Towards Algorithmic Synthesis of Synchronization for Shared-Memory Concurrent Programs

Towards Algorithmic Synthesis of Synchronization for Shared-Memory Concurrent Programs Roopsha Samanta The University of Texas at Austin July 6, 2012 Roopsha Samanta Algorithmic Synthesis of Synchronization 1 / 20

Problem Definition Given a shared-memory concurrent program P, composed of unsynchronized processes P 1,..., P k, and a (branching-time) temporal logic specification φ spec such that P = φ spec, automatically generate synchronized processes P s 1,..., P s k such that P s = φ spec. Roopsha Samanta Algorithmic Synthesis of Synchronization 2 / 20

Example P: x : {0, } with x = 1 P 1 P 2 P 1 : L 1 : while(x < 2) L 2 : x := x + 1; L 3 : while true; P 2 : T 1 : while(x > 0) T 2 : x := x 1; T 3 : while true; φ spec : AF(L 3 T 3 (x = 0 x = 2)) Roopsha Samanta Algorithmic Synthesis of Synchronization 3 / 20

Example P: x : {0, } with x = 1 P 1 P 2 = φ spec : AF(L 3 T 3 (x = 0 x = 2)) Roopsha Samanta Algorithmic Synthesis of Synchronization 3 / 20

Q&A Why shared-memory concurrent programs? Ubiquitous Hard to write Harder to verify Why temporal logic specifications? Why branching-time? Terminating, ongoing programs Safety, liveness Expressivity Why synthesize only the synchronization? Trickiest part Simple specifications, tractable Roopsha Samanta Algorithmic Synthesis of Synchronization 4 / 20

A Vocabulary L L-symbols Variable symbols (x) Function, constant symbols (, +, ) Predicate, proposition symbols (L 1, T 3, >, <) Sort symbols (bool, int0to2) Distinguished equality predicate symbol = (Sorted) L-terms and L-atoms: L-term (t): v or f (t 1,..., t m ) (x, 1, x + 1) L-atom(G): B(t 1,..., t m ) or t 1 = t 2 (x > 1, L 1, x = 1) Sorts mapped to nonempty domains (bool {T, F}, int0to2 {0, }) Symbols, terms, atoms mapped to values in appropriate domains. Roopsha Samanta Algorithmic Synthesis of Synchronization 5 / 20

(Finite-state) Concurrent Programs over L Syntax: P ::= vardeclaration process process vardeclaration ::= v 1 : domain 1,..., v n : domain n process ::= localvardeclaration body localvardeclaration ::= vardeclaration body ::= labeledstmt ; body labeledstmt labeledstmt ::= LOC: atomicstmt atomicstmt ::= assignment conditiontest goto CCR assignment ::= v:=t conditiontest ::= if G LOC if else, LOC else goto ::= goto LOC CCR ::= G body v,loc ::= L-symbol, t ::= L-term, G ::= L-atom Semantics: Finite-state transition system (S, S 0, R) Roopsha Samanta Algorithmic Synthesis of Synchronization 6 / 20

CTL-like Specifications over L (LCTL) Syntax: φ ::= L-atom φ φ φ EX φ EX i φ A[φ U φ] E[φ U φ] Semantics: Defined from semantics of L-atoms, propositional and temporal operators Roopsha Samanta Algorithmic Synthesis of Synchronization 7 / 20

Revisiting Example P: x : {0, } with x = 1 P 1 P 2 P 1 : L 1 : if (x < 2) L 2, L 4 ; L 2 : x := x + 1 ; L 3 : goto L 1 ; L 4 : goto L 4 ; P 2 : T 1 : if (x > 0) T 2, T 4 ; T 2 : x := x 1 ; T 3 : goto T 1 ; T 4 : goto T 4 ; AF(L 4 T 4 (x = 0 x = 2)) Roopsha Samanta Algorithmic Synthesis of Synchronization 8 / 20

Synthesis Algorithm Sketch 1 Formulate an LCTL formula φ P for semantics of P. 2 Construct a tableau T φ for φ : φ P φ spec. If T φ is empty, declare specification as unsatisfiable. 3 If T φ is non-empty, extract a model M for φ from it. 4 Decompose M to obtain CCRs to synchronize each process. 5 Compile CCRs into lower level synchronization primitives ([EmeSam2011]) Roopsha Samanta Algorithmic Synthesis of Synchronization 9 / 20

Formulation of φ P P: x : {0, } with x = 1 P 1 P 2 P 1 : L 1 : if (x < 2) L 2, L 4 ; L 2 : x := x + 1 ; L 3 : goto L 1 ; L 4 : goto L 4 ; P 2 : T 1 : if (x > 0) T 2, T 4 ; T 2 : x := x 1 ; T 3 : goto T 1 ; T 4 : goto T 4 ; At any step, only one process moves AG(L 1 AX 2 (L 1 ))... Roopsha Samanta Algorithmic Synthesis of Synchronization 10 / 20

Formulation of φ P P: x : {0, } with x = 1 P 1 P 2 P 1 : L 1 : if (x < 2) L 2, L 4 ; L 2 : x := x + 1 ; L 3 : goto L 1 ; L 4 : goto L 4 ; P 2 : T 1 : if (x > 0) T 2, T 4 ; T 2 : x := x 1 ; T 3 : goto T 1 ; T 4 : goto T 4 ; A process is always in exactly one of its locations AG(L 1 L 2 L 3 L 4 ) AG(L 1 (L 2 L 3 L 4 ))... Roopsha Samanta Algorithmic Synthesis of Synchronization 10 / 20

Formulation of φ P P: x : {0, } with x = 1 P 1 P 2 P 1 : L 1 : if (x < 2) L 2, L 4 ; L 2 : x := x + 1 ; L 3 : goto L 1 ; L 4 : goto L 4 ; P 2 : T 1 : if (x > 0) T 2, T 4 ; T 2 : x := x 1 ; T 3 : goto T 1 ; T 4 : goto T 4 ; Some process always moves. AG(EX true) Roopsha Samanta Algorithmic Synthesis of Synchronization 10 / 20

Formulation of φ P P: x : {0, } with x = 1 P 1 P 2 P 1 : L 1 : if (x < 2) L 2, L 4 ; L 2 : x := x + 1 ; L 3 : goto L 1 ; L 4 : goto L 4 ; P 2 : T 1 : if (x > 0) T 2, T 4 ; T 2 : x := x 1 ; T 3 : goto T 1 ; T 4 : goto T 4 ; Initial condition L 1 T 1 x = 1 Roopsha Samanta Algorithmic Synthesis of Synchronization 10 / 20

Formulation of φ P P: x : {0, } with x = 1 P 1 P 2 P 1 : L 1 : if (x < 2) L 2, L 4 ; L 2 : x := x + 1 ; L 3 : goto L 1 ; L 4 : goto L 4 ; P 2 : T 1 : if (x > 0) T 2, T 4 ; T 2 : x := x 1 ; T 3 : goto T 1 ; T 4 : goto T 4 ; Local process execution AG((L 1 x < 2) AX 1 (L 2 )) AG((L 1 x 2) AX 1 (L 4 )) AG( c {0,1,2} (L 2 x = c) AX 1 (L 3 x = c + 1)) AG(L 3 AX 1 (L 1 ))... Roopsha Samanta Algorithmic Synthesis of Synchronization 10 / 20

Construction of Tableau T φ Tableau basics: Finite, rooted, directed AND/OR graph Nodes labeled with formulas OR-node formulas valid iff formulas in some AND-node succ. valid AND-node formulas valid iff formulas in all OR-node succ. valid Assume ability to evaluate L-atoms and L-terms Algorithm Sketch: Let root node of T φ be OR-node, labeled φ Repeat until no new nodes can be added: Add appropriate AND-node succ. of OR-nodes and OR-node succ. of AND-nodes, merging equivalent nodes Prune inconsistent nodes to get final T φ Roopsha Samanta Algorithmic Synthesis of Synchronization 11 / 20

Construction of Tableau T φ φ spec = AF(L 4 T 4 (x = 0 x = 2)) L 1 T 1 x = 1 EX 1 T EX 2 T (EX 1 T EX 2 T) φ spec Roopsha Samanta Algorithmic Synthesis of Synchronization 12 / 20

Construction of Tableau T φ φ spec = AF(L 4 T 4 (x = 0 x = 2)) L 1 T 1 x = 1 EX 1 T EX 2 T (EX 1 T EX 2 T) φ spec L 1 T 1 x = 1 AX 1 L 2 AX 1 T 1 AX 2 T 2 AX 2 L 1 EX 1 T φ spec L 1 T 1 x = 1 AX 1 L 2 AX 1 T 1 AX 2 T 2 AX 2 L 1 EX 1 T EX 2 T φ spec L 1 T 1 x = 1 AX 1 L 2 AX 1 T 1 AX 2 T 2 AX 2 L 1 EX 2 T φ spec Roopsha Samanta Algorithmic Synthesis of Synchronization 12 / 20

Construction of Tableau T φ φ spec = AF(L 4 T 4 (x = 0 x = 2)) L 1 T 1 x = 1 EX 1 T EX 2 T (EX 1 T EX 2 T) φ spec L 1 T 1 x = 1 AX 1 L 2 AX 1 T 1 AX 2 T 2 AX 2 L 1 EX 1 T φ spec L 1 T 1 x = 1 AX 1 L 2 AX 1 T 1 AX 2 T 2 AX 2 L 1 EX 1 T EX 2 T φ spec L 1 T 1 x = 1 AX 1 L 2 AX 1 T 1 AX 2 T 2 AX 2 L 1 EX 2 T φ spec 1 1 2 2 L 2 T 1 x = 1 L 1 T 2 x = 1 φ spec φ spec Roopsha Samanta Algorithmic Synthesis of Synchronization 12 / 20

Extraction of Model M from T φ Model-fragment: Rooted DAG of AND-nodes in T φ All eventualities in root node fulfilled Identify model fragments rooted at each AND-node of T φ If multiple model fragments, pick one of minimal size Join model fragments to get model Roopsha Samanta Algorithmic Synthesis of Synchronization 13 / 20

Extraction of Model M from T φ L 1 T 1 x = 2 1 2 L 4 T 1 x = 2 L 1 T 2 x = 2 2 1 L 4 T 2 x = 2 L 1 T 1 x = 1 1 1 2 2 L 2 T 1 x = 1 L 1 T 2 x = 1 1 2 1 2 L 3 T 1 x = 2 L 2 T 2 x = 1 L 1 T 3 x = 0 L 1 T 1 x = 0 1 2 L 2 T 1 x = 0 L 1 T 4 x = 0 2 1 L 2 T 4 x = 0 L 4 T 3 x = 1 1 2 1 2 1 2 L 3 T 4 x = 1 L 3 T 2 x = 2 L 2 T 3 x = 0 L 4 T 1 x = 1 L 1 T 4 x = 1 L 4 T 2 x = 1 L 4 T 4 x = 0 L 4 T 4 x = 2 L 2 T 4 x = 1 L 4 T 3 x = 0 L 4 T 1 x = 0 L 1 T 4 x = 2 L 3 T 4 x = 2 Roopsha Samanta Algorithmic Synthesis of Synchronization 14 / 20

Obtaining P s 1, Ps 2 from M L 1 T 1 x = 2 1 2 L 4 T 1 x = 2 L 1 T 2 x = 2 2 1 2 L 4 T 2 x = 2 L 1 T 1 x = 1 1 1 2 2 L 2 T 1 x = 1 L 1 T 2 x = 1 1 2 1 2 L 3 T 1 x = 2 L 2 T 2 x = 1 L 1 T 3 x = 0 L 1 T 1 x = 0 1 2 L 2 T 1 x = 0 L 1 T 4 x = 0 2 1 L 2 T 4 x = 0 L 4 T 3 x = 1 1 2 1 2 1 2 L 3 T 4 x = 1 L 3 T 2 x = 2 L 2 T 3 x = 0 L 4 T 1 x = 1 2 1 L 1 T 4 x = 1 L 4 T 2 x = 1 L 4 T 4 x = 0 L 4 T 4 x = 2 L 2 T 4 x = 1 L 4 T 3 x = 0 L 4 T 1 x = 0 L 1 T 4 x = 2 L 3 T 4 x = 2 Roopsha Samanta Algorithmic Synthesis of Synchronization 15 / 20

Obtaining P s 1, Ps 2 from M P: x : {0, } with x = 1 P 1 P 2 P s 1: L 1 : if (x < 2) L 2, L 4 ; L 2 : ((x = 0) (T 1 T 3 )) x := x + 1 ; L 3 : goto L 1 ; L 4 : goto L 4 ; P s 2: T 1 : if (x > 0) T 2, T 4 ; T 2 : ((x = 2) (L 1 L 3 )) x := x 1 ; T 3 : goto T 1 ; T 4 : goto T 4 ; AF(L 4 T 4 (x = 0 x = 2)) Roopsha Samanta Algorithmic Synthesis of Synchronization 15 / 20

Assumptions, Correctness, Complexity [Assumptions]: P 1, P 2 are finite-state programs over L All program variables are shared variables, initialized to specific values L-atoms and L-terms can be evaluated [Soundness]: Given unsynchronized processes P 1, P 2 and an LCTL formula φ spec, if the algorithm generates P s, then P s = φ spec. [Completeness]: Given unsynchronized processes P 1, P 2 and an LCTL formula φ spec, the algorithm generates P s such that P s = φ spec if φ = φ spec φ P is satisfiable. [Complexity]: Exponential in the size of φ spec and P variables. Roopsha Samanta Algorithmic Synthesis of Synchronization 16 / 20

Uninitialized Variables φ spec = AF(L 4 T 4 (x = 0 x = 2)) L 1 T 1 x = 1 EX 1 T EX 2 T (EX 1 T EX 2 T) φ spec Roopsha Samanta Algorithmic Synthesis of Synchronization 17 / 20

Uninitialized Variables φ spec = AF(L 4 T 4 (x = 0 x = 2)) L 1 T 1 x = 1 EX 1 T EX 2 T (EX 1 T EX 2 T) φ spec L 1 T 1 x = 1 AX 1 L 2 AX 1 T 1 AX 2 T 2 AX 2 L 1 EX 1 T φ spec L 1 T 1 x = 1 AX 1 L 2 AX 1 T 1 AX 2 T 2 AX 2 L 1 EX 1 T EX 2 T φ spec 1 2 L 1 T 1 x = 1 AX 1 L 2 AX 1 T 1 AX 2 T 2 AX 2 L 1 EX 2 T φ spec 1 2 L 2 T 1 x = 1 φ spec L 1 T 2 x = 1 φ spec Roopsha Samanta Algorithmic Synthesis of Synchronization 17 / 20

Uninitialized Variables φ spec = AF(L 4 T 4 (x = 0 x = 2)) EX(L 1 T 1 x = 0) EX(L 1 T 1 x = 1) EX(L 1 T 1 x = 2) φ spec L 1 T 1 x = 2 EX 1 T EX 2 T (EX 1 T EX 2 T) φ spec L 1 T 1 x = 0 EX 1 T EX 2 T (EX 1 T EX 2 T) φ spec L 1 T 1 x = 1 EX 1 T EX 2 T (EX 1 T EX 2 T) φ spec Roopsha Samanta Algorithmic Synthesis of Synchronization 17 / 20

Uninitialized Variables L 1 T 1 x = 2 1 2 L 4 T 1 x = 2 L 1 T 2 x = 2 2 1 2 L 4 T 2 x = 2 L 1 T 1 x = 1 1 1 2 2 L 2 T 1 x = 1 L 1 T 2 x = 1 1 2 1 2 L 3 T 1 x = 2 L 2 T 2 x = 1 L 1 T 3 x = 0 L 1 T 1 x = 0 1 2 L 2 T 1 x = 0 L 1 T 4 x = 0 1 2 1 L 2 T 4 x = 0 L 4 T 3 x = 1 1 2 1 2 1 2 L 3 T 4 x = 1 L 3 T 2 x = 2 L 2 T 3 x = 0 L 4 T 1 x = 1 2 1 L 1 T 4 x = 1 L 4 T 2 x = 1 L 4 T 4 x = 0 L 4 T 4 x = 2 L 2 T 4 x = 1 L 4 T 3 x = 0 L 4 T 1 x = 0 L 1 T 4 x = 2 L 3 T 4 x = 2 Roopsha Samanta Algorithmic Synthesis of Synchronization 17 / 20

Observability P: x : {0, } with x = 1 P 1 P 2 P s 1: L 1 : if (x < 2) L 2, L 4 ; L 2 : ((x = 0) (T 1 T 3 )) x := x + 1 ; L 3 : goto L 1 ; L 4 : goto L 4 ; P s 2: T 1 : if (x > 0) T 2, T 4 ; T 2 : ((x = 2) (L 1 L 3 )) x := x 1 ; T 3 : goto T 1 ; T 4 : goto T 4 ; AF(L 4 T 4 (x = 0 x = 2)) Roopsha Samanta Algorithmic Synthesis of Synchronization 18 / 20

Observability L 1 T 1 x = 2 1 2 L 4 T 1 x = 2 L 1 T 2 x = 2 2 1 2 L 4 T 2 x = 2 L 1 T 1 x = 1 1 1 2 2 L 2 T 1 x = 1 L 1 T 2 x = 1 1 2 1 2 L 3 T 1 x = 2 L 2 T 2 x = 1 L 1 T 3 x = 0 L 1 T 1 x = 0 1 2 L 2 T 1 x = 0 L 1 T 4 x = 0 1 2 1 L 2 T 4 x = 0 L 4 T 3 x = 1 1 2 1 2 1 2 L 3 T 4 x = 1 L 3 T 2 x = 2 L 2 T 3 x = 0 L 4 T 1 x = 1 2 1 L 1 T 4 x = 1 L 4 T 2 x = 1 L 4 T 4 x = 0 L 4 T 4 x = 2 L 2 T 4 x = 1 L 4 T 3 x = 0 L 4 T 1 x = 0 L 1 T 4 x = 2 L 3 T 4 x = 2 Roopsha Samanta Algorithmic Synthesis of Synchronization 18 / 20

Observability P: x : {0, } with x = 1 t : bool with t = 1 l : bool with l = 1 P 1 P 2 P s 1 : L 1 : if (x < 2) L 2, L 4 ; l := 0 ; L 2 : ((x = 0) (t = 1)) x := x + 1; l := 1 ; L 3 : goto L 1 ; L 4 : goto L 4 ; P s 2 : T 1 : if (x > 0) T 2, T 4 ; t := 0 ; T 2 : ((x = 2) (l = 1)) x := x 1; t := 1 ; T 3 : goto T 1 ; T 4 : goto T 4 ; AF(L 4 T 4 (x = 0 x = 2)) Roopsha Samanta Algorithmic Synthesis of Synchronization 18 / 20

Related Work Inference of high-level synchronization, guarded commands: [EC82, VYY09] Synthesis of low-level synchronization for relaxed memory models: [KVV10] Sketching: [S-LRBE05] Open systems [PR89] Roopsha Samanta Algorithmic Synthesis of Synchronization 19 / 20

Ongoing Work Efficient tableau construction? Infinite-state programs? Pleasant guards? Other synchronization primitives? Roopsha Samanta Algorithmic Synthesis of Synchronization 20 / 20