Rewrite a phase-change rule in Alloy by separating three concerns: the state that can change, the legal current-to-next-state transition, and the temporal property that must hold across a trace. In Alloy 6, declare mutable signatures or fields with var, refer to the next state with a prime mark, express transitions in predicates, and use temporal operators such as always, eventually, after, and until for trace-wide requirements.
Use a macro only to factor repeated syntax, such as frame conditions. A macro does not create a new semantic rule; the Analyzer checks the predicates, facts, and assertions that result after the macro is expanded. Every result is also bounded by the scopes and number of steps in the command.
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What “rewriting” a phase-change rule means in Alloy
Alloy is both a modeling language and an automated exploration tool. The official Alloy project describes it as “a language for expressing structures and a tool for exploring them.” A phase-change model uses that structural logic to describe the possible states of a system and temporal logic to describe how one state may lead to another.
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One free scan finds every outdated or missing driver and matches the right update for your exact hardware.Free scan · exact hardware matchIn a static Alloy model, signatures and fields retain the same value throughout an analysis. Alloy 6 adds native behavioral modeling: a declaration marked var may have a different value in each state of a trace. A phase transition is therefore not an assignment statement. It is a constraint relating the current value of a variable to its value in the next state.
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For example, “solid may become liquid” is a relation between phase and phase'. The prime mark means “the value in the next state.” A correct rewrite must also say what happens to every other object or field. If those frame conditions are omitted, Alloy may change unrelated parts of the model while satisfying the transition.
Declare the phases and mutable state
Use a closed set when the phases are known
If the model has a fixed vocabulary of phases, an abstract signature with one singleton for each phase makes illegal values impossible by construction:
abstract sig Phase {}
one sig Solid, Liquid, Gas extends Phase {}
sig Material {
var phase: one Phase
}Phase, Solid, Liquid, and Gas are static declarations. The phase field is mutable, so each Material may have a different phase at different points in a trace. If membership itself changes—for example, material can enter or leave the system—Alloy 6 also permits mutable signatures with var sig. Make that choice deliberately: mutable membership introduces more possible traces and usually requires additional invariants.
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An initialization predicate describes the first state only. It should not be confused with a transition rule:
pred init {
all m: Material | m.phase = Solid
}Keeping initialization separate lets you run alternative starting conditions without rewriting the transition relation. It also makes counterexamples easier to read: the first state is established by init, while later states are generated by the transition constraints.
Write the legal current-to-next-state transition
Make each event explicit with a predicate
A predicate is the clearest place to state the precondition and result of one phase change. The following rule permits a selected material to melt from solid to liquid:
pred melt[m: Material] {
m.phase = Solid
m.phase' = Liquid
all n: Material - m | n.phase' = n.phase
}
pred freeze[m: Material] {
m.phase = Liquid
m.phase' = Solid
all n: Material - m | n.phase' = n.phase
}The first two lines constrain the selected object across adjacent states. The final line is a frame condition: every other material keeps its phase. Without it, the Analyzer could satisfy melt[m] while silently changing another material as well, producing a trace that does not match the intended physical interpretation.
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Represent no-op steps explicitly
Many systems can remain unchanged for one or more steps. Model that behavior rather than allowing it accidentally:
pred stutter {
all m: Material | m.phase' = m.phase
}
pred step {
stutter
or some m: Material | melt[m]
or some m: Material | freeze[m]
}
fact TraceRules {
always step
}This says that every transition is either a no-op or one permitted event. If simultaneous changes are allowed, write a separate predicate that states exactly which set of materials may change together. Do not rely on the Analyzer to infer whether events are atomic or concurrent.
Use temporal operators for requirements over traces
Transition predicates describe one step. Temporal operators describe obligations over a complete trace. Alloy 6 documentation identifies the following operators as core tools for behavioral models:
| Operator | What it constrains | Typical phase-model use |
|---|---|---|
always P |
P must hold at every state in the trace |
Safety invariants, such as forbidding a direct solid-to-gas jump |
eventually P |
P must hold at some future state |
Progress, such as requiring a liquid state to be reached |
after P |
P is evaluated at the next state |
Checking a postcondition immediately after an event |
P until Q |
P remains true until Q becomes true |
Requiring a material to stay in an intermediate phase until a trigger occurs |
For example, a safety property can forbid a direct jump from solid to gas:
assert NoDirectSolidGas {
always all m: Material |
m.phase = Solid implies m.phase' != Gas
}A liveness requirement can ask whether a liquid state is reachable from the initialized model:
run {
init
eventually some m: Material | m.phase = Liquid
} for 3 Material, 5 stepsUse always for invariants that must never be broken. Use eventually only when progress is genuinely required; adding it to every model can hide the difference between a safe system and a system that is guaranteed to make progress. Use until when the intermediate obligation matters, not merely when you want to mention two phases in the same formula.
Choose between a predicate, fact, and macro
These constructs solve different problems. Rewriting syntax from one form to another does not by itself change the semantics or make the model faster.
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| Construct | Best use | Scope of effect | Main caution |
|---|---|---|---|
| Predicate | A named transition, scenario, or reusable constraint | Applied where invoked, including in commands or other formulas | It has no effect unless a command, fact, or another predicate invokes it |
| Fact | A rule that must hold in every analyzed state or structure | Automatically included in every command | Too many global facts can make debugging and scenario variation difficult |
| Macro | Repeated textual syntax, such as a common frame condition | Expanded at the point of use | Macros have fewer safeguards than functions or predicates; errors and unintended capture can be harder to spot |
Use predicates for semantic events
Keep the meaning of “melt,” “freeze,” or “evaporate” in predicates. Predicates are visible in the model’s logic and can be invoked selectively in a run or check command. They also let you give each event its own preconditions, postconditions, and frame conditions.
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Use facts for universal transition policy
A fact is appropriate when every trace in the model must obey the same transition policy:
fact AllowedTransitions {
always (
stutter
or some m: Material | melt[m]
or some m: Material | freeze[m]
)
}Because facts are implicit in every command, name them clearly and keep them small. If you are debugging one event, temporarily moving a broad rule into an explicitly invoked predicate can make the source of a contradiction more apparent.
Use macros to remove repeated frame clauses
When several events use the same unchanged-elements pattern, a macro can reduce visual noise:
macro frameExcept[x: Material] {
all y: Material - x | y.phase' = y.phase
}
pred melt[m: Material] {
m.phase = Solid
m.phase' = Liquid
frameExcept[m]
}This is textual reuse, not a new kind of Alloy constraint. The Analyzer ultimately checks the expanded expression. Because macros provide fewer safeguards than predicates or functions, avoid hiding important preconditions inside them. A reader should be able to see the event’s semantic effect without expanding several layers of convenience syntax.
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The following model combines initialization, events, frame conditions, and a trace rule. It permits solid-liquid transitions and stuttering, while leaving gas unused until an explicit event is added:
abstract sig Phase {}
one sig Solid, Liquid, Gas extends Phase {}
sig Material {
var phase: one Phase
}
pred init {
all m: Material | m.phase = Solid
}
macro frameExcept[x: Material] {
all y: Material - x | y.phase' = y.phase
}
pred melt[m: Material] {
m.phase = Solid
m.phase' = Liquid
frameExcept[m]
}
pred freeze[m: Material] {
m.phase = Liquid
m.phase' = Solid
frameExcept[m]
}
pred stutter {
all m: Material | m.phase' = m.phase
}
fact Traces {
always (
stutter
or some m: Material | melt[m]
or some m: Material | freeze[m]
)
}
assert NoSolidGasJump {
always all m: Material |
m.phase = Solid implies m.phase' != Gas
}
run {
init
eventually some m: Material | m.phase = Liquid
} for 3 Material, 5 steps
check NoSolidGasJump for 3 Material, 5 stepsTo add evaporation, define a predicate whose precondition is liquid and whose postcondition is gas, then include that predicate in the transition fact. If evaporation requires a temperature or an external trigger, model that trigger as another mutable field or signature and constrain its relationship to the event. Avoid encoding a causal condition only in comments; Alloy checks formulas, not prose.
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Run commands and read what the Analyzer found
Set both object scope and trace length
The command’s object scope limits how many atoms Alloy may create. In for 3 Material, 5 steps, the Analyzer may use up to three Material atoms and searches traces of five states or the corresponding number of transitions defined by Alloy 6’s step-scope syntax. Increase scope when you need interactions among more materials; increase steps when a required phase may occur only after a longer sequence.
Keep solver settings, scope, and step count fixed when comparing two formulations. Otherwise a difference in runtime may come from the search space rather than from the rewrite.
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A run command asks for one instance satisfying its formula and all facts. The resulting instance is a trace, not merely a single snapshot. Inspect the visualizer’s state controls to see which material changed and whether unrelated fields stayed fixed. A satisfying trace demonstrates that the specified behavior is possible within the selected bounds.
Use checks to search for violations
A check command searches for a counterexample to an assertion. If the Analyzer finds one, the displayed trace satisfies the model’s facts and violates the assertion at a particular state transition. That is a concrete defect in the model or in the assertion’s intended meaning, subject to the selected scope.
What an Alloy counterexample proves about a phase transition
A counterexample proves that there is at least one bounded trace in which the assertion fails. For a “no direct solid-to-gas jump” assertion, it shows a material and adjacent states that meet the model’s other constraints while making that jump. Use the trace to determine whether the transition is genuinely illegal or whether the model omitted a frame condition, initialization rule, or environmental constraint.
The reverse interpretation is equally important. If check finds no counterexample, Alloy has not proved the property for every possible number of materials or every unbounded execution. The Analyzer searches finite scopes and a finite number of steps. An absence of an instance means no violation was found within those bounds; it does not establish unrestricted mathematical validity. Increase scopes and steps, and seek a separate inductive argument when you need a stronger guarantee.
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Compare alternative rewrites fairly
There is no published, topic-specific statistic establishing that a macro rewrite, predicate rewrite, or fact rewrite is universally faster. Treat performance as an empirical question for your model.
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Evaluate alternatives using the same:
- model semantics and initial conditions;
- number of atoms and step scope;
- solver and solver options;
- run or check command;
- hardware and measurement method.
Compare more than elapsed time. A formulation that is marginally faster but produces opaque traces may cost more to maintain. Record whether the transition relation is explicit, whether frame conditions are easy to audit, how reusable the event is, and whether counterexamples identify the changed object and state clearly.
Troubleshoot common phase-model failures
The model changes several materials at once
Add a frame condition to each single-material event, or define an explicit simultaneous-event predicate if concurrent changes are intended. A missing frame condition is the usual reason a trace appears to violate the informal story while satisfying the written formula.
No liquid state is ever found
Check that the run includes the initialization predicate and that a transition to liquid is reachable under the global facts. If every step is forced to stutter, or if a fact requires all materials to remain solid, the liveness goal is inconsistent. Increase the step bound only after verifying that the event is actually permitted.
The assertion passes but the model is too weak
Inspect the generated trace and test the opposite behavior with a dedicated run command. A passing bounded check can coexist with an underconstrained model. Add invariants for environmental assumptions, ownership, capacity, or triggers that the physical system requires but the first draft omitted.
Macro expansion makes debugging difficult
Replace the macro temporarily with its expanded expression, or move the important condition into a named predicate. Keep macros focused on mechanical repetition; leave domain meaning in constructs that the reader can invoke and inspect directly.
A practical rewrite workflow
- List the state. Identify phases, mutable fields, external triggers, and any membership that can change. Mark only genuinely changing declarations with
var. - Define the initial condition. Put first-state assumptions in an
initpredicate so alternative scenarios can reuse the same transition rules. - Write one predicate per event. State the current-state precondition, next-state postcondition, and frame conditions explicitly.
- Define the step policy. Decide whether the system may stutter, whether events are exclusive, and whether simultaneous changes are legal. Encode that decision in a fact or an explicitly invoked trace predicate.
- Add temporal requirements. Use
alwaysfor safety,eventuallyfor required progress,afterfor immediate postconditions, anduntilfor sustained intermediate conditions. - Factor only mechanical repetition. Introduce a macro for repeated syntax after the event semantics are correct. Keep critical assumptions visible in predicates and facts.
- Explore both sides. Run commands to demonstrate intended phase changes and check commands to search for forbidden jumps or stalled progress.
- Vary bounds deliberately. Repeat checks with larger object scopes and longer traces, documenting the limits of every conclusion.
- Inspect traces, not just status messages. Confirm that only intended fields change and that the visualized sequence matches the domain’s notion of a phase transition.
Further reading
The Alloy project identifies Daniel Jackson’s Software Abstractions: Logic, Language, and Analysis as the book about Alloy. It provides broader context for relational modeling, constraints, and Analyzer-driven exploration. The Alloy 6 documentation is the appropriate reference for mutable declarations, prime expressions, and temporal operators.
Bottom line
Use a mutable field or signature to represent phase, a predicate to define each legal transition, explicit frame conditions to prevent accidental changes, and temporal operators to state what must hold over the entire trace. A macro can make repeated transition syntax shorter, but it does not replace semantic constraints. Interpret every Alloy result in its stated bounds: a found trace is valid for that model and scope, while no found counterexample is not an unbounded proof.
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