Use of Shape Grammar to Derive Cellular Automata Rule Patterns for System Architecture Thomas H. Speller, Jr. Doctoral Candidate Engineering Systems Division

Use of Shape Grammar to Derive

Cellular Automata Rule Patternsfor System Architecture

NKS 2006 ConferenceWolfram Science

Thomas H. Speller, Jr.Thomas H. Speller, Jr.Doctoral CandidateDoctoral Candidate

Engineering Systems Division Engineering Systems Division MITMIT

June 16, 2006

© 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

Nature’s Creative Process


A. Motivation• System architecture

1. 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

2. 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


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


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


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.


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

2)1r2(kk

256k)1r2(k

3)1r2(kk


Cellular automata parameter space

Cellular Automata Parallel ProcessingFinite State Machine

New CellStates

New System State

Environmentof Neighborhoods

Discrete Cell States In a

Finite State Machine

Controllables: Neighborhood Definition, Neighborhood Dimensions,

or Range of Neighborhood Rule Number of Colors of Cells Total Number of Cells in the Lattice The Shape of the Lattice Number of Steps

Initial Condition


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)


E. System Architecting examples using an SGCA methodology

1. Blocks and LEGO® Bricks

2. Truss

3. Architectural style– Le Pont du Gard bridge-aqueduct


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


The Brick Shape Grammar

• Rectangular brick primitive

• T module

• Shape grammar notation

• Shape rule depiction of the bridge row 1

wt. wt.

wt.9090

momentdirection

Rectangular Brick

A brick 2 cells

I. VT VM.

Row 1

Production

Rule 1-1

RowRule set

I .. . .

Initial Condition

Stage 2

I. VT VM

{., +}


• Shape rule depiction of the basic T module

• Rows 3-5 are either 50% offset left or right and straight combinations

Rule 2-1 . .Row 2 . .

Rule L1

Left OffsetRule set

Rule L2

Rule L3

.

.

.

.

Rule R1

Right OffsetRule set

Rule R2

Rule R3

.

.

.

.

Rule S1

Straight LineRule set

Rule S2

Rule S3

.

.

.

.

L R S

Rules con’t


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

. . . .

1 2 2 1

1 1

1 2 2 1

1 1

Shape Grammar Production

Shape Grammar to Cellular AutomataTranscription

0 0 0 0

EmptySpace

EmptySpace

EmptySpace

EmptySpace


• CA has 6 triplets as lists representing neighborhood rule mappings, which determine the next system state

• CA rule representation for Rows 3 through 5

1 2 2

1

2

1

12 1 2

0

0 1 0

0

2

Neighborhood Rule Mapping using Lists:

1 1

0

2 1 2

0

1

}0{{1,1,2} {0}{2,1,1} }1{}1,2,2{ }1{}2,2,1{ }0{{2,1,0} {0}{0,1,2}


Generating System Architectures for Bridges

• Generating 27 columnar modules

• These columns are combinatorically paired (by a simple program) into 729 higher order modules

Symmetric

Asymmetric

A new primitive emergesfrom some module

combinations(3 cell brick)

a stable higherorder module witha reduced number

of bricks

Stage 4


Combinatorics of Spans and Bridges• 27x27 Modules = 729 Spans

• Replication and Reflection of the 2-Span produces 729 x 2 = 1458 bridge system architectures

Genome[121,212] Genome[113,223] Genome[131,333]

Genome[121,212,121,212]

ex. of a Replication and Reflection being identical

Genome[211,112,221,122]Genome[333,333,333,333]

ex. of Replication ex. of Reflection

I={S,S} D={UR,UL} DC={UR,S}Genome[231,131]DC={S,UR}

{S,S,S,S} {UR,UR,UL,UL} {S,S,S,S}


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.

Genome[113,223,113,223] Genome[221,112,221,112]

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}Genome[133,113,113,133] Genome[222,111,222,111]


Span data samples


Bridge data samples, replicated and reflected


6 Levels of Hierarchy within a Bridge

The Bridge

Span 1 Span 2

Module 1 Module 2 Module 3 Module 4

T Module T Module T Module T Module

Initial ConditionSpecification




Brick Brick Brick Brick

Genome[0]

Genome[113]Genome[221]

Genome[221,113]

Genome[223]

Genome[223,212]

Genome[221,113,223,212]

Genome[112]

Genome[221,113,221,113]

Replication Reflection

Genome[221]

or

Genome[113]

or


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

Rectangular Brick

Free Body Diagramof Truss Forces and Moments

Applied Forcesand Moments

9090

The Line Primitive

Truss:= A triangle whose lines havematerial properties and cross-sectionalarea; these lines are called struts ormembers; these members transmitforce and may be in tension orcompression. The connected ends orstruts are modeled as pin joints. wt.

wt.

wt.

Trianglar Building Block:


Shape Grammar for the Truss Shapes

Row Rule Set

Row 1

Rule 1-1

I VT VM

I

Rule 1-2

Rule 2

Rules

Row 2

Initial Condition

Rule 1-3

Rows 3 and 4

Left Rule

Right Rule

Straight Rule

Rule 5

Row 5 (base)

Marker EraseRule

The Shape Grammar

Row Rule Set

Row 1

Rule 1-1

I VT VM

I

Rule 1-2

Rule 2

Rules

Row 2

Initial Condition

Rule 1-3

Rows 3 and 4

Left Rule

Right Rule

Straight Rule

Rule 5

Row 5 (base)

Marker EraseRule

The Shape Grammar


• 27 Modular columns generated

• Module combinations, example


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


Example 3, Shape Grammar to Cellular Automata Design Variations Based upon the Style of Le Pont du Gard, Nimes Bridge-Aqueduct Roman Style


Variations of design based on Le Pont du Gard, Nimes Bridge-Aqueduct Roman Style

BuildingBlocks

{IC}{I}

Level 3

Level 2

Level 1

{IC} {ICr} {IC}' {ICr}'

{I} {IC} {IC} {IC}'

{IC} {ICr} {IC}' {IC}'

{I}'

Neighborhood 1


{I} {IC} {IC} {IC}' {I}'

Neighborhood 2

Level 3

Level 2

Level 1

{I} {IC} {IC} {IC}' {I}'

Neighborhood 3

Neighborhood 4

{I} {IC} {IC} {IC}' {I}'


Level 3

Level 2

Level 1

{I} {IC} {IC} {IC}' {I}'

Neighborhood 5 Neighborhood 6

{IC} {I} {IC} {I}' {IC}'



{IC} {I} {IC} {I}' {IC}'

{I} {IC} {IC} {IC}' {I}'





Neighborhood 7 Neighborhood 8 Neighborhood 9

{IC} {I} {IC} {I}' {IC}'



{IC} {I} {IC} {I}' {IC}'


{IC} {I} {IC} {I}' {IC}'


{IC} {I} {IC} {I}' {IC}'


Possible neighborhood patterns and interconnectivity (shown as lines)

I = simple independent module IC=complex module composed of 2 simple interconnecting modules IC=symmetrical interconnected module I’ and IC’=reflected module

The generalized algebraic and If-then conditional {m,n} design space expansion:= evolvable neighborhood list structure automata

...

...

...

...

...

E F G H ... A B C D ...

Generating direction example and building block approach: Level 3 IC-IC-IC-IC Level 2 I-IC-IC-IC-I Level 1 IC-IC-IC-IC


Solutions from catalogs of different neighborhood designs

Examples of neighborhoods 1a, {{bd},{bd}} & 1b, {{b,b},{d,d}} {3,861} {13,724} {33,315 }

Examples of neighborhoods 2, 3, 5, 6

{Figure shown with Index number in catalog}


Evolution of System Architectures Generated from the Bottom up

Modular configurations based on architectural style

18858A Seed

Small modules ofprimtive combinations,

some modules areunstable

Larger moduleswith mostly stable combinations and

greater degree of internalinterconnectivity among parts

.

Small higher order modules ofcombinations, some modules are

unstable

Higher order modules ofcombinations, mostmodules are stable,

emergence stabilizes &reduces parts

3783

23473

18858Modular configurations based on

architectural style


Conclusion

A. System Architecture

B. A Method for Generating a Solution Space

C. CA Rule Space

D. Shape Grammar

E. Examples

Thank YouMIT

Engineering Systems Division


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


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


• 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 particle20 patterns of symmetry

1 particle 6 different possible trajectories

of entry exit into and out from the neighborhood

Empty space 5 particle6 patterns of symmetry

(conservation of energy).

Hexagonal star graph (7 vertices)

9 vertex star graphwith unused positions = 0

Nine cellular automata neighborhood in hexagonal format


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

0 c b d 0 a e f 0

Collisionstate

Post-Collisionstate


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.


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


Basis of comparison Shape Grammar approach CA approach Handling of System Architecture (form-function)

Form (shape) emphasis currently; function could be included

Has capability to have form and function encoded

Spatial design representation

A design is a set of shapes generated by rules starting with an initial state

A design is represented by sets of binary (or multicolor) cells on one, two or higher dimensional arrays

Design development Provides ease of design and visualization

Ease of transcription for computation of the design

Intuitive use Visually intuitive rule development for form and function

More difficult to visualize the final form-function to the neighborhood

Interpreter or compiler Generative rules developed and sequenced by hand and applied by hand or machine

Mathematica [22], the chosen programming language for this investigation, will run all rules automatically and manage the combinatorics and graphical output

Computing process Recursive generation produces modules, hierarchies

Same

Pattern analyzing capability

Can be used to analyze design styles manually

Can be used easily as a pattern recognizer and may be suitable for examining cause-effect by reverse engineering


Hierarchyand Genotype to Phenotype Mapping

Phenotype Hierarchy of Moduleswith internal differentiation

Genomic Hierarchywith internal differentiation

Genome[1]

Genome[3]Genome[2]

Genome[11] Genome[12] Genome[13]

Genome[21] Genome[22] Genome[23]Genome[31] Genome[32] Genome[33]

Genome[111] Genome[112] Genome[113] Genome[121] Genome[122] Genome[123] Genome[131] Genome[132] Genome[133]

Genome[0]

Genome[211] Genome[212] Genome[213] Genome[221] Genome[222] Genome[223] Genome[231] Genome[232] Genome[233] Genome[311] Genome[312] Genome[313] Genome[321] Genome[322] Genome[323] Genome[331] Genome[332] Genome[333]

Module 1

T h e G e n o m e i s a S t r i n g o f B l o c k s , A T u r i n g t a p e ( C N C t a p e )

• A T u r i n g M a c h i n e r e a d s t h e t a p e a n d g e n e r a t e s t h e P h e n o t y p e , S y s t e m A r c h i t e c t u r e

• T h e c o m p u t a t i o n a l g e n e r a t i n g p r o c e s s i s b o t h p a r a l l e l a n d s e r i a l

S t a r tC o d o n

E n dC o d o n

. . .I n i t i a lC o n d i t i o n

C A 1 C A 2 C A 3 C A n

B l o c k 1 B l o c k 2 B l o c k 3 B l o c k 4 B l o c k n

M o d u l e 1

Genomic code generation of System Architectures CA 1

CA 2:


Different equivalent representations of the Turing tape

Genome[111]:=

Genome[0] Genome[1] Genome[11] Genome[111]

InitialCondition

simple combinatoricselection program &applied CA[1]



Genome[n]:= {CA[0], simple programs & CA[1], ... , simple programs & CA[n]},

where n = {1, ... ,n}; where n is the CA number and genome number fordevelopment traceability convenience

CA[0]


Observation: emergence of shapes and orientations

• The Line,• Connected Line Open Shapes

• Connected Line Closed Shapes– Triangles

• Squares

• Rectangles Parallelograms

Documents

Use of Shape Grammar to Derive Cellular Automata Rule Patterns for System Architecture Thomas H. Speller, Jr. Doctoral Candidate Engineering Systems Division