Effective Theories for Circuits and Automata

Abstracting an effective theory from a complicated process is central to the study of complexity. Even when the underlying mechanisms are understood, or at least measurable, the presence of dissipation and irreversibility in biological, computational and social systems makes the problem harder. Here we demonstrate the construction of effective theories in the presence of both irreversibility and noise, in a dynamical model with underlying feedback. We use the Krohn-Rhodes theorem to show how the composition of underlying mechanisms can lead to innovations in the emergent effective theory. We show how dissipation and irreversibility fundamentally limit the lifetimes of these emergent structures, even though, on short timescales, the group properties may be enriched compared to their noiseless counterparts.Comment: 11 pages, 9 figure

Statistical Mechanics of Surjective Cellular Automata

Reversible cellular automata are seen as microscopic physical models, and their states of macroscopic equilibrium are described using invariant probability measures. We establish a connection between the invariance of Gibbs measures and the conservation of additive quantities in surjective cellular automata. Namely, we show that the simplex of shift-invariant Gibbs measures associated to a Hamiltonian is invariant under a surjective cellular automaton if and only if the cellular automaton conserves the Hamiltonian. A special case is the (well-known) invariance of the uniform Bernoulli measure under surjective cellular automata, which corresponds to the conservation of the trivial Hamiltonian. As an application, we obtain results indicating the lack of (non-trivial) Gibbs or Markov invariant measures for "sufficiently chaotic" cellular automata. We discuss the relevance of the randomization property of algebraic cellular automata to the problem of approach to macroscopic equilibrium, and pose several open questions. As an aside, a shift-invariant pre-image of a Gibbs measure under a pre-injective factor map between shifts of finite type turns out to be always a Gibbs measure. We provide a sufficient condition under which the image of a Gibbs measure under a pre-injective factor map is not a Gibbs measure. We point out a potential application of pre-injective factor maps as a tool in the study of phase transitions in statistical mechanical models.Comment: 50 pages, 7 figure

Constraint solving over multi-valued logics - application to digital circuits

Due to usage conditions, hazardous environments or intentional causes, physical and virtual systems are subject to faults in their components, which may affect their overall behaviour. In a ‘black-box’ agent modelled by a set of propositional logic rules, in which just a subset of components is externally visible, such faults may only be recognised by examining some output function of the agent. A (fault-free) model of the agent’s system provides the expected output given some input. If the real output differs from that predicted output, then the system is faulty. However, some faults may only become apparent in the system output when appropriate inputs are given. A number of problems regarding both testing and diagnosis thus arise, such as testing a fault, testing the whole system, finding possible faults and differentiating them to locate the correct one. The corresponding optimisation problems of finding solutions that require minimum resources are also very relevant in industry, as is minimal diagnosis. In this dissertation we use a well established set of benchmark circuits to address such diagnostic related problems and propose and develop models with different logics that we formalise and generalise as much as possible. We also prove that all techniques generalise to agents and to multiple faults. The developed multi-valued logics extend the usual Boolean logic (suitable for faultfree models) by encoding values with some dependency (usually on faults). Such logics thus allow modelling an arbitrary number of diagnostic theories. Each problem is subsequently solved with CLP solvers that we implement and discuss, together with a new efficient search technique that we present. We compare our results with other approaches such as SAT (that require substantial duplication of circuits), showing the effectiveness of constraints over multi-valued logics, and also the adequacy of a general set constraint solver (with special inferences over set functions such as cardinality) on other problems. In addition, for an optimisation problem, we integrate local search with a constructive approach (branch-and-bound) using a variety of logics to improve an existing efficient tool based on SAT and ILP

Physical layer network coding based on compute-and-forward

In this thesis, Compute-and-Forward is considered, where the system model consists of multiple users and a single base station. Compute-and-Forward is a type of lattice network coding which is deemed to avoid backhaul load and is therefore an important aspect of modern wireless communications networks. Initially we propose an implementation of construction D into Compute-and-Forward and investigate the implementation of multilayer lattice encoding and decoding strategies. Here we show that adopting a construction D lattice we can implement a practical lattice decoder in Compute-and-Forward. During this investigation and implementation of multilayer lattice encoding and decoding we discover an error floor due to an interaction between code layers in the multilayer decoder. We analyse and describe this interaction with mathematical expressions and give detail using lemmas and proofs. Secondly, we demonstrate the BER performance of the system model for unit valued channels, integer valued channels and complex integer valued channels. We show that using the derived expressions for interaction that the decoders on each code layer are able to indeed decode. The BER results are demonstrated for two scenarios using zero order and second order Reed-Muller codes and first and third order Reed-Muller codes. Finally, we extend our system model using construction D and existing conventional decoders to include coefficient selection algorithms. We employ an exhaustive search algorithm and analyse the throughput performance of the codes. Again, we extend this to both our models. With the throughput of the codes we see that each layer can be successfully decoded considering the interaction expressions. The purpose of the performance results is to show decodability with the extension of using differing codes

Monte Carlo Simulations of Spin Glasses on Cell Broadband Engine

Several large-scale computational scientific problems require high-end computing systems to be solved. In the recent years, design of multi-core architectures delivers on a single chip tens or hundreds Gflops of peak computing performance, with high power dissipation efficiency, and it makes available computational power previously available only on high-end multi-processor systems. The aim of this Ph.D. thesis is to study the capability of multi-core processors for scientific programming, analyzing sustained performance, issues related to multicore programming, data distribution, synchronization, in order to define a set of guideline rules to optimize scientific applications for this class of architectures. As an example of multi-core processor, we consider the Cell Broadband Engine (CBE), developed by Sony, IBM and Toshiba. The CBE is one of the most powerful multi-core CPU current available, integrating eight cores and delivering a peak performance of 200 Gflops in single precision and 100 Gflops in double precision. As case of study, we analyze the performances of CBE for Monte Carlo simulations of the Edwards-Anderson Spin Glass model, a paradigm in theoretical and condensed matter physics, used to describe complex systems characterized by phase transitions (such as the para-ferro transition in magnets) or model “frustrated” dynamics. We descrive several strategies for the distribution of data set among on-chip and off-chip memories and propose analytic models to find out the balance between computational and memory access time as a function of both algorithmic and architectural parameters. We use the analytic models to set the parameters of the algorithm, like for example size of data structures and scheduling of operations, to optimize execution of Monte Carlo spin glass simulations on the CBE architecture

The Phase Diagram of 1-in-3 Satisfiability Problem

We study the typical case properties of the 1-in-3 satisfiability problem, the boolean satisfaction problem where a clause is satisfied by exactly one literal, in an enlarged random ensemble parametrized by average connectivity and probability of negation of a variable in a clause. Random 1-in-3 Satisfiability and Exact 3-Cover are special cases of this ensemble. We interpolate between these cases from a region where satisfiability can be typically decided for all connectivities in polynomial time to a region where deciding satisfiability is hard, in some interval of connectivities. We derive several rigorous results in the first region, and develop the one-step--replica-symmetry-breaking cavity analysis in the second one. We discuss the prediction for the transition between the almost surely satisfiable and the almost surely unsatisfiable phase, and other structural properties of the phase diagram, in light of cavity method results.Comment: 30 pages, 12 figure

Computers from plants we never made. Speculations

We discuss possible designs and prototypes of computing systems that could be based on morphological development of roots, interaction of roots, and analog electrical computation with plants, and plant-derived electronic components. In morphological plant processors data are represented by initial configuration of roots and configurations of sources of attractants and repellents; results of computation are represented by topology of the roots' network. Computation is implemented by the roots following gradients of attractants and repellents, as well as interacting with each other. Problems solvable by plant roots, in principle, include shortest-path, minimum spanning tree, Voronoi diagram, $\alpha$-shapes, convex subdivision of concave polygons. Electrical properties of plants can be modified by loading the plants with functional nanoparticles or coating parts of plants of conductive polymers. Thus, we are in position to make living variable resistors, capacitors, operational amplifiers, multipliers, potentiometers and fixed-function generators. The electrically modified plants can implement summation, integration with respect to time, inversion, multiplication, exponentiation, logarithm, division. Mathematical and engineering problems to be solved can be represented in plant root networks of resistive or reaction elements. Developments in plant-based computing architectures will trigger emergence of a unique community of biologists, electronic engineering and computer scientists working together to produce living electronic devices which future green computers will be made of.Comment: The chapter will be published in "Inspired by Nature. Computing inspired by physics, chemistry and biology. Essays presented to Julian Miller on the occasion of his 60th birthday", Editors: Susan Stepney and Andrew Adamatzky (Springer, 2017