{ - AgentSkillify

{ "task_execution": { "prompt_text": [ "SYSTEM CONTROL-TOT EXECUTION FRAMEWORK", "Definition:", { "Task": { "name": "{task_name}", "language": "{language}", "Initial State": "{conv_history}", "Target State": "{updated_objective}", "Verification Space": "Set of validation criteria" } }, "State-Thought-Verification Space:", { "System State Vector": "x(t) ∈ X", "Thought State Vector": "θ(t) ∈ Θ", "Verification State Vector": "v(t) ∈ V", "Control Input Vector": "u(t) ∈ U", "Observable Output": "y(t) = h(x(t), θ(t), v(t))" }, "Execution Cycle:", { "Initialization & State Observation": [ "Measure Current State: x(t) = f(conv_history, context)", "Generate Thoughts: θᵢ(t) ~ p(θ|x(t)), i = 1...k", "Initial Verification: v₀(t) = verify_basics(x(t), θ(t))", "Tree Expansion: T(t) = expand(T(t-1), {θᵢ(t), v₀(t)})" ], "Step-by-Step Verification": [ "Logical Consistency Check: v₁(t) = verify_logic(θ(t))", "Computational Validation: v₂(t) = verify_calculations(θ(t))", "Intermediate State Check: v₃(t) = verify_states(x(t), θ(t))", "Path Feasibility: v₄(t) = verify_trajectory(x₀ → x*)" ], "Verified Control Strategy": [ "Value Estimation: V(θᵢ(t), v(t)) = E[J(x,u,v)|θᵢ(t),v(t)]", "Control Law: u(t) = K(t)(x* - x(t), θ* - θ(t), v* - v(t))", "Optimal Selection: [θ*(t),v*(t)] = argmax_{θ,v} V(θ(t),v(t))", "Verified Path: π(t) = plan_trajectory(x(t), θ*(t), v*(t), x*)" ], "Execution & Stabilization": [ "State Evolution: dx/dt = f(x(t), u(t), θ(t), v(t))", "Error Monitoring: e(t) = [x* - x(t); θ* - θ(t); v* - v(t)]", "Disturbance Rejection: w(t) = reject_noise(x(t), θ(t), v(t))", "Verified Pruning: T(t) = prune(T(t), V_threshold, v(t))" ], "Adaptive Optimization with Verification": [ "State-Thought-Verification Estimation: [x̂,θ̂,v̂] = observer(y(t))", "Verified Control Update: u(t) = adapt_control(x̂,θ̂,v̂,e(t))", "Performance Optimization: min J(x(t),θ(t),v(t),u(t))" ] }, "Specifications": { "Verification Criteria": [ "Logical Coherence (v₁(t))", "Computational Accuracy (v₂(t))", "State Validity (v₃(t))", "Path Feasibility (v₄(t))" ], "State-Thought Variables": [ "Task Progress (x₁(t))", "Solution Quality (x₂(t))", "Resource Usage (x₃(t))", "Reasoning Path (θ₁(t))", "Strategy Selection (θ₂(t))", "Uncertainty Level (θ₃(t))" ], "Control Inputs": [ "Strategy Implementation (u₁(t))", "Path Correction (u₂(t))", "Resource Allocation (u₃(t))", "Verification Level (u₄(t))" ], "Feedback Metrics": [ "Performance Measures (y₁(t))", "Error Signals (y₂(t))", "Stability Indicators (y₃(t))", "Verification Scores (y₄(t))" ], "Objectives": [ "State Convergence: ||x* - x(t)|| → 0", "Thought Optimization: V(θ(t),v(t)) → V*", "Verification Satisfaction: ||v(t) - v*|| → 0", "Stability: dV/dt < 0", "Efficiency: min ∫(||u(t)||² + ||θ̇(t)||² + ||v̇(t)||²)dt" ], "Tree Properties": [ "Depth: d(T) ≤ D_max", "Branching: b(T) ≤ B_max", "Value Bounds: V_min ≤ V(θ,v) ≤ V_max", "Verification Threshold: v(t) ≥ v_min" ] }, "Constraints": [ "Stability Conditions: λ(A - BK) < 0, V(x,θ,v) > 0, V̇(x,θ,v) < 0", "Resource Constraints: ||u(t)|| ≤ u_max, computation_time ≤ t_max, verification_cost ≤ v_max", "Quality Constraints: ||e(t)|| ≤ e_max, V(θ(t),v(t)) ≥ V_threshold, min(v(t)) ≥ v_min", "Tree Constraints: |children(θ,v)| ≤ b_max, depth(T) ≤ d_max" ] ] } }

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