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Partially Disjunctive Shape Analysis Josh Berdine Byron Cook MSR Cambridge Tal Lev-Ami Roman Manevich Mooly Sagiv Ran Shaham Tel Aviv University Ganesan Ramalingam MSR India Gilad Arnold UC Berkeley John Field IBM Watson

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Motivation Scaling shape analysis Multiple data structures Intricate data structures Concurrency Procedures Complex properties, e.g., linearizability 2

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3 Non-blocking stack [Treiber,86] void push(Stack *S, data_type v) { [1] Node *x = alloc(sizeof(Node)); [2] x->d = v; [3] do { [4] Node *t = S->Top; [5] x->n = t; [6] } while (!CAS(&S->Top,t,x)); [7] } data_type pop(Stack *S){ [8] do { [9] Node *t = S->Top; [10] if (t == NULL) [11] return EMPTY; [12] Node *s = t->n; [13] data_type r = t->d; [14] } while (!CAS(&S->Top,t,s)); [15] return r; [16] } benign data races unbounded number of threads x points to valid memory? does list remain acyclic? stack linearizable? Automatic proof of linearizability for an unbounded number of threads

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push2(4,5) pop2():8,5push2(7,8) 4 Non-linearizable pairs stack void push2(Stack *S, data_type v1, data_type * v2) { push(s, v1); push(s, v2); } void pop2(Stack *S, data_type * v1, data_type * v2) { *v2 = pop(s); *v1 = pop(s); } time push2(4,5) pop2():8,5push2(7,8) illegal sequential execution

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push2(4,5) pop2():8,5push2(7,8) 5 Non-linearizable pairs stack void push2(Stack *S, data_type v1, data_type * v2) { push(s, v1); push(s, v2); } void pop2(Stack *S, data_type * v1, data_type * v2) { *v2 = pop(s); *v1 = pop(s); } time push2(4,5) pop2():8,5push2(7,8) illegal sequential execution

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Motivation Scaling shape analysis Multiple data structures Intricate data structures Concurrency Procedures Complex properties, e.g., linearizability Develop abstraction techniques to Handle state space blow-ups Handle unbounded number of threads More generally, unbounded subheaps Handle states in a more modular/local way 6

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Developed techniques Partial join [SAS04] Fuse similar abstract heaps Approximate disjunction abstract heaps Intersecting heap abstractions [VMCAI06] Approximate conjunction of abstract heaps Heap decomposition [TACAS07, SAS08] Localize heap abstractions Universal quantified abstractions [CAV08] Represent unbounded copies of abstract (sub)heap 7

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9 Thread-modular analysis Single global resource invariant [Flanagan & Qadeer, SPIN 03] pc=1 Separated resource invariants [Gotsman et al., PLDI 07] Coarse-grained concurrency pc=1 Non-disjoint resource invariants [SAS 08, CAV 08] Fine-grained concurrency pc=1

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10 Concurrent heaps [Yahav, POPL01] Heaps contain both threads and objects Logical structure, or Formula in subset of FO TC [Yorsh et al., TOCL07] thread object with program counter thread-local variable list field list object pc=6 pc=2 x n x Top t

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11 Heaps contain both threads and objects Logical structure, or Formula in subset of FO TC [Yorsh et al., TOCL07] pc=6 pc=2 x n x Top t pc(tr 1 )=6 pc(tr 2 )=2 v 1,v 2,v 3. Top(v 1 ) x(tr 1,v 2 ) t(tr 1,v 1 ) x(tr 2,v 3 ) n(v 2,v 1 ) … v1v1 v3v3 v2v2 tr 1 tr 2 Concurrent heaps [Yahav, POPL01]

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12 Outline Heap decomposition Universally quantified heap abstractions Checking linearizability for an unbounded number of threads Experimental results Partial join

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Heap Decomposition [R. Manevich, T. Lev-Ami, G. Ramalingam, M. Sagiv, J. Berdine, SAS08] 13

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14 Select subheaps Parametrically State-sensitive selection Top x t n pc=7 pc=14 t pc(tr 1 )=7 pc(tr 2 )=14… v 1,v 2. t(tr 1,v 1 ) x(tr 1,v 2 ) t(tr 2,v 1 ) n(v 2,v 1 )… v1v1 v2v2 tr 1 tr 2

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15 Abstraction by decomposition Represent a heap as a conjunction of subheaps (may overlap) Top x t n pc=7 pc(tr 1 )=7 v 1. t(tr 1,v 1 ) x(tr 1,v 2 ) n(v 2,v 1 ) … pc(tr 2 )=14 v 2. t(tr 2,v 1 ) … Top pc=14 t v1v1 tr 2 v1v1 v2v2 tr 1 t n pc=?? n t n n

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Universally Quantified Shape Abstractions [J. Berdine, T. Lev-Ami, R. Manevich, G. Ramalingam, M. Sagiv, CAV08] 16

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Unbounded concurrent heaps 17 void push(Stack *S, data_type v) { [1] Node *x = alloc(sizeof(Node)); [2] x->d = v; [3] do { [4] Node *t = S->Top; [5] x->n = t; [6] } while (!CAS(&S->Top,t,x)); [7] } pc=6 pc=4 x n x Top pc=1 pc=2 x x t pc=4 x t pc=6 x n t t pc=1 Unbounded parallel composition: push(Top,?) ||... || push(Top,?) n n

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Local heaps Each local heap Presents a view of heap relative to one thread Can be instantiated 0 times 18 pc=4 t pc=2 x x pc=1 Top pc=6 t n x Top n n n n n n n n

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Bounded local heaps Each local heap Presents a view of heap relative to one thread Can be instantiated 0 times Bounded by finitary abstraction (Canonical Abstraction) 19 pc=4 t pc=2 x x pc=1 Top pc=6 t n x Top n n n n n n n n

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20 Full heap Top x t n pc=6 pc=2 t pc(tr 1 )=7 pc(tr 2 )=14… v 1,v 2. t(tr 1,v 1 ) x(tr 1,v 2 ) t(tr 2,v 1 ) n(v 2,v 1 )… v1v1 v2v2 tr 1 tr 2

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Decomposed abstract heap pc=2 x Top pc=6 x n Top t tr 1 tr 2 v1v1 v1v1 v2v2 v3v3 pc(tr 1 )=6 v 1,v 2. Top(v 1 ) x(tr 1,v 2 ) t(tr 1,v 1 ) n(v 2,v 1 ) … pc(tr 2 )=2 v 1,v 3. Top(v 1 ) x(tr 2,v 3 ) … 21

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pc=2 x Top pc(t)=6 v 1,v 2. Top(v 1 ) x(t,v 2 ) t(t,v 1 ) n(v 2,v 1 ) … t. pc(t)=2 v 1,v 3. Top(v 1 ) x(t,v 3 ) … Universally quantified local heaps pc=6 x n Top t overlapping local heaps 22 t t v1v1 v1v1 v2v2 v3v3 symbolic thread

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pc(t)=6 v 1,v 2. Top(v 1 ) x(t,v 2 ) t(t,v 1 ) n(v 2,v 1 ) … t. pc(t)=2 v 1,v 3. Top(v 1 ) x(t,v 3 ) … Meaning of quantified invariant pc=6 x n Top t x pc=1 pc=6 pc=2 t Information maintained (dis)equalities between local variables of each thread and global variables Objects reachable from global variables Information lost Number of threads (dis)equalities between local variables of different threads 23 pc=2 x Top x pc=1 pc=6 pc=3 t pc=1 ×m×m n×n×

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Loss of non-aliasing information pc(t)=6 v 1,v 2. Top(v 1 ) x(t,v 2 ) t(t,v 1 ) n(v 2,v 1 ) … t. pc=6 x n Top pc=6 x n t t x n t x t unwanted aliasing consider x->n=t Remedy: record non-aliasing information explicitly 24 n

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Adding non-aliasing information pc=6 P x n Top pc=6 P x n t t x n t x Referenced by exactly one thread pc(t)=6 v 1,v 2. Top(v 1 ) x(t,v 2 ) t(t,v 1 ) n(v 2,v 1 ) Private(v 1 ) Private(v 2 ) … t. P t 25 n

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Adding non-aliasing information pc(t)=6 v 1,v 2. Top(v 1 ) x(t,v 2 ) t(t,v 1 ) n(v 2,v 1 ) Private(v 1 ) Private(v 2 ) … t. pc=6 P x n Top pc=6 P x n t t x n t P x P t Operation on private objects invisible to other threads 26 n

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Recap Decompose heap into conjunction of subheaps Add universal quantification on top of finitary heap abstractions Handle unbounded number of threads Local heaps can overlap Handle fine-grained concurrency Strengthen local heaps by Private predicate Private objects cannot be affected by actions of other threads Missing: transformers (see papers) 27

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Checking linearizability for an unbounded number of threads 28

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Verification of fixed linearization points [Amit et al., CAV07] Compare each concurrent execution to a specific sequential execution Show that every (terminating) concurrent operation returns the same result as its sequential counterpart linearization point operation Concurrent Execution Sequential Execution compare results... linearization point Conjoined Execution compare results 29

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30 Linearization points for Treibers stack void push(Stack *S, data_type v) { [1] Node *x = alloc(sizeof(Node)); [2] x->d = v; [3] do { [4] Node *t = S->Top; [5] x->n = t; [6] } while (!CAS(&S->Top,t,x)); // @LINEARIZE on CAS [7] } data_type pop(Stack *S){ [8] do { [9] Node *t = S->Top; // @LINEARIZE [10] if (t == NULL) [11] return EMPTY; [12] Node *s = t->n; [13] data_type r = t->d; [14] } while (!CAS(&S->Top,t,s)); // @LINEARIZE on CAS [15] return r; [16] }

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Shape analysis with delta abstraction [Amit et al., CAV07] Tracks bounded differences between concurrent and sequential execution Abstracts two heaps together Limited to bounded number of threads Tracks correlations between all threads Feasible up to 4 threads 31 What about an unbounded number of threads?

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32 Our approach Tracks bounded differences between concurrent and sequential execution per thread Handles unbounded number of threads Abstracts correlations between threads Thread-modular characteristics

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Top pc=1 33 Conjoined execution for push concurrent state sequential view isomorphism relation Top void push(Stack *S, data_type v) { [1] Node *x = alloc(sizeof(Node)); [2] x->d = v; [3] do { [4] Node *t = S->Top; [5] x->n = t; [6] } while (!CAS(&S->Top,t,x)); // LINEARIZE on CAS [7] }

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Top pc=1 34 Conjoined execution for push conjoined state void push(Stack *S, data_type v) { [1] Node *x = alloc(sizeof(Node)); [2] x->d = v; [3] do { [4] Node *t = S->Top; [5] x->n = t; [6] } while (!CAS(&S->Top,t,x)); // LINEARIZE on CAS [7] } duo-object

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35 Conjoined execution for push Top pc=2 P x delta object Top pc=1 void push(Stack *S, data_type v) { [1] Node *x = alloc(sizeof(Node)); [2] x->d = v; [3] do { [4] Node *t = S->Top; [5] x->n = t; [6] } while (!CAS(&S->Top,t,x)); // LINEARIZE on CAS [7] }

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36 Conjoined execution for push void push(Stack *S, data_type v) { [1] Node *x = alloc(sizeof(Node)); [2] x->d = v; [3] do { [4] Node *t = S->Top; [5] x->n = t; [6] } while (!CAS(&S->Top,t,x)); // LINEARIZE on CAS [7] } Top pc=2 P x Top pc=1 Top pc=5 P x t … Top pc=6 P x t n if (S Top == t) S Top = x; evaluate to true; else evaluate to false; Top pc=7 n

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37 Run operation sequentially void push(Stack *S, data_type v) { [1] Node *x = alloc(sizeof(Node)); [2] x->d = v; [3] do { [4] Node *t = S->Top; [5] x->n = t; [6] } while (!CAS(&S->Top,t,x)); // LINEARIZE on CAS [7] } Top pc=7 n Top pc=7 n x Top pc=7 n x t Top pc=7 n x t n Top pc=7 nn

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38 Run operation sequentially Top pc=7 n Top pc=7 n x Top pc=7 n x t Top pc=7 n x t n Top pc=7 n But how do you handle unboundedness due to recursive data structures? Employ Canonical Heap Abstraction void push(Stack *S, data_type v) { [1] Node *x = alloc(sizeof(Node)); [2] x->d = v; [3] do { [4] Node *t = S->Top; [5] x->n = t; [6] } while (!CAS(&S->Top,t,x)); // LINEARIZE on CAS [7] }

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39 An unbounded state void push(Stack *S, data_type v) { [1] Node *x = alloc(sizeof(Node)); [2] x->d = v; [3] do { [4] Node *t = S->Top; [5] x->n = t; [6] } while (!CAS(&S->Top,t,x)); LINEARIZE on CAS [7] } pc=6 pc=4 P x n P x Top pc=1 pc=2 P x P x t pc=4 P x t pc=6 P x n t t pc=1 unbounded number of delta objects n n

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Top pc=1 n n Top pc=2 P x n n pc=4 P x Top t n n pc=6 P x n Top t n n 40 Bounded local states number of delta objects per local heap bounded

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Observations used Unbounded number of heap objects Number of delta objects created per thread is bounded Objects in recursive data structures bounded by known shape abstractions Delta objects always referenced by local variables + global variables Captured by local heaps Threads mutate data structure near global access points 41

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42 HeDec: system for Heap Decomposition Parametric: allows experimenting with different decompositions Analysis designer specifies decomposition Subheaps not necessarily disjoint Applicable for states with threads Soundness automatically guaranteed for Any decomposition specification Any transformer specification

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43 Verified Programs#statestime (sec.) Treibers stack [1986] 764 7 Two-lock queue [Michael & Scott, PODC96] 3,415 17 Non-blocking queue [Doherty & Groves, FORTE04] 10,333 252 Experimental results First automatic verification of linearizability for unbounded number of threads

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Partial Join [R. Manevich, M. Sagiv, G. Ramalingam, J. Field, SAS04] 44

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Similar abstract heaps pc=6 P x n Top t 45 n pc=6 P x n Top t n

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Fused heap pc=6 P x n Top t 46 n

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Non-similar abstract heaps pc=6 P x n Top t 47 n pc=5 P x Top t n

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Principle 48 u,v. 1 (u) 1 (u,v) t. u,v. 2 (u) 2 (u,v)

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Principle 49 u,v. (u) (u,v) 1 (u,v) t. u,v. (u) (u,v) 2 (u,v)

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Principle 50 u,v. (u) (u,v) t.

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51 Related work [Yahav, POPL01] Shape analysis with counter abstraction [Gotsman et al., PLDI07] Thread-modular shape analysis for coarse-grained concurrency [Amit et al., CAV07] Linearizability for a bounded number of threads [Vafeiadis et al.,06,07,08] Linearizability for an unbounded number of threads with Rely-Guarantee reasoning w. separation logic Requires user annotations [Gulwani et al., POPL08] Lifting abstract interpreters to quantified logical domains [Pnueli et al., TACAS01] [Clarke et al., TACAS08] [Namjoshi, VMCAI07] Model checking concurrent systems shape analysis model checking concurrency +

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52 Conclusion Three abstraction techniques Heap decomposition Thread quantification Partial join Parametric system Integrated into TVLA Handle combination of Unbounded number of threads Dynamically-allocated memory Automatically proves linearizability Also useful for sequential programs

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Thanks! 53

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54 i : v: j } i, j (v) } v, w: m,n } i,m (v) i, n (w) i, m, n (v, w) } Canonical Heaps

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55 t: i : v: j { i, j (t, v)} v,w: m,n } i,m (t, v) i, n (t, w) i, m, n (t, v, w) } Lifted Canonical Heaps

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