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**Cellular Automata Rule Patterns for System Architecture**

Use of Shape Grammar to Derive Cellular Automata Rule Patterns for System Architecture NKS 2006 Conference Wolfram Science Thomas H. Speller, Jr. Doctoral Candidate Engineering Systems Division MIT F5 June 16, 2006

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**A Method for Generating a Solution Space CA Rule Space Shape Grammar **

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

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**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

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**B. SGCA Methodology for System Architecting**

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

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**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

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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

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**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 3 dimensional CA << >> © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology

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**Cellular automata parameter space**

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

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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

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**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

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**Example 1: Brick ‘bridge’**

Given Specification:= a stable, efficient span of supported bricks Stage 1 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. © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology

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**The Brick Shape Grammar**

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

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**Shape rule depiction of the basic T module **

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

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**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

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**CA rule representation for Rows 3 through 5**

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

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**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

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**Combinatorics of Spans and Bridges**

27x27 Modules = 729 Spans Replication and Reflection of the 2-Span produces 729 x 2 = 1458 bridge system architectures I={S,S} D={UR,UL} DC={UR,S} DC={S,UR} 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

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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 {S,UR,UR,S} {U,U,U,U} © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology

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

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**Bridge data samples, replicated and reflected**

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

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**6 Levels of Hierarchy within a Bridge**

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

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**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

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**Shape Grammar for the Truss Shapes**

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

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**27 Modular columns generated**

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

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**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

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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

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**Possible neighborhood patterns and interconnectivity (shown as lines)**

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

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**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

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**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

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**A Method for Generating a Solution Space CA Rule Space Shape Grammar **

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

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**MIT Engineering Systems Division**

Thank You MIT Engineering Systems Division

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**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

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**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

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**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) 9 vertex star graph with unused positions = 0 Nine cellular automata neighborhood in hexagonal format © Thomas H. Speller, Jr. 2006, Engineering Systems Division (ESD), Massachusetts Institute of Technology

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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

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**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

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**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

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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**

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

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**Different equivalent representations of the Turing tape**

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

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**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

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