972 research outputs found
Methods of class field theory to separate logics over finite residue classes and circuit complexity
This is a pre-copyedited, author-produced version of an article accepted for publication in Journal of logic and computation following peer review.Separations among the first-order logic Res(0,+,×) of finite residue classes, its extensions with generalized quantifiers, and in the presence of a built-in order are shown in this article, using algebraic methods from class field theory. These methods include classification of spectra of sentences over finite residue classes as systems of congruences, and the study of their h-densities over the set of all prime numbers, for various functions h on the natural numbers. Over ordered structures, the logic of finite residue classes and extensions are known to capture DLOGTIME-uniform circuit complexity classes ranging from AC to TC. Separating these circuit complexity classes is directly related to classifying the h-density of spectra of sentences in the corresponding logics of finite residue classes. General conditions are further shown in this work for a logic over the finite residue classes to have a sentence whose spectrum has no h-density. A corollary of this characterization of spectra of sentences is that in Res(0,+,×,<)+M, the logic of finite residue classes with built-in order and extended with the majority quantifier M, there are sentences whose spectrum have no exponential density.Peer ReviewedPostprint (author's final draft
Non-equilibrium phase transitions in biomolecular signal transduction
We study a mechanism for reliable switching in biomolecular
signal-transduction cascades. Steady bistable states are created by system-size
cooperative effects in populations of proteins, in spite of the fact that the
phosphorylation-state transitions of any molecule, by means of which the switch
is implemented, are highly stochastic. The emergence of switching is a
nonequilibrium phase transition in an energetically driven, dissipative system
described by a master equation. We use operator and functional integral methods
from reaction-diffusion theory to solve for the phase structure, noise
spectrum, and escape trajectories and first-passage times of a class of minimal
models of switches, showing how all critical properties for switch behavior can
be computed within a unified framework
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Symmetric Circuits and Model-Theoretic Logics
The question of whether there is a logic that characterises polynomial-time is arguably the
most important open question in finite model theory. The study of extensions of fixed-point
logic are of central importance to this question. It was shown by Anderson and Dawar that
fixed-point logic with counting (FPC) has the same expressive power as uniform families of
symmetric circuits over a basis with threshold functions.
In this thesis we prove a far-reaching generalisation of their result and establish an
analogous circuit characterisation for each from a broad range of extensions of fixed-point
logic. In order to do so we fist develop a very general framework for defining and studying
extensions of fixed-point logics, which we call generalised operators. These operators generalise
Lindström quantifiers as well as the counting and rank operators used to define FPC and
fixed-point logic with rank (FPR).
We also show that in order to define a symmetric circuit model that goes beyond FPC
we need to consider circuits with gates that are allowed to compute non-symmetric functions.
In order to do so we develop a far more general framework for studying circuits. We also
show that key notions, such as the notion of a symmetric circuit, can be analogously defined
in this more general framework. The characterisation of FPC in terms of symmetric circuits,
and the treatment of circuits generally, relies heavily on the assumption that the gates in
the circuit compute symmetric functions. We develop a broad range of new techniques and
approaches in order to study these more general symmetric circuit models.
As a corollary of our main result we establish a circuit characterisation of FPR. We also
show that the question of whether there is a logic that characterises polynomial-time can
be understood as a question about the symmetry property of circuits. We lastly propose
a number of new approaches that might exploit this new-found connection between circuit
complexity and descriptive complexity.Gates Cambridge Scholarship
Foundations of Software Science and Computation Structures
This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.
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