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Detecting Temporal Logic Predicates on Distributed Computations. Vinit Ogale ( [email protected] ) and Vijay K. Garg ( [email protected] ) Parallel and Distributed Systems Lab. Predicate Detection. - PowerPoint PPT Presentation
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Detecting Temporal Logic Predicates onDistributed Computations
Vinit Ogale ([email protected])and Vijay K. Garg ([email protected])
Parallel and Distributed Systems Lab
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Predicate DetectionPredicate: A property expressed using variables
on processes. e.g., more than one process is in critical section
Predicate detection: Determining if an execution trace satisfies the predicate
Predicate detection
trace
predicate
Yes
No
Program
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Trace Model: Total Order Total order: interleaving of events in a trace
Temporal Rover [Drusinsky 03], Java-MaC [Kim, Kannan, Lee, Sokolsky, and Viswanathan 04], JPaX [Havelund and Rosu 04]
PET [Gunter, Kurshan, Peled 00] Low computational complexity
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Trace Model: Partial Order Partial order: Lamport’s happened-before
model [Lamport 78] suitable for concurrent and distributed programs encodes exponential number of total orders )
captures bugs that may not be found with a total order
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{}
Partial Order Traces
Predicate Detection Exponential time algorithm for general predicate [Cooper and Marzullo 91] NP-complete for simple boolean expressions (2-CNF) [Mittal and Garg 01]
{e1} {f1}
{e1, f1}
{e2, e1, f1}
{e2, e1, f2, f1}
{e1, f2, f1}
e1 e2
f1 f2
P1
P2 {e2,e1}
Computation
Corresponding computation lattice
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Structure of the Computation Lattice Frontier notation for cuts {e2,f3}
instead of {e1,e2,f1,f2,f3} Meet/join of two cuts is their
intersection/union Join irreducible cut: cannot be
expressed as the join of two other cuts
Ideal is a set of cuts: with a unique maximum contains all cuts less than the maximum
e3, f2
e3, f3
e3,f1 e2, f2
e2, f3
e2,f1 e1, f2
e1, f3
e1,f1 f2
f3
f1
{}
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Stable predicate: once it becomes true, it
stays true, e.g., deadlock [Chandy and Lamport 85]
Meet/join closed: global states are closed
under meet/join (intersection/union)
Regular: meet and join closed [Garg
and Mittal 01]
e3, f2
e3, f3
e3,f1 e2, f2
e2, f3
e2,f1 e1, f2
e1, f3
e1,f1 f2
f3
f1
{}
Special Classes of Predicates
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subset of CTL [Clarke and Emerson 81]
Examples:violation of mutual exclusion: EF(critical1 Æ critical2) starvation freedom : : EF(request Æ EG( : granted))resettability, AG(EF(restart))
Specification: Temporal Logic
EF (p) / p
EG (p) AG (p)
p is true
C C C
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POTA [A. Sen and V.K. Garg 03] Efficient detection of RCTL
predicates EF(), AG, EG, Æ
JMPax (exponential time)[K. Sen, G. Rosu, G. Agha 05]
e3, f2
e3, f3
e3,f1 e2, f2
e2, f3
e2,f1 e1, f2
e1, f3
e1,f1 f2
f3
f1
{}
Previous Work: Temporal Predicates
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BTL (Basis Temporal Logic) A predicate l in BTL is defined recursively as
follows: l AP (AP is the set of atomic propositions, consists of local
predicates) If P and Q are BTL predicates then the following
are BTL predicates P Q , P Q , P P (also called EF(P))
AG(P) can be represented in BTL as P)
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Basis of a Computation Compact representation of the set of cuts
which satisfy the given predicate
Efficient detection of membership
Computation latticePredicate
Basis
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Simple Example of Basis Predicate class : Any ideal
in the computational lattice
The basis consists of the maximal element that satisfies the predicate
e3, f2
e3, f3
e3,f1 e2, f2
e2, f3
e2,f1 e1, f2
e1, f3
e1,f1 f2
f3
f1
{}
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Regular Predicates Slice: [Mittal, Garg 01] All
join irreducible cuts of the smallest sublattice satisfying the predicate
For regular predicates the slice is a basis
e3, f2
e3, f3
e3,f1 e2, f2
e2, f3
e2,f1 e1, f2
e1, f3
e1,f1 f2
f3
f1
{}
A join irreducible element can not be expressed as the join of any other elements
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Representing Stable Predicates
P : set of ideals I ´ {C1, C2}
Cut C ² P iff 8 C’ 2 I:
: (C < C’)
Note: This representation is not necessarily a basis
Ideal with max c2
Ideal with max c1
Stable Predicate P
c2
c1
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Semiregular Predicates Can be expressed as:
regular predicate Æ stable predicate
Generalizes regular and stable predicates
E.g. There is no token and all processes are red
Semiregular Structure Tuple h slice, I i Regular Predicate
Stable Predicate
Predicate is true
e3, f2
e3, f3
e3,f1 e2, f2
e2, f3
e2,f1 e1, f2
e1, f3
e1,f1 f2
f3
f1
{}
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Main Idea of Algorithm A BTL predicate can be represented as a
disjunction of semiregular predicates Basis for BTL predicates consists of a set of
semiregular structures
BTL predicate
Semiregular predicates with efficient representation
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AlgorithmInputs: Predicate Pin, Computation C
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Predicate: pa Ç pb
e3, f2
e3, f3
e3,f1 e2, f2
e2, f3
e2,f1 e1, f2
e1, f3
e1,f1 f2
f3
f1
{}
S[pa ] = { h slice1 , { [e3,f1], [f2]} i }
S[pb ] = { h slice2 , { [f3 ], [e1, f1]} i }
S[pa Ç pb] = S[pa] S[pb] = { h slice1 , { [e3,f1], [f2]} i , h slice2 , { [f3 ], [e1, f1]} i }
papb
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Predicate: pa Æ pb
e3, f2
e3, f3
e3,f1 e2, f2
e2, f3
e2,f1 e1, f2
e1, f3
e1,f1 f2
f3
f1
{}
S[pa ] = { h slice1 , { [e3,f1], [f2]} i }
S[pb ] = { h slice2 , { [f3 ], [e1, f1 ]} i }
pa pb
S[pa Æ pb] = { h slice1 Å slice2 , { [e3, f1] ,[ f2 ], [f3], [ e1, f1 ] } i }
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Predicate: }(pa Ç pb )
e3, f2
e3, f3
e3,f1 e2, f2
e2, f3
e2,f1 e1, f2
e1, f3
e1,f1 f2
f3
f1
{}
pa pb
S[}(pa Æ pb)] = { h slice (}pa),{} i , h slice (}pb),{}i }
} pa
} pb
Algorithm to compute slice[pa] by Sen and Garg 03
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Predicate: :(pc Ç pd) max(slice[pd]) max(slice[pc])
slice[pc] slice[pd]
S[(pc pd )] ={ slice[original computation], { max(slice[pc], max(slice[pd] } }
e3, f2
e3, f3
e3,f1 e2, f2
e2, f3
e2,f1 e1, f2
e1, f3
e1,f1 f2
f3
f1
{}
S[pc pd] ={slice[pc], {} , slice[pd], {} }
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Important Results
Theorem 1: The basis for a BTL predicate has at most 2k semiregular structures
Theorem 2: The total number of ideals in the basis is at most 2k
Time Space Complexity: O(2k |E|n) |E| is the total number of events n is the number of processes
Where k is the size of the predicate
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Experimental ResultsBasis-based Trace Verifier (BTV) Offline trace verifier implemented in Java
Dining Philosophers
BTV (Basis based trace verifier)POTA(with SPIN)
Number of processes
Time(in seconds)
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Conclusion Polynomial time predicate detection on
partially ordered traces Properties composed of the following (nested)
operators: Æ , Ç , : , . Implemented and tested on Java programs
and hardware system level designs (SystemC)
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Extra Slides
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Properties of Semiregular Predicates All regular and stable predicates are
semiregular Join-closed Closed under conjunction Closed under } and
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Basis of a Computation Given a computational lattice C, corresponding
to a computation E, and a predicate P, a subset S[P] of C is a basis of P is (Compactness) The size of S[P] is polynomial in the
size of the computation (Efficient membership) Given any cut C C, there
exists a polynomial time algorithm that takes S[P], E and C as inputs and determines if (C,E) ² P
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Riv73 , Riv74 Any co-regular predicate can be expressed as
an union of intervals, each interval defined by a join-irreducible element and a meet-irreducible element
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Predicate: pb Ç pc
pc pb e3, f2
e3, f3
e3,f1 e2, f2
e2, f3
e2,f1 e1, f2
e1, f3
e1,f1 f2
f3
f1
{}
Computation
Lattice
e1 Process 1 e2 e3
f1 f2 f3 Process 2
OriginalComputati
on
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Predicate: pb Ç pc
slice[pb] slice[pc]
e3, f2
e3, f3
e3,f1 e2, f2
e2, f3
e2,f1 e1, f2
e1, f3
e1,f1 f2
f3
f1
{}
S[pb] ={ slice[pb], {} }
S[pc] ={ slice[pc], {} }
S[pb pc] ={slice[pb], {} , slice[pc], {} }
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AlgorithmInputs: Predicate Pin, Computation C
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Slicer Module : Big Picture
trace
slice
state explosion
keep all red consistent cutsslicing
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Slice: Definitionslice: a sub-trace such that: it contains all consistent cuts of the trace
satisfying the given predicate it contains the least number of consistent
cuts[Garg and Mittal 01, Mittal and Garg 01] predicate
trace
sliceSlicer
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x=2 x=1
y=3 y=4
x=0
y=0>
x=2 x=1
y=3 y=4
x=0
y=0>
Slice: More Edges Less States
x=2,y=0
x=0,y=0
x=0,y=3
x=2,y=3x=1,y=0
x=1,y=3
x=1,y=4
x=2,y=4
x=2,y=0 x=0,y=3
x=2,y=3
x=1,y=4
x=2,y=4
C
C
C
x=1,y=3
x=0,y=0
Observation: edges in G µ edges in H iff cuts in H µ cuts in G.
Idea: To obtain the slice, add edges to the trace.
?
?
Trace G
Trace H : ( (x = 1) Æ (y = 0) )slicing
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x=2 x=1
y=3 y=4
x=0
y=0>
x=2 x=1
y=3 y=4
x=0
y=0>
Slice: Example
no message in transit
x=2,y=0
x=0,y=0
x=0,y=3
x=2,y=3x=1,y=0
x=1,y=3
x=1,y=4
x=2,y=4
slicing
x=2,y=0
x=0,y=0
x=0,y=3
x=2,y=3x=1,y=0
x=1,y=3
x=1,y=4
x=2,y=4
?
?
D
D
D
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Detect Bugs from Successful Tracesf2e1
: ordering, dependency CS2 CS1
f1 e2
Process 1
Process 2
Partial Order Trace
CS2
CS1
e1 e2
f1 f2
e2e1
CS1CS2
f1 f2
Trace
Specification:CS1 Æ CS2
: CS2: CS1
: CS1
: CS2
: CS1 : CS2
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Trace Model: Partial OrderPartial Order Trace
e1, f0
e0, f0
e0, f1
e1, f1e2, f0
e2, f1
e2, f2
e1, f2
State Space
1 Partial Order Tracecaptures
5 Total Order Traces
Total Order Trace ´Path in the State Space
Process 1
Process 2 CS2
CS1
e1 e2
f1 f2
Initiallye0, f0
CS1 Æ CS2
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Characterizing PredicatesPredicates are generally characterized in terms
of The computation (e.g. local predicates) or The computational lattice (e.g. semiregular
predicates)
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Characterizing Predicates We aim to relate these two characterizations
E.g. Our algorithm for predicate detection can support any atomic proposition that is regular as well as co-regular. This is a more general class than local predicates and precisely characterizing such a class will automatically make the algorithm more general.
Focus on locality, co-regularity, co-semiregularity and observer independence