Inequalities

An Inequality is a symbolic statement comparing two numeric expressions built from your data.

These statements take the form:

\[L(G) \;\le\; R(G) \quad\text{or}\quad L(G) \;\ge\; R(G)\]

where (L) and (R) are expressions over graph invariants (or other structured data features). They serve as the building blocks for numerical conjectures in txgraffiti.

Concrete Example

Here’s a worked example using the built-in graph_data dataset:

from txgraffiti.playground    import ConjecturePlayground
from txgraffiti.generators    import convex_hull, linear_programming, ratios
from txgraffiti.heuristics    import dalmatian_accept, morgan_accept
from txgraffiti.processing    import remove_duplicates, sort_by_touch_count
from txgraffiti.example_data  import graph_data
from txgraffiti               import Predicate

ai = ConjecturePlayground(
    graph_data,
    object_symbol='G',
    base='connected',
)

# Define custom predicate: G is cubic
regular = Predicate('regular', lambda df: df['min_degree'] == df['max_degree'])
cubic   = regular & (ai.max_degree == 3)

# Run discovery using three methods
ai.discover(
    methods         = [ratios, convex_hull, linear_programming],
    features        = [ai.independence_number],
    target          = ai.zero_forcing_number,
    heuristics      = [dalmatian_accept, morgan_accept],
    hypothesis      = [cubic],
    post_processors = [remove_duplicates, sort_by_touch_count],
)

# Print resulting inequalities
for idx, conj in enumerate(ai.conjectures[:3], start=1):
    print(f"Conjecture {idx}: {ai.forall(conj)}")

Output:

Conjecture 1: ∀ G: ((connected) ∧ (regular) ∧ (max_degree == 3)) → (zero_forcing_number >= 3)

Conjecture 2: ∀ G: ((connected) ∧ (regular) ∧ (max_degree == 3)) → (zero_forcing_number >= (-2 + independence_number))

Conjecture 3: ∀ G: ((connected) ∧ (regular) ∧ (max_degree == 3)) → (zero_forcing_number >= ((2 * independence_number) + -8))

Each of these is an Inequality object automatically constructed from symbolic arithmetic and evaluated row-by-row on the data.

Touch and Slack

For any Inequality, you can compute:

Slack: the numeric distance between the LHS and RHS.
Touch count: how many rows satisfy the inequality exactly (i.e., with zero slack).

ineq = ai.conjectures[1].conclusion
print("Touch count:", ineq.touch_count(graph_data))
print("Slack values:", ineq.slack(graph_data).tolist())

Integration

Inequalities can be used directly as:

Predicate objects for filtering or logic composition
Conclusions inside a Conjecture statement
Targets for filtering, ranking, and exporting