Frames as Distributions over Concepts

@cite{fillmore-1982} @cite{schank-abelson-1977}

A frame/scenario is a coherent situation type that makes certain concepts more or less likely.

structure Semantics.Probabilistic.Scenarios.Scenario (Concept : Type) : Type

A scenario is a named distribution over concepts.

This models the "scenario constraint" in SDS: P(concept | scenario)

• name : String

  Name of the scenario (e.g., "SPORTS", "ASTRONOMY")

• conceptDist : Concept → ℚ

  Distribution: P(concept | scenario)

structure Semantics.Probabilistic.Scenarios.ScenarioMix (Scenario : Type) : Type

A scenario mixture: distribution over scenarios.

When the scenario itself is uncertain, we have a mixture: P(concept) = Σ_s P(s) × P(concept | s)

• dist : Scenario → ℚ

  Distribution over scenarios: P(scenario)

Frame Elements

A frame defines a set of roles (frame elements) and their typical fillers.

COMMERCIAL_TRANSACTION frame:

• Buyer: human
• Seller: human or organization
• Goods: artifact or service
• Money: currency
• etc.

structure Semantics.Probabilistic.Scenarios.FrameElement (Concept : Type) : Type

A frame element (role within a frame).

• name : String

  Name of the element (e.g., "Buyer", "Goods")

• typicalFillers : Concept → ℚ

  Typical fillers: P(concept | this element)

structure Semantics.Probabilistic.Scenarios.FrameDef (Concept : Type) : Type

A full frame definition with its elements.

• name : String

  Frame name

• elements : List (FrameElement Concept)

  Frame elements with their filler distributions

• scenarioBase : Concept → ℚ

  Overall scenario distribution (for concepts not tied to specific elements)

Scenario Induction from Context

Context words activate scenarios:

• "player" activates SPORTS
• "astronomer" activates ASTRONOMY
• "cave" activates WILDLIFE/NATURE

This is modeled as P(scenario | context-words).

See Phenomena.Polysemy.Studies.ErkHerbelot2024 for worked disambiguation examples using these types.

structure Semantics.Probabilistic.Scenarios.ContextCue (Scenario : Type) : Type

A context cue: a word that provides evidence for scenarios.

• word : String

  The cue word

• likelihood : Scenario → ℚ

  Likelihood: P(word | scenario)

def Semantics.Probabilistic.Scenarios.inferScenario {S : Type} [BEq S] (prior : S → ℚ) (cues : List (ContextCue S)) (scenarios : List S) : S → ℚ

Infer scenario distribution from context cues (Bayesian update).

P(scenario | cues) ∝ P(scenario) × Π_cue P(cue | scenario)

Equations

• One or more equations did not get rendered due to their size.

Frame Evocation

Words evoke frames. "buy" evokes COMMERCIAL_TRANSACTION. Once evoked, the frame provides expectations about:

• What other elements will appear
• What concepts are likely for each element

structure Semantics.Probabilistic.Scenarios.FrameEvocation (Frame : Type) : Type

Frame evocation: which frames does a word evoke?

• word : String

  The evoking word

• evokedFrames : Frame → ℚ

  Distribution: P(frame | word)

def Semantics.Probabilistic.Scenarios.frameExpectations {Concept : Type} (frame : FrameDef Concept) (filledElements : List String) : List (FrameElement Concept)

Given an evoked frame, get expectations for unfilled elements.

Equations

• One or more equations did not get rendered due to their size.

Full SDS-Style Disambiguation

Combining selectional preferences with scenario constraints:

P(concept | role, context) ∝ P_sel(concept | role) × P_scen(concept | context)

def Semantics.Probabilistic.Scenarios.sdsDisambiguate {Concept : Type} (selectional scenario : Concept → ℚ) (concepts : List Concept) : Concept → ℚ

Full SDS disambiguation combining selectional and scenario constraints.

Equations

• Semantics.Probabilistic.Scenarios.sdsDisambiguate selectional scenario concepts = Core.ProductOfExperts.poe2 selectional scenario concepts

def Semantics.Probabilistic.Scenarios.sdsDisambiguateWithUncertainty {Concept Scenario : Type} (selectional : Concept → ℚ) (scenarioConceptDist : Scenario → Concept → ℚ) (scenarioPosterior : Scenario → ℚ) (concepts : List Concept) (scenarios : List Scenario) : Concept → ℚ

With scenario uncertainty: marginalize over scenarios first.

Equations

• One or more equations did not get rendered due to their size.

Scenarios as RSA Goals/QUDs

In RSAScenario, the Goal type can encode scenarios:

• Each goal corresponds to a scenario
• The goal prior = scenario prior
• Goal projection filters by scenario-relevance

Mapping

SDS	RSA
Scenario s	Goal g
P(s | context)	goalPrior
P(concept | s)	World structure filtered by g

Implementation

We can create RSA scenarios where:

1. Worlds encode concept assignments
2. Goals encode scenarios
3. goalProject filters worlds compatible with scenario

structure Semantics.Probabilistic.Scenarios.ScenarioWorld (Concept Scenario : Type) : Type

A scenario-structured world: pairs (concept-assignment, scenario).

• concept : Concept
• scenario : Scenario

def Semantics.Probabilistic.Scenarios.scenarioGoalFilter {Concept Scenario : Type} [BEq Scenario] (targetScenario : Scenario) : ScenarioWorld Concept Scenario → Bool

Create a "scenario goal" that filters worlds by scenario membership.

Equations

• Semantics.Probabilistic.Scenarios.scenarioGoalFilter targetScenario w = (w.scenario == targetScenario)