Syntaxe, smantique, calculabilit

Playing with Time and Spacein Circuits and ProgramsGrard BerryCollge de FranceChaire Algorithmes, machines et langages PPS, 24/04/2013

125/04/2013 2G. Berry, PPS, Agenda2-adic numbers and space / time exchange in synchronous circuitsNever determinize non-deterministic automata !Use hierarchical automata for another exponential gain in space and timing optimization25/04/2013 3G. Berry, PPS, Agenda2-adic numbers and space / time exchange in synchronous circuitsNever determinize non-deterministic automata !Use hierarchical automata for another exponential gain in space and timing optimization25/04/2013 4G. Berry, PPS, Source of the 2-adic PartJ. Vuillemin.On circuits and numbers, IEEE Trans. on Computers, 43:8:868-79, 1994

25/04/2013 5G. Berry, PPS, 2-adic Numbers (Hensel, ~1900)R is a completion de Q. Is it the only one ? No : p-adic numbers for p prime infinite numbers written low-order bits first2-adic numbers unify computable arithmeticwith Boolean logicBeautiful, but physical ? cf. Matire Pense, p. 32 Alain Connes / JP ChangeuxJean Vuillemin : 2-adiques integers are the right modelJean Vuillemin : of arithmetic digital circuits Jean Vuillemin : Let us create their physics!

25/04/2013 6G. Berry, PPS, 2 Z : the Ring of 2-adic Numbersx 2 x0 x1 x2 low-order bits firstoperations and from left to rightx 2101010... 2 (10)x 1 4xx 1/3x 2100000... 2001010... 0 200000... 2 (0)1 210000... 21(0) 2 201000... 201(0) 1 211111... 2 (1)2 201111... 2 0(1) y 2010101... 2 (01)y 2 xy 2/3or x y 125/04/2013 7G. Berry, PPS, 2 Z : the Ring of 2-adic Numbers1 / 2 does not existbecause x0x0 cannot have value 1 p / q exists for all integer p, q iff q est odd(cf. Euclide)No order compatible with the operations1 0 125/04/2013 8G. Berry, PPS, 2 Z as a Boolean Algebra2-adic x seen as the set { i | xi 1 } example: 1/3 2101010... { i | i even }pointwise Boolean operations x y x y x (x y)n x n yn etc.2100011... 2011100... 2111111... x x 1Fundamental arithmetico-logical equality :25/04/2013 9G. Berry, PPS, Cantor Metric Spaced(x,x) 0d(x,y) 2n n minimal s.t. xn ynLemma : 2 Z is ultrametric : d(x,z) max (d(x,y), d(y,z))Example : d (201111..., 201101... 1/8)xyzd(x,z) min (d(x,y), d(y,z))925/04/2013 10G. Berry, PPS, Cantor Metric SpaceCompact very different from reals !ex.: open set for 210010 Open set basis : finite prefixes x0 x1...xn { 2 x0 x1...xn y0 y1...yn... | y 2 Z }110001025/04/2013 11G. Berry, PPS, Continuous functionsLemma : f : 2 Z 2 Z continuous iff f(x)n depends on a finite number of xm2 x0 x1......xm... 2 y0 y1...yn... Continuity preservation of information finiteness25/04/2013 12G. Berry, PPS, Synchronous Functionsxf(x)x0 x1...xn...0 x0 x1...xn...

25/04/2013 13G. Berry, PPS, Synchronous and Contracting FunctionsTheorem : f : 2 Z 2 Z is synchronous iff f(x)n only depends on x0x1...xn, i.e., iff f is contracting x,y. d(f(x),f(y)) d(x,y)Definition : f : 2 Z 2 Z synchronous iff computable by a synchronous circuit (with finite or infinite memory)Preuve : only if trivial, Preuve : for if see SDD construction later on xf(x)25/04/2013 14G. Berry, PPS, Moore Circuits and Strict ContractionA function f : 2 Z 2 Z is strictly contracting iff f(x)n only depends on x0x1...xn1 x,y. d(f(x),f(y)) d(x,y)A Moore synchronous circuit is such that any wire between an input and an output traverses a registerTheorem : a function is computable by a Moore circuit if and only if it is contractingMooreCircuit25/04/2013 15G. Berry, PPS, Feedbacks in Moore Circuits are LegalMooreCircuitf(x)x25/04/2013 16G. Berry, PPS, Feedbacks in Moore Circuits are Legal x,y. d(f(x),f(y)) d(x,y)Lipschitz x,y. d(f(x),f(y)) 0,6 d(x,y)Banach theorem: any Lipschitzian function over a compact set has a unique fixpointMooreCircuitf(x) xf(x)25/04/2013 17G. Berry, PPS, Full Adderabcsrs a xor b xor cr (a and b) or (b and c) or (c and a)++abscrxororandandandbitsEinsten et lpaisseur de linstant.Passer du continu au discret et du non-dterminisme au dtermininme

1725/04/2013 18G. Berry, PPS, Addition in Space++a0b0s0++1a1b1sr1++2a2b2s2r3rr = 00s a bbut within infinite time!continuity:cut at n bits for n output bitss2n1 a2n b2nx2n x mod 2n

25/04/2013 19G. Berry, PPS, Full Adderabcsrs a xor b xor cr (a and b) or (b and c) or (c and a)++abscrxororandandandbitsa b c s 2 r Einsten et lpaisseur de linstant.Passer du continu au discret et du non-dterminisme au dtermininme

1925/04/2013 20G. Berry, PPS, Basic 2-adic Operators++abscrx2 xa b c s 2 r x1 2 x2 x0 x1...xn...2 0 x0 x1...xn...2 x0 x1...xn...2 1 x0 x1...xn...25/04/2013 21G. Berry, PPS, Addition and Subtraction Over Time++absr2rtick ! a b 2 r s 2 r s a bsame equationas over space!a b 1 2 r s 2 r b 1 bs a b12 r++absrb b 1a b s

25/04/2013 22G. Berry, PPS, Mixed Space / Time Additionro

++aebe

se

2ro

tick ! s a bstill the sameequation !++ao

bo

so

re

a ae ao

b be bo

s se so

x y 2 x0 y0 x1 y1...

Same source codefor any space / time tradeoff25/04/2013 23G. Berry, PPS, Stereo Addition++absrstereoadderAlternates 2 additions over time (even / odd bits)stereo left / right channelsse so (ae be) (ao bo)

25/04/2013 24G. Berry, PPS, Addition and Subtraction Over Time++abstick ! ++absabsabs25/04/2013 25G. Berry, PPS, Multiplication and division by a constantx3 x_xy x / 3proof : y x 2 ydivision onlyby odd integers!proof : x 2 x 3 x25/04/2013 26G. Berry, PPS, Quasi-inversey 1 2 x y 1 / (12 x)xy 2 x y 1y 1 2 x yy 1 / (12 x)?contracting synchronousbut infinite memory(cf. SDD construction) 25/04/2013 27G. Berry, PPS, Quasi Square Rooty 1 4 zz x 2 z 2y 2 1 8 z 16 z 2y 1 8 xxzyy 2 1 8 x 16 z 2 16 z 2... but tells us nothingabout bit transformations!

25/04/2013 28G. Berry, PPS, Spatio-Temporal Decomposition of f Synchronousf 0 0-predictor : f 0w = f (w0) for any word wf 0 first bit output by f for inputs 0...f 1 ... 1...f 1 1-predictor : f 1w = f (w1)f w last bit output by f for the finite word wf u u-predictor : f uw = f (wu) for any words w, u

25/04/2013 29G. Berry, PPS, Automaton of x 3 x+x3 x001001110 / 01 / 10 / 11 / 10 / 10 / 01 / 01 / 00 /1 / 10 / 00 /25/04/2013 30G. Berry, PPS, Predictor 0 of x 3 x+x3 x001001110 / 01 / 10 / 11 / 10 / 10 / 01 / 01 / 0001001111 /0 /1 /0 /1 /1 /0 /01 / 00 / 11 / 01 / 00 / 025/04/2013 31G. Berry, PPS, SDD Decomposition Stepf(x) = mux(x, f 1 2 f 1(x), f 0 2 f 0(x))x10f 1f 0f (x)f 1f 025/04/2013 32G. Berry, PPS, SDD Space/Time Normal Form of fTruth-table in space and timeultra-fast : critical path one muxHalf of the bits disappear at each cycle10f11f10x10f01f00xf 01f 00

f 11f 10

f 1f 0x10f1f0f (x)............25/04/2013 33G. Berry, PPS, Shared SDD of f with Finite Memory10f11f10x10f01f00xf 01f 00

f 11f 10

f 1f 0x10f1f0f (x)............

f finite memory finitely many distinct predictors f uf with n registers SDD(f ) may have22 registersn25/04/2013 34G. Berry, PPS, From continuous functions to circuitsf continuous but not synchronous: over space : trivial if infinite spaceover time : expand the time2-adic number : < value, validity >0 0 1 0 1 1 0 1 1 1 0 0 ...0 1 1 0 0 1 1 0 0 1 1 0 ... 0 1 1 0 1 Theorem : every continuous function can berealized by a synchronous circuit with validity25/04/2013 35G. Berry, PPS, Trace of a Synchronous FunctionTr(f) 2 f0 f1 f00 f01 f10 f11 f000 f001 ... f0 2 f1 4 (Tr(f 0) Tr(f1 ))Power series over Z/2Z : S(f) = n Tr(f)n znTheorem : f : 2 Z 2 Z synchronous has finite memory iff S(f) is algebraic over Z/2ZApplication of a trace Tr(f) to an argument x is continuous calculus ?Theorem (Van der Porten) : if f has finite memory, them the real number 0, f0 f1 f00 f01 f10 f11 f000 f001 ...is either rational or transcendantal25/04/2013 36G. Berry, PPS, From Synchronous Traces to Transcendental NumbersAutomatic Sequences: Theory, Applications, GeneralizationsJean-Paul Allouche et Jeffrey ShallitCambridge University Press (21 juillet 2003)

Almost any finite automaton generates a transcendental number!2-adic numbers are the good model of arithmetic synchronous circuits (only?)the 2-adic metric, continuity, and synchronism are fundamental notions to explore further The structure of the predictor space is largely unknownThe relation between continuous functions and validity-circuits remains to be studied (calculus?)25/04/2013 37G. Berry, PPS, Conclusion

Thanks to Jean Vuillemin25/04/2013 38G. Berry, PPS, Agenda2-adic numbers and space / time exchange in synchronous circuitsNever determinize non-deterministic automata !Use hierarchical automata for another exponential gain in space and timing optimization25/04/2013 39G. Berry, PPS, From Deterministic Automata to Circuits(ab+b)*bae1e2e3e0abbabb

r0r2r1r3abbbabok1-hot encoding(only one ri to 1)size explosion!

25/04/2013 40G. Berry, PPS, The Non-Deterministic Case(ab+b)*bano size explosion much better!

bbbaaoks0s1s2s3abbab25/04/2013 41G. Berry, PPS, Electrical Subset Construction

bbbaaokas0s1s2s3abbab(ab+b)*ba25/04/2013 42G. Berry, PPS, Electrical Subset Constructions0s1s2s3abbab(ab+b)*ba

bbbaaokatick!25/04/2013 43G. Berry, PPS, Electrical Subset Constructions0s1s2s3abbab(ab+b)*ba

bbbaaokab25/04/2013 44G. Berry, PPS, Electrical Subset Constructions0s1s2s3abbab(ab+b)*ba

bbbaaokabtick!25/04/2013 45G. Berry, PPS, Electrical Subset Constructions0s1s2s3abbab(ab+b)*ba

abbbbbaaok25/04/2013 46G. Berry, PPS, Electrical Subset Constructions0s1s2s3abbab

abbbbbaaoktick!25/04/2013 47G. Berry, PPS, Electrical Subset Constructions0s1s2s3abbab(ab+b)*ba

abbabbbaaok25/04/2013 48G. Berry, PPS, Electrical Subset Constructions0s1s2s3abbab(ab+b)*ba

abbabbbaaoktick!25/04/2013 49G. Berry, PPS, Esterel v7 implementationmodule Autom :input a, b ;output ok; local {r0, r1, r2, r3} : reg;refine r0 : init true;sustain { next r0