Programming Spatial Computers. Jonathan Bachrach & Jacob Beal October, 2006
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1 Programming Spatial Computers Jonathan Bachrach & Jacob Beal October, 2006
2 Space-filling Computers Biological Computing Robot Swarms Sensor networks Distributed Control Systems PGAs Programmable Matter
3 Amorphous Medium Approach
4 Amorphous Medium Approach
5 Amorphous Medium Approach? Program the space... approximate with a network
6 Amorphous Medium Approach The discretization hardly matters!
7 Global v. Local v. Discrete Program Global Compiler Local Kernel Discrete
8 Global v. Local v. Discrete Program Global Compiler Local Kernel Discrete
9 Discrete Model Dozens to billions of simple, unreliable agents Distributed through space, communicating by local broadcast Agents may be added or removed No guaranteed global services (e.g. time, naming, routing, coordinates) Relatively cheap power, memory, processing Partial synchrony
10 Kernel Responsibilities: Emulate amorphous medium Time evolution Interface with sensors, actuators Viral reprogramming Current platforms: Simulator Mica2 Mote McLurkin's Swarm
11 Amorphous Medium Manifold (locally Euclidean space) Assume Riemannian, smooth, compact Simple locally, complex globally
12 Amorphous Medium Time Space Points access past values in their neighborhood Information propagates at a fixed rate c Evaluation is repeated at fixed intervals
13 Neighborhood Abstraction Program exposed state neighbor values half-phase UID Timer Area Range Lag Exposed State send ME N/A receive decay Aggregate access to best-effort estimate of neighbor state, space-time properties Neighbors decay without updates
14 Kernel Open Questions What is the optimal replacement policy when there are more neighbors than table memory? What is the optimal decay rate? How much energy can be saved by throttling update and decay rates? What are good ways to expose the cost/responsiveness tradeoff to the programmer?
15 Global v. Local v. Discrete Program Global Compiler Local Kernel Discrete
16 Expressions rgn An expression maps a manifold to a field rgn: M (M R)
17 Expressions rgn An expression maps a manifold to a field rgn: M (M R)
18 Operators rgn = T Operators map fields to fields (= rgn 2) =: (M R) (M R) (M B)
19 Composition & Abstraction unctional composition: operator expressions = expression operator operators = operator scope expression = operator } Lambda! Purely functional pointwise computation
20 Computation over Neighborhoods x T (or ((n xor) (nbrval x) (local x)))
21 Computation over Neighborhoods x T nbrval local T T (or ((n xor) (nbrval x) (local x))) local, nbrval select fields of neighborhood fields
22 Computation over Neighborhoods T T (n xor) T (or ((n xor) (nbrval x) (local x))) n applies an operator to neighborhood fields
23 Computation over Neighborhoods T or T T (or ((n xor) (nbrval x) (local x))) Measures (e.g. or, integral) reduce fields to values Sugar: (reduce nbrs or (xor x (local x)))
24 Neighborhood Open Questions Are the summary operations and, or, min, max, and integral sufficient for all approximatable continuous neighborhood computations? Are the field primitives local, nbrval nbrrange, nbr lag, and nbr bearing sufficient sources of neighborhood data? What is the discretization error of arbitrary composite neighborhood computations?
25 Conditional Computation x vent T (mux x (or (nbrval (restrict x vent)) #)
26 Conditional Computation x vent T restrict (mux x (or (nbrval (restrict x vent)) #) restrict limits the domain of a field
27 Conditional Computation nbrval T or T T (mux x (or (nbrval (restrict x vent)) #) operations proceed normally in the restricted field
28 Conditional Computation x T T T # mux T T (mux x (or (nbrval (restrict x vent)) #) mux constructs a field piecewise from inputs Sugar: (if x (or (nbrval vent)))
29 Computation with State 3 f delay 3 (delay default init) Previous values, current domain
30 Computation with State 3 f delay (delay default init) Previous values, current domain
31 Computation with State 3 f 3 2 delay 4 5 (delay default init) Previous values, current domain
32 Computation with State 3 f 3 1 delay 3 2 (delay default init) Previous values, current domain
33 Computation with State delay (letfed ((n 0 (+ n 1))) n)
34 Putting it all together State chains neighborhoods to arbitrary regions Example: relaxation to calculate distance (lambda (src) (letfed ((d (mux src 0 (reduce nbrs min (+ d nbr range))))) d)) 0
35 Putting it all together State chains neighborhoods to arbitrary regions Example: relaxation to calculate distance (lambda (src) (letfed ((d (mux src 0 (reduce nbrs min (+ d nbr range))))) d)) 0
36 Putting it all together State chains neighborhoods to arbitrary regions Example: relaxation to calculate distance (lambda (src) (letfed ((d (mux src 0 (reduce nbrs min (+ d nbr range))))) d)) 0
37 Putting it all together State chains neighborhoods to arbitrary regions Example: relaxation to calculate distance (lambda (src) (letfed ((d (mux src 0 (reduce nbrs min (+ d nbr range))))) d)) 0
38 Program Open Questions Under what conditions does continuous convergence imply discrete convergence? How do convergence properties compose? Given a continuous program and desired error bounds, what discretization will suffice? Given a continuous program and a discretization, what will the error bounds be?
39 Programs scale gracefully 100 nodes (and (green (dilate (sense 1) 30)) (blue (dilate (sense 2) 20)))
40 Programs scale gracefully 1,000 nodes (and (green (dilate (sense 1) 30)) (blue (dilate (sense 2) 20)))
41 Programs scale gracefully 10,000 nodes (and (green (dilate (sense 1) 30)) (blue (dilate (sense 2) 20)))
42 Target Tracking
43 Target Tracking (def local-average (v) (/ (reduce-nbrs v integral) (reduce-nbrs integral 1))) (def gradient (src) (letfed ((n infinity (+ 1 (mux src 0 (reduce-nbrs min (+ n nbr-range)))))) (- n 1))) (def grad-value (src v) (let ((d (gradient src))) (letfed ((x 0 (mux src v (2nd (reduce-nbrs min (tup d x)))))) x))) (def distance (p1 p2) (grad-value p1 (gradient p2))) (def channel (src dst width) (let* ((d (distance src dst)) (trail (<= (+ (gradient src) (gradient dst)) d))) (dilate width trail))) (def track (target dst coord) (let ((point (if (channel target dst 10) (grad-value target (mux target (tup (local-average (1st coord)) (local-average (2nd coord))) (tup 0 0))) (tup 0 0)))) (mux dst (vsub point coord) (tup 0 0))))
44 Threat Avoidance
45 Threat Avoidance (def exp-gradient (src d) (letfed ((n src (max (* d (reduce-nbrs max n)) src))) n)) (def sq (x) (* x x)) (def dist (p1 p2) (sqrt (+ (sq (- (1st p1) (1st p2))) (sq (- (2nd p1) (2nd p2)))))) (def l-int (p1 v1 p2 v2) (pow (/ (- 2 (+ v1 v2)) 2) (+ 1 (dist p1 p2)))) (def max-survival (dst v p) (letfed ((ps 0 (mux dst 1 (reduce-nbrs max (* (l-int p v (local p) (local v)) ps))))) ps)) (def greedy-ascent (v coord) (- (2nd (reduce-nbrs max (tup v coord))) coord)) (def avoid-threats (dst coords) (greedy-ascent (max-survival dst (exp-gradient (sense :threat) 0.8) coords) coords))
46 uture Directions Continuous time evaluation Analysis of distortion from space discretization Evaluation on a changing manifold Actuation of the manifold Applications!
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