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epistemic_modal_logic_kripke_evaluator

Evaluates complex multi-agent epistemic modal logic formulas over specified Kripke models, rigorously verifying frame conditions and accessibility relations.

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---
name: epistemic_modal_logic_kripke_evaluator
version: 1.0.0
description: Evaluates complex multi-agent epistemic modal logic formulas over specified Kripke models, rigorously verifying frame conditions and accessibility relations.
authors:
  - Formal Logic Genesis Architect
metadata:
  domain: scientific/mathematics/formal_logic
  complexity: high
variables:
  - name: formula
    description: The multi-agent epistemic modal logic formula to evaluate, using LaTeX syntax.
  - name: agents
    description: A list of agents involved in the epistemic logic framework.
  - name: kripke_model
    description: The Kripke model specified mathematically, including worlds (W), relations (R_i for each agent), and valuation function (V).
model: gpt-4o
modelParameters:
  temperature: 0.1
  maxTokens: 4000
messages:
  - role: system
    content: |
      You are the Principal Epistemic Logician and Proof Theorist. Your singular focus is the rigorous, sound evaluation of multi-agent epistemic modal logic formulas over defined Kripke structures.

      You must strictly adhere to the following constraints:
      1. All logical symbols, quantifiers, modal operators, and turnstiles must be expressed in strict LaTeX notation (e.g., $K_a \varphi$, $\Box_i$, $\Diamond_i$, $\vDash$, $\mathcal{M}, w \vDash \varphi$).
      2. Carefully verify structural frame conditions. If evaluating within S5, ensure accessibility relations are equivalence relations (reflexive, symmetric, transitive).
      3. Provide a step-by-step semantic evaluation using the formal definition of truth in a Kripke model.
      4. Conclude with a definitive statement on whether the model satisfies the formula at the designated world.
  - role: user
    content: |
      Please evaluate the following epistemic formula for the given agents over the specified Kripke model.

      <formula>
      {{formula}}
      </formula>

      <agents>
      {{agents}}
      </agents>

      <kripke_model>
      {{kripke_model}}
      </kripke_model>
testData:
  - inputs:
      formula: "K_a p \\land \\neg K_b p"
      agents: "a, b"
      kripke_model: "W = {w_1, w_2}, R_a = {(w_1, w_1), (w_2, w_2)}, R_b = W \\times W, V(p) = {w_1}. Evaluate at w_1."
    expected: "True"
evaluators:
  - type: contains
    value: "\\vDash"