5. Adalog solver rewrite¶

5.1. Global goal¶

Rewrite Adalog into a way that allows us to:

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

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

5.2. 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.

5.2.1. 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.

5.2.2. 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.

5.2.3. 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)

5.3. 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)

or, with variadic logic operations:

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.

5.3.1. 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 Anys afterwards.
Take the first Any, and “inline” every other relation (including subsequent Anys) 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

5.3.2. Exponential time resolution optimization (1)¶

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:

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.
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.

5.4. 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.

5.5. 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:

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

digraph g{
  y_to_x -> x_to_y
  x_to_y -> y_to_x
  y_to_x [label="X ← Val (Y)",shape=box,style=rounded];
  x_to_y [label="Y ← Val (X)",shape=box,style=rounded];
}

5.6. 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:

Take every all branch one after the other.
Execute every atomic relation in it one after the other. If one returns false, switch to the next all branch, and reset all variables.
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 :)

5.6.1. Exponential time resolution optimization (2)¶

Following the removal of the eq_prop feature of propagate relations, predicates generally used in Libadalang equations work with two logic variables instead of a single one, which essentially cancels the optimization (1). Indeed, it is easily implementable for predicates with one logic var (the var-to-constraint map can reflect relations that use only one var), and much harder for predicates taking more variables.

As a consequence, the (1) optimization is not enough to efficiently solve the following real-world equation:

procedure Test is
   type Rec is record
      X : Integer;
   end record;

   type Arr is array (Positive range <>) of Integer;

   function Foo (X : Integer) return Rec is ((X => X));
   function Foo (X : Integer) return Arr is (1 => X);
   function Foo (X : Integer) return Integer is (X);
   function Foo (X : Integer) return Float is (1.0);
   function Foo (X : Integer) return Duration is (1.0);

   V : Integer;
begin
   V := Foo (Foo (Foo (1 + 2 * 3).X) (4 + 5 + 6 + 7));
   pragma Test_Statement;
end Test;

The inefficiency is the same as before: the equation is mostly a conjuction of disjunctions, and we spend a lot of time exploring all combinations.

Instead of stopping early the exploration of a branch in the combination exploration recursion, a variation of the optimization would be to simplify the big relation before exploring all combinations of anys.

For instance:

All(
    R1,
    Any(R_A1, R_A2, R_A3),
    Any(R_B1, R_B2, R_B3)
)

We could first look for a contradiction between R1 and in any of the R_* relations, and remove relations that do have a contradiction. If both R1 & R_A1 and R1 & R_A3 are false, we can simplify the above big relation to:

All(
    R1,
    R_A2,
    Any(R_B1, R_B2, R_B3)
)

From there, we can search for contradictions between R1, R_A2 and the remaining R_* relations. Assuming R_B1 and R_B3 do have a contradiction, we are now left with:

All(R1, R_A2, R_B3)

i.e. the search space is much reduced.

In general, the simplification may not be able to remove all Anys, but the hope is that it would be able to remove most of them, and reduce their size otherwise, and thus restrict a lot the time taken to explore all combinations (because there are much fewer combinations left). However, for Anys that stay with more than 1 element, this algorithm will not be able to use constraints in their alternatives to simplify other parts of the relation.

If this new optimization is too weak for equations that Libadalang produces in practice (too many/too big Anys left), it may be valuable to run this optimization later, at various stages during the development of the equation, i.e. when exploring combinations from an All relation: at this point, the current list of atoms may contain atoms from alternatives of already explored Anys, and thus may provide more opportunities to run simplifications.

Here is pseudo Python code for the simplification algorithm:

def has_contradiction(atoms: List[Relation]) -> bool:
    """Return whether the given list of atoms contains a contradiction.

    This is similar to the regular resolution of a list of atoms, with an
    important difference: one atom uses a variable that no other atom
    defines will not cause this function to return True (the regular solver
    considers that the list of atoms has no solution in that case, i.e.
    "solve()" returns False in that case).
    """
    ...

def split_all(relation: Relation) -> Tuple[List[Relation], List[Relation]]:
    """Split LogicAny from atoms in the given LogicAll relation."""
    ...

def simplify(relation: Relation, outer_atoms: List[Relation]) -> Relation:
    """
    Try to simplify the given relation (a LogicAll relation). Return the
    simplified relation.
    """
    anys, self_atoms = split_all(relation)
    atoms = outer_atoms + self_atoms

    # Run a fixed-point algorithm: as long as "atoms" keeps growing, look
    # for new contradictions in "anys". Every time a LogicAny relation is
    # is simplified enough to
    atoms_changed = True
    while atoms_changed:
        atoms_changed = False
        if has_contradiction(atoms):
            return LogicFalse()

        # In every Any relation, go through alternatives ("alt_rel") and
        # recursively try to simplify them.
        any_rels = list(enumerate(anys))
        for i, any_rel in reversed(any_rels):
            alt_rels = list(enumerate(any_rel.sub_relations))
            for j, alt_rel in reversed(alt_rels):

                if isinstance(alt_rel, LogicAll):
                    alt_rel = simplify(alt_rel, atoms)
                elif has_contradiction(atoms + [any_rel]):
                    alt_rel = LogicFalse()

                # If this alternative always lead to a contradiction: we
                # can remove it from "any_rel"...
                if isinstance(alt_rel, LogicFalse):
                    any_rel.sub_relations.pop(j)

                    # ... and if we have only one alternative left in
                    # "any_rel", this is no longer a disjunction: unwrap it
                    # and move its items to "anys"/"atoms".
                    if len(any_rel.sub_relations) == 1:
                        alt_rel = any_rel.sub_relations[0]
                        anys.pop(i)
                        if isinstance(alt_rel, LogicAll):
                            sub_anys, sub_atoms = split_all(alt_rel)
                            anys.extend(sub_anys)
                            atoms.extend(sub_atoms)
                            if sub_atoms:
                                atoms_changed = True
                        else:
                            atoms.append(alt_rel)
                            atoms_changed = True

                else:
                    # Now use the simplified sub-relation
                    any_rel.sub_relations[j] = alt_rel

    return LogicAll(atoms + anys)

5. Adalog solver rewrite¶

5.1. Global goal¶

5.2. Expand high level operations to low level ones¶

5.2.1. Introduce a `Val` operation¶

5.2.2. Expand `Unify` to `Assign` and `Val`¶

5.2.3. Expand `Member` into `Any` and `Assign`¶

5.3. Develop relations¶

5.3.1. The algorithm¶

5.3.2. Exponential time resolution optimization (1)¶

5.4. Check completeness¶

5.5. Topo sort/check out for cycles¶

5.6. Execute the resulting relation¶

5.6.1. Exponential time resolution optimization (2)¶

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5. Adalog solver rewrite¶

5.1. Global goal¶

5.2. Expand high level operations to low level ones¶

5.2.1. Introduce a Val operation¶

5.2.2. Expand Unify to Assign and Val¶

5.2.3. Expand Member into Any and Assign¶

5.3. Develop relations¶

5.3.1. The algorithm¶

5.3.2. Exponential time resolution optimization (1)¶

5.4. Check completeness¶

5.5. Topo sort/check out for cycles¶

5.6. Execute the resulting relation¶

5.6.1. Exponential time resolution optimization (2)¶

5.2.1. Introduce a `Val` operation¶

5.2.2. Expand `Unify` to `Assign` and `Val`¶

5.2.3. Expand `Member` into `Any` and `Assign`¶