reference to proof complexity
notions as follows:
* There exist a p-time function stretching each input by one bit such
that its range intersects all infinite NP sets.
We consider several facets of this conjecture, including its links to bounded
arithmetic (witnessing and independence results), to time-bounded Kolmogorov
complexity, to feasible disjunction property of propositional proof systems and
to complexity of proof search. We argue that a specific gadget generator from
K.2009 is a good candidate for . We define a new hardness property of
generators, the -hardness, and shows that one specific gadget
generator is the -hardest (w.r.t. any sufficiently strong proof
system). We define the class of feasibly infinite NP sets and show, assuming a
hypothesis from circuit complexity, that the conjecture holds for all feasibly
infinite NP sets.Comment: preliminary version August 2022, revised July 202
Algebraic and Combinatorial Methods in Computational Complexity
At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The PCP characterization of NP and the Agrawal-Kayal-Saxena polynomial-time primality test are two prominent examples. Recently, there have been some works going in the opposite direction, giving alternative combinatorial proofs for results that were originally proved algebraically. These alternative proofs can yield important improvements because they are closer to the underlying problems and avoid the losses in passing to the algebraic setting. A prominent example is Dinur's proof of the PCP Theorem via gap amplification which yielded short PCPs with only a polylogarithmic length blowup (which had been the focus of significant research effort up to that point). We see here (and in a number of recent works) an exciting interplay between algebraic and combinatorial techniques. This seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic and combinatorial methods in a variety of settings
Bibliography on Realizability
AbstractThis document is a bibliography on realizability and related matters. It has been collected by Lars Birkedal based on submissions from the participants in âA Workshop on Realizability Semantics and Its Applicationsâ, Trento, Italy, June 30âJuly 1, 1999. It is available in BibTEX format at the following URL: http://www.cs.cmu.edu./~birkedal/realizability-bib.html
Networks become navigable as nodes move and forget
We propose a dynamical process for network evolution, aiming at explaining
the emergence of the small world phenomenon, i.e., the statistical observation
that any pair of individuals are linked by a short chain of acquaintances
computable by a simple decentralized routing algorithm, known as greedy
routing. Previously proposed dynamical processes enabled to demonstrate
experimentally (by simulations) that the small world phenomenon can emerge from
local dynamics. However, the analysis of greedy routing using the probability
distributions arising from these dynamics is quite complex because of mutual
dependencies. In contrast, our process enables complete formal analysis. It is
based on the combination of two simple processes: a random walk process, and an
harmonic forgetting process. Both processes reflect natural behaviors of the
individuals, viewed as nodes in the network of inter-individual acquaintances.
We prove that, in k-dimensional lattices, the combination of these two
processes generates long-range links mutually independently distributed as a
k-harmonic distribution. We analyze the performances of greedy routing at the
stationary regime of our process, and prove that the expected number of steps
for routing from any source to any target in any multidimensional lattice is a
polylogarithmic function of the distance between the two nodes in the lattice.
Up to our knowledge, these results are the first formal proof that navigability
in small worlds can emerge from a dynamical process for network evolution. Our
dynamical process can find practical applications to the design of spatial
gossip and resource location protocols.Comment: 21 pages, 1 figur
Universal Prediction
In this thesis I investigate the theoretical possibility of a universal method of prediction. A prediction method is universal if it is always able to learn from data: if it is always able to extrapolate given data about past observations to maximally successful predictions about future observations. The context of this investigation is the broader philosophical question into the possibility of a formal specification of inductive or scientific reasoning, a question that also relates to modern-day speculation about a fully automatized data-driven science.
I investigate, in particular, a proposed definition of a universal prediction method that goes back to Solomonoff (1964) and Levin (1970). This definition marks the birth of the theory of Kolmogorov complexity, and has a direct line to the information-theoretic approach in modern machine learning. Solomonoff's work was inspired by Carnap's program of inductive logic, and the more precise definition due to Levin can be seen as an explicit attempt to escape the diagonal argument that Putnam (1963) famously launched against the feasibility of Carnap's program.
The Solomonoff-Levin definition essentially aims at a mixture of all possible prediction algorithms. An alternative interpretation is that the definition formalizes the idea that learning from data is equivalent to compressing data. In this guise, the definition is often presented as an implementation and even as a justification of Occam's razor, the principle that we should look for simple explanations.
The conclusions of my investigation are negative. I show that the Solomonoff-Levin definition fails to unite two necessary conditions to count as a universal prediction method, as turns out be entailed by Putnam's original argument after all; and I argue that this indeed shows that no definition can. Moreover, I show that the suggested justification of Occam's razor does not work, and I argue that the relevant notion of simplicity as compressibility is already problematic itself
Universal Prediction
In this dissertation I investigate the theoretical possibility of a universal method of prediction. A prediction method is universal if it is always able to learn what there is to learn from data: if it is always able to extrapolate given data about past observations to maximally successful predictions about future observations. The context of this investigation is the broader philosophical question into the possibility of a formal specification of inductive or scientific reasoning, a question that also touches on modern-day speculation about a fully automatized data-driven science.
I investigate, in particular, a specific mathematical definition of a universal prediction method, that goes back to the early days of artificial intelligence and that has a direct line to modern developments in machine learning. This definition essentially aims to combine all possible prediction algorithms. An alternative interpretation is that this definition formalizes the idea that learning from data is equivalent to compressing data. In this guise, the definition is often presented as an implementation and even as a justification of Occam's razor, the principle that we should look for simple explanations.
The conclusions of my investigation are negative. I show that the proposed definition cannot be interpreted as a universal prediction method, as turns out to be exposed by a mathematical argument that it was actually intended to overcome. Moreover, I show that the suggested justification of Occam's razor does not work, and I argue that the relevant notion of simplicity as compressibility is problematic itself
On Randomness Extraction in AC0
We consider randomness extraction by AC0 circuits. The main parameter, n, is the length of the source, and all other parameters are functions of it. The additional extraction parameters are the min-entropy bound k=k(n), the seed length r=r(n), the output length m=m(n), and the (output) deviation bound epsilon=epsilon(n).
For k = r+1) is possible if and only if k * r > n/poly(log(n)). For k >= n/log^(O(1))(n),
we show that AC0-extraction of r+Omega(r) bits is possible when r=O(log(n)), but leave open the question of whether more bits can be extracted in this case.
The impossibility result is for constant epsilon, and the possibility result supports epsilon=1/poly(n). The impossibility result is for (possibly) non-uniform AC0, whereas the possibility result hold for uniform AC0. All our impossibility results hold even for the model of bit-fixing sources, where k coincides with the number of non-fixed (i.e., random) bits.
We also consider deterministic AC0 extraction from various classes of restricted sources. In particular, for any constant , we give explicit AC0 extractors for poly(1/delta) independent sources that are each of min-entropy rate delta; and four sources suffice for delta=0.99. Also, we give non-explicit AC0 extractors for bit-fixing sources of entropy rate 1/poly(log(n)) (i.e., having n/poly(log(n)) unfixed bits). This shows that the known analysis of the "restriction method" (for making a circuit constant by fixing as few variables as possible) is tight for AC0 even if the restriction is picked deterministically depending on the circuit
Applications of Derandomization Theory in Coding
Randomized techniques play a fundamental role in theoretical computer science
and discrete mathematics, in particular for the design of efficient algorithms
and construction of combinatorial objects. The basic goal in derandomization
theory is to eliminate or reduce the need for randomness in such randomized
constructions. In this thesis, we explore some applications of the fundamental
notions in derandomization theory to problems outside the core of theoretical
computer science, and in particular, certain problems related to coding theory.
First, we consider the wiretap channel problem which involves a communication
system in which an intruder can eavesdrop a limited portion of the
transmissions, and construct efficient and information-theoretically optimal
communication protocols for this model. Then we consider the combinatorial
group testing problem. In this classical problem, one aims to determine a set
of defective items within a large population by asking a number of queries,
where each query reveals whether a defective item is present within a specified
group of items. We use randomness condensers to explicitly construct optimal,
or nearly optimal, group testing schemes for a setting where the query outcomes
can be highly unreliable, as well as the threshold model where a query returns
positive if the number of defectives pass a certain threshold. Finally, we
design ensembles of error-correcting codes that achieve the
information-theoretic capacity of a large class of communication channels, and
then use the obtained ensembles for construction of explicit capacity achieving
codes.
[This is a shortened version of the actual abstract in the thesis.]Comment: EPFL Phd Thesi
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