Constraints

Formal theory of constraints as structure-preserving functors over deontic modalities

Core Definition

A constraint is a functor mapping structural contexts to deontic outcomes:

C : (Σ × A) → Δ

Where:
  Σ = Simplicial complex (structural context from ontolog)
  A = Action/state space
  Δ = Deontic modality {P, O, F, I, ?}

Constraints shape possibility spaces without adding energy—they are rate-independent causes that determine what can happen without forcing any particular outcome.

The Constraint Equation

C(σ, a) = δ ⟺ Within context σ, action a has deontic status δ

Composition: Constraints form a category with:

Objects: Deontic positions
Morphisms: Constraint transformations
Identity: Trivial permission (P by default)
Composition: Sequential constraint application

Deontic Modalities (Δ)

Four fundamental operators from modal deontic logic:

Operator	Symbol	Semantics	Dual
Permitted	P(a)	a is allowed	¬F(a)
Obligated	O(a)	a is required	¬P(¬a)
Forbidden	F(a)	a is prohibited	¬P(a)
Impossible	I(a)	a cannot occur	¬◇a

Axioms (Standard Deontic Logic):

D: O(a) → P(a)           — Ought implies may
K: O(a→b) → (O(a)→O(b))  — Distribution
N: If ⊢a then ⊢O(a)      — Necessitation for tautologies

Interdefinitions:
P(a) ≡ ¬O(¬a)            — Permission as non-obligatory negation
F(a) ≡ O(¬a)             — Prohibition as obligated negation

Constraint Trichotomy (Juarrero)

Constraints differ by how they shape possibility:

┌─────────────────────────────────────────────────────────────────┐
│                    CONSTRAINT TRICHOTOMY                        │
├──────────────┬──────────────────────────────────────────────────┤
│   ENABLING   │ Creates new possibilities, opens pathways        │
│              │ Examples: catalysts, bridges, APIs, permissions   │
│              │ Effect: Expands action space                      │
│              │ Hohfeldian: Privilege, Power                      │
├──────────────┼──────────────────────────────────────────────────┤
│  GOVERNING   │ Regulates without participating, modulates rate  │
│              │ Examples: thermostats, regulatory genes, duties   │
│              │ Effect: Channels existing possibilities          │
│              │ Hohfeldian: Claim-Duty                            │
├──────────────┼──────────────────────────────────────────────────┤
│ CONSTITUTIVE │ Defines identity, creates closure                │
│              │ Examples: rules of chess, cell membranes, types  │
│              │ Effect: Determines what something IS             │
│              │ Hohfeldian: Immunity-Disability                   │
└──────────────┴──────────────────────────────────────────────────┘

Causal flow: Enabling → Constitutive → Governing

Enabling constraints induce coherence
Constitutive constraints maintain coherence
Governing constraints regulate coherent behaviour

Hohfeldian Rights Structure

Eight fundamental jural positions in two squares:

First-Order (Conduct)

    Claim ←─correlative─→ Duty
      ↕                     ↕
   opposite             opposite
      ↕                     ↕
  No-Right ←─correlative─→ Privilege

Second-Order (Normative Change)

    Power ←─correlative─→ Liability
      ↕                     ↕
   opposite             opposite
      ↕                     ↕
 Disability ←─correlative─→ Immunity

Position semantics:

Position	Definition	Constraint Type
Claim	X has claim that Y φ	Governing
Duty	Y must φ toward X	Governing
Privilege	X may φ (no duty not to)	Enabling
No-Right	Y cannot demand X not φ	Enabling
Power	X can change Y's positions	Enabling
Liability	Y's positions changeable by X	Governing
Immunity	X's positions unchangeable by Y	Constitutive
Disability	Y cannot change X's positions	Constitutive

Correlative inference: If X has Claim against Y, then Y has Duty to X (and vice versa).

The KROG Theorem

Valid actions satisfy all constraint layers within governance:

Valid(a) ⟺ K(a) ∧ R(a) ∧ O(a) ∧ (a ∈ G)

Where:
  K(a) = Knowledge constraint (a is knowable/queryable)
  R(a) = Rights constraint (agent has right to a)
  O(a) = Obligation constraint (a satisfies duties)
  G    = Governance boundary (meta-rules)

Implication: No valid action can violate knowledge transparency, rights structure, obligation compliance, OR governance bounds.

ontolog Integration

Constraints map to λ-calculus primitives:

ontolog	Constraint Role	Mapping
ο (base)	Constraint domain	Agents, states subject to C
λ (operation)	Constrained action	Actions C evaluates
τ (terminal)	Satisfied state	C(σ,a) = P ∨ O fulfilled
Σ (complex)	Context	Structural scope of C
H (holon)	Authority scope	Nested jurisdiction

Constraint as λ-abstraction:

C = λσ.λa.δ    — Constraint abstracts context and action to modality

Reduction:

(C σ₀) a₀ →β δ₀    — Applying C to specific context and action yields modality

Constraint Composition

Logical Operations

C₁ ∧ C₂ : Both constraints must permit
C₁ ∨ C₂ : Either constraint permits
C₁ → C₂ : If C₁ permits, C₂ must permit
¬C      : Opposite constraint (correlative)

Category-Theoretic Operations

C₁ ∘ C₂ : Sequential (C₂ then C₁)
C₁ ⊗ C₂ : Parallel independent
C₁ + C₂ : Coproduct (choice)

Functor Laws

C(id) = id                    — Identity preservation
C(f ∘ g) = C(f) ∘ C(g)       — Composition preservation

Temporal Extension

Constraints over time using temporal operators:

□C     : C holds always (invariant)
◇C     : C holds eventually
C U C' : C until C' (deadline)
○C     : C holds next (sequence)

Common patterns:

O(a) U t        — Obligated to do a before time t
□P(a)           — Always permitted to do a
F(a) U C(b)     — Forbidden until condition b claimed

Rigidity Classification

Rigidity	Can Change?	Ontological Type	Example
Constitutional	Never	Fundamental identity	"Persons have rights"
Static	By governance only	Kind, Category	Type definitions
Dynamic	By power holders	Role, Phase	Employment status
Contextual	By situation	Circumstantial	Weather-dependent rules

Validation Protocol

def validate_constraint(C, σ, a):
    """Validate constraint application."""
    
    # 1. Scope check
    if not in_scope(a.agent, C.holon):
        return Invalid("Agent outside constraint scope")
    
    # 2. Context check
    if not C.domain.contains(σ):
        return Invalid("Context outside constraint domain")
    
    # 3. Modality check
    δ = C(σ, a)
    
    # 4. Correlative consistency
    for correlative in C.correlatives:
        if not consistent(δ, correlative(σ, a)):
            return Invalid("Correlative inconsistency")
    
    # 5. KROG theorem
    if not (K(a) and R(a) and O(a) and in_governance(a)):
        return Invalid("KROG violation")
    
    return Valid(δ)

Inference Rules

Correlative Inference

Claim(X, Y, φ)  ⊢  Duty(Y, X, φ)
Power(X, Y, φ)  ⊢  Liability(Y, X, φ)

Transitivity (where applicable)

C(a, b) ∧ C(b, c) ∧ C.transitive  ⊢  C(a, c)

Governing Propagation

Governing(C) ∧ C(parent, child) ∧ scope(H)  ⊢  C(parent, descendants(H))

Constitutional Immutability

Constitutive(C)  ⊢  ¬◇modify(C)   — Constitutive constraints cannot be modified

Quick Reference

Constraint Declaration

constraint:
  id: "hiring_authority"
  type: enabling
  modality: P
  domain: [Manager]
  action: hire
  scope: department_holon
  correlative: liability_to_be_hired
  rigidity: dynamic

Hohfeldian Position Check

has_position(agent, position, target, action) → Boolean
correlative_of(position) → Position
opposite_of(position) → Position

KROG Compliance

krog_valid(action) := K(action) ∧ R(action) ∧ O(action) ∧ G(action)

File Structure

constraints/
├── SKILL.md              # This file
├── references/
│   ├── deontic.md        # Full deontic logic reference
│   ├── hohfeld.md        # Complete Hohfeldian analysis
│   └── composition.md    # Category-theoretic details
└── scripts/
    └── validate.py       # Constraint validation

Integration Points

Skill	Integration
ontolog	ο/λ/τ mapping, Σ context, H scope
graph	Constraint edges, validation rules
reason	Constraint-guided branching bounds
agency	Agent authority via Hohfeldian positions

Core insight: Constraints are not limitations imposed from outside but structure-preserving functors that shape possibility spaces. The trichotomy (enabling/governing/constitutive) combined with deontic modality (P/O/F/I) and Hohfeldian relational structure provides complete formal vocabulary for any rule-based system.