Distributed On-the-Fly Resolution of Boolean Equation Systems

Distributed On-the-Fly Resolution of Boolean Equation Systems C. Joubert

R. Mateescu

INRIA Rhône-Alpes / VASY http://www.inrialpes.fr/vasy SENVA’2005 (Montbonnot, France) November 16-17

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Outline

1. Boolean Equation Systems (B ES) Local Resolution of B ES Related Work 2. Distributed Local Resolution of B ES Distributed Algorithm DS OLVE Implementation and Experiments

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Boolean Equation Systems (B ES)

B ESs, Boolean Graphs, and Local Sequential Resolution x1 x2 x3 x4 x5 x6

µ

= x2 ∨ x3 µ = x4 ∨ x5 µ = x2 ∧ x5 ∧ x6 µ = x4 ∧ x5 µ =T µ = x3 ∨ x1

x1 ∨ x2

x3 ∧

∨ ∧ x4

C. Joubert, R. Mateescu

∧x5

x6 ∨

I

σ

B ES {xi = opi Xi }1≤i≤n Set of boolean fixed point equations Pure disjunctive or conjunctive formulas Absence of negations to ensure monotonicity of least fixed point

I

Boolean graph G = (V , E, L) associated to a B ES V = {x1 , ..., xn } E = {(xi , xj )|xj Xi } L : V → {∨, ∧}, L(xi ) = opi SENVA’2005

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Boolean Equation Systems (B ES)

B ES Application Areas

Verification of concurrent systems I

Equivalence checking [Andersen-Vergauwen-95]

I

Model-checking [Andersen-94]

I

Partial order reduction [Pace-Lang-Mateescu-03]

Propositional logic programming I

Horn clauses satisfiability [Liu-Smolka-98]

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Boolean Equation Systems (B ES)

Motivation for Distributed B ES Resolution 1

Limitation of Sequential B ES Resolution Methods Memory I

B ES with more than 108 variables to solve (current sequential machines swap around 107 variables)

Time I

x1 ∨

Traversals of very large B ES (the larger is the B ES, the more resolution tasks will have the process)

1

x3 ∧

x4 ∧

2 x2 ∨

x5 ∧

3 x6 ∨

communication network

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Reasons for Distribution Running faster with few memory used per machine Regular problem prone to balanced distribution of tasks and data SENVA’2005

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Boolean Equation Systems (B ES)

Related Work

Distributed Local Resolution using Game Graphs [Bollig-Leucker-Weber-02] I

I

Algorithm specialized for model checking properties expressed in a fragment of the modal µ-calculus Operates on game graphs (⇒ Alternative representation to boolean graphs) Vertices = configurations in two-player games Edges = moves

I

Effective speedups obtained for model checking

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Distributed Local Resolution of B ES

The DS OLVE Algorithm Computation model I Distributed memory architecture (message passing): N OW , cluster of P C I P S PMD processes and 1 coordinator process I Each process solves a subset of boolean variables determined by a static hash function Distributed algorithm I Forward exploration of boolean graph (V , E, L) starting at a variable of interest xV I Backward propagation of stable (computed) variables I Distribution (communication) of variables through remote dependencies I Termination detection when x stable or boolean graph entirely solved C. Joubert, R. Mateescu

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Distributed Local Resolution of B ES

Execution of DS OLVE on previous B ES example 1

Initialization (variable of interest x1 )

2

Local expansion and remote expansion (message E XP)

3

Conjunctive variable whithout successor (i.e., constant true)

4

Local and remote (message E VL) back-propagation of stabilized (i.e., computed) variables

5

If variable of interest stabilizes, then resolution terminates

x 2) 1,

x1 true P(x ∨ EX x3 , x2 ) x2 true E V L(x1 ∧ ∨ E XP(x2 , x4 )

5)

x 6)

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x2

,x

5)

,x

x2

P(

EV

x 3,

P(

P3

EX

EX

x4 ∧ x5 true ∧

P2

L(

P1

x6 ∨

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Distributed Local Resolution of B ES

Distributed Termination Detection Algorithm Principle I Two waves of global inactivity detection between the coordinator and the resolution processes

K(

AC

) ) mp ) sta stamp mp K( sta AC K( AC

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K(

1 Inactivity detection

Inactivity detection and confirmation PP 1 inactivei = P ∧ Pi=1 P i=0 senti − recvi = 0

AC

[C]

v 3) ) rec recv 2 − v 1) − rec nt 3 − (se (sent 2 nt 1 IDL (se IDL

[P1 ]

IDL

[P2 ]

ACK(stamp) AC K(sta mp) sta mp )

[P3 ]

AC

T

2 Inactivity confirmation

2

nb(ACK(stamp)) = P ∧ stamp = current_stamp

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Distributed Local Resolution of B ES

Complexity Results For a boolean graph (V , E, L) and P running processes: I

Worst-case time complexity = O(|V | + |E|) 2 intertwined graph traversals (forward and backward)

I

Worst-case memory complexity = O(|V | + |E|) Dependencies stored during graph exploration

I

Worst-case message complexity = O(2 · |E| · (P − 1)/P) 2 messages (expansion and stabilization) at most exchanged by edges

I

Distributed termination detection = O(|E| · 3 · P) 2 waves with 3 · P messages at most exchanged by edge

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Distributed Local Resolution of B ES

Distributed Software Architecture DS OLVE (10,000 lines of C code) is based on: I

Prototype generic communication library (4000 lines of C code) enabling communication through T CP/I P sockets

I

Generic O PEN /C ÆSAR environment for on-the-fly graph exploration [Garavel-98], part of the C ADP verification toolbox (www.inrialpes.fr/vasy/cadp)

I

Generic boolean resolution A PI given by the library C ÆSAR _S OLVE [Mateescu-03]

I

Available under S OLARIS, L INUX, W INDOWS and M AC OS operating systems

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Distributed Local Resolution of B ES

Platform Architecture Experiments performed on homogeneous parallel architectures (N OW and clusters of P C), among which:

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I

I DPOT (http://idpot.imag.fr) 48 Bi-Xeon 2.5 GHz 1.5 Gb

I

IC LUSTER (http://icluster.imag.fr) 216 PIII 733 MHz 256 Mb

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Distributed Local Resolution of B ES

Experiments Random B ES Solver O PEN /C ÆSAR and C ÆSAR _S OLVE libraries

C compiler

random B ES solver (.exe)

true / false coordinator communication network (T CP /I P sockets)

random B ES solver (.exe)

I

Example of small application of DS OLVE (1000 lines of C code)

I

Enables to test the performance of DS OLVE

I

Produces B ES (represented by the successor function of their corresponding boolean graph) according to various parameters which vary randomly in a given domain

random B ES solver (implicit boolean graph (.c))

random B ES solver (.exe)

random B ES solver (.exe) 1

2

3 nodes

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Distributed Local Resolution of B ES

Speedup in the Worst Case I

almost no alternation (2%), hence long path of ∨ (or ∧) variables ended by a constant T (or F) (1%) encountered in model checking (C TL, ACTL, P DL) encountered in equivalence checking involving deterministic graphs

18 2e+06 4e+06 6e+06 8e+06 1e+07 1e+06 100000 10000 1.2e+07 1.4e+07 1.6e+07 Ideal speedup

16

14

Speedup

12

Worst case class of B ES:

10

8

6

4

2

I

0 0

2

4

6

8

10

Number of nodes

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14

16

18

Quasi-linear speedup compared to the sequential breadth-first search algorithm of C ÆSAR _S OLVE SENVA’2005

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Distributed Local Resolution of B ES

Scalability I

Good scalability with the number of machines, as well as with the size of the BES to solve

I

Experimented DS OLVE on more than 80 machines on IC LUSTER

I

Solved 240 · 106 of variables and 1 · 109 operators in 28 minutes with 17 machines for a BES with 0% constant and 0% alternation

2.5e+08 p=3 p=5 p=7 p=9 p=11 p=13 p=15 p=17

2e+08

Variables

1.5e+08

1e+08

5e+07

0 0

200

400

600

800

1000

Time (sec)

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1400

1600

1800

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Distributed Local Resolution of B ES

Memory and Communication Cost

I

Low memory overhead of distributed resolution compared to memory allocated for data

I

Perfect load balancing achieved by the static hash function

I

High communication cost due to numerous cross-dependencies ((P − 1)/P) · |E|

I

Low percentage (0, 01%) of termination detection messages over all exchanged messages

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Summary

Summary I

DS OLVE: a new distributed local B ES resolution algorithm

I

Generic implementation within the C ADP verification toolbox

I

Linear worst-case time and memory complexity DS OLVE algorithm

I

Extensive set of experiments showing linear speedups and good scalability

I

Ongoing work Generalizing DS OLVE to B ES having several blocks of equations with acyclic inter-block dependencies Developing other applications over DS OLVE, e.g. resolution of Horn clauses, model-checking and test case generation

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Appendix

For Further Reading D. Bergamini, N. Descoubes, C. Joubert and R. Mateescu. B ISIMULATOR: A Modular Tool for On-the-Fly Equivalence Checking. TACAS’2005, LNCS 3440:581–585. C. Joubert and R. Mateescu. Distributed Local Resolution of Boolean Equation Systems. PDP’2005, p264–271, IEEE Computer Society Press. C. Joubert and R. Mateescu. Distributed On-the-Fly Equivalence Checking. PDMC’2004, ENTCS 128(3):47–62. R. Mateescu. A Generic On-the-Fly Solver for Alternation-Free Boolean Equation Systems. TACAS’2003, LNCS 1443:53–66.

http://www.inrialpes.fr/vasy

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