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Use of Shape Grammar to Derive Cellular Automata Rule Patterns for System Architecture Thomas H. Speller, Jr. Doctoral Candidate Engineering Systems Division MIT NKS 2006 Conference Wolfram Science Thomas H. Speller, Jr. Doctoral Candidate Engineering Systems Division MIT June 16, 2006

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Outline A.System Architecture B.A Method for Generating a Solution Space C.CA Rule Space D.Shape Grammar E.Examples Image; Courtesy of NASA/JPL/Caltech Natures Creative Process

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology A. Motivation System architecture 1.To generate a creative space of system architectures that are physically legitimate and satisfy a given specification inspired by natures bottom-up self- generative processes –using a shape grammar and cellular automata approach 2.To better understand natures self-generative process and to contribute normative principles for system architecture & engineering of systems –Part of the challenge in bottom-up system architecting is finding or choosing the CA rule(s) –This talk is a report on one possible way to derive the CA rule using shape grammar in modeling complex, nonlinear physical phenomena; Science Application

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology B. SGCA Methodology for System Architecting 1.Concepting the system architecting process for the given specification –Using the human brain as a computational system 2.Defining a shape grammar (SG) 3.Transcribing to cellular automata (CA) and determining accompanying simple programs 4.Generating a creative solution space and graphically outputting system architectures for stakeholder selection 4 Stages

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Importance of Stage 1: Human cognitive tasks in the SGCA approach Conceptualize/visualize the general specification solution Analyze from the whole to constituents that can be represented in a shape grammar Identify relevant rules (laws and constraints) to be encoded by the shape grammar Logically construct the shape grammar to synthesize higher order systems from sequential and combinatoric applications of the rules to the constituents

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Paradigm of Thinking In the system-of-systems approach everything is in connected neighborhoods This is a paradigm shift from a single rule applied repeatedly to generate an entire system Imagine here a string of concatenated rules and simple programs: a Turing tape of CAs, combinatorics and simple programs in variable length block format When the Turing tape is read, the system architecture(s) is generated –from genome (emulated as Turing tape) phenotype The CA evolves according to rules expressed as list mappings (See CAEvolveList 1 ) 1 Refer to Chapter 2 and its notes, Wolfram, S., A New Kind of Science.2002, Champaign, Ill.: Wolfram Media, p. 865.

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology C. Examples CA rule space size problem 1 dimensional CA; if k = 2 and r = 1, then the rule space is –where k represents the color possibilities for each state and r is the range or radius of the neighborhood. It is interesting to notice that merely increasing the r from 1 to 2 and maintaining the colors at two increases the rule space from 256 to ~4.3 billion. 2 dimensional CA If k = 2 and r = 1, then the rule space is 1340780792994259709957402499820584612747936582059239337772356144372176403007354697680187429 8166903427690031858186486050853753882811946569946433649006084096 3 dimensional CA 1196380724997376356710237763087067030291123782412927478906332372851427131104586655217888862 991011094264665383712016988668915930291 > 5363566618500230909965343379755515881097161429702187245535577191250779599631027489243844027 055777730755772876157409694792816787456

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Cellular automata parameter space

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology D. Shape Grammar Based on transformational grammars [N. Chomsky 1957], which generate a language of one dimensional strings Shape grammars (Stiny, 1972; Knight, 1994, Stiny 2006) –are systems of rules for characterizing the composition of designs in spatial languages –The grammar is unrestricted having the capability of producing languages that are recursively enumerable –defined by a quadruple SG = (V T, V M, R, I), generate a language of two or even three dimensional objects that are composed of an assemblage of terminal shapes, where V T is a set of terminal shapes (i.e., terminal symbols) V M is a set of markers (i.e., variables) R is a set of shape rules (addition/subtraction and Euclidean transformations), u v is the shape rule (i.e., productions; a production set of rules specifies the sequence of shape rules used to transform an initial shape to a final state and thus constitutes the heart of the grammar) –u is in (V M V T ) + and v is in (V M V T )* I is the initial shape to which the first rule is applied (i.e., start variable)

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology E. System Architecting examples using an SGCA methodology 1.Blocks and LEGO® Bricks 2.Truss 3.Architectural style –Le Pont du Gard bridge-aqueduct

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Example 1: Brick bridge A single block shape was selected as the primitive, and experiments were conducted by hand Specification is decomposed into horizontal row of blocks and vertical supporting columns Mechanical statics were applied to determine the block configuration with the minimum mechanical action Basic building modules were created to address the specification The block was changed to a LEGO® brick due to its additional connective force, which effectively expanded the diversity of columnar shapes and allowed for emerging interconnectivity. Stage 1 Given Specification:= a stable, efficient span of supported bricks

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology The Brick Shape Grammar Rectangular brick primitive T module Shape grammar notation Shape rule depiction of the bridge row 1 moment direction Stage 2

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Shape rule depiction of the basic T module Rows 3-5 are either 50% offset left or right and straight combinations L RS Rules cont

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Transcription of the 5 Row Design Space into Cellular Automata Transcribing the shape grammar of rows 1 and 2 into a cellular automaton Stage 3

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology CA has 6 triplets as lists representing neighborhood rule mappings, which determine the next system state CA rule representation for Rows 3 through 5

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Generating System Architectures for Bridges Generating 27 columnar modules These columns are combinatorically paired (by a simple program) into 729 higher order modules Stage 4

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Combinatorics of Spans and Bridges 27x27 Modules = 729 Spans Replication and Reflection of the 2-Span produces 729 x 2 = 1458 bridge system architectures ex. of a Replication and Reflection being identical ex. of Replicationex. of Reflection I={S,S} D={UR,UL}DC={UR,S}DC={S,UR} {S,S,S,S}{UR,UR,UL,UL}{S,S,S,S}

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Possible Emergence: A natural consequence of certain combinations is shared, or interconnecting parts, requiring less energy (more efficient) by elimination of a brick. The red bricks are nonlinear interdependencies that create an unanticipated stable form-function from unstable modules, including a new beam primitive. ex. of 2 totally unstable modules becoming stable after interconnection {UR,UL,UR,UL}{UL,UR,UL,UR} {U,U,U,U} new primitive creation {S,UR,UR,S}

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Span data samples

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Bridge data samples, replicated and reflected

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology 6 Levels of Hierarchy within a Bridge Replication Reflection or

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Example 2: SGCA Shape, Truss Same as example 1 except the brick primitive is replaced by a truss, To assure that replacement trusses are in equilibrium, additional structural support members (struts) are required A line can create any polygon; a line serves as a component to the truss primitive just as the lattice cell serves as a component of the brick primitive

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Shape Grammar for the Truss Shapes

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology 27 Modular columns generated Module combinations, example

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Algebraically mapping the shapes into cellular automata for computing the 27 modules {{0, 0}, {0, 1}, {1, 1}, {0, 0}} a {{0, 0}, {1, 0}, {1, 1}, {0, 0}} b {{0, 0}, {1, 0}, {0, 1}, {0, 0}} c {{0, 1}, {1, 1}, {1, 0}, {0, 1}} d

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Example 3, Shape Grammar to Cellular Automata Design Variations Based upon the Style of Le Pont du Gard, Nimes Bridge-Aqueduct Roman Style

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Variations of design based on Le Pont du Gard, Nimes Bridge-Aqueduct Roman Style Possible neighborhood patterns and interconnectivity (shown as lines) The generalized algebraic and If-then conditional {m,n} design space expansion:= evolvable neighborhood list structure automata

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Solutions from catalogs of different neighborhood designs Examples of neighborhoods 1a, {{bd},{bd}} & 1b, {{b,b},{d,d}} Examples of neighborhoods 2, 3, 5, 6 {Figure shown with Index number in catalog}

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Evolution of System Architectures Generated from the Bottom up Modular configurations based on architectural style

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Conclusion A.System Architecture B.A Method for Generating a Solution Space C.CA Rule Space D.Shape Grammar E.Examples

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Thank You MIT Engineering Systems Division

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Example 4, the Lattice Gas, the Navier- Stokes equations The case example of the lattice gas used a single particle as the primitive. Previous researchers [Frisch, Hasslacher, Pomeau, 1986] had discovered by iterative experimentation that 0 to 6 particles represented in a hexagonal star graph properly matched the Navier-Stokes equations. The interactive behavior of these particles was represented in this study as shapes in the form of picture graphs depicting states at time, t, and then state changes at time, t+1. Stage 1

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Shape grammar for a lattice gas Shape:= a point representing an indestructible particle Hexagonal neighborhood, Rules of particle interaction (64) 0, Empty condition: the pattern (condition) of zero particles present in the neighborhood 1 particle present has 6 different possible trajectories of entry & exit into and out from the neighborhood

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Example of Shape Grammar Application to Lattice Gas –Particle conservation of mass and momentum can be represented in shape grammar with 64 pictures of particle interaction 3 particle 20 patterns of symmetry 1 particle 6 different possible trajectories of entry exit into and out from the neighborhood Empty space 5 particle 6 patterns of symmetry (conservation of energy). Hexagonal star graph (7 vertices) 9 vertex star graph with unused positions = 0 Nine cellular automata neighborhood in hexagonal format

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology SG CA Convert the hexagonal neighborhood graph to an adapted 2D Moore 9 cell to 6 cell Transcribe shape rules (patterns) to list structure, Ex. The 5 particle pattern can be represented as the lists Collision state Post-Collision state

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Five particle shape rule The 5 particle pattern can be represented as the lists or equivalently in the 2-dimension 9-neighborhood matrix with the directionalities of the incoming particles reversed per the symmetry of reflection of those particles then departing the neighborhood after collision. (The 0 cells have no effect on the neighborhood.) The other 63 particle collision patterns can be depicted in the same manner. The CA rules are executed in parallel on a grid wherein the particles move according to their correct physical properties and conform to the Navier-Stokes equations for fluids and gases at slow velocities relative to Mach.

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology System Architecture Includes –Function (purpose, what the system is to do) –Form (primitives, elements, parts, simple modules) –Structure (the interface, links among elements of form and organization: hierarchy, layered or network) –Properties Stability, robustness Flexibility, extensibility, reconfigurability Aesthetics Other ilities Cost Complexity –Environment –Creative space generation –Stakeholder choice of system architecture –Life cycle

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology

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Hierarchy and Genotype to Phenotype Mapping Phenotype Hierarchy of Modules with internal differentiation Genomic Hierarchy with internal differentiation Genomic code generation of System Architectures CA 1 CA 2:

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Different equivalent representations of the Turing tape

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© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology Observation: emergence of shapes and orientations The Line, Connected Line Open Shapes Connected Line Closed Shapes –Triangles Squares Rectangles Parallelograms

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