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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations1 Detecting and measuring randomness in process-algebraic computations Tommaso Bolognesi CNR - ISTI - Pisa

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations2 Abstract A technique is introduced for visually characterizing, by 2-D plots, the complexity of process-algebraic computations -- an area that has not been directly invested, so far, by NKS research. An extension of the familiar notion of functional derivation is proposed, that reveals some interesting properties of this class of plots. Pseudo-random features seem to emerge even for simple subsets of process algebra that are regarded as uncapable of universal computations, at least w.r.t. the standard notion of Turing universality. This apparently surprising result suggests to complement the visual inspection of process-algebraic diagrams with refined, numeric techniques for measuring their degree of randomness. In particular we explore the application of a density- dependent compressibility measure, based on pointer-encoding. We have tested this technique by applying it also to the family of elementary cellular automata, where it does indeed prove useful for discriminating between computations, most notably within Class 3.

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations3 Contents Process Algebra (PA) as a citizen of the World of Simple Programs A set of universal PA operators and a useful 2D visual indicator Emergent features from PA subclasses –particles - derivatives - exponential and fibonacci - emulation by compression - emergence of randomness How much random? How much universal? –PEncode compression values for PA computations –Non universality of a PA class with randomness-capability Next

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations4 Process Algebra(s) Formal methods for Software Engineering Models/languages for the specification and verification of concurrent systems (interaction, communication) –CCS, ACP, CSP, LOTOS, … behavior expression: an algebraic term built by operators P := a; [b; stop [+] c; P] A process definition … …its standard interpretation as a Labeled Transition System (LTS) a a b b a c c SOS rules syntaxsemantics

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations5 Seven PA operators and their SOS rules… …yielding a special multiway, symbolic, non-local rewrite system easily coded in Mathematica: B ===[SOS rules]===> {(a1, B1), …, (an, Bn)}

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations6 PA under NKS light Not a typical citizen of the World of Simple Programs –Syntax: more parameters than average NKS citizen ==> huge spaces, hard to structure and to explore –Semantics: several options (operational, denotational, axiomatic -- LTS, true concurrency, refusal sets…) reactivity (interactive systems)/concurrency/nondeterminism event-based and state-based Some universality results available Objectives: –find good 2D visual indicators –spot emergent features for different PA sub-classes –spot randomness before universality ?

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations7 Non standard interpretation of deterministic specs (no branching) a a a a P := (a; a; P) |[a]| (a; P) behavior expression with parallel composition op. operators in prefix form PA plots: different grey levels for different operators

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations8 Proof: PA spec using all 7 operators for emulating ECA 110 Theorem 1 - The chosen operator set is universal compression

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations9 Regular plots: costant or linear growth

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations10 Exponential growth - derivatives a n = a n-1 + a n-1 (* Base 2 *) a n = a n-1 + a n-2 (* Fibonacci *) analogy with (parallel) substitution systems [NKS, p. 82]

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations11 Emulation by compression (3) P := a[a[par[P, {a}, P]]] (4) P := a[a[par[a[a[P]], {a}, P]]]

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations12 first and second derivative (5) P := a[a[par[P, {a}, a[P]]]]

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations13 Quadratic growth derivatives (6) P := a[par[P, {a}, choice[Q, stop]]] Q := a[hideB[choice[choice[Q, Q], Q]]] (7) P := a[par[P, {a}, choice[Q, stop]]] Q := a[par[par[stop, {}, Q], {}, stop]] (8) P := par[a[Q], {b}, b[P]] Q := b[choice[stop, hideA[Q]]] (9) P := a[par[choice[Q, Q], {a}, P]] Q := a[par[stop, {}, Q]] (10) P := a[par[par[stop, {}, Q], {a}, P]] Q := par[a[par[stop, {}, Q]], {}, stop]

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations14 Emergence of randomness (?) (11) P := a[hideB[par[Q, {a}, hideB[Q]]]] Q := a[par[P, {}, stop]] (12) P := a[hideB[par[Q, {a}, Q]]] Q := a[par[P, {}, stop]]

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations15 (13) P := c[hideA[par[Q,{a,b,c,d},choice[Q,P]]]] Q := d[swap1[choice[P,stop]]] R := a[hideC[choice[c[P],P]]] (14) P := b[par[b[swap2[choice[stop,R]]],{a,b,d},P]] Q := b[hideA[choice[stop,P]]] R := d[swap2[par[hideC[P],{a,b,c,d},P]]],

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations16 (15) P := a[swap1[par[stop,{a,c},swap1[R]]]] Q := a[par[c[stop],{b},R]] R := a[par[P,{a},Q]] swap1 is a particular instance of the relabeling operator

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations17 How much random? How much universal? Measure randomness via Pencode-compressibility (thanks to S. Wolfram!) Find simplest PA subclass exhibiting maximum randomness (*), and …question its universality (*) relative to selected measure and sample size

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations18 Pencode compression of spec (15) - 1st derivative - 23 rows row length compress. value for equivalent (*) random row of reference compression value for actual row (*) of same length, alphabet, and distribution … rows are bad

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations19 = actual compression values = compression values of one equivalent random tuple of reference compress rows and columns of a square region of spec (15) … columns are good

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations20 Pencode compression of a fragment of spec (5) - 1st derivative = column lengths = actual compression values = averaged compression values of equivalent random tuple of reference … these columns are good too…

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations21 spec(5) is based on a PA subset with only –action prefix a[…] –parallel par[{a},…,…] –instantiationP Theorem 2: the above PA operator subset is not universal –even if we add the inaction operator (stop) Proof –by induction on the structure of behavior expressions –shows that any such PA specification is equivalent to a context-free rewrite system, which cannot be universal [] How much random? How much universal?

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations22 Conclusions Useful 2D visual indicators for PA identified –similarity with NKS symbolic system / combinator diagrams [p. 103] –original notion of derivative Emergence of randomness –spotted visually –measured by Pencode compressibility - for limited size data sets Elements collected for questioning class 3 universality conjecture –but: is randomness-capability a clear cut property? Next: –Other measures of randomness degree –Random-like features in derivative plots of context-free rewrite systems –Other notions of universality, e.g. intermediate degrees (Davis, Sutner - thanks to M. Szudzik!) –Maximize non-universal operator set in th. 2

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations23 References Bolognesi, T.: Process Algebra under the light of Wolframs NKS. In: A. Gordon and L. Aceto (editors) - Proceed. of APC Electronic Notes in Theoretical Computer Science, Elsevier, Davis, M.: The definition of universal Turing machines, Proc. of the American Mathematical Society, 8: , Sutner, K.: Universality and cellular automata. In Proceedings of MCU LNCS 3354, pp , Springer-Verlag, Wolfram, S.: A New Kind of Science, Wolfram Media, Inc., 2002.

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations24 rows C[row]D[row] C[Rand[Length[row], D[row]]] ± StDev D[row] = (num. of 1s in row)/Length[row] Density C[row] = Length[PEncode[row]]Compression value Rand[length, prob1] = random bit tuple of given length and probability of 1 Pencode compressibility in ECAs

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations25 18, * 22,151106,120,169,225 * 30,86,135,149 *122,161 45,101,75,89 *126,129 60,102,195,153 *146,182 90,165 *150 * ECA Class symmetry families - 30 members Bold: density = 1/2; Italics: Pencode-ottimale * = family multiple of 15

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations26 Class ,22,30, 45,60,90…

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations27 Class 3 - …105,106,122, 126,146,150

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations28 remarks ECA is pencode-optimal ===> ECA keeps = 1/2 The converse is false (122, 126) ECA is Pencode-optimal ===> the family includes a multiple of 15 Overall, there are 66 Pencode-optimal ECAs, that include all 18 multiples of 15

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NKS06 - Washington DC - June 2006T. Bolognesi - Randomness in Process algebraic computations29 Class 4 - (54,147), ( 110,124,137,193)

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