Temporal Forge Overview

Temporal mode extends Forge with temporal operators to ease specification and checking of dynamic systems. This mode draws heavily on the Electrum work by INESC TEC and ONERA, and the newer Alloy 6 version, but there are (slight) differences.

To use Temporal Forge, just use #lang forge/temporal. This will cause a number of changes in how Forge processes your models, the largest one being that it will now always search for lasso traces (see below) rather than arbitrary instances. The maximum length of the trace is given by option max_tracelength <k> and defaults to 5. The minimum length of the trace is given by option min_tracelength <k> and defaults to 1.

Using Minimum Trace Length

The temporal solver works by iterative deepening: try length 1, then length 2, ... and so sometimes (not always) a longer minimum trace length can speed up the process.

Variable state (var)

Electrum mode allows var annotations to be placed on sig and field definitions. Such an annotation indicates that the contents of the corresponding relation may vary over time. Any sigs or fields that are not declared var will not vary over time.

Importance of var

If no sigs or fields are declared var, the temporal solver cannot change anything from state to state.

var declarations

Writing sig Vertex { var edges: set Vertex } defines a directed graph whose edge relation may vary.

Writing var sig Student {} denotes that the set of students need not remain constant.

Temporal Mode Semantics (Informally)

When the temporal solver is enabled, instances are always traces, and a trace is an infinite object, containing a state for every natural number . However, the solver underlying Forge needs to finitize the problem in order to solve it. The solution to this, given a finite state-space, is to invoke the Pigeonhole Principle and seek traces that end in a loop: "lassos".

Lassos must loop back!

If your system needs more steps before it loops back than the maximum trace length (e.g., an integer counter that overflows with a small trace length), Forge may tell you that there are no traces. Always pay attention to your max_tracelength option, and be convinced that it is sufficient to include the lassos you're interested in.

Temporal formulas and expressions are always evaluated with respect to a state index. At the top level of any formula, the index is (the first state of the trace). Temporal operators either change or "fan out" across multiple state indexes to let you express things like:

  • "in the next state, ..." (add 1 to the state index)
  • at some point in the future, ... (search for some state index)
  • at all points in the future, ... (check all future indexes)
  • ...

Added relational operators: priming

Electrum mode adds one relational operator: priming ('). Any relational expression that is primed implicitly means "this expression in the next state". Thus, you can use priming to concisely write transition effects.


Writing cookies' in cookies would mean that in every transition, the set of cookies never grows over time (will either shrink or remain the same).

Added formula operators

Electrum mode adds a number of formula operators, corresponding to those present in Linear Temporal Logic (LTL) and Past-Time Linear Temporal Logic.

Traditional LTL (future time)

Next State

  • next_state <fmla> is true in a state i if and only if: fmla holds in state i+1.


In a model with one sig Nim { var cookies: set Cookie }, writing next_state no Nim.cookies expresses the sad fact that, in the next state, Nim has no cookies.

Let and temporal operators

Don't try to write something like:

let oldCount = Counter.count | 
    next_state Counter.count = add[oldCount, 1]

The let construct is implemented with substitution, and so the above will be rewritten to:

next_state Counter.count = add[Counter.count, 1]

which expresses that, in the next state, the counter must be one greater than itself. This will be unsatisfiable.

Always (now and in the future)

  • always <fmla> is true in a state i if and only if: fmla holds in every state >=i.


In a model with one sig Nim { var cookies: set Cookie }, writing always no Nim.cookies expresses an alarming universal truth: from now on, Nim will never have any cookies.

Of course, operators like always can be used inside other operators. So this lamentable destiny might be avoided. E.g., not always no Nim.cookies would mean the reverse: at some point, Nim will in fact have cookies.

Eventually (now or sometime in the future)

  • eventually <fmla> is true in a state i if and only if: fmla holds in some state >=i.


In a model with one sig Nim { var cookies: set Cookie }, writing eventually no Nim.cookies expresses that at some point either now or in the future, Nim will lack cookies. Before and after that point, nothing prevents Nim from acquiring cookies.

Until and Release (obligations)

  • <fmla-a> until <fmla-b> is true in a state i if and only if: fmla-b holds in some state j>=i and fmla-a holds in all states k where i <= k < j. Note that the obligation to see fmla-a ceases in the state where fmla-b holds.


In a model with one sig Nim { var cookies: set Cookie }, writing no Nim.cookies until some Nim.vegetables expresses that, starting at this point in time, Nim must have no cookies until they also have vegetables. Once Nim obtains vegetables (including at this point in time), the obligation ends (and thus after that point Nim may have only cookies but no vegetables).

A natural question to ask is: what happens if fmla-b holds at multiple points in the future? However, this turns out not to matter: the definition says that the until formula is true in the current state if fmla-b holds at some point in the future (and fmla-a holds until then).

There is an alternative way to express obligations, which we won't use much:

  • <fmla-a> releases <fmla-b> is true in a state i if and only if: fmla-a holds in some state j>=i and fmla-b holds in all states k where i <= k <= j, or fmla-a never occurs in a later state and fmla-b holds forever after. Note that the intuitive role of the two formulas is reversed between until and releases, and that, unlike until, the obligation extends to the state where fmla-a holds.

Past-time LTL

Past-time operators are useful for concisely expressing some constraints. They don't add more expressive power to the temporal language, but they do sometimes make it easier to avoid adding state, etc.

Previous State

  • prev_state <fmla> is true in a state i if and only if: fmla holds in state i-1.

There is nothing before the first state

The formula prev_state <fmla> is canonically false if i=0, since there is no prior state. This holds regardless of what <fmla> contains; prev_state asserts the existence of a prior state.

Historically (always in the past)

  • historically <fmla> is true in a state i if and only if: fmla holds in every state <=i.

Once (at some point in the past)

  • once <fmla> is true in a state i if and only if: fmla holds in some state <=i.

Note for Alloy Users

Alloy 6 and Electrum use the keywords after and before where Forge uses next_state and prev_state. We changed Forge to use these (admittedly more verbose) keywords in the hopes they are more clear. For example, after sounds like it could be a binary operator; in English, we might say "turn left after 3 stops". Also, next_state is definitely a formula about the following state whereas after does not communicate the same sense of immediacy.