## 13.3.1. Global goal¶

1. Express analyses about whether an equation is sound or not
2. Refactor an equation into the list of all its disjunctions.
3. Easily weed out branches that cannot provide a solution (because they cannot provide a sound solution where all variables are defined)
4. Write a much simpler interpreter

In this document I will try to summarize how we expect to do that.

## 13.3.2. Expand high level operations to low level ones¶

The first pass we want to go through, is to get rid of some of the existing Adalog relations, by expanding them to simpler bricks.

One of the goals is to be able to clearly express which variables a relation defines or uses.

### 13.3.2.1. Introduce a `Val` operation¶

We want to introduce a `Val` low level operation that simply accesses the value of a logic variable. The operation `Val (X)` uses `X`.

### 13.3.2.2. Expand `Unify` to `Assign` and `Val`¶

We want to transform

```Unify X <=> Y
```

into

```X <= Val (Y) or Y <= Val (X)
```

This way, we have two assign operations, one that uses `X` and depends on `Y`, the other that uses `Y` and depends on `X`.

### 13.3.2.3. Expand `Member` into `Any` and `Assign`¶

This was the original implementation of `Member`, that was removed because of performance in the old execution scheme. In the new scheme, this will be necessary in order to do full disjunction expansion.

What we want to do is to then expand

```Member (X, {1, 2, 3, 4, 5})
```

into

```Any (X <= 1, X <= 2, X <= 3, X <= 4, X <= 5)
```

## 13.3.3. Develop relations¶

Here, we want to apply a development rule, similar to arithmetic development rules, which is

```A and (B or C)  => (A and B) or (B and C)
```

```All(A, B, C, ..., Any(D, E, F))

<=>

Any(
All(D, A, B, C, ...),
All(E, A, B, C, ...),
All(F, A, B, C, ...)
)
```

Applying this transformation recursively, until we reach a point where there is only one top-level disjunction, and all other relations are either conjunctions or non-composite relations. Since an All(All) can be inlined, the expected max depth of the relation is 3. At this stage the relation is effectively a list of either conjunctions or non-composite relations. The latter being a very unlikely and pretty simple corner case, let’s assume that we have a disjunction list of all possible conjunctions.

### 13.3.3.1. The algorithm¶

Note: Here we assume that every `Any` can only contain `All` relations or atomic relations, because other `Any` relations would have been inlined.

Given an `All` relation that contains number of `Any` between `1 .. N`, we want to:

• Sort the relation to have the atomic relations first, and the `Any`s afterwards.
• Take the first `Any`, and “inline” every other relation (including subsequent `Any`s) of the parent `All` inside each of its branches.
• If there were remaining `Any` relations, run this algorithm recursively on every resulting `All` relations that is inside the first `Any`.
• Return the first `Any`, destroy self

### 13.3.3.2. Exponential time resolution optimization¶

It is likely that it is during that expansion phase that we want to optimize out exponential time resolution problems. Assuming we start with an `All` relation that contains an `Any`, we can try to do an optimization similar to the one described in the Exponential resolution in Adalog document.

The above document informally describes the construction of a constraint map, mapping specific variables to “constraints”. We can define more formally those constraints using the terms defined in this document:

A constraint is any atomic relation that is part of the All relation, and that uses the value of a logic variable.

We can then create a mapping from logic variables to constraints, mapping the variable used by a relation to the relation, for every atomic relation that uses a logic variable.

Then, before inlining Self’s relations in the `Any` sub-branches, we can iterate over every of those sub-branches:

1. If the sub-branch is an atomic relation, check if it defines the value of a variable, and also that it doesn’t use any other variable. If that is the case, check if there is one or several constraints for this variable in the constraint set. If that is the case, execute the relation, then the constraints. If the result is negative, we can get rid of the relation right away.
2. If the sub-branch is an `All`, run the above algorithm over all of its atomic components.

When transforming children `All` relations, we pass down the existing set of constraints. It will then be added to the one that will be constructed in the recursive call.

## 13.3.4. Check completeness¶

We define an equation as being complete if at least one of the `All` branches in the toplevel `Any` binds every variables involved in the equation. After development, we want to check that this condition is satisfied. If not, return a special error.

## 13.3.5. Topo sort/check out for cycles¶

Once we’ve transformed a relation - any relation, not necessarily the top level one - into the form above, we can do a topological sort of the relations inside a conjunction.

After this transformation, every relation inside a conj is assumed to be an atomic relation. Every relation uses a logic variable, or defines a logic variable, or both (but not for the same variable). This *allows us to build a dependency graph. If we do a topological sort of this dependency graph, it allows us to:

1. Check for cycles. For example, such a conjunction would be detected and flagged as incorrect
```All(X <= Val (Y), Y <= Val (X))
```

because, expanded into a dependency list it gives the following:

```[(Uses(X), Defines(Y)), (Uses(Y), Defines(X))]
```

which in turns forms the following dependency graph

## 13.3.6. Execute the resulting relation¶

Executing the resulting relations should be extremely simple, since you have a structure like this:

```Any(
All(A11, A12, A13, ...),
All(A21, A22, A23, ...),
All(A31, A32, A33, ...),
...
)
```

Where every `A..` relation is an atomic relation that you can execute in order. The execution algorithm is then to:

1. Take every all branch one after the other.
2. Execute every atomic relation in it one after the other. If one returns false, switch to the next all branch, and reset all variables.
3. If we get to the end of the All without a failure, then we have a solution.

This is very easy to execute, and almost completely stateless, unlike the previous interpreter. Writing an interpreter should be trivial, and writing a JIT should even be possible if needed :)