Weighted frameworks and β

Weighted argumentation assigns a real-valued strength to each attack. A budget β lets you drop attacks whose total weight fits in the budget — turning acceptance from a binary question into a continuous one.

Introduced by Dunne, Hunter, McBurney, Parsons & Wooldridge (2011), weighted argument systems are the mechanism that lets scene tension modulate outcome. In our bridge, β is the scene intensity dial.

The semantic chain

1. A framework has attacks with weights w(a, b) > 0.
2. Given a budget β, you consider all residuals — frameworks obtained by dropping some subset of attacks whose total cost ≤ β.
3. An argument is credulously accepted at β if it is credulously accepted in at least one such residual.

At β = 0 you can drop nothing (equivalent to ordinary Dung semantics). At β large enough to cover every attack, every non-self-attacking argument becomes acceptable.

The boundary is inclusive: attacks whose weight exactly equals β are droppable.

As scene intensity

Think of β as how much the scene can "let slide":

• β = 0.0 — every counter bites. Tense courtroom energy.
• β = 0.4 — mid-range; some counters survive, some are waved off.
• β = 1.0 — counters slide off; amicable boardroom cordiality.

Here's the east-wall example at four discrete β values:

East-wall scene across β

β:0.00

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Drag the slider. At β = 0 both characters accept everything (no attacks survive the budget check below zero — so neither argument is credulous, no boost fires, both fall back to the same action and accept each other). At β = 0.4 and above, alice's argument becomes credulous — the scorer boosts it, she switches to her signature action, and bob rejects with his accepted counter.

Weighted support (bipolar)

Supports can also carry weights. They don't consume budget — they reinforce the supported argument in the weighted-bipolar semantics of argumentation-weighted-bipolar.

In our library

• argumentation_weighted::types::Budget — a validated β value in [0, 1].
• argumentation_weighted_bipolar::WeightedBipolarFramework — the underlying graph.
• EncounterArgumentationState::set_intensity — set β mid-scene through a shared reference.
• EncounterArgumentationState::is_credulously_accepted — acceptance at current β.

Further reading

• Dunne, Hunter, McBurney, Parsons, Wooldridge (2011) — weighted argument systems.
• Amgoud, Ben-Naim (2016) — graded semantics in weighted frameworks.