Copilot.Library.LTL

Bounded Linear Temporal Logic (LTL) operators. For a bound n, a property p holds if it holds on the next n transitions (between periods). If n == 0, then the trace includes only the current period. For example,

  eventually 3 p

holds if p holds at least once every four periods (3 transitions).

Interface: See Examples/LTLExamples.hs in the Copilot repository.

You can embed an LTL specification within a Copilot specification using the form:

  operator spec

For some properties, stream dependencies may not allow their specification. In particular, you cannot determine the "future" value of an external variable.

Formulas defined with this module require that predicates have enough history, which is only true if they have an append directly in front of them. This results in a limited ability to nest applications of temporal operators, since simple application of pointwise combinators (e.g., always n1 (p ==> eventually n2 p2)) will hinder Copilot's ability to determine that there is enough of a buffer to carry out the necessary drops to look into the future.

In general, the Copilot.Library.PTLTL library is probably more useful.

Documentation

next :: Stream Bool -> Stream Bool

Property s holds at the next period. For example:

          0 1 2 3 4 5 6 7
s      => F F F T F F T F ...
next s => F F T F F T F ...

Note: s must have sufficient history to drop a value from it.

eventually

Arguments

:: Integral a

=> a

n

-> Stream Bool

s

-> Stream Bool

Property s holds at some period in the next n periods. If n == 0, then s holds in the current period. We require n >= 0. E.g., if p = eventually 2 s, then we have the following relationship between the streams generated:

s => F F F T F F F T ...
p => F T T T F T T T ...

Note: s must have sufficient history to drop n values from it.

always :: Integral a => a -> Stream Bool -> Stream Bool

Property s holds for the next n periods. We require n >= 0. If n == 0, then s holds in the current period, e.g., if p = always 2 s, then we have the following relationship between the streams generated:

     0 1 2 3 4 5 6 7
s => T T T F T T T T ...
p => T F F F T T ...

Note: s must have sufficient history to drop n values from it.

until :: Integral a => a -> Stream Bool -> Stream Bool -> Stream Bool

until n s0 s1 means that eventually n s1, and up until at least the period before s1 holds, s0 continuously holds.

Note: Both argument streams must have sufficient history to drop n values from them.

release :: Integral a => a -> Stream Bool -> Stream Bool -> Stream Bool

release n s0 s1 means that either always n s1, or s1 holds up to and including the period at which s0 becomes true.

Note: Both argument streams must have sufficient history to drop n values from them.