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NKS 2006 Conference Wolfram Science Thomas H. Speller, Jr. Doctoral Candidate Engineering Systems Division MIT

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##### NKS 2006 Conference Wolfram Science Thomas H. Speller, Jr. Doctoral Candidate Engineering Systems Division MIT

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**Use of Shape Grammar**to Derive Cellular Automata Rule Patterns for System Architecture NKS 2006 ConferenceWolfram ScienceThomas H. Speller, Jr.Doctoral CandidateEngineering Systems Division MIT June 16, 2006**Outline**• System Architecture • A Method for Generating a Solution Space • CA Rule Space • Shape Grammar • Examples Nature’s Creative Process Image; Courtesy of NASA/JPL/Caltech © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**A. Motivation**• System architecture • To generate a creative space of system architectures that are physically legitimate and satisfy a given specification inspired by nature’s bottom-up self-generative processes • using a shape grammar and cellular automata approach • To better understand nature’s 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 © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**B. SGCA Methodology for System Architecting**• Concepting the system architecting process for the given specification • Using the human brain as a computational system • Defining a shape grammar (SG) • Transcribing to cellular automata (CA) and determining accompanying simple programs • Generating a creative solution space and graphically outputting system architectures for stakeholder selection 4 Stages © 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 © 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 CA’s, 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 CAEvolveList1) 1Refer to Chapter 2 and its notes, Wolfram, S., A New Kind of Science. 2002, Champaign, Ill.: Wolfram Media, p. 865. © 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 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096 • 3 dimensional CA 1196380724997376356710237763087067030291123782412927478906332372851427131104586655217888862991011094264665383712016988668915930291 <<40403303>> 5363566618500230909965343379755515881097161429702187245535577191250779599631027489243844027055777730755772876157409694792816787456 © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**Cellular automata parameter space**© 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 = (VT, VM, R, I), generate a language of two or even three dimensional objects that are composed of an assemblage of terminal shapes, where • VT is a set of terminal shapes (i.e., terminal symbols) • VM is a set of markers (i.e., variables) • R is a set of shape rules (addition/subtraction and Euclidean transformations), uv 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 (VM VT)+ and v is in (VM VT)* • I is the initial shape to which the first rule is applied (i.e., start variable) © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**E. System Architecting examples using an SGCA methodology**• Blocks and LEGO® Bricks • Truss • Architectural style • Le Pont du Gard bridge-aqueduct © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**Example 1: Brick ‘bridge’**Given Specification:= a stable, efficient span of supported bricks • 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 © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**moment**direction The Brick Shape Grammar Stage 2 • Rectangular brick primitive • T module • Shape grammar notation • Shape rule depiction of the bridge row 1 © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**Rules con’t**• Shape rule depiction of the basic T module • Rows 3-5 are either 50% offset left or right and straight combinations L R S © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**Transcription of the 5 Row Design Space into Cellular**Automata Stage 3 • Transcribing the shape grammar of rows 1 and 2 into a cellular automaton © 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 © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**Generating System Architectures for Bridges**Stage 4 • Generating 27 columnar modules • These columns are combinatorically paired (by a simple program) into 729 higher order modules © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**Combinatorics of Spans and Bridges**I={S,S} D={UR,UL} DC={UR,S} DC={S,UR} • 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 Replication ex. of Reflection {S,S,S,S} {UR,UR,UL,UL} {S,S,S,S} © 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. {UR,UL,UR,UL} {UL,UR,UL,UR} ex. of 2 totally unstable modules becoming stable after interconnection new primitive creation {U,U,U,U} {S,UR,UR,S} © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**Span data samples**© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**Bridge data samples, replicated and reflected**© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**Replication**Reflection or or 6 Levels of Hierarchy within a Bridge © 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 © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**Shape Grammar for the Truss Shapes**© Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**27 Modular columns generated**• Module combinations, example © 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 © 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 © 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 © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**Solutions from catalogs of different neighborhood designs**{Figure shown with Index number in catalog} Examples of neighborhoods 1a, {{bd},{bd}} & 1b, {{b,b},{d,d}} Examples of neighborhoods 2, 3, 5, 6 © 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 © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**Conclusion**• System Architecture • A Method for Generating a Solution Space • CA Rule Space • Shape Grammar • Examples © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**Thank You**MIT Engineering Systems Division**Example 4, the Lattice Gas, the Navier-Stokes equations**Stage 1 • 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. © 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 © 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 5 particle 6 patterns of symmetry (conservation of energy). Empty space 1 particle 6 different possible trajectories of entry exit into and out from the neighborhood 3 particle 20 patterns of symmetry Hexagonal star graph (7 vertices) Nine cellular automata neighborhood in hexagonal format 9 vertex star graph with unused positions = 0 © 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 © 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. © 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 © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**© Thomas H. Speller, Jr. 2006, Engineering Systems Division**(ESD), Massachusetts Institute of Technology**Hierarchyand Genotype to Phenotype Mapping**Genomic Hierarchy with internal differentiation CA 1 Genomic code generation of System Architectures CA 2: Phenotype Hierarchy of Modules with internal differentiation © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology**Different equivalent representations of the Turing tape**© 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 © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology