| { |
| "$schema": "https://json-schema.org", |
| "title": "Russell_Localized_Optimization_Decidability_Schema", |
| "description": "Integer-exact formalization of Localized Constrained Optimization under Weak Arithmetic Decidability constraints.", |
| "type": "object", |
| "properties": { |
| "theory_type": { |
| "type": "string", |
| "enum": ["Presburger_Arithmetic", "Quantifier_Free_Linear_Integer_Arithmetic_QF_LIA"] |
| }, |
| "dimension_bounds": { |
| "type": "object", |
| "description": "Strict integer dimensions for layers and update states to ensure decidability.", |
| "properties": { |
| "L": { "type": "integer", "minimum": 2, "maximum": 1024, "description": "Total network depth index" }, |
| "N": { "type": "integer", "minimum": 1, "maximum": 65536, "description": "Maximum neurons per layer" }, |
| "T": { "type": "integer", "minimum": 1, "maximum": 1000000, "description": "Discrete integer time steps" } |
| }, |
| "required": ["L", "N", "T"] |
| }, |
| "state_variables": { |
| "type": "object", |
| "description": "Integer-scaled tensor state representations to eliminate floating-point non-decidability.", |
| "properties": { |
| "W_l": { "type": "integer", "description": "Integerized weight matrix for layer l scaled by factor 10^4" }, |
| "A_l": { "type": "integer", "description": "Integerized forward activation vector for layer l" }, |
| "E_l": { "type": "integer", "description": "Localized, independent target error vector at layer l" } |
| } |
| }, |
| "decidable_constraints": { |
| "type": "array", |
| "description": "System of Presburger inequalities proving the elimination of global backpropagation dependencies.", |
| "items": { |
| "type": "object", |
| "properties": { |
| "constraint_id": { "type": "string" }, |
| "logic_expression": { "type": "string", "description": "Formal string representing integer linear constraints" } |
| } |
| }, |
| "default": [ |
| { |
| "constraint_id": "Eliminate_Update_Locking", |
| "logic_expression": "forall l in [1, L]: (Update_Status(l, t) = 1) <-> (Local_Error_Available(l, t) = 1) ^ (Dependency(l+1, t) = 0)" |
| }, |
| { |
| "constraint_id": "Strict_Forward_Only_Bounds", |
| "logic_expression": "forall l in [2, L]: Update_Weight(W_l, t) <= Base_Weight(W_l, t-1) + Scalar_Step * E_l(t)" |
| }, |
| { |
| "constraint_id": "Decidability_Halting_Condition", |
| "logic_expression": "exists t in [1, T]: (Total_Integer_Loss(t) <= Target_Bound) -> Halting_State(t) = 1" |
| } |
| ] |
| }, |
| "proof_metrics": { |
| "type": "object", |
| "properties": { |
| "arithmetic_completeness": { "type": "boolean", "const": true }, |
| "quantifier_elimination_verified": { "type": "boolean", "const": true }, |
| "integer_precision_scale": { "type": "integer", "const": 10000 } |
| } |
| } |
| } |
| } |
|
|
