Formal Proof of Absence of Memory Leaks in Multithreaded Environments

This section extends the single-threaded proof to multithreaded scenarios, incorporating inter-thread communication, shared immutable objects, and thread ownership rules. The proof maintains the core invariants while addressing concurrency and object transfers.

1. Definitions and Setup

Threads: Let Θ be the set of threads in the system. Each thread θ ∈ Θ maintains:

• A local set of mutable objects M_θ, disjoint from M_θ' for θ ≠ θ' (Invariant I10: Mutable Exclusivity).
• A local stack for temp pointers T_θ, scoped to θ's execution.
• Access to a global set of immutable objects F.
• Local outgoing and incoming queues for inter-thread communication.

Object Types (extended):

1. Mutable Objects (M): Partitioned into {M_θ | θ ∈ Θ}. Each o ∈ M_θ is owned and accessed exclusively by θ.
2. Immutable Objects (F): Shared across all threads, immutable post-creation.

Pointer Types (extended):

1. Composite Pointer (C_θ): Exclusive ownership within M_θ. |C_θ| ≤ 1 per object in M_θ.
2. Temp Pointer (T_θ): Stack-bound in θ, extending lifetime of objects in M_θ.
3. Weak Pointer (W): Non-owning references to any M object. Cross-thread W (to M_θ' from θ ≠ θ') cannot be dereferenced (returns None on lock-and-check, Invariant I11: Weak Isolation).
4. Shared Pointer (S): References to F, contributing to global reference count.

Graph Representation (extended):

• Local mutable graph per thread: G_θ = (M_θ, E_Cθ ∪ E_Tθ ∪ E_Wθ), where E_Cθ forms an acyclic tree, E_Tθ ⊆ Stack_θ × M_θ.
• Global immutable graph: G_F = (F, E_S), a DAG.
• Weak edges E_W may span threads but do not affect reference counts or allow cross-thread dereferencing.

Reference Counting (extended):

• For o ∈ M_θ: RC_Mθ(o) = |C_θ| + |T_θ| (local, non-atomic, as access is single-threaded).
• For f ∈ F: RC_F(f) = Σ |S_θ| over θ ∈ Θ (global, atomic for thread-safety).
• Deallocation: o ∈ M_θ deallocated when RC_Mθ(o) = 0; f ∈ F when RC_F(f) = 0.
• Retain/release on F: May be batched per thread (retains before releases) and applied atomically to prevent races (Invariant I12: Batch Ordering).

Inter-Thread Communication:

• Queues: Thread-safe, using shared pointers for immutable transfers and ownership transfer for mutable hierarchies.
• Mutable Transfer: Objects must be unowned (no C_θ) and unreferenced (no T_θ). Performed in two stages: a. Place in the outgoing queue (holds temporary ownership). b. Post to receiver's incoming queue after current operation ends (ensuring no residual T_θ).
• This transfers the object hierarchy from M_θ to M_θ', maintaining disjointness.

Thread Objects: Special mutable objects (sys_Thread) encapsulating thread state (M_θ, queues). Owned within other threads' mutable hierarchies. A root thread θ_root is owned by the runtime. Copy/freeze/transfer operations on sys_Thread produce empty objects to prevent cycles (Invariant I13: Thread Acyclicity).

Invariants (extended from single-threaded):

• I10: Mutable Exclusivity – M_θ ∩ M_θ' = ∅ for θ ≠ θ'.
• I11: Weak Isolation – Lock-and-check on cross-thread W returns None.
• I12: Batch Ordering – Per-thread batches on RC_F ensure retains precede releases; applied atomically.
• I13: Thread Acyclicity – sys_Thread ownership forms a tree rooted at θ_root, with no cycles.
• All prior invariants (I1–I9) hold locally per thread for M_θ.

Goal: Prove no memory leaks in the multithreaded system, i.e., all objects are deallocated when no longer referenced, with no cycles or lost references due to concurrency or transfers.

2. Proof Structure

The proof builds on the single-threaded case, showing that multithreading introduces no leaks through local isolation, safe transfers, and global DAG management.

Statement 1: Local Mutable Objects Remain Leak-Free Per Thread

Claim: For each θ, the subgraph G_θ is leak-free as in the single-threaded proof.

Proof:

• By Invariants I1–I9 applied locally to M_θ, E_Cθ forms a tree, E_Tθ is stack-bound and acyclic.
• Access to M_θ is exclusive (I10), so no concurrent modifications.
• RC_Mθ is local and accurate, deallocating when =0.
• Weak pointers (local or cross) do not affect RC_Mθ.
• Conclusion: No leaks within each thread's mutable hierarchy (from single-threaded Statements 1, 2, 4).

Statement 2: Shared Immutable Objects Are Leak-Free

Claim: The global immutable graph G_F has no cycles and deallocates correctly under concurrent access.

Proof:

• By I3, G_F is a DAG (acyclic).
• RC_F is global and atomic, incremented/decremented via shared pointers S.
• Batching (I12) ensures logical order: retains before releases, preventing premature deallocation.
• Traversal safety: Holding an S to a root f ensures RC_F(f) ≥1; since E_S forms a DAG, all descendants have RC_F ≥1 from ancestors.
• Thus, traversal without additional retains is safe—no use-after-free.
• Deallocation only when RC_F=0, with no cycles, ensures all unreferenced subgraphs are reclaimed.
• Conclusion: Standard ref-counted DAG guarantees no leaks (extended from Statement 3).

Statement 3: Inter-Thread Transfers Introduce No Leaks

Claim: Transferring objects (shared or mutable) between threads preserves leak-freedom and disjointness.

Proof:

• Shared Transfers: Sender retains (atomic increment RC_F), enqueues S. Receiver acquires S (no increment needed, as already retained). Later release by any thread decrements RC_F. No lost references; standard ref-counting applies.
• Mutable Transfers: Object o ∈ M_θ must have |C_θ|=0 and |T_θ|=0 (unowned, no stack refs).
  • Stage 1: Move to the outgoing queue (queue holds effective C, RC_Mθ ≥1).
  • Wait for operation end: Unwinds stack, releasing any residual T_θ (RC_Mθ=1 from queue).
  • Stage 2: Post to receiver θ' incoming queue, transferring ownership (move C to θ').
• This ensures no references remain in θ (I10 preserved). o moves to M_θ', integrated into G_θ' without cycles (by local invariants).
• No intermediate loss: Queue holds reference throughout.
• Conclusion: Transfers maintain global invariants, no leaks introduced.

Statement 4: Cross-Thread Weak Pointers Introduce No Leaks

Claim: Cross-thread W pointers do not cause leaks or violate isolation.

Proof:

• By definition, W do not contribute to RC (anywhere).
• Cross-thread lock-and-check returns None (I11), preventing T_θ to M_θ' (no data races).
• Usage scenarios: a. Pass back: Enqueue W to owner thread θ'; dereference there as local W (reduces to single-threaded). b. Existence check: Atomic read (bool), no reference held. c. Async target: Task enqueued to θ', executed locally (single-threaded).
• All reduce to proven leak-free cases; no cross-thread references created.
• On deallocation in θ, cross-thread W auto-nullified (I9 extended atomically).
• Conclusion: Cross-thread set of W maintains safety without affecting leak-freedom.

Statement 5: Thread Objects Form a Leak-Free Hierarchy

Claim: sys_Thread objects are leak-free and acyclic.

Proof:

• Each sys_Thread ∈ M_θ' for some owner thread θ'.
• Root θ_root owned by runtime (effective global reference).
• Ownership via C pointers in mutable hierarchies.
• Copy/freeze/transfer: Produce empty objects (no state transfer), preventing self-ownership or cycles (I13).
• Thus, sys_Thread ownership forms a tree: acyclic, rooted at θ_root.
• Termination: Thread state (M_θ, queues) released locally; sys_Thread object released by owner when unreferenced.
• No cycles ensure deallocation when RC=0.
• Conclusion: Thread hierarchy integrates into mutable proofs without leaks.

3. Overall Proof of Leak Freedom

Theorem: The Argentum ownership model ensures no memory leaks in multithreaded contexts.

Proof:

• Mutable objects: Per-thread leak-free (Statement 1), with transfers preserving invariants (Statement 3).
• Immutable objects: Global DAG with atomic ref-counting, leak-free (Statement 2).
• Weak pointers: No impact on RC, cross-thread usage reduces to single-threaded (Statement 4).
• Threads: Acyclic tree, integrated without leaks (Statement 5).
• Concurrency: Mutable isolation (I10) prevents races; immutable batching (I12) ensures correct RC_F.
• All operations (assignments, scopes, transfers) maintain accurate reference counts and acyclicity.
• Exceptions and terminations: Release held references (T_θ, queues), preserving integrity.

Conclusion: The model is leak-free multithreaded, with mutable isolation enabling single-thread proofs and shared immutable DAG ensuring global safety.