E-graphs as Circuits, and Optimal Extraction via Treewidth

Overview

Our main contribution is an equivalence:

e-graphs $\leftrightarrow$ (cyclic) monotone circuits

This allows us to:

• Apply [KTX17] to efficiently solve extraction, parameterized by treewidth.
• More easily simplify e-graphs, which significantly reduces treewidth.

The algorithm is very similar to [GLP24].

Monotone circuits

A monotone circuit is a circuit with only AND and OR gates.

e-node $\leftrightarrow$ all children true $\leftrightarrow$ AND

e-class $\leftrightarrow$ at least one child true $\leftrightarrow$ OR

What about cycles?

$x$	old output	new output
0	0	0
0	1	0
1	0	0
1	1	1

But e-graphs researchers never have to think about this. Why not?

What about cycles?

$x$	old output	new output
0	0	0
0	1	0
1	0	0
1	1	1

But e-graphs researchers never have to think about this. Why not?

What about cycles?

$x$	old output	new output
0	0	0
0	1	0
1	0	0
1	1	1

But e-graphs researchers never have to think about this. Why not?

• To find the minimum cost extraction, only need to consider minimal solutions.
• For this particular circuit construction, every choice of minimally satisfying inputs (what to extract) has a unique way to set the rest of the circuit.

Converting to circuits

To convert an e-graph $G$ to a circuit $C$:

• Replace every e-class with an OR gate.
• Replace every e-node with an AND gate and a variable connected to it.

$\implies$ Extraction is equivalent to weighted monotone circuit SAT!

Minimum weight monotone circuit SAT

[KTX17] already solved this, albeit stated only for circuits that do not have cycles.

Proposition 4.7 in [KTX17]. Let $C$ be a monotone circuit (with no cycles) and $G$ be the underlying undirected graph with $n$ nodes. If a tree decomposition for $G$ and treewidth $k$ is given, then a minimum weight satsifying assignment of $C$ can be computed in time $2^{O(k)}\text{poly}(n, k)$.

The algorithm works unchanged for circuits with cycles, if we allow cyclic extraction!

• Intuition: The undirected graph $G$ already "forgot" whether $C$ is a DAG.

For acyclic extraction, can maintain the transitive closure of the subgraph of 1-edges to make sure no 1-cycles are formed, achieving $2^{O(k^2)}\text{poly}(n, k)$.

Circuit simplification

E-graph simplification before extraction has been used with ILP (i.e. see https://github.com/egraphs-good/extraction-gym/pull/16)

With circuits, we can leverage existing circuit minimization tools that support sequential circuits*.

• These preserve the whole $\{\text{input}\} \times \{\text{old state}\} \to \{\text{new state}\}$ table.
• Takes care of simplifications like the previous slide.

But we only care about minimal satisfying solutions, sometimes acyclic.

• Can write additional specialized rules to take advantage of this!

Custom simplification rules

Because we are only interested in minimal extractions:

$\longrightarrow$

Custom simplification rules

If we are only interested in acyclic extractions:

Custom simplification rules

If we are only interested in acyclic extractions:

$0$

delete this gate and all out-neighbors that are AND gates recursively

Custom simplification rules

If we are only interested in acyclic extractions:

$\downarrow$

Simplification results

We applied several ad-hoc rules like this to a test set of e-graphs used with egg. Depending on which applications the e-graphs came from, we observed:

• Up to 65% reduction in treewidth.
• 42-97% reduction in $|V|$.
• 57-96% reduction in $|E|$.

A more systematic approach might improve this further.

Open directions

• Can the treewidth-based extraction algorithm be adapted to support more general cost functions?
• Cost of an e-node is often written as a function of the cost of children.
• Current algorithm (both [GLP24] and us) has no way to deal with the cost of a yet-unknown child.
• How much simplification can you achieve if you actually adapted modern circuit simplification software that is used in practice?
• Can we find more applications of the connection between e-graphs and circuits?

Thank you!