TLA+

TLA+ is a formal specification language and tool ecosystem for describing and analyzing concurrent and distributed systems. It provides a concrete realization substrate for selected temporal-logic and state-machine models; it is not the semantic notion of temporal logic itself and is not a production execution runtime.

TLA+ combines the Temporal Logic of Actions with mathematical definitions based on set theory and first-order logic. A specification commonly separates:

An initial-state predicate Init over declared variables.
A next-state relation Next composed from actions over current and primed next-state variables.
A behavior specification such as Init /\ [][Next]_vars, including admitted stuttering steps.
Safety properties such as type-correctness predicates and invariants.
Liveness properties and weak or strong fairness assumptions where progress depends on eligible actions eventually occurring.

Modules, operators, constants, variables, sets, functions, records, tuples, sequences, and quantification support abstraction over a system's relevant state. PlusCal provides an algorithm-oriented notation that translates into TLA+; the translated TLA+ specification remains the analysis target.

Tooling and Evidence

The TLA+ ecosystem includes several distinct mechanisms:

TLC explores reachable states for a configured, bounded model and checks invariants, deadlock conditions, action properties, and supported temporal properties. A successful run establishes no counterexample within the explored model and tool assumptions; it is not automatically a proof for all parameter sizes or an implementation.
TLAPS supports deductive proofs of TLA+ obligations. A proof establishes the stated theorem from definitions, assumptions, and the trusted proof stack; it does not establish that deployed code refines the specification.
PlusCal translates imperative- or algorithm-shaped descriptions into TLA+, making control state and atomic actions explicit for analysis.

Simulation, model checking, and proof provide different evidence. Each result supports a judgement scoped to the specification, model configuration, property, tool, and assumptions actually checked; it does not silently judge implementation conformance or a larger parameter space. Each specification must name its abstraction boundary, environmental actions, failure assumptions, fairness assumptions, and the state or history omitted from the model.

Realization Boundary

TLA+ can realize an analyzable specification of system behavior. It does not by itself:

Execute the production process, protocol, state machine, or infrastructure.
Prove that source code, configuration, storage, networks, or operators implement the specification.
Convert a domain invariant into a system guarantee without a traceable encoding and implementation correspondence.
Supply the fairness, timing, durability, delivery, or failure assumptions used by a liveness claim.
Turn one bounded TLC result into an unbounded theorem.

Compiler-like realization therefore needs an explicit correspondence from Cohesive concepts and system-graph structure into TLA+ variables, actions, behaviors, and properties, followed by a separate correspondence from that specification to the implementation. Refinement mappings, conformance checks, generated specifications, implementation-derived models, and trace validation can strengthen this chain, but their boundaries and trusted assumptions must remain visible.

Cohesive Mapping

A public mapping may relate:

State and system-graph projections to TLA+ variables and values.
Transitions, commands, events, and environmental moves to TLA+ actions.
State-machine initial conditions and transition relations to Init and Next.
Invariants and safety properties to state or action predicates over all admitted behaviors.
Process progress and protocol completion to liveness formulas under explicit fairness and failure assumptions.
Abstraction boundaries to hidden variables, stuttering steps, and refinement mappings.

These are correspondence choices, not identities. A TLA+ variable may combine several graph roles for analysis, while one semantic role may require several variables or history structures. The mapping should record which distinctions it preserves and which it intentionally abstracts away.

Formal relations

may_realize: Temporal Logic — Provides a concrete language and analysis toolchain for action-based temporal specifications, bounded model checking, and deductive proof without defining the broader family of temporal logics.
may_realize: State Machines — Encodes initial states, next-state actions, stuttering, and behavior properties for analysis without asserting that a production runtime conforms to the encoded machine.

External References

Leslie Lamport, Specifying Systems: The TLA+ Language and Tools for Hardware and Software Engineers, Addison-Wesley, 2002.
TLA+ project.
TLA+ tools.