3.1.5. cnfgen.families.pebbling module

Implementation of the pigeonhole principle formulas

PebblingFormula(digraph)

Pebbling formula

Build a pebbling formula from the directed graph. If the graph has an ordered_vertices attribute, then it is used to enumerate the vertices (and the corresponding variables).

Arguments: - digraph: directed acyclic graph.

SparseStoneFormula(D, B)

Sparse Stone formulas

This is a variant of the StoneFormula(). See that for a description of the formula. This variant is such that each vertex has only a small selection of which stone can go to that vertex. In particular which stones are allowed on each vertex is specified by a bipartite graph \(B\) on which the left vertices represent the vertices of DAG \(D\) and the right vertices are the stones.

If a vertex of \(D\) correspond to the left vertex \(v\) in \(B\), then its neighbors describe which stones are allowed for it.

The vertices in \(D\) do not need to have the same name as the one on the left side of \(B\). It is only important that the number of vertices in \(D\) is the same as the vertices in the left side of \(B\).

In that case the element at position \(i\) in the ordered sequence enumerate_vertices(D) corresponds to the element of rank \(i\) in the sequence of left side vertices of \(B\) according to the output of Left, Right = bipartite_sets(B).

Standard StoneFormula() is essentially equivalent to a sparse stone formula where \(B\) is the complete graph.

Parameters:

D : a directed acyclic graph

it should be a directed acyclic graph.

B : bipartite graph

Raises:

ValueError

if \(D\) is not a directed acyclic graph

ValueError

if \(B\) is not a bipartite graph

ValueError

when size differs between \(D\) and the left side of \(B\)

See also

StoneFormula

StoneFormula(D, nstones)

Stone formulas

The stone formulas have been introduced in [2] and generalized in [1]. They are one of the classic examples that separate regular resolutions from general resolution [1].

A “Stones formula” from a directed acyclic graph \(D\) claims that each vertex of the graph is associated with one on \(s\) stones (not necessarily in an injective way). In particular for each vertex \(v\) in \(V(D)\) and each stone \(j\) we have a variable \(P_{v,j}\) that claims that stone \(j\) is associated to vertex \(v\).

Each stone can be either red or blue, and not both. The propositional variable \(R_j\) if true when the stone \(j\) is red and false otherwise.

The clauses of the formula encode the following constraints. If a stone is on a source vertex (i.e. a vertex with no incoming edges), then it must be red. If all stones on the predecessors of a vertex are red, then the stone of the vertex itself must be red.

The formula furthermore enforces that the stones on the sinks (i.e. vertices with no outgoing edges) are blue.

Parameters:

D : a directed acyclic graph

it should be a directed acyclic graph.

nstones : int

the number of stones.

Raises:

ValueError

if \(D\) is not a directed acyclic graph

ValueError

if the number of stones is negative

References

[1]	(1, 2) M. Alekhnovich, J. Johannsen, T. Pitassi and A. Urquhart An Exponential Separation between Regular and General Resolution. Theory of Computing (2007)

[2]	R. Raz and P. McKenzie Separation of the monotone NC hierarchy. Combinatorica (1999)