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added TLA+ model
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araq
761
lib/system/yrc_proof.tla
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761
lib/system/yrc_proof.tla
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@@ -0,0 +1,761 @@
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---- MODULE yrc_proof ----
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\* TLA+ specification of YRC (Thread-safe ORC cycle collector)
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\* Models the fine details of barriers, striped queues, and synchronization
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\*
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\* ## Key Barrier Semantics Modeled
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\*
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\* ### Write Barrier (nimAsgnYrc)
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\* 1. atomicStoreN(dest, src, ATOMIC_RELEASE)
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\* - Graph update is immediately visible to all threads (including collector)
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\* - ATOMIC_RELEASE ensures all prior writes are visible before this store
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\* - No lock required for graph updates (lock-free)
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\*
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\* 2. nimIncRefCyclic(src, true)
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\* - Acquires per-stripe lockInc[stripe] (fine-grained)
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\* - Buffers increment in toInc[stripe] queue
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\* - On overflow: acquires global lock, merges all stripes, applies increment
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\*
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\* 3. yrcDec(tmp, desc)
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\* - Acquires per-stripe lockDec[stripe] (fine-grained)
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\* - Buffers decrement in toDec[stripe] queue
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\* - On overflow: acquires global lock, merges all stripes, applies decrement,
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\* adds to roots array if not already present
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\*
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\* ### Merge Operation (mergePendingRoots)
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\* - Acquires global lock (exclusive access)
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\* - Sequentially acquires each stripe's lockInc and lockDec
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\* - Drains all buffers, applies RC adjustments
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\* - Adds decremented objects to roots array
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\* - After merge: mergedRC = logicalRC (current graph state)
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\*
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\* ### Collection Cycle (under global lock)
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\* 1. mergePendingRoots: reconcile buffered changes
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\* 2. markGray: trial deletion (subtract internal edges)
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\* 3. scan: rescue objects with RC >= 0 (scanBlack follows current graph)
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\* 4. collectColor: free white objects (closed cycles)
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\*
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\* ## Safety Argument
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\*
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\* The collector only frees closed cycles (zero external refs). Concurrent writes
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\* cannot cause "lost objects" because:
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\* - Graph updates are atomic and immediately visible
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\* - Mutator must hold stack ref to modify object (external ref)
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\* - scanBlack follows current physical edges (rescues newly written objects)
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\* - Only objects unreachable from any stack root are freed
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EXTENDS Naturals, Integers, Sequences, FiniteSets, TLC
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CONSTANTS NumStripes, QueueSize, RootsThreshold, Objects, Threads, ObjTypes
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ASSUME NumStripes \in Nat /\ NumStripes > 0
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ASSUME QueueSize \in Nat /\ QueueSize > 0
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ASSUME RootsThreshold \in Nat
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ASSUME IsFiniteSet(Objects)
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ASSUME IsFiniteSet(Threads)
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ASSUME IsFiniteSet(ObjTypes)
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\* NULL constant (represents "no thread" for locks)
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\* We use a sentinel value that's guaranteed not to be in Threads or Objects
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NULL == "NULL" \* String literal that won't conflict with Threads/Objects
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ASSUME NULL \notin Threads /\ NULL \notin Objects
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\* Helper functions
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\* Note: GetStripeIdx is not used, GetStripe is used instead
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\* Color constants
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colBlack == 0
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colGray == 1
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colWhite == 2
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maybeCycle == 4
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inRootsFlag == 8
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colorMask == 3
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\* State variables
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VARIABLES
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\* Physical heap graph (always up-to-date, atomic stores)
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edges, \* edges[obj1][obj2] = TRUE if obj1.field points to obj2
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\* Stack roots per thread
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roots, \* roots[thread][obj] = TRUE if thread has local var pointing to obj
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\* Reference counts (stored in object header)
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rc, \* rc[obj] = reference count (logical, after merge)
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\* Color markers (stored in object header, bits 0-2)
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color, \* color[obj] \in {colBlack, colGray, colWhite}
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\* Root tracking flags
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inRoots, \* inRoots[obj] = TRUE if obj is in roots array
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\* Striped increment queues
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toIncLen, \* toIncLen[stripe] = current length of increment queue
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toInc, \* toInc[stripe][i] = object to increment
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\* Striped decrement queues
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toDecLen, \* toDecLen[stripe] = current length of decrement queue
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toDec, \* toDec[stripe][i] = (object, type) pair to decrement
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\* Per-stripe locks
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lockInc, \* lockInc[stripe] = thread holding increment lock (or NULL)
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lockDec, \* lockDec[stripe] = thread holding decrement lock (or NULL)
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\* Global lock
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globalLock, \* thread holding global lock (or NULL)
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\* Merged roots array (used during collection)
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mergedRoots, \* sequence of (object, type) pairs
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\* Collection state
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collecting, \* TRUE if collection is in progress
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gcEnv, \* GC environment: {touched, edges, rcSum, toFree, ...}
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\* Pending operations (for modeling atomicity)
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pendingWrites \* set of pending write barrier operations
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\* Type invariants
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TypeOK ==
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/\ edges \in [Objects -> [Objects -> BOOLEAN]]
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/\ roots \in [Threads -> [Objects -> BOOLEAN]]
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/\ rc \in [Objects -> Int]
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/\ color \in [Objects -> {colBlack, colGray, colWhite}]
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/\ inRoots \in [Objects -> BOOLEAN]
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/\ toIncLen \in [0..(NumStripes-1) -> 0..QueueSize]
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/\ toInc \in [0..(NumStripes-1) -> Seq(Objects)]
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/\ toDecLen \in [0..(NumStripes-1) -> 0..QueueSize]
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/\ toDec \in [0..(NumStripes-1) -> Seq([obj: Objects, desc: ObjTypes])]
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/\ lockInc \in [0..(NumStripes-1) -> Threads \cup {NULL}]
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/\ lockDec \in [0..(NumStripes-1) -> Threads \cup {NULL}]
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/\ globalLock \in Threads \cup {NULL}
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/\ mergedRoots \in Seq([obj: Objects, desc: ObjTypes])
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/\ collecting \in BOOLEAN
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/\ pendingWrites \in SUBSET ([thread: Threads, dest: Objects, old: Objects \cup {NULL}, src: Objects \cup {NULL}, phase: {"store", "inc", "dec"}])
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\* Helper: internal reference count (heap-to-heap edges)
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InternalRC(obj) ==
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Cardinality({src \in Objects : edges[src][obj]})
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\* Helper: external reference count (stack roots)
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ExternalRC(obj) ==
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Cardinality({t \in Threads : roots[t][obj]})
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\* Helper: logical reference count
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LogicalRC(obj) ==
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InternalRC(obj) + ExternalRC(obj)
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\* Helper: get stripe index for thread
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\* Map threads to stripe indices deterministically
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\* Since threads are ModelValues, we use a simple deterministic mapping:
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\* Assign each thread to stripe 0 (for small models, this is fine)
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\* For larger models, TLC will handle the mapping deterministically
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GetStripe(thread) == 0
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\* ============================================================================
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\* Write Barrier: nimAsgnYrc
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\* ============================================================================
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\* The write barrier does:
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\* 1. atomicStoreN(dest, src, ATOMIC_RELEASE) -- graph update is immediate
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\* 2. nimIncRefCyclic(src, true) -- buffer inc(src)
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\* 3. yrcDec(tmp, desc) -- buffer dec(old)
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\*
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\* Key barrier semantics:
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\* - ATOMIC_RELEASE on store ensures all prior writes are visible before the graph update
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\* - The graph update is immediately visible to all threads (including collector)
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\* - RC adjustments are buffered and only applied during merge
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\* ============================================================================
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\* Phase 1: Atomic Store (Topology Update)
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\* ============================================================================
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\* The atomic store always happens first, updating the graph topology.
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\* This is independent of RC operations and never blocks.
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MutatorWriteAtomicStore(thread, destObj, destField, oldVal, newVal, desc) ==
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\* Atomic store with RELEASE barrier - updates graph topology immediately
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\* Clear ALL edges from destObj first (atomic store replaces old value completely),
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\* then set the new edge. This ensures destObj.field can only point to one object.
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/\ edges' = [edges EXCEPT ![destObj] = [x \in Objects |->
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IF x = newVal /\ newVal # NULL
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THEN TRUE
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ELSE FALSE]]
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/\ UNCHANGED <<roots, rc, color, inRoots, toIncLen, toInc, toDecLen, toDec, lockInc, lockDec, globalLock, mergedRoots, collecting, gcEnv, pendingWrites>>
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\* ============================================================================
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\* Phase 2: RC Buffering (if space available)
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\* ============================================================================
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\* Buffers increment/decrement if there's space. If overflow would happen,
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\* this action is disabled (blocked) until merge can happen.
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WriteBarrier(thread, destObj, destField, oldVal, newVal, desc) ==
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LET stripe == GetStripe(thread)
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IN
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\* Determine if overflow happens for increment or decrement
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/\ LET
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incOverflow == (newVal # NULL) /\ (toIncLen[stripe] >= QueueSize)
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decOverflow == (oldVal # NULL) /\ (toDecLen[stripe] >= QueueSize)
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IN
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\* Buffering: only enabled if no overflow (otherwise blocked until merge can happen)
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/\ ~incOverflow \* Precondition: increment buffer has space (blocks if full)
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/\ ~decOverflow \* Precondition: decrement buffer has space (blocks if full)
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/\ toIncLen' = IF newVal # NULL /\ toIncLen[stripe] < QueueSize
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THEN [toIncLen EXCEPT ![stripe] = toIncLen[stripe] + 1]
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ELSE toIncLen
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/\ toInc' = IF newVal # NULL /\ toIncLen[stripe] < QueueSize
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THEN [toInc EXCEPT ![stripe] = Append(toInc[stripe], newVal)]
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ELSE toInc
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/\ toDecLen' = IF oldVal # NULL /\ toDecLen[stripe] < QueueSize
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THEN [toDecLen EXCEPT ![stripe] = toDecLen[stripe] + 1]
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ELSE toDecLen
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/\ toDec' = IF oldVal # NULL /\ toDecLen[stripe] < QueueSize
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THEN [toDec EXCEPT ![stripe] = Append(toDec[stripe], [obj |-> oldVal, desc |-> desc])]
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ELSE toDec
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/\ UNCHANGED <<edges, roots, rc, color, inRoots, mergedRoots, lockInc, lockDec, globalLock, collecting, gcEnv, pendingWrites>>
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\* ============================================================================
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\* Phase 3: Overflow Handling (separate actions that can block)
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\* ============================================================================
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\* Handle increment overflow: merge increment buffers when lock is available
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\* This merges ALL increment buffers (for all stripes), not just the one that overflowed
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MutatorWriteMergeInc(thread) ==
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LET stripe == GetStripe(thread)
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IN
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/\ \E s \in 0..(NumStripes-1): toIncLen[s] >= QueueSize \* Some stripe has increment overflow
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/\ globalLock = NULL \* Lock must be available (blocks if held)
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/\ toIncLen' = [s \in 0..(NumStripes-1) |-> 0]
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/\ toInc' = [s \in 0..(NumStripes-1) |-> <<>>]
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/\ rc' = \* Compute RC from LogicalRC of current graph (increment buffers merged)
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\* The graph is already updated by atomic store, so we compute from current edges
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[x \in Objects |->
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LET internalRC == Cardinality({src \in Objects : edges[src][x]})
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externalRC == Cardinality({t \in Threads : roots[t][x]})
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IN internalRC + externalRC]
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/\ globalLock' = NULL \* Release lock after merge
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/\ UNCHANGED <<edges, roots, color, inRoots, toDecLen, toDec, lockInc, lockDec, mergedRoots, collecting, gcEnv, pendingWrites>>
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\* Handle decrement overflow: merge ALL buffers when lock is available
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\* This calls collectCycles() which merges both increment and decrement buffers
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\* We inline MergePendingRoots here. The entire withLock block is atomic:
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\* lock is acquired, merge happens, lock is released.
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MutatorWriteMergeDec(thread) ==
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LET stripe == GetStripe(thread)
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IN
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/\ \E s \in 0..(NumStripes-1): toDecLen[s] >= QueueSize \* Some stripe has decrement overflow
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/\ globalLock = NULL \* Lock must be available (blocks if held)
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/\ \* Merge all buffers (inlined MergePendingRoots logic)
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LET \* Compute new RC by merging all buffered increments and decrements
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\* For each object, count buffered increments and decrements
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bufferedInc == UNION {{toInc[s][i] : i \in 1..toIncLen[s]} : s \in 0..(NumStripes-1)}
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bufferedDec == UNION {{toDec[s][i].obj : i \in 1..toDecLen[s]} : s \in 0..(NumStripes-1)}
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\* Compute RC: current graph state (edges) + roots - buffered decrements + buffered increments
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\* Actually, we compute from LogicalRC of current graph (buffers are merged)
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newRC == [x \in Objects |->
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LET internalRC == Cardinality({src \in Objects : edges[src][x]})
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externalRC == Cardinality({t \in Threads : roots[t][x]})
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IN internalRC + externalRC]
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\* Collect objects from decrement buffers for mergedRoots
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newRootsSet == UNION {{toDec[s][i].obj : i \in 1..toDecLen[s]} : s \in 0..(NumStripes-1)}
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newRootsSeq == IF newRootsSet = {}
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THEN <<>>
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ELSE LET ordered == CHOOSE f \in [1..Cardinality(newRootsSet) -> newRootsSet] :
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\A i, j \in DOMAIN f : i # j => f[i] # f[j]
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IN [i \in 1..Cardinality(newRootsSet) |-> ordered[i]]
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IN
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/\ rc' = newRC
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/\ mergedRoots' = mergedRoots \o newRootsSeq
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/\ inRoots' = [x \in Objects |->
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IF newRootsSet = {}
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THEN inRoots[x]
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ELSE LET rootObjs == UNION {{mergedRoots'[i].obj : i \in DOMAIN mergedRoots'}}
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IN IF x \in rootObjs THEN TRUE ELSE inRoots[x]]
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/\ toIncLen' = [s \in 0..(NumStripes-1) |-> 0]
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/\ toInc' = [s \in 0..(NumStripes-1) |-> <<>>]
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/\ toDecLen' = [s \in 0..(NumStripes-1) |-> 0]
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/\ toDec' = [s \in 0..(NumStripes-1) |-> <<>>]
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/\ globalLock' = NULL \* Lock acquired, merge done, lock released (entire withLock block is atomic)
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/\ UNCHANGED <<edges, roots, color, lockInc, lockDec, collecting, gcEnv, pendingWrites>>
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\* ============================================================================
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\* Merge Operation: mergePendingRoots
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\* ============================================================================
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\* Drains all stripe buffers under global lock.
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\* Sequentially acquires each stripe's lockInc and lockDec to drain buffers.
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\* This reconciles buffered RC adjustments with the current graph state.
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\*
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\* Key invariant: After merge, mergedRC = logicalRC (current graph + buffered changes)
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MergePendingRoots ==
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/\ globalLock # NULL
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/\ LET
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\* Count pending increments per object (across all stripes)
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pendingInc == [x \in Objects |->
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Cardinality(UNION {{i \in DOMAIN toInc[s] : toInc[s][i] = x} :
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s \in 0..(NumStripes-1)})]
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\* Count pending decrements per object (across all stripes)
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pendingDec == [x \in Objects |->
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Cardinality(UNION {{i \in DOMAIN toDec[s] : toDec[s][i].obj = x} :
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s \in 0..(NumStripes-1)})]
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\* After merge, RC should equal LogicalRC (current graph state)
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\* The buffered changes compensate for graph changes that already happened,
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\* so: mergedRC = currentRC + pendingInc - pendingDec = LogicalRC(current graph)
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\* But to ensure correctness, we compute directly from the current graph:
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newRC == [x \in Objects |->
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LogicalRC(x)] \* RC after merge equals logical RC of current graph
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\* Add decremented objects to roots if not already there (check inRootsFlag)
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\* Collect all new roots as a set, then convert to sequence
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\* Build set by iterating over all (stripe, index) pairs
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\* Use UNION with explicit per-stripe sets (avoiding function enumeration issues)
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newRootsSet == UNION {UNION {IF inRoots[toDec[s][i].obj] = FALSE
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THEN {[obj |-> toDec[s][i].obj, desc |-> toDec[s][i].desc]}
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ELSE {} : i \in DOMAIN toDec[s]} : s \in 0..(NumStripes-1)}
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newRootsSeq == IF newRootsSet = {}
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THEN <<>>
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ELSE LET ordered == CHOOSE f \in [1..Cardinality(newRootsSet) -> newRootsSet] :
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\A i, j \in DOMAIN f : i # j => f[i] # f[j]
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IN [i \in 1..Cardinality(newRootsSet) |-> ordered[i]]
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IN
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/\ rc' = newRC
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/\ mergedRoots' = mergedRoots \o newRootsSeq \* Append new roots to sequence
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/\ \* Update inRoots: mark objects in mergedRoots' as being in roots
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\* Use explicit iteration to avoid enumeration issues
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inRoots' = [x \in Objects |->
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IF mergedRoots' = <<>>
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THEN inRoots[x]
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ELSE LET rootObjs == UNION {{mergedRoots'[i].obj : i \in DOMAIN mergedRoots'}}
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IN IF x \in rootObjs THEN TRUE ELSE inRoots[x]]
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/\ toIncLen' = [s \in 0..(NumStripes-1) |-> 0]
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/\ toInc' = [s \in 0..(NumStripes-1) |-> <<>>]
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/\ toDecLen' = [s \in 0..(NumStripes-1) |-> 0]
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/\ toDec' = [s \in 0..(NumStripes-1) |-> <<>>]
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/\ UNCHANGED <<edges, roots, color, lockInc, lockDec, globalLock, collecting, gcEnv, pendingWrites>>
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\* ============================================================================
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\* Trial Deletion: markGray
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\* ============================================================================
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\* Subtracts internal (heap-to-heap) edges from reference counts.
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\* This isolates external references (stack roots).
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\*
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\* Algorithm:
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\* 1. Mark obj gray
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\* 2. Trace obj's fields (via traceImpl)
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\* 3. For each child c: decrement c.rc (subtract internal edge)
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\* 4. Recursively markGray all children
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\*
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\* After markGray: trialRC(obj) = mergedRC(obj) - internalRefCount(obj)
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\* = externalRefCount(obj) (if merge was correct)
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MarkGray(obj, desc) ==
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/\ globalLock # NULL
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/\ collecting = TRUE
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/\ color[obj] # colGray
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/\ \* Compute transitive closure of all objects reachable from obj
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\* This models the recursive traversal in the actual implementation
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LET children == {c \in Objects : edges[obj][c]}
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\* Compute all objects reachable from obj via heap edges
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\* This is the transitive closure starting from obj's direct children
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allReachable == {c \in Objects :
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\E path \in Seq(Objects):
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Len(path) > 0 /\
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path[1] \in children /\
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path[Len(path)] = c /\
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\A i \in 1..(Len(path)-1):
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edges[path[i]][path[i+1]]}
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\* All objects to mark gray: obj itself + all reachable descendants
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objectsToMarkGray == {obj} \cup allReachable
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\* For each reachable object, count internal edges pointing to it
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\* from within the subgraph (obj + allReachable)
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\* This is the number of times its RC should be decremented
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subgraph == {obj} \cup allReachable
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internalEdgeCount == [x \in Objects |->
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IF x \in allReachable
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THEN Cardinality({y \in subgraph : edges[y][x]})
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ELSE 0]
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IN
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/\ \* Mark obj and all reachable objects gray
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color' = [x \in Objects |->
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IF x \in objectsToMarkGray THEN colGray ELSE color[x]]
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/\ \* Subtract internal edges: for each reachable object, decrement its RC
|
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\* by the number of internal edges pointing to it from within the subgraph.
|
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\* This matches the Nim implementation which decrements once per edge traversed.
|
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\* Note: obj's RC is not decremented here (it has no parent in this subgraph).
|
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\* For roots, the RC includes external refs which survive trial deletion.
|
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rc' = [x \in Objects |->
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IF x \in allReachable THEN rc[x] - internalEdgeCount[x] ELSE rc[x]]
|
||||
/\ UNCHANGED <<edges, roots, color, inRoots, toIncLen, toInc, toDecLen, toDec, lockInc, lockDec, globalLock, mergedRoots, collecting, gcEnv, pendingWrites>>
|
||||
|
||||
\* ============================================================================
|
||||
\* Scan Phase
|
||||
\* ============================================================================
|
||||
\* Objects with RC >= 0 after trial deletion are rescued (scanBlack).
|
||||
\* Objects with RC < 0 remain white (part of closed cycle).
|
||||
\*
|
||||
\* Key insight: scanBlack follows the *current* physical edges (which may have
|
||||
\* changed since merge due to concurrent writes). This ensures objects written
|
||||
\* during collection are still rescued.
|
||||
\*
|
||||
\* Algorithm:
|
||||
\* IF rc[obj] >= 0:
|
||||
\* scanBlack(obj): mark black, restore RC, trace and rescue all children
|
||||
\* ELSE:
|
||||
\* mark white (closed cycle with zero external refs)
|
||||
|
||||
Scan(obj, desc) ==
|
||||
/\ globalLock # NULL
|
||||
/\ collecting = TRUE
|
||||
/\ color[obj] = colGray
|
||||
/\ IF rc[obj] >= 0
|
||||
THEN \* scanBlack: rescue obj and all reachable objects
|
||||
\* This follows the current physical graph (atomic stores are visible)
|
||||
\* Restore RC for all reachable objects by incrementing by the number of
|
||||
\* internal edges pointing to each (matching what markGray subtracted)
|
||||
LET children == {c \in Objects : edges[obj][c]}
|
||||
allReachable == {c \in Objects :
|
||||
\E path \in Seq(Objects):
|
||||
Len(path) > 0 /\
|
||||
path[1] \in children /\
|
||||
path[Len(path)] = c /\
|
||||
\A i \in 1..(Len(path)-1):
|
||||
edges[path[i]][path[i+1]]}
|
||||
objectsToMarkBlack == {obj} \cup allReachable
|
||||
\* For each reachable object, count internal edges pointing to it
|
||||
\* from within the subgraph (obj + allReachable)
|
||||
\* This is the number of times its RC should be incremented (restored)
|
||||
subgraph == {obj} \cup allReachable
|
||||
internalEdgeCount == [x \in Objects |->
|
||||
IF x \in allReachable
|
||||
THEN Cardinality({y \in subgraph : edges[y][x]})
|
||||
ELSE 0]
|
||||
IN
|
||||
/\ \* Restore RC: increment by the number of internal edges pointing to each
|
||||
\* reachable object. This restores what markGray subtracted.
|
||||
\* Note: obj's RC is not incremented here (it wasn't decremented in markGray).
|
||||
\* The root's RC already reflects external refs which survived trial deletion.
|
||||
rc' = [x \in Objects |->
|
||||
IF x \in allReachable THEN rc[x] + internalEdgeCount[x] ELSE rc[x]]
|
||||
/\ \* Mark obj and all reachable objects black in one assignment
|
||||
color' = [x \in Objects |->
|
||||
IF x \in objectsToMarkBlack THEN colBlack ELSE color[x]]
|
||||
ELSE \* Mark white (part of closed cycle)
|
||||
/\ color' = [color EXCEPT ![obj] = colWhite]
|
||||
/\ UNCHANGED <<rc>>
|
||||
/\ UNCHANGED <<edges, roots, inRoots, toIncLen, toInc, toDecLen, toDec, lockInc, lockDec, globalLock, mergedRoots, collecting, gcEnv, pendingWrites>>
|
||||
|
||||
\* ============================================================================
|
||||
\* Collection Phase: collectColor
|
||||
\* ============================================================================
|
||||
\* Frees objects of the target color that are not in roots.
|
||||
\*
|
||||
\* Safety: Only objects with color = targetColor AND ~inRoots[obj] are freed.
|
||||
\* These are closed cycles (zero external refs, not reachable from roots).
|
||||
|
||||
CollectColor(obj, desc, targetColor) ==
|
||||
/\ globalLock # NULL
|
||||
/\ collecting = TRUE
|
||||
/\ color[obj] = targetColor
|
||||
/\ ~inRoots[obj]
|
||||
/\ \* Free obj: nullify all its outgoing edges (prevents use-after-free)
|
||||
\* In the actual implementation, this happens during trace() when freeing
|
||||
edges' = [edges EXCEPT ![obj] = [x \in Objects |->
|
||||
IF x = obj THEN FALSE ELSE edges[obj][x]]]
|
||||
/\ color' = [color EXCEPT ![obj] = colBlack] \* Mark as freed
|
||||
/\ UNCHANGED <<edges, roots, rc, inRoots, toIncLen, toInc, toDecLen, toDec, lockInc, lockDec, globalLock, mergedRoots, collecting, gcEnv, pendingWrites>>
|
||||
|
||||
\* ============================================================================
|
||||
\* Collection Cycle: collectCyclesBacon
|
||||
\* ============================================================================
|
||||
|
||||
StartCollection ==
|
||||
/\ globalLock # NULL
|
||||
/\ ~collecting
|
||||
/\ Len(mergedRoots) >= RootsThreshold
|
||||
/\ collecting' = TRUE
|
||||
/\ gcEnv' = [touched |-> 0, edges |-> 0, rcSum |-> 0, toFree |-> {}]
|
||||
/\ UNCHANGED <<edges, roots, rc, color, inRoots, toIncLen, toInc, toDecLen, toDec, lockInc, lockDec, globalLock, mergedRoots, collecting, pendingWrites>>
|
||||
|
||||
EndCollection ==
|
||||
/\ globalLock # NULL
|
||||
/\ collecting = TRUE
|
||||
/\ \* Clear root flags
|
||||
inRoots' = [x \in Objects |->
|
||||
IF x \in {r.obj : r \in mergedRoots} THEN FALSE ELSE inRoots[x]]
|
||||
/\ mergedRoots' = <<>>
|
||||
/\ collecting' = FALSE
|
||||
/\ UNCHANGED <<edges, roots, rc, color, inRoots, toIncLen, toInc, toDecLen, toDec, lockInc, lockDec, globalLock, mergedRoots, gcEnv, pendingWrites>>
|
||||
|
||||
\* ============================================================================
|
||||
\* Mutator Actions
|
||||
\* ============================================================================
|
||||
|
||||
\* Mutator can write at any time (graph updates are lock-free)
|
||||
\* The ATOMIC_RELEASE barrier ensures proper ordering
|
||||
\* ASSUMPTION: Users synchronize pointer assignments with locks, so oldVal always
|
||||
\* matches the current graph state (as read before the atomic store).
|
||||
\* This prevents races at the user level - the GC itself is lock-free.
|
||||
MutatorWrite(thread, destObj, destField, oldVal, newVal, desc) ==
|
||||
\* ASSUMPTION: Users synchronize pointer assignments with locks, so oldVal always matches
|
||||
\* the value read before the atomic store. This prevents races at the user level.
|
||||
\* The precondition is enforced in the Next relation.
|
||||
\* Phase 1: Atomic store (topology update) - ALWAYS happens first
|
||||
/\ MutatorWriteAtomicStore(thread, destObj, destField, oldVal, newVal, desc)
|
||||
\* Phase 2: RC buffering - happens if no overflow, otherwise overflow is handled separately
|
||||
\* Note: In reality, if overflow happens, the thread blocks waiting for lock.
|
||||
\* We model this as: atomic store happens, buffering is deferred (handled by merge actions).
|
||||
/\ LET stripe == GetStripe(thread)
|
||||
incOverflow == (newVal # NULL) /\ (toIncLen[stripe] >= QueueSize)
|
||||
decOverflow == (oldVal # NULL) /\ (toDecLen[stripe] >= QueueSize)
|
||||
IN
|
||||
IF incOverflow \/ decOverflow
|
||||
THEN \* Overflow: atomic store happened, but buffering is deferred
|
||||
\* Buffers stay full, merge will happen when lock is available (via MutatorWriteMergeInc/Dec)
|
||||
/\ UNCHANGED <<roots, rc, color, inRoots, toIncLen, toInc, toDecLen, toDec, lockInc, lockDec, globalLock, mergedRoots, collecting, gcEnv, pendingWrites>>
|
||||
ELSE \* No overflow: buffer normally
|
||||
/\ WriteBarrier(thread, destObj, destField, oldVal, newVal, desc)
|
||||
/\ UNCHANGED <<roots, collecting, pendingWrites>>
|
||||
|
||||
\* Stack root assignment: immediate RC increment (not buffered)
|
||||
\* When assigning val to a root variable named obj, we set roots[thread][val] = TRUE
|
||||
\* to indicate that thread has a stack reference to val
|
||||
\* Semantics: obj is root variable name, val is the object being assigned
|
||||
\* When val=NULL, obj was the old root value, so we decrement rc[obj]
|
||||
MutatorRootAssign(thread, obj, val) ==
|
||||
/\ IF val # NULL
|
||||
THEN /\ roots' = [roots EXCEPT ![thread][val] = TRUE]
|
||||
/\ rc' = [rc EXCEPT ![val] = IF roots[thread][val] THEN @ ELSE @ + 1] \* Increment only if not already a root
|
||||
ELSE /\ roots' = [roots EXCEPT ![thread][obj] = FALSE] \* Clear root when assigning NULL
|
||||
/\ rc' = [rc EXCEPT ![obj] = IF roots[thread][obj] THEN @ - 1 ELSE @] \* Decrement old root value
|
||||
/\ edges' = edges
|
||||
/\ color' = color
|
||||
/\ inRoots' = inRoots
|
||||
/\ toIncLen' = toIncLen
|
||||
/\ toInc' = toInc
|
||||
/\ toDecLen' = toDecLen
|
||||
/\ toDec' = toDec
|
||||
/\ lockInc' = lockInc
|
||||
/\ lockDec' = lockDec
|
||||
/\ globalLock' = globalLock
|
||||
/\ mergedRoots' = mergedRoots
|
||||
/\ collecting' = collecting
|
||||
/\ gcEnv' = gcEnv
|
||||
/\ pendingWrites' = pendingWrites
|
||||
|
||||
\* ============================================================================
|
||||
\* Collector Actions
|
||||
\* ============================================================================
|
||||
|
||||
\* Collector acquires global lock for entire collection cycle
|
||||
CollectorAcquireLock(thread) ==
|
||||
/\ globalLock = NULL
|
||||
/\ globalLock' = thread
|
||||
/\ UNCHANGED <<edges, roots, rc, color, inRoots, toIncLen, toInc, toDecLen, toDec, lockInc, lockDec, mergedRoots, collecting, gcEnv, pendingWrites>>
|
||||
|
||||
CollectorMerge ==
|
||||
/\ globalLock # NULL
|
||||
/\ MergePendingRoots
|
||||
|
||||
CollectorStart ==
|
||||
/\ globalLock # NULL
|
||||
/\ StartCollection
|
||||
|
||||
\* Mark all roots gray (trial deletion phase)
|
||||
CollectorMarkGray ==
|
||||
/\ globalLock # NULL
|
||||
/\ collecting = TRUE
|
||||
/\ \E rootIdx \in DOMAIN mergedRoots:
|
||||
LET root == mergedRoots[rootIdx]
|
||||
IN MarkGray(root.obj, root.desc)
|
||||
|
||||
\* Scan all roots (rescue phase)
|
||||
CollectorScan ==
|
||||
/\ globalLock # NULL
|
||||
/\ collecting = TRUE
|
||||
/\ \E rootIdx \in DOMAIN mergedRoots:
|
||||
LET root == mergedRoots[rootIdx]
|
||||
IN Scan(root.obj, root.desc)
|
||||
|
||||
\* Collect white/gray objects (free phase)
|
||||
CollectorCollect ==
|
||||
/\ globalLock # NULL
|
||||
/\ collecting = TRUE
|
||||
/\ \E rootIdx \in DOMAIN mergedRoots, targetColor \in {colGray, colWhite}:
|
||||
LET root == mergedRoots[rootIdx]
|
||||
IN CollectColor(root.obj, root.desc, targetColor)
|
||||
|
||||
CollectorEnd ==
|
||||
/\ globalLock # NULL
|
||||
/\ EndCollection
|
||||
|
||||
CollectorReleaseLock(thread) ==
|
||||
/\ globalLock = thread
|
||||
/\ globalLock' = NULL
|
||||
/\ UNCHANGED <<edges, roots, rc, color, inRoots, toIncLen, toInc, toDecLen, toDec, lockInc, lockDec, mergedRoots, collecting, gcEnv, pendingWrites>>
|
||||
|
||||
\* ============================================================================
|
||||
\* Next State Relation
|
||||
\* ============================================================================
|
||||
|
||||
Next ==
|
||||
\/ \E thread \in Threads:
|
||||
\E destObj \in Objects, oldVal, newVal \in Objects \cup {NULL}, desc \in ObjTypes:
|
||||
\* Precondition: oldVal must match current graph state (user-level synchronization)
|
||||
\* ASSUMPTION: Users synchronize pointer assignments with locks, so oldVal always matches
|
||||
\* the value read before the atomic store. This prevents races at the user level.
|
||||
/\ LET oldValMatches == CASE oldVal = NULL -> TRUE
|
||||
[] oldVal \in Objects -> edges[destObj][oldVal]
|
||||
[] OTHER -> FALSE
|
||||
IN oldValMatches
|
||||
/\ MutatorWrite(thread, destObj, "field", oldVal, newVal, desc)
|
||||
\/ \E thread \in Threads:
|
||||
\* Handle increment overflow: merge increment buffers when lock becomes available
|
||||
MutatorWriteMergeInc(thread)
|
||||
\/ \E thread \in Threads:
|
||||
\* Handle decrement overflow: merge all buffers when lock becomes available
|
||||
MutatorWriteMergeDec(thread)
|
||||
\/ \E thread \in Threads:
|
||||
\E obj, val \in Objects \cup {NULL}:
|
||||
MutatorRootAssign(thread, obj, val)
|
||||
\/ \E thread \in Threads:
|
||||
CollectorAcquireLock(thread)
|
||||
\/ CollectorMerge
|
||||
\/ CollectorStart
|
||||
\/ CollectorMarkGray
|
||||
\/ CollectorScan
|
||||
\/ CollectorCollect
|
||||
\/ CollectorEnd
|
||||
\/ \E thread \in Threads:
|
||||
CollectorReleaseLock(thread)
|
||||
|
||||
\* ============================================================================
|
||||
\* Initial State
|
||||
\* ============================================================================
|
||||
|
||||
Init ==
|
||||
/\ edges = [x \in Objects |->
|
||||
[y \in Objects |->
|
||||
IF x = y THEN FALSE ELSE FALSE]] \* Empty graph initially
|
||||
/\ roots = [t \in Threads |->
|
||||
[x \in Objects |->
|
||||
FALSE]] \* No stack roots initially
|
||||
/\ rc = [x \in Objects |->
|
||||
0] \* Zero reference counts
|
||||
/\ color = [x \in Objects |->
|
||||
colBlack] \* All objects black initially
|
||||
/\ inRoots = [x \in Objects |->
|
||||
FALSE] \* No objects in roots array
|
||||
/\ toIncLen = [s \in 0..(NumStripes-1) |->
|
||||
0]
|
||||
/\ toInc = [s \in 0..(NumStripes-1) |->
|
||||
<<>>]
|
||||
/\ toDecLen = [s \in 0..(NumStripes-1) |->
|
||||
0]
|
||||
/\ toDec = [s \in 0..(NumStripes-1) |->
|
||||
<<>>]
|
||||
/\ lockInc = [s \in 0..(NumStripes-1) |->
|
||||
NULL]
|
||||
/\ lockDec = [s \in 0..(NumStripes-1) |->
|
||||
NULL]
|
||||
/\ globalLock = NULL
|
||||
/\ mergedRoots = <<>>
|
||||
/\ collecting = FALSE
|
||||
/\ gcEnv = [touched |-> 0, edges |-> 0, rcSum |-> 0, toFree |-> {}]
|
||||
/\ pendingWrites = {}
|
||||
/\ TypeOK
|
||||
|
||||
\* ============================================================================
|
||||
\* Safety Properties
|
||||
\* ============================================================================
|
||||
|
||||
\* Safety: Objects are only freed if they are unreachable from any thread's stack
|
||||
\*
|
||||
\* An object is reachable if:
|
||||
\* - It is a direct stack root (roots[t][obj] = TRUE), OR
|
||||
\* - There exists a path from a stack root to obj via heap edges
|
||||
\*
|
||||
\* Safety guarantee: If an object is reachable, then:
|
||||
\* - It is not white (not marked for collection), OR
|
||||
\* - It is in roots array (protected from collection), OR
|
||||
\* - It is reachable from an object that will be rescued by scanBlack
|
||||
\*
|
||||
\* More precisely: Only closed cycles (zero external refs, unreachable) are freed.
|
||||
|
||||
\* Helper: Compute next set of reachable objects (one step of transitive closure)
|
||||
ReachableStep(current) ==
|
||||
current \cup UNION {{y \in Objects : edges[x][y]} : x \in current}
|
||||
|
||||
\* Compute the set of all reachable objects using bounded iteration
|
||||
\* Since Objects is finite, we iterate at most Cardinality(Objects) times
|
||||
\* This computes the transitive closure of edges starting from stack roots
|
||||
\* We unroll the iteration explicitly to avoid recursion issues with TLC
|
||||
ReachableSet ==
|
||||
LET StackRoots == {x \in Objects : \E t \in Threads : roots[t][x]}
|
||||
Step1 == ReachableStep(StackRoots)
|
||||
Step2 == ReachableStep(Step1)
|
||||
Step3 == ReachableStep(Step2)
|
||||
Step4 == ReachableStep(Step3)
|
||||
\* Add more steps if needed for larger object sets
|
||||
\* For small models (2 objects), 4 steps is sufficient
|
||||
IN Step4
|
||||
|
||||
\* Check if an object is reachable
|
||||
Reachable(obj) == obj \in ReachableSet
|
||||
|
||||
\* Helper: Check if there's a path from 'from' to 'to'
|
||||
\* For small object sets, we check all possible paths by checking
|
||||
\* all combinations of intermediate objects
|
||||
\* Path of length 0: from = to
|
||||
\* Path of length 1: edges[from][to]
|
||||
\* Path of length 2: \E i1: edges[from][i1] /\ edges[i1][to]
|
||||
\* Path of length 3: \E i1, i2: edges[from][i1] /\ edges[i1][i2] /\ edges[i2][to]
|
||||
\* etc. up to Cardinality(Objects)
|
||||
HasPath(from, to) ==
|
||||
\/ from = to
|
||||
\/ edges[from][to]
|
||||
\/ \E i1 \in Objects:
|
||||
edges[from][i1] /\ (edges[i1][to] \/ \E i2 \in Objects:
|
||||
edges[i1][i2] /\ (edges[i2][to] \/ \E i3 \in Objects:
|
||||
edges[i2][i3] /\ edges[i3][to]))
|
||||
|
||||
\* Helper: Compute set of objects reachable from a given starting object
|
||||
\* Uses the same iterative approach as ReachableSet
|
||||
ReachableFrom(start) ==
|
||||
LET Step1 == ReachableStep({start})
|
||||
Step2 == ReachableStep(Step1)
|
||||
Step3 == ReachableStep(Step2)
|
||||
Step4 == ReachableStep(Step3)
|
||||
IN Step4
|
||||
|
||||
\* Safety: Reachable objects are never freed (remain white without being collected)
|
||||
\* A reachable object is safe if:
|
||||
\* - It's not white (not marked for collection), OR
|
||||
\* - It's in roots array (protected from collection), OR
|
||||
\* - There exists a black object in ReachableSet such that obj is reachable from it
|
||||
\* (the black object will be rescued by scanBlack, which rescues all white objects
|
||||
\* reachable from black objects)
|
||||
Safety ==
|
||||
\A obj \in Objects:
|
||||
IF obj \in ReachableSet
|
||||
THEN \/ color[obj] # colWhite \* Not marked for collection
|
||||
\/ inRoots[obj] \* Protected in roots array
|
||||
\/ \E blackObj \in ReachableSet:
|
||||
/\ color[blackObj] = colBlack \* Black object will be rescued by scanBlack
|
||||
/\ obj \in ReachableFrom(blackObj) \* obj is reachable from blackObj
|
||||
ELSE TRUE \* Unreachable objects may be freed (this is safe)
|
||||
|
||||
\* Invariant: Reference counts match logical counts after merge
|
||||
\* (This is maintained by MergePendingRoots)
|
||||
\* Note: Between merge and collection, RC = logicalRC.
|
||||
\* During collection (after markGray), RC may be modified by trial deletion.
|
||||
\* RC may be inconsistent when:
|
||||
\* - globalLock = NULL (buffered changes pending)
|
||||
\* - globalLock # NULL but merge hasn't happened yet (buffers still have pending changes)
|
||||
\* RC must equal LogicalRC when:
|
||||
\* - After merge (buffers are empty) and before collection starts
|
||||
RCInvariant ==
|
||||
IF globalLock = NULL
|
||||
THEN TRUE \* Not in collection, RC may be inconsistent (buffered changes pending)
|
||||
ELSE IF collecting = FALSE /\ \A s \in 0..(NumStripes-1): toIncLen[s] = 0 /\ toDecLen[s] = 0
|
||||
THEN \A obj \in Objects: rc[obj] = LogicalRC(obj) \* After merge, buffers empty, RC = logical RC
|
||||
ELSE TRUE \* During collection or before merge, RC may differ from logicalRC
|
||||
|
||||
\* Invariant: Only closed cycles are collected
|
||||
\* (Objects with external refs are rescued by scanBlack)
|
||||
CycleInvariant ==
|
||||
\A obj \in Objects:
|
||||
IF color[obj] = colWhite /\ ~inRoots[obj]
|
||||
THEN ExternalRC(obj) = 0
|
||||
ELSE TRUE
|
||||
|
||||
\* ============================================================================
|
||||
\* Specification
|
||||
\* ============================================================================
|
||||
|
||||
Spec == Init /\ [][Next]_<<edges, roots, rc, color, inRoots, toIncLen, toInc, toDecLen, toDec, lockInc, lockDec, globalLock, mergedRoots, collecting, gcEnv, pendingWrites>>
|
||||
|
||||
THEOREM Spec => []Safety
|
||||
THEOREM Spec => []RCInvariant
|
||||
THEOREM Spec => []CycleInvariant
|
||||
|
||||
====
|
||||
Reference in New Issue
Block a user