4,264 research outputs found
Phase Diagram for Anderson Disorder: beyond Single-Parameter Scaling
The Anderson model for independent electrons in a disordered potential is
transformed analytically and exactly to a basis of random extended states
leading to a variant of augmented space. In addition to the widely-accepted
phase diagrams in all physical dimensions, a plethora of additional, weaker
Anderson transitions are found, characterized by the long-distance behavior of
states. Critical disorders are found for Anderson transitions at which the
asymptotically dominant sector of augmented space changes for all states at the
same disorder. At fixed disorder, critical energies are also found at which the
localization properties of states are singular. Under the approximation of
single-parameter scaling, this phase diagram reduces to the widely-accepted one
in 1, 2 and 3 dimensions. In two dimensions, in addition to the Anderson
transition at infinitesimal disorder, there is a transition between two
localized states, characterized by a change in the nature of wave function
decay.Comment: 51 pages including 4 figures, revised 30 November 200
Polynomial Path Orders: A Maximal Model
This paper is concerned with the automated complexity analysis of term
rewrite systems (TRSs for short) and the ramification of these in implicit
computational complexity theory (ICC for short). We introduce a novel path
order with multiset status, the polynomial path order POP*. Essentially relying
on the principle of predicative recursion as proposed by Bellantoni and Cook,
its distinct feature is the tight control of resources on compatible TRSs: The
(innermost) runtime complexity of compatible TRSs is polynomially bounded. We
have implemented the technique, as underpinned by our experimental evidence our
approach to the automated runtime complexity analysis is not only feasible, but
compared to existing methods incredibly fast. As an application in the context
of ICC we provide an order-theoretic characterisation of the polytime
computable functions. To be precise, the polytime computable functions are
exactly the functions computable by an orthogonal constructor TRS compatible
with POP*
An exponential lower bound for Individualization-Refinement algorithms for Graph Isomorphism
The individualization-refinement paradigm provides a strong toolbox for
testing isomorphism of two graphs and indeed, the currently fastest
implementations of isomorphism solvers all follow this approach. While these
solvers are fast in practice, from a theoretical point of view, no general
lower bounds concerning the worst case complexity of these tools are known. In
fact, it is an open question whether individualization-refinement algorithms
can achieve upper bounds on the running time similar to the more theoretical
techniques based on a group theoretic approach.
In this work we give a negative answer to this question and construct a
family of graphs on which algorithms based on the individualization-refinement
paradigm require exponential time. Contrary to a previous construction of
Miyazaki, that only applies to a specific implementation within the
individualization-refinement framework, our construction is immune to changing
the cell selector, or adding various heuristic invariants to the algorithm.
Furthermore, our graphs also provide exponential lower bounds in the case when
the -dimensional Weisfeiler-Leman algorithm is used to replace the standard
color refinement operator and the arguments even work when the entire
automorphism group of the inputs is initially provided to the algorithm.Comment: 21 page
Polynomial Path Orders
This paper is concerned with the complexity analysis of constructor term
rewrite systems and its ramification in implicit computational complexity. We
introduce a path order with multiset status, the polynomial path order POP*,
that is applicable in two related, but distinct contexts. On the one hand POP*
induces polynomial innermost runtime complexity and hence may serve as a
syntactic, and fully automatable, method to analyse the innermost runtime
complexity of term rewrite systems. On the other hand POP* provides an
order-theoretic characterisation of the polytime computable functions: the
polytime computable functions are exactly the functions computable by an
orthogonal constructor TRS compatible with POP*.Comment: LMCS version. This article supersedes arXiv:1209.379
Short Proofs for Slow Consistency
Let denote the finite
consistency statement "there are no proofs of contradiction in with
symbols". For a large class of natural theories , Pudl\'ak
has shown that the lengths of the shortest proofs of
in the theory
itself are bounded by a polynomial in . At the same time he conjectures that
does not have polynomial proofs of the finite consistency
statements . In contrast we show that Peano arithmetic
() has polynomial proofs of
,
where is the slow consistency statement for
Peano arithmetic, introduced by S.-D. Friedman, Rathjen and Weiermann. We also
obtain a new proof of the result that the usual consistency statement
is equivalent to iterations
of slow consistency. Our argument is proof-theoretic, while previous
investigations of slow consistency relied on non-standard models of arithmetic
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
A Step-indexed Semantics of Imperative Objects
Step-indexed semantic interpretations of types were proposed as an
alternative to purely syntactic proofs of type safety using subject reduction.
The types are interpreted as sets of values indexed by the number of
computation steps for which these values are guaranteed to behave like proper
elements of the type. Building on work by Ahmed, Appel and others, we introduce
a step-indexed semantics for the imperative object calculus of Abadi and
Cardelli. Providing a semantic account of this calculus using more
`traditional', domain-theoretic approaches has proved challenging due to the
combination of dynamically allocated objects, higher-order store, and an
expressive type system. Here we show that, using step-indexing, one can
interpret a rich type discipline with object types, subtyping, recursive and
bounded quantified types in the presence of state
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