Formal Specification: Probabilistic Points and Resolutions

Abstract

This document formalizes a new paradigm for text processing that treats all textual elements as inherently uncertain “Points” and all operations as probabilistic “Resolutions”. Unlike traditional deterministic text processing, this system acknowledges that meaning in natural language is fundamentally probabilistic and context-dependent.

Core Concepts

1. Points

A Point is a variable representing textual content with inherent uncertainty.

Definition

Point := {
    content: Text,
    certainty: Probability ∈ [0, 1],
    context: ContextSpace,
    temporal_state: TimeStamp
}

Properties

Examples

point₁ = {
    content: "bank",
    certainty: 0.7,
    context: "financial_document",
    temporal_state: now()
}

point₂ = {
    content: "bank", 
    certainty: 0.6,
    context: "nature_description",
    temporal_state: now()
}

2. Resolutions

A Resolution is a probabilistic operation that processes points along with supporting evidence to produce new probabilistic outputs.

Definition

Resolution := Function(
    primary_point: Point,
    affirmations: Set<Evidence>,
    contentions: Set<CounterEvidence>,
    context: ContextSpace
) → ProbabilisticResult

Input Components

Primary Point: The main textual element being processed

Affirmations: Evidence that supports certain interpretations

Contentions: Evidence that challenges or weakens certain interpretations

Output: ProbabilisticResult

ProbabilisticResult := {
    interpretations: List<WeightedInterpretation>,
    confidence_bounds: (lower: Probability, upper: Probability),
    resolution_strength: Probability,
    remaining_uncertainty: Probability,
    evidence_synthesis: EvidenceMap
}

WeightedInterpretation := {
    meaning: InterpretedContent,
    probability: Probability,
    supporting_evidence: List<Evidence>,
    conflicting_evidence: List<CounterEvidence>
}

Mathematical Foundation

Probability Space

All operations occur within a probability space Ω where:

Uncertainty Propagation

When multiple points or evidence pieces are combined:

Combined_Uncertainty = f(
    point_uncertainty,
    affirmation_strength,
    contention_strength,
    evidence_conflicts
)

Where f is a function that accounts for:

Resolution Strength

Each resolution has an inherent strength based on:

Operational Framework

1. Point Creation

create_point(text_content, initial_context) → Point

2. Evidence Gathering

gather_affirmations(point, knowledge_base) → Set<Evidence>
gather_contentions(point, knowledge_base) → Set<CounterEvidence>

3. Resolution Process

resolve(point, affirmations, contentions, context) → ProbabilisticResult

4. Result Interpretation

extract_most_likely(result) → WeightedInterpretation
extract_all_plausible(result, threshold) → List<WeightedInterpretation>
calculate_ambiguity(result) → Probability

Types of Resolutions

1. Semantic Resolution

Purpose: Determine meaning of ambiguous text

semantic_resolve(
    point: "bank statement",
    affirmations: ["financial context", "numerical data present"],
    contentions: ["near river mentioned"],
    context: "business_document"
) → "financial record" (p=0.89), "river documentation" (p=0.11)

2. Sentiment Resolution

Purpose: Determine emotional or attitudinal content

sentiment_resolve(
    point: "That's interesting",
    affirmations: ["positive context", "engaged tone"],
    contentions: ["formal setting", "brief response"],
    context: "professional_email"
) → "genuine_interest" (p=0.45), "polite_dismissal" (p=0.55)

3. Intent Resolution

Purpose: Determine intended action or purpose

intent_resolve(
    point: "Could you help me with this?",
    affirmations: ["urgent tone", "specific problem context"],
    contentions: ["indirect phrasing", "casual setting"],
    context: "colleague_communication"
) → "direct_request" (p=0.7), "casual_inquiry" (p=0.3)

4. Temporal Resolution

Purpose: Determine time-related meaning

temporal_resolve(
    point: "We'll do this soon",
    affirmations: ["deadline context", "urgency indicators"],
    contentions: ["vague phrasing", "no specific timeframe"],
    context: "project_management"
) → "within_week" (p=0.4), "within_month" (p=0.5), "indefinite" (p=0.1)

Evidence Types

Affirmations (Supporting Evidence)

  1. Contextual Affirmations
    • Domain-specific terminology present
    • Consistent thematic elements
    • Appropriate register/formality level
  2. Structural Affirmations
    • Grammatical patterns supporting interpretation
    • Discourse markers indicating meaning
    • Punctuation and formatting cues
  3. Semantic Affirmations
    • Related concepts in surrounding text
    • Coherent conceptual framework
    • Logical argument structure
  4. Pragmatic Affirmations
    • Cultural context alignment
    • Situational appropriateness
    • Speaker/writer intention indicators

Contentions (Counter-Evidence)

  1. Contextual Contentions
    • Mixed domain signals
    • Inconsistent terminology
    • Conflicting contextual cues
  2. Ambiguity Contentions
    • Multiple valid interpretations available
    • Unclear referents
    • Polysemous terms present
  3. Coherence Contentions
    • Logical inconsistencies
    • Contradictory statements
    • Incomplete information
  4. Reliability Contentions
    • Source credibility issues
    • Historical inaccuracy
    • Contradicts established knowledge

Implementation Philosophy

Design Principles

  1. Uncertainty is Fundamental: Never pretend certainty where none exists
  2. Evidence-Driven: All conclusions must be supported by explicit evidence
  3. Context-Aware: Same text can have different meanings in different contexts
  4. Probabilistic Throughout: No binary true/false - everything is probabilistic
  5. Transparent Reasoning: Show how conclusions were reached
  6. Updateable: New evidence can change previous conclusions

Human-Like Reasoning

This system mimics how humans actually process text:

Example Workflow

Scenario: Processing “The solution is optimal”

  1. Point Creation 
    point = create_point("The solution is optimal", context="technical_document")
// Initial uncertainty: 0.3 (due to ambiguous terms)
  2. Evidence Gathering ``` affirmations = [ “mathematical context present”, “optimization terminology used”, “technical audience” ]

contentions = [ “no specific metrics provided”, “subjective term ‘optimal’”, “could be business optimization vs mathematical” ]


3. **Resolution**

result = resolve(point, affirmations, contentions, “technical_document”)

// Output: interpretations = [ {meaning: “mathematically optimal solution”, probability: 0.6}, {meaning: “best practical choice”, probability: 0.3}, {meaning: “subjective quality assessment”, probability: 0.1} ]

confidence_bounds = (0.45, 0.75) resolution_strength = 0.7 remaining_uncertainty = 0.3


4. **Application**

// For mathematical processing: if extract_most_likely(result).meaning contains “mathematical” apply_mathematical_optimization_analysis()

// For uncertainty-aware display: display(“Solution optimality: “ + result.confidence_bounds + “ confidence”) ```

Advantages Over Traditional Approaches

  1. Honest About Uncertainty: Acknowledges when we don’t know
  2. Evidence-Based: Conclusions are grounded in explicit reasoning
  3. Context-Sensitive: Same text processed differently in different contexts
  4. Updatable: Can incorporate new evidence
  5. Nuanced: Captures subtle degrees of meaning
  6. Transparent: Shows reasoning process
  7. Human-Like: Reflects how humans actually understand text

Future Research Directions

  1. Machine Learning Integration: Train systems to better identify affirmations and contentions
  2. Context Modeling: Develop sophisticated context representation systems
  3. Evidence Networks: Model complex relationships between evidence pieces
  4. Temporal Dynamics: Handle how meaning changes over time
  5. Multi-Modal: Extend to non-textual evidence (images, audio, etc.)
  6. Collaborative Resolution: Multiple agents contributing evidence and perspectives
  7. Meta-Resolution: Resolutions about the quality of other resolutions

Conclusion

This formal specification establishes a foundation for text processing that embraces uncertainty rather than hiding it. By treating text as inherently probabilistic and using evidence-based reasoning, we can build systems that more accurately reflect the complexity and nuance of human language understanding.

The key insight is that traditional deterministic text processing is fundamentally flawed because it assumes certainty where none exists. By building uncertainty into the foundation of our text processing systems, we can create more honest, nuanced, and ultimately more useful tools for working with natural language.