1 / 17

Introduction to Probabilistic Process Algebra: Key Concepts and Bisimulation Theory

This document explores key principles of Probabilistic Process Algebra (PPA) and its application in modeling Probabilistic Labelled Transition Systems (PLTS). It covers fundamental components such as states, transitions, and probabilistic choices, illustrating how they can be used to calculate total probabilities and establish bisimulation relations. Additionally, it provides axioms for PBPA and exercises for developing algebraic specifications of PLTSs, enhancing understanding of non-deterministic and probabilistic behaviors in complex systems.

shay
Télécharger la présentation

Introduction to Probabilistic Process Algebra: Key Concepts and Bisimulation Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Process Algebra (2IF45)Probabilistic extension Dr. Suzana Andova

  2. Probabilistic LTS • Basic ingredients of a PLTS: • states • non-detereministic states set N • probabilistic states set P • transitions • action transitions labelled with actions and t P • probabilistic transitions labelled with probabilities and t N • For a probabilistic state s, •   = 1 a s  t  s  t sel_d sel_c 5/6  s  t ½ not_six ½ 1/6 t six 1 h h pass t Process Algebra (2IF45)

  3. Probabilistic LTS sel_d sel_c 1 5/6 2 9 10 ½ not_six ½ 1/6 6 six t 3 5 h 1 1 pass 11 1 4 7  s  t, t  C 8 1 h t For a probabilistic state sP and a set of states C  S we define the total probability of s to reach C as Prob(s,C) =   Process Algebra (2IF45)

  4. Probabilistic LTS sel_d sel_c 1 5/6 2 9 10 ½ not_six ½ 1/6 6 six t 3 5 h 1 1 pass 11 1 4 7 8 1 h Examples: Prob(9, {3,4}) = 1/6 Prob(9, {1}) = 0 Prob(10,{5,6}) = 1 Process Algebra (2IF45)

  5. Probabilistic LTS 1 C1 = {9, 13}, C2 = {10,11,12}, Prob(1, {2}) = 1/3 Prob(1, {3}) = 2/3 Prob(1,C1) = Prob(1,C2) = 0 Prob(5,C2) = 1 Prob(6,C2) = 1 Prob(7,C2) = 1/3 Prob(5,C1) = 0 Prob(6,C1) = 0 Prob(7,C1) = 2/3 Prob(4,C1) = 1 Prob(8,C1) = 1 Prob(4,C2) = 0 Prob(8,C2) = 0 2/3 1/3 2 3 a b a b b 4 5 6 7 8 1 1 2/3 1 1 1/3 9 10 12 11 13 c c c c c c c Process Algebra (2IF45)

  6. Probabilistic LTS 1 E1 = {9,10,11,12,13}, E2={2,3} Prob(1, E2) = 1 Prob(1,E1) = Prob(1,E2) = 0 Prob(5,E1) = 1 Prob(6,E1) = 1 Prob(7,E1) = 1/3 + 2/3 = 1 Prob(4,E1) = 1 Prob(8,E1) = 1 Prob(4,E2) = …=Prob(8,E2)=0 2/3 1/3 2 3 a b a b b 4 5 6 7 8 1 1 2/3 1 1 1/3 9 10 12 11 13 c c c c c c c Process Algebra (2IF45)

  7. Probabilistic LTS 1 E1 = {9,10,11,12,13}, E2={2,3} Prob(1, E2) = 1 Prob(1,E1) = Prob(1,E2) = 0 Prob(5,E1) = 1 Prob(6,E1) = 1 Prob(7,E1) = 1/3 + 2/3 = 1 Prob(4,E1) = 1 Prob(8,E1) = 1 Prob(4,E2) = …=Prob(8,E2)=0 2/3 1/3 2 3 a b a b b 4 5 6 7 8 1 1 2/3 1 1 1/3 9 10 12 11 13 c c c c c c c Process Algebra (2IF45)

  8. Probabilistic LTS 1 E1 = {9,10,11,12,13}, E2={2,3} Prob(1, E2) = 1 Prob(1,E1) = Prob(1,E2) = 0 Prob(5,E1) = 1 Prob(6,E1) = 1 Prob(7,E1) = 1/3 + 2/3 = 1 Prob(4,E1) = 1 Prob(8,E1) = 1 Prob(4,E2) = …=Prob(8,E2)=0 2/3 1/3 2 3 a b a b b 4 5 6 7 8 1 1 2/3 1 1 1/3 9 10 12 11 13 c c c c c c c 2 –a-> 4 2-b->5 3 –a-> 8 3-b->6 3-b->7 Process Algebra (2IF45)

  9. Bisimulation on LTSs • An equivalence relation R on the set of state S = N  P of an PLTS is probabilistic strong bisimulation iff the following conditions hold: • for all states s, tN, whenever (s, t)  R and s –a-> s’ for some a  A and s’ S, then there is a state t’ S such that t –a-> t’ and (s’, t’)  R; • for all states s, tP, Prob(s, C) = Prob(t, C) for all C S/R • Two PLTSs s and t are probabilistic strong bisimilar, s p t, iff there is a probabilistic strong bisimulation R such that (s, t)  R Process Algebra (2IF45)

  10. Probabilistic LTS 1 2/3 1/3 1 2 3 a b a b a b b 4 5 6 7 8 1 1 1 1 2/3 1 1 1/3 9 10 12 11 13 c c c c c c c c Process Algebra (2IF45)

  11. Composing PLTSs 1/2 1 1/2 1 1/2 = b b a a 1/6 1/3 1/3 1/2 1/3 1/2 2/3 1/6 + = d c b a b c d a b d a c

  12. Composing PLTSs: Parallel composition Assume two persons flipping a coin (independently), one coin per person. If they both have the same sides appeared then they can continue (whatever with). If they have different sides thrown they stop. Model the persons and model the system as a composition.

  13. Composing PLTSs 1/6 1/6 1/3 1/3 1/3 1/2 2/3 1/2 || = c c b d a b d a e b d a c c a b c d b d a where a and c communicate in e, and no other communication is defined (in this examples)

  14. Equational theory. Language • Specify processes that can execute certain actions from a given set A • The language of the Probabilistic Basic Process Algebra, namely, the operators in the signature • 0 deadlock constant (inaction) • a._ action prefix for a in A • + non-deterministic choice •  probabilistic choice for   (0,1) Process Algebra (2IF45)

  15. Axioms of PBPA(A) PBPA(A) Signature: 0, a._ , _+_ ,  • (A1) x+ y = y+x • (A2) (x+y) + z = x+ (y + z) • (A3) x + x = x • (A4) x+ 0 = x Process Algebra (2IF45)

  16. Axioms of PBPA(A) PBPA(A) Signature: 0, a._ , _+_ ,  • (PA1) x  y = y 1- x • (PA2) x(y  z) = (xy)  z • where = /( + - ) and  =  + -  • (PA3) x  x = x • (PA4) (x  y) + z = (x + z)  (y + z) Process Algebra (2IF45)

  17. Exercises • Write the process algebraic specification of the PLTSs from the examples. • For the PLTSs on slide 9 derive using the axioms the equality between the two PBPA specifications. Process Algebra (2IF45)

More Related