1. Detecting Temporal Logic Predicates on Distributed Computations
Vinit Ogale
and Vijay K. Garg
Parallel and Distributed Systems Lab

2. Predicate Detection
Predicate: 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
Program
trace
Yes
Predicate
detection
predicate No

3. 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

4. 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

5. Partial Order Traces Predicate Detection Computation
{e2, e1, f2, f1} P1
{e2, e1, f1}
{e1, f2, f1} P2
{e2,e1}
{e1, f1} f1 f2
{e1}
{f1}
Computation {}
Let’s look at the partial order traces a little closer.
To detect predicates on partial orders, we need to enumerate all the possible global states from the computation poset.
f1 and f2 is not a possible global state, since for f2 to be present in the global state, e1 should also be included.
these enumerated global states forms a distributive lattice which we call computation lattice.
As seen from the example, the number global states might be exponential.
hence, using partial order model also comes with its own difficulty – state explosion.
Cooper and Marzullo gave an algorithm for general predicates in 91, later Mittal and Garg showed the NP-completeness of the problem.
To overcome the state explosion problem, we follow two methods;
Restricting predicate structures, and
Using computation slicing method.
From now on, to represent computation lattices, instead of using full global states, we’ll just keep a frontier from each process,
So for this computation, frontier lattice representation looks like this.
symbolic methods [McMillan 93, Stoller and Liu 98]
partial order methods [Godefroid 90, Valmari 90, Peled 93, Esparza 94, Stoller et al. 00]
: exploits the structure of the predicate by imposing restrictions on it
stable [Chandy and Lamport 85] conjunctive [Garg and Waldecker 94 96, Hurfin et al. 96] observer-independent [Charron-Bost et. al. 95] linear [Chase and Garg 95] relational [Tomlinson and Garg 94]
slicing method: concise representation of all states satisfying the predicate
Slice for an arbitrary predicate NP-hard, in general. Efficient algorithms for linear predicates and Ç, Æ operators. [Mittal and Garg 01]
Corresponding computation lattice
Predicate Detection
Exponential time algorithm for general predicate [Cooper and Marzullo 91]
NP-complete for simple boolean expressions (2-CNF) [Mittal and Garg 01]

6. 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, f3
e2, f3
e3, f2
e1, f3
e2, f2
e3,f1
e1, f2 f3
e2,f1 f2
e1,f1
The bottom element is not generally considered to be a JIRR but for ease of notation we consider it to be a JIRR in this presentation f1 {}

7. Special Classes of Predicates
e3, f3
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]
e2, f3
e3, f2
e1, f3
e2, f2
e3,f1
e1, f2 f3
e2,f1 f2
e1,f1 f1 {}

8. Specification: Temporal Logic
EG (p)
AG (p) C C C
EF (p) /  p
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))
p is true
safety and liveness properties
specification detection: whether the initial state satisfies specification.
path: a sequence of states st the next state is obtained from the current state by adding a single state.
F: future, finally, eventually
non-temporal

9. Previous Work: Temporal Predicates
e3, f3
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]
e2, f3
e3, f2
e1, f3
e2, f2
e3,f1
e1, f2 f3
e2,f1 f2
e1,f1 f1 {}

10. 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)
Sematics

11. Basis of a Computation
Compact representation of the set of cuts which satisfy the given predicate
Efficient detection of membership
Computation lattice
Predicate
Basis

12. Simple Example of Basis
Predicate class : Any ideal in the computational lattice
The basis consists of the maximal element that satisfies the predicate
e3, f3
e2, f3
e3, f2
e1, f3
e2, f2
e3,f1
e1, f2 f3
e2,f1 f2
e1,f1 f1 {}

13. 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, f3
e2, f3
e3, f2
e1, f3
e2, f2
e3,f1
e1, f2 f3
e2,f1 f2
e1,f1
Backup Slides of a predicate and the slice for that f1 {}
A join irreducible element can not be expressed as the join of any other elements

14. 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
Stable Predicate P c1 c2
Ideal with max c1
Ideal with max c2

15. 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
e3, f3
e2, f3
e3, f2
e1, f3
e2, f2
e3,f1
e1, f2 f3
e2,f1 f2
e1,f1 f1 {}
Regular Predicate
Stable Predicate
Predicate is true

16. 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

17. Algorithm Inputs: Predicate Pin, Computation C

18. Predicate: pa Ç pb S[pa Ç pb] = S[pa]  S[pb]
e3, f3
S[pa ] = { h slice1 , { [e3,f1], [f2]} i }
e2, f3
e3, f2
S[pb ] = { h slice2 , { [f3 ], [e1, f1]} i }
e1, f3
e2, f2
e3,f1
e1, f2 f3
e2,f1 f2
e1,f1 pb pa f1 {}
S[pa Ç pb] = S[pa]  S[pb]
= { h slice1 , { [e3,f1], [f2]} i , h slice2 , { [f3 ], [e1, f1]} i }

19. Predicate: pa Æ pb
e3, f3
S[pa ] = { h slice1 , { [e3,f1], [f2]} i }
e2, f3
e3, f2
S[pb ] = { h slice2 , { [f3 ], [e1, f1 ]} i }
e1, f3
e2, f2
e3,f1
e1, f2 f3
e2,f1 pa f2
e1,f1 pb f1 {}
S[pa Æ pb] = { h slice1 Å slice2 , { [e3, f1] ,[ f2 ], [f3], [ e1, f1 ] } i }

20. Algorithm to compute slice[pa] by Sen and Garg 03
Predicate: }(pa Ç pb )
e3, f3
e2, f3
e3, f2
e1, f3
e2, f2
e3,f1
e1, f2 f3
e2,f1 pa f2
e1,f1 pb
} pa f1
} pb {}
S[}(pa Æ pb)] = { h slice (}pa),{} i , h slice (}pb),{}i }
Algorithm to compute slice[pa] by Sen and Garg 03

21. Predicate: :(pc Ç pd) max(slice[pc]) max(slice[pd]) S[pc  pd] ={
e3, f3
S[pc  pd] ={
slice[pc], {} ,
 slice[pd], {}  }
e2, f3
e3, f2
e1, f3
e2, f2
e3,f1
e1, f2 f3
e2,f1
S[(pc  pd )] ={
slice[original computation],
{ max(slice[pc], max(slice[pd] }  }
\not \dia is stable predicate f2
e1,f1 f1
slice[pc]
slice[pd] {}

22. 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
(NP-Hard) w.r.t k
Typically, fo rmost predicates it will be better than 2^k
Where k is the size of the predicate

23. Experimental Results Basis-based Trace Verifier (BTV)
Offline trace verifier implemented in Java
POTA
(with SPIN)
BTV (Basis based trace verifier)
Dining Philosophers
Time
(in seconds)
(eat_0 \and eat_1) \or (eat1 \and eat2) \and (\Diamond ( (eat_2 \and eat_3) \or (eat_3 \and eat_4)))
Violation of mutex now on some processes and violation of mutex on others processes any time in the future
Number of processes

24. 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)

25. Extra Slides

26. Properties of Semiregular Predicates
All regular and stable predicates are semiregular
Join-closed
Closed under conjunction
Closed under } and 

27. 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

28. 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

29. Predicate: pb Ç pc e1 e2 e3 pb pc f1 f2 f3 Process 1 Process 2
e3, f2
e3, f3
e3,f1
e2, f2
e2, f3
e2,f1
e1, f2
e1, f3
e1,f1 f2 f3 f1 {} e1 e2 e3 pb pc
Process 1 f1 f2 f3
Process 2
Original
Computation
Let’s look at disjunction of two semi-regular predicates;
Let’s say yellow represents the states that hold pb,
purple represents the states that hold pb.
Disjunction is trivial.
Computation
Lattice

30. Predicate: pb Ç pc S[pb] ={ slice[pb], {} }
e3, f3
e2, f3
e3, f2
S[pc] ={ slice[pc], {}  }
e1, f3
e2, f2
e3,f1
e1, f2 f3
e2,f1
S[pb  pc] ={
slice[pb], {} ,
 slice[pc], {}  } f2
e1,f1
Then, we first find slice for pb , then for pc,
Then disjunction is trivial, mainly union of structure pb and pc. f1
slice[pb]
slice[pc] {}

31. Algorithm Inputs: Predicate Pin, Computation C
I will not go over the algorithm in detail,
But given a computation and a predicate, k-slicing algorithm works recursively.
Initially, we push all the negations inside the predicate. But we don’t push further when there’s a diamond.
We have different approaches for different cases, such as local predicate , disjunction, conjunction, diamond P and not diamond P.
I’ll go over some of them as an example.

32. Slicer Module : Big Picture
state explosion
trace
keep all red
consistent cuts
slicing
Given a computation a slice gives a concise representation of a subset of consistent cuts that satisfy a certain predicate
Our goal is to spend polynomial-time effort to obtain the slice
slice

33. Slice: Definition slice: 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]
trace
Slicer
slice
Computation, trace, slice same
among all such directed graphs it contains the least number of consistent cuts
predicate

34. Slice: More Edges Less States
Trace G C
x=1,y=4
x=2
x=1
y=3
y=4
x=0
y=0
>
x=1,y=3
x=2,y=4 ?
x=1,y=0
x=2,y=3
x=2,y=0
x=0,y=3
x=0,y=0
slicing
: ( (x = 1) Æ (y = 0) )
Trace H
x=1,y=4
x=2
x=1
y=3
y=4
x=0
y=0
>
x=1,y=3
x=2,y=4
:: We will explain the algorithm that adds edges later. it has E^2 complexity
Notes:
We have an algorithm to compute slice for local predicates and for disjunction.
Predicate is not symmetric since y != 1 is satisfied at \bot ?
x=2,y=3
x=2,y=0
x=0,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.

35. Slice: Example D > D no message in transit slicing D > x=2 x=1
y=3
y=4
x=0
y=0
>
x=1,y=4
x=1,y=3
x=2,y=4 ? D
x=1,y=0
x=2,y=3
x=2,y=0
x=0,y=3
no message in transit
x=0,y=0
slicing D
x=2
x=1
y=3
y=4
x=0
y=0
>
x=1,y=4
Two main ideas: 1) add Additional edges to obtain the slice 2) A cut Z that does not belong to the slice is such that for any additional edge (e, f) in the slice, Z includes f but not e.
Notes:
We have an algorithm to compute slice for local predicates and for disjunction.
Predicate is not symmetric since y != 1 is satisfied at \bot
x=1,y=3
x=2,y=4 ?
x=1,y=0
x=2,y=3
x=2,y=0
x=0,y=3
x=0,y=0

36. Detect Bugs from Successful Traces
: CS1
CS2
: CS2
CS1
: ordering, dependency e1 f1 f2 e2
Partial Order Trace
: CS1
CS1
Process 1
Specification:
CS1 Æ CS2 e1 e2
CS2
: CS2
Process 2
Consider an example trace as follows: …. Note that the trace does not have to be from a message passing program, it could be from a shared memory program or a hardware program. In fact, later on we will see an example of our technique on a hardware protocol.
Let Trace 1 be the actual trace. In fact, there are several other possible traces that we can infer from the observed trace.
from one trace to all traces that are consistent with the actual execution trace.
Goal: predict bugs from successful traces.
Runtime: may not even observe the bug
Simulation: may observe the bug with exhaustive simulation, then the complexity is high f1 f2
: CS1
CS2
CS1
: CS2 e1 f1 e2 f2

37. Trace Model: Partial Order
CS1
Process 1
Initially
e0, f0 e1 e2
Partial Order Trace
CS2
Process 2 f1 f2
e1, f0
e0, f0
e0, f1
e1, f1
e2, f0
e2, f1
e2, f2
e1, f2
State Space
1 Partial Order Trace
captures
5 Total Order Traces
Total Order Trace ´
Path in the State Space
CS1 Æ CS2
1) Preserves concurrency 2) Encodes exponential number of total orders
3) Catches bugs not found using total orders 4) The best model for message passing programs, asynchronous circuits

38. Characterizing Predicates
Predicates are generally characterized in terms of
The computation (e.g. local predicates) or
The computational lattice (e.g. semiregular predicates)

39. 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