# Module `LustreSlicing`

Cone of influence reduction and dependency ordering of equations

- author
- Christoph Sticksel

## Dependency order

`val order_equations : bool -> (LustreIdent.t * (LustreIndex.index list LustreIndex.t * LustreIndex.index list LustreIndex.t)) list -> LustreNode.t -> LustreNode.equation list * LustreIndex.index list LustreIndex.t`

Return equations of node in topological order of their dependencies

`order_equations i d n`

takes as input a Lustre node`n`

and an association list`d`

of node names to a map from output indexes to the list of input indexes the output variable depends on. Return the equations of`n`

ordered such that each equation comes after the equations defining the variables occurring on its right-hand.Along with the list of equations return a map that for each output identified by its position indicates which positions in the inputs it depends on. The input parameter

`d`

must contain such a map for each called node.If the flag

`i`

is true, consider the equations for the inital state only, otherwise consider the equations for post-initial states only.Fail with a parse error if a variable definition is cyclic, that is, the variable on the left-hand side of the equation occurs in the definition of a variable on the right-hand side.

For the detection of cycles also consider definitions of called nodes by using the map of outputs to their dependent inputs. If in the node

`node N (i: t) returns (o: t)`

the output

`o`

depends on the input`i`

, then the node call`N(x)`

must not occur in the definition of`x`

. If the output`o`

does not depend on the input`i`

,`N(x)`

may occur in the definition of`x`

.In particular, if a node is viewed as its contract only, the implemention is omitted and node calls never cause cycles in defintions. One could argue that since the implementation is not known, the outputs should depend on all inputs, but this might trigger spurious cycles. We view this more as a realizability issue and we could implement separately a check if a node as specified by its contract can be implemented without introducing a depency on the inputs.

Array typed variables are not considered in the cycle detection, and this may lead to actual cycles not rejected. This is difficult, because we would need to consider each occurrence of an array typed variable together with its indexes. Then we would need to compare indexes if a state variable occurs on a path with the same index. A syntactic comparison would again miss some cycles, and we would need to evaluate the indexes for a precise comparison. This is probably too much effort for what it is worth.

## Cone of influence reduction

`val node_is_abstract : Analysis.param -> LustreNode.t -> bool`

Return

`true`

if the node is flagged as abstract in`abstraction_map`

. Default to`false`

if the node is not in the map.

`val slice_to_abstraction : ?preserve_sig:bool -> bool -> Analysis.param -> LustreNode.t SubSystem.t -> LustreNode.t SubSystem.t`

Return a node hierarchy reduced to the cone of influence of properties and contracts, given a list of which nodes should be abstracted.

`slice_to_abstraction r a s`

takes the parameters of an analysis, which contains an association map`m`

of scopes to a flag that is`true`

if the node of that name is to be abstracted to its contract by omitting its implementation. Return a node hierarchy based on`s`

, reduced to the cone of influence of the properties and contracts if`r`

is`true`

. Otherwise, only abstraction is applied.For each node, start with a copy of the node without equations and node calls. Maintain a set of state variables, called the roots of the cone of influence, and move all equations and node calls that define a root to the copy of the node. Add all state variables as new roots that occur on the right-hand side of a moved equation, or are an inputs, oracles or the clock of a moved node call. Slice called nodes before the calling nodes. If an equation defines a record, tuple or array, the equations for each element are kept as soon as one of the elements is in the cone of influence.

The roots of a node always contain its outputs and the state variables occurring in contracts, except for the top node. The top node is never called and we can safely change its signature. It would be possible to analyze each call of a node to see exactly which of its inputs and outputs of are in the cone of influence, then modify the signature appropriately, and adjust all calls to this modified signature. However, this non-local analyisis is probably more complicated and error-prone than it is worth.

If a node is sliced to its implementation, add the property state variables and all state variables in assertions.

State variables in assertions are considered to be roots, because they potentially constrain a state variable in the cone of influence together with a state variable not in the cone of influence. In this case, both state variables are in the cone of influence. We may add a better analysis later.

`val slice_to_abstraction_and_property : ?preserve_sig:bool -> Analysis.param -> StateVar.StateVarSet.t -> LustreNode.t SubSystem.t -> LustreNode.t SubSystem.t`