Program Analysis and Synthesis of Parallel Systems

Program Analysis and Synthesis of Parallel Systems

Roman Manevich Ben-Gurion University

Three papers

1. A Shape Analysis for Optimizing Parallel Graph Programs [POPL’11]

2. Elixir: a System for Synthesizing Concurrent Graph Programs [OOPSLA’12]

3. Parameterized Verification of Transactional Memories [PLDI’10]

What’s the connection?

A Shape Analysisfor Optimizing ParallelGraph Programs [POPL’11]

Elixir: a System for SynthesizingConcurrent Graph Programs [OOPSLA’12]

Parameterized Verificationof Transactional Memories [PLDI’10]

From analysisto language design

Creates opportunitiesfor more optimizations.Requires other analyses

Similarities betweenabstract domains

What’s the connection?

A Shape Analysisfor Optimizing ParallelGraph Programs [POPL’11]

Elixir: a System for SynthesizingConcurrent Graph Programs [OOPSLA’12]

Parameterized Verificationof Transactional Memories [PLDI’10]

From analysisto language design

Creates opportunitiesfor more optimizations.Requires other analyses

Similarities betweenabstract domains

A Shape Analysis for Optimizing Parallel Graph Programs

Dimitrios Prountzos1

Keshav Pingali1,2

Roman Manevich2

Kathryn S. McKinley1

1: Department of Computer Science, The University of Texas at Austin2: Institute for Computational Engineering and Sciences, The

University of Texas at Austin

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Motivation• Graph algorithms are ubiquitous

• Goal: Compiler analysis for optimization of parallel graph algorithms

Computational biology Social Networks

Computer Graphics

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Minimum Spanning Tree Problem

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Minimum Spanning Tree Problem

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Boruvka’s Minimum Spanning Tree Algorithm

Build MST bottom-uprepeat { pick arbitrary node ‘a’ merge with lightest neighbor ‘lt’ add edge ‘a-lt’ to MST} until graph is a single node

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Parallelism in Boruvka

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Non-conflicting iterations

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Non-conflicting iterations


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Conflicting iterations

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Optimistic parallelization in Galois• Programming model

– Client code has sequential semantics– Library of concurrent data structures

• Parallel execution model– Thread-level speculation (TLS)– Activities executed speculatively

• Conflict detection– Each node/edge has associated exclusive lock– Graph operations acquire locks on

read/written nodes/edges– Lock owned by another thread conflict

iteration rolled back– All locks released at the end

• Two main overheads– Locking– Undo actions

i1

i2

i3

Generic optimization structure

Program AnnotatedProgram

ProgramAnalyzer

ProgramTransformer

OptimizedProgram

Overheads (I): locking• Optimizations– Redundant locking elimination– Lock removal for iteration private data– Lock removal for lock domination

• ACQ(P): set of definitely acquired locks per program point P• Given method call M at P:

Locks(M) ACQ(P) Redundant Locking

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Overheads (II): undo actions

Lockset Grows

Lockset Stable

Failsafe

…

foreach (Node a : wl) {

…

…

}

foreach (Node a : wl) { Set<Node> aNghbrs = g.neighbors(a); Node lt = null; for (Node n : aNghbrs) { minW,lt = minWeightEdge((a,lt), (a,n)); } g.removeEdge(a, lt); Set<Node> ltNghbrs = g.neighbors(lt); for (Node n : ltNghbrs) { Edge e = g.getEdge(lt, n); Weight w = g.getEdgeData(e); Edge an = g.getEdge(a, n); if (an != null) { Weight wan = g.getEdgeData(an); if (wan.compareTo(w) < 0) w = wan; g.setEdgeData(an, w); } else { g.addEdge(a, n, w); } } g.removeNode(lt); mst.add(minW); wl.add(a);}

Program point P is failsafe if: Q : Reaches(P,Q) Locks(Q) ACQ(P)

GSet<Node> wl = new GSet<Node>();wl.addAll(g.getNodes());GBag<Weight> mst = new GBag<Weight>();


Lockset analysis

• Redundant Locking• Locks(M) ACQ(P)

• Undo elimination• Q : Reaches(P,Q)

Locks(Q) ACQ(P)

• Need to compute ACQ(P) : Runtime

overhead

The optimization technically

• Each graph method m(arg1,…,argk, flag) contains optimization level flag• flag=LOCK – acquire locks • flag=UNDO – log undo (backup) data• flag=LOCK_UNO D – (default) acquire locks and

log undo• flag=NONE – no extra work

• Example:Edge e = g.getEdge(lt, n, NONE)

Analysis challenges

• The usual suspects: – Unbounded Memory Undecidability – Aliasing, Destructive updates

• Specific challenges:– Complex ADTs: unstructured graphs– Heap objects are locked– Adapt abstraction to ADTs

• We use Abstract Interpretation [CC’77]– Balance precision and realistic performance

Shape analysis overview

HashMap-Graph

Tree-based Set

……

Graph { @rep nodes @rep edges …}

Graph Spec

Concrete ADTImplementationsin Galois library

Predicate Discovery

Shape Analysis

Boruvka.javaOptimizedBoruvka.java

Set { @rep cont …}

Set Spec

ADT Specifications

ADT specification

Graph<ND,ED> {

@rep set<Node> nodes @rep set<Edge> edges

Set<Node> neighbors(Node n);

}

Graph Spec

...Set<Node> S1 = g.neighbors(n);

...

Boruvka.java

Abstract ADT state by virtual set fields

@locks(n + n.rev(src) + n.rev(src).dst + n.rev(dst) + n.rev(dst).src)@op( nghbrs = n.rev(src).dst + n.rev(dst).src , ret = new Set<Node<ND>>(cont=nghbrs) )

Assumption: Implementation satisfies Spec

Graph<ND,ED> {

@rep set<Node> nodes@rep set<Edge> edges

@locks(n + n.rev(src) + n.rev(src).dst + n.rev(dst) + n.rev(dst).src)@op( nghbrs = n.rev(src).dst + n.rev(dst).src , ret = new Set<Node<ND>>(cont=nghbrs) ) Set<Node> neighbors(Node n);}

Modeling ADTs

c

a bGraph Spec

dst

src

srcdst

dst

src

Modeling ADTs

c

a b

nodes edges

Abstract State

cont

ret nghbrs

Graph Spec

dst

src

srcdst

dst

src

Graph<ND,ED> {

@rep set<Node> nodes@rep set<Edge> edges

@locks(n + n.rev(src) + n.rev(src).dst + n.rev(dst) + n.rev(dst).src)@op( nghbrs = n.rev(src).dst + n.rev(dst).src , ret = new Set<Node<ND>>(cont=nghbrs) ) Set<Node> neighbors(Node n);}

Abstraction scheme

cont cont

S1 S2L(S1.cont) L(S2.cont)

(S1 ≠ S2) ∧ L(S1.cont) ∧ L(S2.cont)

• Parameterized by set of LockPaths: L(Path) o . o ∊ Path Locked(o)– Tracks subset of must-be-locked objects

• Abstract domain elements have the form: Aliasing-configs 2LockPaths …

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( L(y.nd) ) )( ( L(x.nd))

( L(y.nd) ) ( L(y.nd) L(x.rev(src)) ) )( ( L(x.nd))

Joining abstract states

( L(y.nd) ) ( () L(x.nd) )

Aliasing is crucial for precisionMay-be-locked does not enable our optimizations

#Aliasing-configs : small constant (6)

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lt

GSet<Node> wl = new GSet<Node>();wl.addAll(g.getNodes());GBag<Weight> mst = new GBag<Weight>();


Example invariant in Boruvka The immediate neighbors of a and lt are locked

a

( a ≠ lt ) ∧ L(a) L(a.rev(src)) L(a.rev(dst))∧ ∧ ∧ L(a.rev(src).dst) L(a.rev(dst).src) ∧ ∧ L(lt) L(lt.rev(dst)) L(lt.rev(src)) ∧ ∧ ∧ L(lt.rev(dst).src) L(lt.rev(src).dst)∧

…..

Heuristics for finding LockPaths

• Hierarchy Summarization (HS)– x.( fld )*– Type hierarchy graph acyclic

bounded number of paths

– Preflow-Push: • L(S.cont) L(S.cont.nd)∧• Nodes in set S and their data are locked

Set<Node>

S

Node

NodeData

cont

nd

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Footprint graph heuristic• Footprint Graphs (FG)[Calcagno et al. SAS’07]

– All acyclic paths from arguments of ADT method to locked objects– x.( fld | rev(fld) )* – Delaunay Mesh Refinement: L(S.cont) L(S.cont.rev(src)) L(S.cont.rev(dst)) ∧ ∧ ∧ L(S.cont.rev(src).dst) L(S.cont.rev(dst).src)∧– Nodes in set S and all of their immediate neighbors are

locked

• Composition of HS, FG– Preflow-Push: L(a.rev(src).ed)

FG HS

Experimental evaluation• Implement on top of TVLA– Encode abstraction by 3-Valued Shape Analysis

[SRW TOPLAS’02]• Evaluation on 4 Lonestar Java benchmarks

• Inferred all available optimizations• # abstract states practically linear in program size

Benchmark Analysis Time (sec)

Boruvka MST 6Preflow-Push Maxflow 7Survey Propagation 12Delaunay Mesh Refinement 16

Impact of optimizations for 8 threads

Boruvka MST Delaunay Mesh Refinement

Survey Propa-gation

Preflow-Push Maxflow

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Baseline Optimized

Tim

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5.6× 4.7×11.4×

2.9× 8-core Intel Xeon @ 3.00 GHz

Note 1

• How to map abstract domain presented so far to TVLA?– Example invariant: (x≠y L(y.nd)) (x=y L(x.nd))– Unary abstraction predicate x(v) for pointer x– Unary non-abstraction predicate

L[x.p] for pointer x and path p– Use partial join– Resulting abstraction similar to the one shown

Note 2

• How to come up with abstraction for similar problems?1. Start by constructing a manual proof• Hoare Logic

2. Examine resulting invariants and generalize into a language of formulas• May need to be further specialized for a given

program – interesting problem (machine learning/refinement)

– How to get sound transformers?

Note 3

• How did we avoid considering all interleavings?

• Proved non-interference side theorem

Elixir : A System for Synthesizing Concurrent Graph Programs

Dimitrios Prountzos1

Roman Manevich2

Keshav Pingali1

1. The University of Texas at Austin2. Ben-Gurion University of the Negev

GoalAllow programmer to easily implement

correct and efficientparallel graph algorithms

• Graph algorithms are ubiquitousSocial network analysis, Computer graphics, Machine learning, …

• Difficult to parallelize due to their irregular nature

• Best algorithm and implementation usually

– Platform dependent– Input dependent

• Need to easily experiment with different solutions• Focus: Fixed graph structure

• Only change labels on nodes and edges• Each activity touches a fixed number of nodes

• Problem Formulation– Compute shortest distance

from source node S to every other node• Many algorithms

– Bellman-Ford (1957)– Dijkstra (1959)– Chaotic relaxation (Miranker 1969)– Delta-stepping (Meyer et al. 1998)

• Common structure– Each node has label dist

with known shortest distance from S• Key operation

– relax-edge(u,v)

Example: Single-Source Shortest-Path

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if dist(A) + WAC < dist(C) dist(C) = dist(A) + WAC

Scheduling of relaxations:• Use priority queue of nodes,

ordered by label dist• Iterate over nodes u in priority

order• On each step: relax all

neighbors v of u – Apply relax-edge to all (u,v)

Dijkstra’s algorithm2 5

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<C,3> <B,5><B,5> <E,6> <D,7><B,5>

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Chaotic relaxation

• Scheduling of relaxations:• Use unordered set of edges• Iterate over edges (u,v) in any

order• On each step:– Apply relax-edge to edge (u,v)

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(S,A) (B,C) (C,D)(C,E)

Insights behind Elixir

What should be done How it should

be done

Unordered/Ordered algorithms

Operator Delta

: activity

Parallel Graph Algorithm

Operators Schedule

Order activity processing

Identify new activities

Static Schedule

Dynamic Schedule

“TAO of parallelism”PLDI 2011

Insights behind ElixirParallel Graph

Algorithm

Operators Schedule



Static Schedule

Dynamic Schedule

Dijkstra-style Algorithm

q = new PrQueueq.enqueue(SRC)

while (! q.empty ){ a = q.dequeue

for each e = (a,b,w){ if dist(a) + w < dist(b){

dist(b) = dist(a) + w q.enqueue(b)

} } }

Contributions

• Language– Operators/Schedule separation– Allows exploration of

implementation space• Operator Delta Inference

– Precise Delta required for efficient fixpoint computations

• Automatic Parallelization– Inserts synchronization to atomically

execute operators– Avoids data-races / deadlocks– Specializes parallelization based on

scheduling constraints


Operators Schedule



Static Schedule

Dynamic Schedule

Synchronization

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SSSP in ElixirGraph [ nodes(node : Node, dist : int) edges(src : Node, dst : Node, wt : int)]

relax = [ nodes(node a, dist ad) nodes(node b, dist bd)

edges(src a, dst b, wt w) bd > ad + w ] ➔ [ bd = ad + w ]

sssp = iterate relax schedule≫

Graph type

OperatorFixpointStatement

OperatorsGraph [ nodes(node : Node, dist : int) edges(src : Node, dst : Node, wt : int)]

relax = [ nodes(node a, dist ad) nodes(node b, dist bd) edges(src a, dst b, wt w) bd > ad + w ] ➔ [ bd = ad + w ]

sssp = iterate relax schedule ≫

Redex patternGuardUpdate

ba if bd > ad + w

adw

bd

ba

adw

ad+w

Cautiousby construction – easy to generalize

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Fixpoint statementGraph [ nodes(node : Node, dist : int) edges(src : Node, dst : Node, wt : int)]



Apply operator until fixpoint

Scheduling expression

Scheduling examplesGraph [ nodes(node : Node, dist : int) edges(src : Node, dst : Node, wt : int)]



Locality enhanced Label-correctinggroup b unroll 2 approx metric ad ≫ ≫Dijkstra-style

metric ad group b ≫

q = new PrQueueq.enqueue(SRC)while (! q.empty ) { a = q.dequeue for each e = (a,b,w) { if dist(a) + w < dist(b) { dist(b) = dist(a) + w q.enqueue(b) } }}

Operator Delta Inference


Operators Schedule



Static Schedule

Dynamic Schedule

Identifying the delta of an operator

b

a

relax1

??

Delta Inference Example

ba

SMT Solver

SMT Solver

assume (da + w1 < db)

assume ¬(dc + w2 < db)

db_post = da + w1

assert ¬(dc + w2 < db_post)Query Program

relax1

c

w2

w1

relax2

(c,b) does not become active

assume (da + w1 < db)

assume ¬(db + w2 < dc)

db_post = da + w1

assert ¬(db_post + w2 < dc)Query Program

Delta inference example – active

SMT Solver

SMT Solver

ba

relax1

cw1

relax2

w2

Apply relax on all outgoing edges (b,c) such that:

dc > db +w2

and c a≄

Influence patterns

b=cad

ba=c

d

a=dc

b

b=da=c b=ca=d

b=da

c

System architecture

Elixir

Galois/OpenMP Parallel Runtime

Algorithm Spec

Parallel Thread-PoolGraph ImplementationsWorklist Implementations

Synthesize codeInsert synchronization

C++Program

ExperimentsExplored Dimensions

Grouping Statically group multiple instances of operator

Unrolling Statically unroll operator applications by factor K

Dynamic Scheduler Choose different policy/implementation for the dynamic worklist

...

Compare against hand-written parallel implementations

SSSP results

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Elixir

Lonestar

Threads

Tim

e (m

s)

• 24 core Intel Xeon @ 2 GHz• USA Florida Road Network (1 M nodes, 2.7 M Edges)

Group + Unroll improve localityImplementation Variant

Breadth-First search results

Scale-Free Graph1 M nodes, 8 M edges

USA road network24 M nodes, 58 M edges

1 2 4 8 16 20 240

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Elixir (Variant 1)LonestarCilk

Threads

Tim

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s)

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Elixir (Variant 2)LonestarCilk

Threads

Tim

e (m

s)

Conclusion

• Graph algorithm = Operators + Schedule– Elixir language :

imperative operators + declarative schedule• Allows exploring implementation space• Automated reasoning for efficiently computing

fixpoints• Correct-by-construction parallelization • Performance competitive with hand-parallelized

code

Parameterized Verificationof Software Transactional Memories

Michael Emmi Rupak MajumdarRoman Manevich

Motivation• Transactional memories [Herlihy ‘93]

– Programmer writes code w. coarse-grained atomic blocks – Transaction manager takes care of conflicts providing illusion of

sequential execution• Strict serializability – correctness criterion

– Formalizes “illusion of sequential execution”• Parameterized verification

– Formal proof for given implementation– For every number of threads– For every number of memory objects– For every number and length of transactions

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STM terminology

• Statements: reads, writes, commit, abort• Transaction: reads and writes of variables followed by

commit (committing transaction)or abort (aborting transaction)

• Word: interleaved sequence of transactions of different threads

• Conflict: two statements conflict if– One is a read of variable X and other is a commit of a

transaction that writes to X– Both are commits of transactions that write to X

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Safety property: strict serializability

• There is a serialization for the committing threads such that order of conflicts is preserved

• Order of non-overlapping transactions remains the same

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Safety property: strict serializability• There is a serialization for the committing threads such

that order of conflicts is preserved• Order of non-overlapping transactions remains the same• Example word: (rd X t1), (rd Y t2), (wr X t2), (commit t2), (commit t1)

=>

Can be serialized to : (rd X t1), (commit t1), (rd Y t2), (wr X t2), (commit t2)

conflict

conflict62

Main results

• First automatic verification ofstrict serializability for transactional memories– TPL, DSTM, TL2

• New proof technique:– Template-based invisible invariant generation– Abstract checking algorithm to check inductive

invariants

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Challenging – requires reasoning on both

universal and existential properties

Outline

• Strict serializability verification approach• Automating the proof• Experiments• Conclusion• Related work

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Proof roadmap 1

Goal: prove model M is strictly serializable1. Given a strict-serializability reference model

RSS reduce checking strict-serializabilityto checking that M refines RSS

2. Reduce refinement to checking safety– Safety property SIM: whenever M can execute a

statement so can RSS– Check SIM on product system M RSS

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Proof roadmap 2

3. Model STMs M and RSS in first-order logic• TM models use set data structures and typestate bits

4. Check safety by generating strong enough candidate inductive invariant and checking inductiveness– Use observations on structure of transactional

memories– Use sound first-order reasoning

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Reference strict serializability model

• Guerraoui, Singh, Henzinger, Jobstmann [PLDI’08]• RSS : Most liberal specification of strictly

serializable system– Allows largest language of strictly-serializable

executions• M is strictly serializable iff every word of M is also

a word of RSS– Language(M) Language(RSS)– M refines RSS

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Modeling transactional memories

• Mn,k=(predicates,actions)– Predicate : ranked relation symbol

p(t), q(t,v),…– Binary predicates used for sets so instead of rs(t,v)

I’ll write vrs(t)– Action :

a(t,v) = if pre(a) then p’(v)=…, q’(u,v)=…• Universe = set of k thread individuals and n memory

individuals• State S = a valuation to the predicates

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Reference model (RSS) predicates

• Typestates:– RSS.finished(t), RSS.started(t), RSS.pending(t),

RSS.invalid(t)• Read/write sets– RSS.rs(t,v), RSS.ws(t,v)

• Prohibited read/write sets– RSS.prs(t,v), RSS.pws(t,v)

• Weak-predecessor– RSS.wp(t1,t2)

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DSTM predicates

• Typestates:– DSTM.finished(t), DSTM.validated(t),

DSTM.invalid(t), DSTM.aborted(t)• Read/own sets– DSTM.rs(t,v), DSTM.os(t,v)

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RSS commit(t) action

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if RSS.invalid(t) RSS.wp(t,t) thent1,t2 . RSS.wp’(t1,t2) t1t t2t (RSS.wp(t1,t2) RSS.wp(t,t2) (RSS.wp(t1,t) v . vRSS.ws(t1) vRSS.ws(t)))…

post-statepredicate

current-statepredicate

action precondition

write-write conflict

executing thread

DSTM commit(t) action

if DSTM.validated(t) thent1 . DSTM.validated’(t1) t1t DSTM.validated(t1) v . vDSTM.rs(t1) vDSTM.os(t1)…

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read-own conflict

FOTS states and execution

v

rd v t1

DSTMt1

t2

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memory locationindividual

threadindividual

state S1


v

rd v t1

DSTM.rs

predicate evaluationDSTM.rs(t1,v)=1

DSTMt1

t2

DSTM.started

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predicate evaluationDSTM.started(t1)=1

state S2


v

wr v t2

DSTM.rs DSTMt1

t2 DSTM.ws

DSTM.started

DSTM.started

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state S3

Product system

• The product of two systems: AB • Predicates = A.predicates B.predicates• Actions =

commit(t) = { if (A.pre B.pre) then … }rd(t,v) = { if (A.pre B.pre) then … }…

• M refines RSS iff on every executionSIM holds: action a M.pre(a) RSS.pre(a)

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Checking DSTM refines RSS

• The only precondition in RSS is for commit(t)• We need to check SIM =

t.DSTM.validated(t) RSS.invalid(t) RSS.wp(t,t)

holds for DSTM RSS for all reachable states• Proof rule:

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DSTMRSS SIM

DSTM refines RSS

how do we check this safety

property?

Checking safety by invisible invariants

• How do we prove that property holdsfor all reachable states of system M?

• Pnueli, Ruah, Zuck [TACAS’01]• Come up with inductive invariant that

contains reachable states of M and strengthens SIM:

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I1: Initial I2: transition ’ I3:

M

Strict serializability proof rule

Proof roadmap:1. Divine candidate invariant 2. Prove I1, I2, I3

I1: Initial I2: transition ’ I3: SIM

DSTM RSS SIM

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Two challenges

Proof roadmap:1. Divine candidate invariant 2. Prove I1, I2, I3But

how do we find a candidate ?infinite space of possibilitiesgiven candidate how do we check the proof rule? checking AB is undecidable for first-order logic


DSTM RSS SIM

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Our solution

Proof roadmap:1. Divine candidate invariant 2. Prove I1, I2, I3But

how do we find a candidate ?use templates and iterative weakeninggiven candidate how do we check the proof rule? use abstract checking


DSTM RSS SIM

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utilize insightson transactional memory

implementations

Invariant for DSTMRSSP1: t, t1 . RSS.wp(t,t1) RSS.invalid(t) RSS.pending(t)

v . vRSS.ws(t1) vRSS.ws(t)

P2: t, v . vRSS.rs(t) DSTM.aborted(t) vDSTM.rs(t)

P3: t, v . vRSS.ws(t) DSTM.aborted(t) vDSTM.os(t)

P4: t . DSTM.validated(t) RSS.wp(t,t)

P5: t . DSTM.validated(t) RSS.invalid(t)

P6: t . DSTM.validated(t) RSS.pending(t)

Inductive invariant involving only RSS – can use for all future proofs

Templates for DSTMRSSP1: t, t1 . 1(t,t1) 2(t) 3(t)

v . v4(t1) v5(t)

P2: t, v . v1(t) 2(t) v3(t)

P3: t, v . V1(t) 2(t) v3(t)

P4: t . 1(t) 2(t,t)

P5: t . 1(t) 2(t)

P6: t . 1(t) 2(t)

Templates for DSTMRSSt, t1 . 1(t,t1) 2(t) 3(t) v . v4(t1) v5(t)

t, v . v1(t) 2(t) v3(t)

t . 1(t) 2(t,t)

Why templates?• Makes invariant separable• Controls complexity of invariants• Adding templates enables refinement

Mining candidate invariants

• Use predefined set of templates to specify structure of candidate invariants– t,v . 1 2 3

– 1, 2, 3 are predicates of M or their negations– Existential formulas capturing 1-level conflicts

v . v4(t1) v5(t2)

• Mine candidate invariants from concrete execution

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Iterative invariant weakening

• Initial candidate invariant C0=P1 P2 … Pk• Try to prove I2: transition ’

C1 = { Pi | I0 transition Pi for PiI0}

• If C1=C0 then we have an inductive invariant• Otherwise, compute

C2 = { Pi | C1 transition Pi for PiC1}• Repeat until either– found inductive invariant – check I3: Ck SIM– Reached top {} – trivial inductive invariant


DSTM RSS SIM

86

Weakening illustration

P1 P2 P3 P1 P2 P3

Abstract proof rule

I1: (Initial) I2: abs_transition(()) ’ I3: () SIM

DSTM RSS SIM


DSTM RSS SIM

Formulaabstraction

Abstracttransformer

Approximateentailment

88

Conclusion

• Novel invariant generation using templates – extends applicability of invisible invariants

• Abstract domain and reasoning to check invariants without state explosion

• Proved strict-serializability for TPL, DSTM, TL2– BLAST and TVLA failed

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Verification resultsproperty TPL DSTM TL2 RSS

Bound for invariant gen.

(2,1) (2,1) (2,1) (2,1)

No. cubes 8 184 344 7296

Bounded time 4 8 10 23

Invariant mining time

6 13 26 57

#templates 28 28 28 28

#candidates 22 53 97 19

#proved 22 30 68 14

#minimal 4 8 5 -

avg, time per invariant

3.7 20.3 36 43.4

avg. abs. size 31.7 256.9 1.19k 2.86k

Total time 3.5m 54.3m 129.3m 30.9m

90

Insights on transactional memories

• Transition relation is symmetric – thread identifiers not usedp’(t,v) … t1 … t2

• Executing thread t interacts only with arbitrary thread or conflict-adjacent thread

• Arbitrary thread:v . vTL2.rs(t1) vTL2.ws(t1)

• Conflict adjacent:v . vDSTM.rs(t1) vDSTM.ws(t)

91

v1

readset

writeset

v2

t2

t

t3

writeset

read-writeconflict

write-writeconflict

Conflict adjacency

92

v . vrs(t) vDSTM.ws(t2)v . vws(t1) vDSTM.ws(t2)

......

t2

t

t3

Conflict adjacency

93

Related work

• Reduction theoremsGuerarroui et al. [PLDI’08,CONCUR’08]

• Manually-supplied invariants – fixed number of threads and variables + PVSCohen et al. [FMCAS’07]

• Predicate Abstraction + Shape Analysis– SLAM, BLAST, TVLA

94

Related work

• Invisible invariantsArons et al. [CAV’01] Pnueli et al. [TACAS’01]

• Templates – for arithmetic constraints• Indexed predicate abstraction

Shuvendu et al. [TOCL’07]• Thread quantification

Berdine et al. [CAV’08]Segalov et al. [APLAS’09]

95

96

Thank You!

Date post:	24-Feb-2016
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Program Analysis and Synthesis of Parallel Systems

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