everything you wanted to know about computers* John Alex *but were too afraid to askOverview • Functions as representation • Function optimization methods, issues – Steepest Descent – Simulated Annealing – Genetic Algorithms • Analysis of shape grammars • PossibilitiesOde to Functions • Math is based on them • Computers are based on them • Very general representation: a mapping • Helpful as intermediate object too – aid to formalization, rigor • Limited – only maps numbers to numbers – is mapping it?Functions •y = f(x) • x,y: vector of parameters (‘parametric’?) • “Form function” – Vertices = f(dimensions, key pts, etc) • “Fitness function” – Quality = f(vertices)Functions •y = f(x) • x,y are each a vector of parameters • Each parameter can be either – discrete (combinatorial): 0, 1, 2, 3, 4 – continuous: 0-4Functions •y = f(x) • such that c(x) = 0 • Constraints: valid parameter combinationsTrouble with Functions in Design • Pre-Optimization questions: – how to define a useful form function? • Vertices = f(dimensions, key pts, etc) – how to define a useful fitness function? • fitness = f(geometry only)? – generality vs. specificity – myth: computer functions can be random – myth: designers’ functions are randomTrouble with Functions in Design • Optimization question: – how to find the most fit form? • Pre, mid, post-optimization question: – how to handle emergence? • changing form, fitness functions during design • changing question in middle of trying to answer itArchitectural Function Optimization • Vertices = f(model parameters) • Quality = f(vertices) • Quality = f(model parameters) • Optimization = vary model parameters to maximize goodnessFunction Optimization max q(p0, p1, …, pn) Keeping c0(p0, p1, …, pn) = 0 and c1(p0, p1, …, pn) = 0 and… max q (p) keeping c(p) = 0 given some initial state p0Function Optimization max q (p) keeping c(p) = 0 p0: initial state • Goodness must be a single number – “multi-objective optimization” • The problem: given start, where to go next? – in which direction? – how far?Functions – Moving Around • Move in a direction for a certain distance • Start: (0,0) • Move to (2,2): moving in (1,1) direction for a distance of 2. • ‘Direction’ = a change in each parameter • ‘Distance’ = multiplier of a directionFunctions – Continuity • Changing continuous parameters: nicer – correlation: direction (1,1) roughly equals combination of effects from directions (1,0) and (0,1) – correlation: moving further along (.5, 4.2) tends to produce more of what that direction does – derivatives: can determine effect of parameter change without trying all possible changes • Combinatorial: no correlation between directions, between distance and outcome – (usually) – shape grammars are combinatorialOptimization Techniques • Steepest Descent • Simulated Annealing • Genetic AlgorithmsHill-Climbing/Steepest Descent • Go downhill – best downhill direction: downhill for each parameter – distance: select best along direction – can’t go downhill? Stop. • Local or global extrema? – fundamental problemSimulated Annealing • Jump around… • Jump around… • Jump up, jump up…. • …and get down.Simulated Annealing • Jump around/up... – provisionally jump around a distance j ( ‘taking’ the new position if it’s better - or not much worse) • …and get down – steadily lower jump distance j and uphill tolerance t • lower the energy of the system • steel cooling, water flooding – t = 0, small j: must get better at each step = hill-fallingGenetic Algorithms • GA concepts can be thought of functionally: – Phenotype = f(genotype) • form = f(dimensions) – Fitness = f(phenotype) • fitness = f(form) – Genotype = dimensions – Phenotype = formGenetic Algorithms • Evolution as functional optimization – arbitrary directions • generated by: – interleaving parameters of parents (crossover) – ‘random’ change of parameter (mutation) • ignore direction, distance correlations • assume crossover groups are independent of each other • generates invalid parameter sets • locking down substrings: when? how many? for how long? – multiple kids: parallel optimization • i.e., multiple directions at once – unknown improvement in kids • have to evaluate fitness of each kid after generating it – possibly no improvement in kids (convergence?)Shape Grammars - Generation Generate all shapes s(rules, s0, n) s0: initial shape rules: shape rules n: number of iterations • Fitness function inside human operator • Human optimizer changes parameters to increase fitness • Implicit fitness/recognition functionShape Grammars - Generation •(r0,r1,r2,…) = recognizer(shape, set of all rules) – recognizer compares left sides, allowing for translation, scale, and rotation – recognizer as local fitness function: only certain rules are fit for this situation •(r0,r1,r2,…) are a set of (equally good!) directions • Go in all directions, distance 1 (apply all valid rules once), producing shape0, shape1, …, shapen • Recurse on all the kid shapesShape Grammars - Optimization max q (s0,r0, r1, r2, .., rn) keeping c(s0, r0,r1,r2,..,rn) = 0 s0: initial shape n: number of rules to apply • Combinatorial representation – regardless of fitness function, no good way to search the spaceThe Tough Questions • Form function: What bogus forms are allowed? What useful forms are not allowed? What framework is embodied in the function? • Fitness function: What does ‘fitness’ mean? (Does it encode architectural knowledge? How?) • Representation of emergence? • Optimization: Is it finding the global minimum (by starting near solution or being a bowl-shaped function)? If not, what is it finding? • Recognition = binning based on sliding qualities – ‘x is awfully chair-like’ -> ‘x is a chair’ – ‘x is somewhat chair-like’ -> ‘x is a chair’?? – When is x not a chair? Who’s deciding, and how?Computers in Design: Future • support quick, narrow optimizations – support form changes • quick specification of formwork as it changes – brainstorming – show all combinations • more fine-grained study of frameworks and how they