Detecting and measuring randomness in processalgebraic computations

1
Detecting and measuring randomnessin
process-algebraic computations

• Tommaso Bolognesi
• CNR - ISTI - Pisa

2
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.

3
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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Process Algebra(s)

• Formal methods for Software Engineering
• Models/languages for the specification and
  verification of concurrent systems
  (interaction, communication)
• CCS, ACP, CSP, LOTOS,

syntax
semantics
its standard interpretation as a Labeled
Transition System (LTS)
A process definition
a
SOS rules
b
c
P a b stop c P
a
b
c
behavior expression an algebraic term built by
operators
a
5
Seven PA operators and their SOS rules
yielding a special multiway, symbolic, non-local
rewrite system easily coded in Mathematica B
SOS rulesgt (a1, B1), , (an, Bn)
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PA under NKS light

• Not a typical citizen of the World of Simple
  Programs
• Syntax
• more parameters than average NKS citizen
• gt huge spaces, hard to structure and to explore
• Semantics
• several options (operational, denotational,
  axiomatic -- LTS, true concurrency, refusal
  sets)
• reactivity (interactive systems)/concurrency/nonde
  terminism
• 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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Non standard interpretation of deterministic specs
a
P (a a P) a (a P)
a
a
behavior expression with parallel composition op.

• (no branching)

a
operators in prefix form
PA plots different grey levels for different
operators
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ProofPA spec using all 7 operators for
emulating ECA 110
Theorem 1 - The chosen operator set is universal
compression
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Regular plots costant or linear growth
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Exponential growth - derivatives
an an-1 an-1 ( Base 2 ) an an-1
an-2 ( Fibonacci )
analogy with (parallel) substitution systems
NKS, p. 82
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Emulation by compression
(3) P aaparP, a, P
(4) P aaparaaP, a, P
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first and second derivative
(5) P aaparP, a, aP
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Quadratic growth
(6) P aparP, a, choiceQ, stop Q
ahideBchoicechoiceQ, Q, Q
(7) P aparP, a, choiceQ, stop Q
aparparstop, , Q, , stop
(8) P paraQ, b, bP Q
bchoicestop, hideAQ (9) P
aparchoiceQ, Q, a, P Q
aparstop, , Q (10) P aparparstop,
, Q, a, P Q paraparstop, ,
Q, , stop

• derivatives

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Emergence of randomness (?)
(11) P ahideBparQ, a, hideBQ
Q aparP, , stop
(12) P ahideBparQ, a, Q Q
aparP, , stop
15

(13) P chideAparQ,a,b,c,d,choiceQ,P
Q dswap1choiceP,stop R
ahideCchoicecP,P
(14) P bparbswap2choicestop,R,a,b,d,P
Q bhideAchoicestop,P R
dswap2parhideCP,a,b,c,d,P,
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(15) P aswap1parstop,a,c,swap1R
Q aparcstop,b,R R
aparP,a,Q
swap1 is a particular instance of the relabeling
operator
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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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Pencode compression of spec (15) - 1st derivative
- 23 rows
row lengthcompress. value for equivalent ()
random row of referencecompression value for
actual row
rows are bad
() of same length, alphabet, and distribution
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compress rows and columns of a square region of
spec (15)

• actual compression values
• compression values of one equivalent random
  tuple of reference

columns are good
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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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How much random? How much universal?

• spec(5) is based on a PA subset with only
• action prefix a
• parallel para,,
• instantiation P
• 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

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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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References

• Bolognesi, T. Process Algebra under the light of
  Wolframs NKS. In A. Gordon and L. Aceto
  (editors) - Proceed. of APC 2005 - Electronic
  Notes in Theoretical Computer Science, Elsevier,
  2005.
• Davis, M. The definition of universal Turing
  machines, Proc. of the American Mathematical
  Society, 8 1125-1126, 1957.
• Sutner, K. Universality and cellular automata.
  In Proceedings of MCU 2005 - LNCS 3354, pp.
  50-59, Springer-Verlag, 2005.
• Wolfram, S. A New Kind of Science, Wolfram
  Media, Inc., 2002.

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Pencode compressibility in ECAs
CRandLengthrow, Drow StDev
Crow
Drow
rows
rows
rows

• Drow (num. of 1s in row)/Lengthrow
  Density
• Crow LengthPEncoderow Compression value
• Randlength, prob1 random bit tuple of given
  length and probability of 1

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ECA Class 3 - 12 symmetry families - 30 members
Bold density 1/2 Italics Pencode-ottimale
ltgt family multiple of 15
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Class 3 - 18,22,30, 45,60,90
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Class 3 - 105,106,122, 126,146,150
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remarks

• ECA is pencode-optimal gt ECA keeps 1/2
• The converse is false (122, 126)
• ECA is Pencode-optimal gt 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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Class 4 - (54,147), (110,124,137,193)