DOCSLIB.ORG

Control of Spatio-Temporal Pattern Formation Governed by Geometrical Models of Interface Evolution

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Control of Spatio-Temporal Pattern Formation Governed by Geometrical Models of Interface Evolution Control of spatio-temporal pattern formation governed by geometrical models of interface evolution Adrian Bouaru A thesis submitted to the University of Sheffield for the degree of Doctor of Philosophy Department of Automatic Control and Systems Engineering University of Sheffield Sheffield, UK 2015 Abstract Numerous natural phenomena are characterized by spatio-temporal dynamics which give rise to time evolving spatial patterns. Although studies that address the problem of modelling these complex dynamics exist, a model based control approach for such systems is still a challenging task. The work in this thesis is concerned with the development of control methods for such spatio-temporal systems, where interface growth is represented using a geometric evolution law. In particular, the focus is set on the control of dendritic crystal growth and wind aided wildfire spread. In the first section of the thesis several modeling approaches used to describe spatio-temporal systems are discussed and partial differential equation models are selected as a suitable class of models for describing the system dynamics. Following this, the complex nonlinear dynamics found in dendritic crystal growth and wind aided wildfire evolution are analyzed through the scope of geometrical partial differential equation models of interface evolution. In the next part of the thesis, a new method for controlling the shape of evolving interfaces, such as the boundaries of dendritic crystals, described by a temperature-dependent geometrical evolution law, where the normal velocity of the growing interface depends on both local curvature and temperature, is presented. A study of the control method in both two and three dimensions is provided, using both linear and non-linear interface dynamics, to demonstrate the effectiveness of the proposed method. Following this, a novel model-based approach for controlling the evolving complex-shaped fire front using a level-set formulation of the model is presented. The work shows how key influencing parameters such as terrain topography and fuel loading, extracted directly from satellite imaging data, can be incorporated in the model. Simulation results show the numerical method is computationally efficient and examples are given with both synthetic and real data to illustrate the robustness of the containment method against changing wind speed, fuel loading and terrain topography. Based on the developments made in the previous chapters, a new optimization method for controlling wildfire evolution is presented. Using a level set formulation of the evolving interface, a cost function is derived that allows the selection of a spatial location, such that the firefighting action has the best results with respect to a predefined target (minimum fire area increase etc.). Several tuning parameters are incorporated within the optimization algorithm, enabling a wide range of fire combat scenarios to be implemented. Numerical simulations provided demonstrate the effectiveness of the method. Acknowledgments I would like to thank my supervisor Daniel Coca for offering me the opportunity to become part of the exciting research area of spatio-temporal systems. I am grateful for his constant guidance and support and his extremely valuable insight into the study of spatio-temporal systems throughout the PhD. I would also like to thank my colleagues from Office Room 316, with whom I’ve had numerous enjoyable discussions. My thanks and appreciations go to the Department of Automatic Control and Systems Engineering which has provided the support and resources I have needed to complete my thesis. I gratefully acknowledge that this work has been supported by the UK Engineering and Physical Sciences Research Council under the grant award EP/G042209/1. In addition I would like to thank my friends and family who have supported me through the entire period. Most of all I would like to thank my loving wife, who helped me enormously and with whom I’ve shared most of the stress of going through a PhD. Her support has been invaluable throughout this entire period. Table of contents Abstract Acknowledgements List of figures List of tables 1. Introduction ............................................................................................................................................ 1 1.1. Background to the project and Motivation ........................................................................................ 3 1.2. Dendritic crystal growth .................................................................................................................... 3 1.3. Wild fire propagation ........................................................................................................................ 5 1.4. Aims and objectives .......................................................................................................................... 9 1.5. Overview of the thesis ....................................................................................................................... 9 2. Mathematical representations of spatio-temporal systems ............................................................... 13 2.1. Introduction ...................................................................................................................................... 13 2.2. Cellular automata models................................................................................................................. 13 2.3. Coupled map lattice models ............................................................................................................. 16 2.4. Partial differential equations ............................................................................................................ 17 2.5. Conclusions ...................................................................................................................................... 21 3. Models of dendritic crystal growth ...................................................................................................... 23 3.1. Introduction ...................................................................................................................................... 23 3.2 Models of interface evolution ........................................................................................................... 23 3.2.1. Geometrical models of dendritic growth .................................................................................. 23 3.2.2. The Stefan problem .................................................................................................................. 27 3.2.3. Cellular automata models for dendritic crystal growth ........................................................... 29 3.3. Numerical methods for solving interface evolution problems ......................................................... 30 3.4. Numerical studies of crystal growth in 2D ...................................................................................... 34 3.4.1. Time and space discretization .................................................................................................. 34 3.4.2. Simulation of geometric models of dendritic crystal growth .................................................. 35 3.4.2.1. Motion of level sets in the normal direction ................................................................ 35 3.4.2.2. Motion of level sets under curvature flow .................................................................. 38 3.4.2.3. Motion of level sets under curvature flow with advection in the normal direction ..... 39 vii 3.4.2.4. Motion of level sets under the second derivative of curvature..................................... 40 3.4.2.5. Numerical difficulties of geometric models of dendritic crystal growth .................... 42 3.4.3. Simulation of the modified Stefan problem ............................................................................ 43 3.4.3.1. Velocity extension ........................................................................................................ 43 3.4.3.2. Re-initialization of the level set function ..................................................................... 44 3.4.3.3. Temperature update ...................................................................................................... 45 3.4.3.4. Simulation results ......................................................................................................... 47 3.4.4. CML simulation of dendritic crystal growth ........................................................................... 49 3.4.4.1. Model description ....................................................................................................... 49 3.4.4.2. Model simulations ...................................................................................................... 50 3.5. Conclusions ...................................................................................................................................... 52 4. Models of wildfire propagation ........................................................................................................... 53 4.1. Introduction ...................................................................................................................................... 53 4.2. Wind aided fire evolution models .................................................................................................... 53 4.3. Numerical studies of forest fire propagation ...................................................................................
Load more

Recommended publications
  • Automatic Detection of Interesting Cellular Automata
    Automatic Detection of Interesting Cellular Automata Qitian Liao Electrical Engineering and Computer Sciences University of California, Berkeley Technical Report No. UCB/EECS-2021-150 http://www2.eecs.berkeley.edu/Pubs/TechRpts/2021/EECS-2021-150.html May 21, 2021 Copyright © 2021, by the author(s). All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission. Acknowledgement First I would like to thank my faculty advisor, professor Dan Garcia, the best mentor I could ask for, who graciously accepted me to his research team and constantly motivated me to be the best scholar I could. I am also grateful to my technical advisor and mentor in the field of machine learning, professor Gerald Friedland, for the opportunities he has given me. I also want to thank my friend, Randy Fan, who gave me the inspiration to write about the topic. This report would not have been possible without his contributions. I am further grateful to my girlfriend, Yanran Chen, who cared for me deeply. Lastly, I am forever grateful to my parents, Faqiang Liao and Lei Qu: their love, support, and encouragement are the foundation upon which all my past and future achievements are built. Automatic Detection of Interesting Cellular Automata by Qitian Liao Research Project Submitted to the Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, in partial satisfaction of the requirements for the degree of Master of Science, Plan II.
    [Show full text]
  • Complexity Properties of the Cellular Automaton Game of Life
    Portland State University PDXScholar Dissertations and Theses Dissertations and Theses 11-14-1995 Complexity Properties of the Cellular Automaton Game of Life Andreas Rechtsteiner Portland State University Follow this and additional works at: https://pdxscholar.library.pdx.edu/open_access_etds Part of the Physics Commons Let us know how access to this document benefits ou.y Recommended Citation Rechtsteiner, Andreas, "Complexity Properties of the Cellular Automaton Game of Life" (1995). Dissertations and Theses. Paper 4928. https://doi.org/10.15760/etd.6804 This Thesis is brought to you for free and open access. It has been accepted for inclusion in Dissertations and Theses by an authorized administrator of PDXScholar. Please contact us if we can make this document more accessible: [email protected]. THESIS APPROVAL The abstract and thesis of Andreas Rechtsteiner for the Master of Science in Physics were presented November 14th, 1995, and accepted by the thesis committee and the department. COMMITTEE APPROVALS: 1i' I ) Erik Boaegom ec Representativi' of the Office of Graduate Studies DEPARTMENT APPROVAL: **************************************************************** ACCEPTED FOR PORTLAND STATE UNNERSITY BY THE LIBRARY by on /..-?~Lf!c-t:t-?~~ /99.s- Abstract An abstract of the thesis of Andreas Rechtsteiner for the Master of Science in Physics presented November 14, 1995. Title: Complexity Properties of the Cellular Automaton Game of Life The Game of life is probably the most famous cellular automaton. Life shows all the characteristics of Wolfram's complex Class N cellular automata: long-lived transients, static and propagating local structures, and the ability to support universal computation. We examine in this thesis questions about the geometry and criticality of Life.
    [Show full text]
  • Dipl. Eng. Thesis
    A Framework for the Real-Time Execution of Cellular Automata on Reconfigurable Logic By NIKOLAOS KYPARISSAS Microprocessor & Hardware Laboratory School of Electrical & Computer Engineering TECHNICAL UNIVERSITY OF CRETE A thesis submitted to the Technical University of Crete in accordance with the requirements for the DIPLOMA IN ELECTRICAL AND COMPUTER ENGINEERING. FEBRUARY 2020 THESIS COMMITTEE: Prof. Apostolos Dollas, Technical University of Crete, Thesis Supervisor Prof. Dionisios Pnevmatikatos, National Technical University of Athens Prof. Michalis Zervakis, Technical University of Crete ABSTRACT ellular automata are discrete mathematical models discovered in the 1940s by John von Neumann and Stanislaw Ulam. They constitute a general paradigm for massively parallel Ccomputation. Through time, these powerful mathematical tools have been proven useful in a variety of scientific fields. In this thesis we propose a customizable parallel framework on reconfigurable logic which can be used to efficiently simulate weighted, large-neighborhood totalistic and outer-totalistic cellular automata in real time. Simulating cellular automata rules with large neighborhood sizes on large grids provides a new aspect of modeling physical processes with realistic features and results. In terms of performance results, our pipelined application-specific architecture successfully surpasses the computation and memory bounds found in a general-purpose CPU and has a measured speedup of up to 51 against an Intel Core i7-7700HQ CPU running highly optimized £ software programmed in C. i ΠΕΡΙΛΗΨΗ α κυψελωτά αυτόματα είναι διακριτά μαθηματικά μοντέλα που ανακαλύφθηκαν τη δεκαετία του 1940 από τον John von Neumann και τον Stanislaw Ulam. Αποτελούν ένα γενικό Τ υπόδειγμα υπολογισμών με εκτενή παραλληλισμό. Μέχρι σήμερα τα μαθηματικά αυτά εργαλεία έχουν χρησιμεύσει σε πληθώρα επιστημονικών τομέων.
    [Show full text]
  • Asymptotic Densities for Packard Box Rules Janko Gravner Mathematics
    Asymptotic densities for Packard Box rules Janko Gravner Mathematics Department University of California Davis, CA 95616 e-mail: [email protected] David Griffeath Department of Mathematics University of Wisconsin Madison, WI 53706 e-mail: [email protected] (Revised version, February 2009) Abstract. This paper studies two-dimensional totalistic solidification cellular automata with Moore neighborhood such that a single occupied neighbor is sufficient for a site to join the occupied set. We focus on asymptotic density of the final configuration, obtaining rigorous results mainly in cases with an additive component. When sites with two occupied neighbors also solidify, additivity yields a comprehensive result for general finite initial sets. In particular, the density is independent of the seed. More generally, analysis is restricted to special initializations, cases with density 1, and various experimental findings. In the case that requires either one or three occupied neighbors for solidification, we show that different seeds may produce many different densities. For each of two particularly common densities we present a scheme that recognizes a seed in its domain of attraction. 2000 Mathematics Subject Classification. Primary 37B15. Secondary 68Q80, 11B05. Keywords: additive dynamics, asymptotic density, cellular automaton, growth model. Acknowledgments. Experimental aspects of this research were conducted by a Collaborative Undergraduate Research Lab (CURL) at the University of Wisconsin-Madison Mathematics Department during the 2006–7 academic year. The CURL was sponsored by a National Science Foundation VIGRE award. We are particularly grateful to lab participants Charles D. Brummitt and Chad Koch for empirical analysis of the Box 12 rule. Support. JG was partially supported by NSF grant DMS–0204376 and the Republic of Slove- nia’s Ministry of Science program P1-285.
    [Show full text]
  • Fifty Years of Research on Self-Replication: an Overview
    Fifty Years of Research on Moshe Sipper Logic Systems Laboratory Self-Replication: An Overview Swiss Federal Institute of Technology IN-Ecublens CH-1015 Lausanne Switzerland [email protected]fl.ch http://lslwww.epfl.ch/ Abstract The study of artificial self-replicating structures or moshes machines has been taking place now for almost half a » century. My goal in this article is to present an overview of research carried out in the domain of self-replication over the past 50 years, starting from von Neumann’s work in the late 1940s and continuing to the most recent research efforts. I Keywords shall concentrate on computational models, that is, ones that self-replication, cellular automata, self-replicating programs, self-repli- have been studied from a computer science point of view, be cating strings, self-replicating ma- it theoretical or experimental. The systems are divided into chines four major classes, according to the model on which they are based: cellular automata, computer programs, strings (or strands), or an altogether different approach. With the advent of new materials, such as synthetic molecules and nanomachines, it is quite possible that we shall see this somewhat theoretical domain of study producing practical, real-world applications. 1 Introduction In the late 1940s eminent mathematician and physicist John von Neumann had become interested in the question of whether a machine can self-replicate, that is, produce copies of itself. Von Neumann wished to investigate the logic necessary for replication— he was not interested in, nor did he have the tools for, building a working machine at the biochemical or genetic level.
    [Show full text]
  • Lattice-Gas Cellular Automata and Lattice Boltzmann Models - an Introduction
    Dieter A. Wolf-Gladrow Alfred Wegener Institute for Polar and Marine Research, Postfach 12 01 61 D-27515 Bremerhaven Germany e-mail: [email protected] Version 1.05 Lattice-Gas Cellular Automata and Lattice Boltzmann Models - An Introduction June 26, 2005 Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo Contents 1 Introduction ............................................... -1 1.1 Preface ................................................ 0 1.2 Overview............................................... 2 1.3 The basic idea of lattice-gas cellular automata and lattice Boltzmannmodels ...................................... 5 1.3.1 TheNavier-Stokesequation......................... 5 1.3.2 Thebasicidea.................................... 7 1.3.3 Top-downversusbottom-up........................ 9 1.3.4 LGCAversusmoleculardynamics................... 9 2 Cellular Automata ......................................... 13 2.1 What are cellular automata? . 13 2.2 A short history of cellular automata . 14 2.3 One-dimensional cellular automata . 15 2.3.1 Qualitative characterization of one-dimensional cellular automata . 22 2.4 Two-dimensional cellular automata . 28 2.4.1 Neighborhoodsin2D.............................. 28 2.4.2 Fredkin’s game . 29 2.4.3 ‘Life’ ............................................ 30 2.4.4 CA: what else? Further reading . 34 2.4.5 FromCAtoLGCA................................ 35 VI Contents 3 Lattice-gas cellular automata .............................. 37 3.1 The HPP lattice-gas cellular automata . 37 3.1.1 Modeldescription................................. 37 3.1.2 Implementation of the HPP model: How to code lattice-gas cellular automata? . 42 3.1.3 Initialization...................................... 46 3.1.4 Coarse graining . 48 3.2 The FHP lattice-gas cellular automata . 51 3.2.1 The lattice and the collision rules . 51 3.2.2 MicrodynamicsoftheFHPmodel .................. 57 3.2.3 The Liouville equation . 62 3.2.4 Massandmomentumdensity......................
    [Show full text]
  • Arxiv:1708.09301V1 [Nlin.CG] 29 Aug 2017
    A GAME OF LIFE ON PENROSE TILINGS DUANE A. BAILEY AND KATHRYN A. LINDSEY Abstract. We define rules for cellular automata played on quasiperiodic tilings of the plane arising from the multigrid method in such a way that these cellular automata are isomorphic to Conway's Game of Life. Although these tilings are nonperiodic, determining the next state of each tile is a local computation, requiring only knowledge of the local structure of the tiling and the states of finitely many nearby tiles. As an example, we show a version of a \glider" moving through a region of a Penrose tiling. This constitutes a potential theoretical framework for a method of executing computations in non-periodically structured substrates such as quasicrystals. 1. Introduction Inspired by Conway's Game of Life ([3]), various researchers have investigated properties of similarly- defined cellular automata played on Penrose tilings (e.g. [4, 7, 8, 10, 11]). Since Penrose tilings are not periodic ([6]), the challenge is to define a finite state cellular automaton which is locally computable { meaning that the next state of a tile depends only on the current states of finitely many \nearby" tiles { and yet has interesting emergent global properties. In particular, the existence of \gliders" and the ability to support universal computation is of interest. In this note, we present a natural method for embedding the original Game of Life in any tiling of the plane that arises via the multigrid method, a class of tilings that include Penrose rhomb tilings. The resulting cellular automata, defined on quasiperiodic tilings, are locally computable and precisely mimic the behavior of the original Game of Life.
    [Show full text]
  • Cellular Automataautomata
    CellularCellular AutomataAutomata Moreno Marzolla Dip. di Informatica—Scienza e Ingegneria (DISI) Università di Bologna http://www.moreno.marzolla.name/ Complex Systems 2 Cellular Automata ● Originally developed in the 1940s by Stanislaw Ulam and John von Neumann ● Interest in CAs expanded in 1970 after Conway's Game of Life Stanislaw Ulam John von Neumann (1909—1984) (1903—1957) Complex Systems 3 Cellular Automata (CA) ● A regular grid of cells, each in one of a finite number k of states (e.g., k = 2 for binary states) ● At time t = 0 an initial state for each cell is selected ● A new generation is created (advancing t by 1), according to some fixed rule that determines the new state of each cell in terms of the current state of the cell and the states of the cells in its neighborhood – The rule for updating cell states is the same for each cell and does not change over time, though exceptions are known (e.g., stochastic CAs, asynchronous CAs...) Complex Systems 4 Cellular Automata ● We will generally consider CAs over a linear square grid lattice (i.e., an array or matrix) ● Other topologies can be considered – Irregular – Hexagonal – … ● Example: Game of Life on an hexagonal lattice Complex Systems 5 Elementary Cellular Automata Complex Systems 6 1-Dimensional CA ● “Infinite” linear array of cells ● Each cell can be in any of k states ● The neighborhood of each cell in general consists of r cells on the left and on the right – Usually r = 1, immediate neighbors – More complex neighborhoods can be considered Cell ... ... Neighbors (r
    [Show full text]
  • A Colorful Introduction to Cellular Automata
    A Colorful Introduction to Cellular Automata Silvio Capobianco February 5, 2011 Revised: February 10, 2011 Silvio Capobianco () February 5, 2011 1 / 37 Overview Cellular automata (ca) are local presentations of global dynamics They are powerful tools for qualitative analysis They display several interesting theoretical features We will set some of them in action Silvio Capobianco () February 5, 2011 2 / 37 History of cellular automata von Neumann, 1950s: mechanical model of self-reproduction Moore and Myhill, 1962: the Garden of Eden problem Hedlund, 1969: shift dynamical systems Hardy, de Pazzis, Pomeau 1976: lattice gas automata Amoroso and Patt, 1972; Kari, 1990: the invertibility problem Mach`ıand Mignosi, 1993: cellular automata on Cayley graphs Silvio Capobianco () February 5, 2011 3 / 37 Life is a Game Ideated by John Horton Conway (1960s) popularized by Martin Gardner. The checkboard is an infinite square grid. Each case (cell) of the checkboard is \surrounded" by those within a chess' king's move, and can be \living" or \dead". 1 A dead cell surrounded by exactly three living cells, becomes living. 2 A living cell surrounded by two or three living cells, survives. 3 A living cell surrounded by less than two living cells, dies of isolation. 4 A living cell surrounded by more than three living cells, dies of overpopulation. Silvio Capobianco () February 5, 2011 4 / 37 Game of Life situations Silvio Capobianco () February 5, 2011 5 / 37 Cellular automata Conway's Game of Life is an example of cellular automaton. Definition A cellular
    [Show full text]
  • Justine Seastres Cellular Automata and the Game of Life
    Justine Seastres Saint Mary's College of California Department of Mathematics May 16, 2016 Cellular Automata and the Game of Life Supervisors: Professor Porter Professor Sauerberg 2 Contents 1 Introduction 1 2 A Closer Look at Cellular Automata 3 2.1 Key Characteristics . .3 2.2 State . .3 2.3 Dimension . .3 2.4 Neighborhoods . .3 2.5 Local Update Function . .3 2.6 Conclusion . .4 3 One-Dimensional Cellular Automata 5 3.1 Definition . .5 3.2 Local Transition Function . .5 4 Two Dimensional Cellular Automata 6 5 Example: Conway's Game of Life 7 5.1 History . .7 5.2 Description . .7 5.3 Rules . .7 5.4 Mathematics of the Game of Life . .9 5.5 Drawing out a Life Board . 10 5.6 Pattern Emergence . 11 5.6.1 Still Life . 11 5.6.2 Oscillators/Life Forms . 11 5.6.3 Spaceship . 12 5.7 Garden of Eden . 12 5.7.1 Definitions . 13 5.7.2 Moore-Myhill Theorem . 13 5.8 Applications in the Real World . 15 5.8.1 Physical Reality of the Universe . 15 5.8.2 Biology . 15 5.8.3 Computers and Cryptography . 16 5.9 Bibliography . 18 1 Introduction \Cellular automata may be viewed as computers, in which data represented by initial configura- tions is processed by time evolution" -Stephen Wolfram Cellular Automata is an idea formulated in the 1940s by John von Neumann, a Hungarian- American mathematician. He had previously attended the University of Budapest to study chemical engineering, later earning his doctorate in mathematics. Through his interest in both fields of study, Von Neumann began to develop this idea of self-reproduction.
    [Show full text]
  • Tutorial on Cellular Automata
    Tutorial on Cellular Automata Nicolas Ollinger LIF, Aix-Marseille Université, CNRS UC 2011, Turku — June 2011 Theory of self-reproducing automata Cellular automata emerged in the late 40s from the work of Ulam and von Neumann. 1/108 Cellular Automata A cellular automaton (CA) is a discrete dynamical model. Space is discrete and consists of an infinite regular grid of cells. Each cell is described by a state among a common finite set. Time is discrete. At each clock tick cells change their state deterministically, synchronously and uniformly according to a common local update rule. 2/108 Conway’s famous Game of Life The Game of Life is a 2D CA invented by Conway in 1970. Space is an infinite chessboard of alive or dead cells. The local update rule counts the number of alive cells among the eight surrounding cells : • exactly three alive cells give birth to dead cells ; • less than three alive cells kill by loneliness ; • more than four alive cells kill by overcrowding ; • otherwise the cell remains in the same state. 3/108 von Neumann self-reproducing CA A 29 states CA with von Neumann neighborhood with wires and construction/destruction abilities. Self-reproduction using Universal Computer + Universal Constructor. (Theory of Self-Reproducing Automata, edited by Burks, 1966) 4/108 Codd self-reproducing CA A 8 states self-reproducing CA with von Neumann neighborhood using sheathed wires. Implemented by Hutton in 2009, several millions cells, self-reproducing in 1:7 × 1018 steps (estimated). (Cellular Automata, Codd, 1968) 5/108 Codd self-reproducing CA A 8 states self-reproducing CA with von Neumann neighborhood using sheathed wires.
    [Show full text]
  • Propositional Proof Complexity and Rewriting
    Univerzita Karlova v Praze Matematicko – Fyzikaln´ ´ı Fakulta Propositional Proof Complexity and Rewriting Stefano Cavagnetto Doktorsk´aDisertaˇcn´ı Pr´ace 2008 Skolitel:ˇ Prof. RNDr. Jan Kraj´ıˇcek, DrSc. Obor M1 – Algebra, teorie ˇc´ısel a matematick´a logika Charles University in Prague Faculty of Mathematics and Physics Propositional Proof Complexity and Rewriting Stefano Cavagnetto Doctoral Dissertation 2008 Thesis Advisor: Prof. RNDr. Jan Kraj´ıˇcek, DrSc. Branch M1 – Algebra, Number Theory and Mathematical Logic Propositional Proof Complexity 3 Abstract In this work we want to find a new framework for propositional proofs (and in particular for resolution proofs) utilizing rewriting techniques. We interpret the well-known propositional proof system resolution using string rewriting ∗ systems (semi-Thue system [70], [71]) Σn and Σn corresponding to tree-like ∗ proofs and sequence-like proofs, respectively. We prove that the system Σn is complete and sound with respect to tree-like Resolution R∗ (and we show how it is possible to obtain the same result for R). Using this interpretation ∗ we give a representation of Σn using planar diagrams in van Kampen style. In this representation we show how the classical complexity measures for Resolution (size, width and space) can be interpreted. Subsequently, we consider rewriting in a synchronous, parallel fashion as it is used in the theory of cellular automata. In this respect, we give a new proof of Richardson theorem [63](a global function GA of a cellular automaton A is injective if and only if the inverse of GA is a global function of a cellular automaton), a classical result in the field, exploiting only propositional logic.
    [Show full text]