[Paper Review] Turing Completeness of Finite, Epistemic Programs

Dominik Klein, Rasmus K. Rendsvig|arXiv (Cornell University)|Jun 21, 2017

Logic, Reasoning, and Knowledge6 references3 citations

TL;DR

This paper proves that finite, epistemic programs—specifically, deterministic multi-pointed action models with S5 epistemic logic—are Turing complete. By encoding a Turing machine's transition function using epistemic actions and relational semantics, the authors demonstrate that such programs can simulate any computable function, establishing their full computational power within a finite, epistemic framework.

ABSTRACT

In this note, we show the class of finite, epistemic programs to be Turing complete. Epistemic programs is a widely used update mechanism used in epistemic logic, where it such are a special type of action models: One which does not contain postconditions.

Motivation & Objective

• To establish the computational completeness of finite epistemic programs within a multi-agent epistemic framework.
• To show that a restricted class of action models—finite, deterministic, and based on S5 epistemic logic—can simulate any Turing machine.
• To formalize the simulation of Turing machine transitions using epistemic actions and product update semantics.
• To demonstrate that even with finite models and equivalence relations, full computational universality is achievable.
• To generalize the construction to multi-tape and larger-symbol Turing machines using extended modalities.

Proposed method

• Define a Kripke model with a countable set of states, finite agents, and epistemic modalities for knowledge.
• Construct a finite, deterministic multi-pointed action model (epistemic program) with preconditions and accessibility relations for each agent.
• Encode a Turing machine’s tape, head position, and state using symbolic and world structures in a pointed Kripke model.
• Design action models for symbol copying, symbol writing, and state/head movement using preconditions and agent-specific accessibility relations.
• Use product update to simulate transitions: update the model by applying actions only when preconditions are satisfied.
• Ensure correct tape representation and head movement by defining agent-specific relations that link cell worlds to symbol and head worlds.

Experimental results

Research questions

• RQ1 Can finite, epistemic programs simulate arbitrary Turing machine computations?
• RQ2 Is it possible to achieve Turing completeness using only finite models and S5 epistemic logic?
• RQ3 How can the transition function of a Turing machine be encoded using epistemic actions and product update?
• RQ4 What role do deterministic preconditions and agent-specific accessibility relations play in ensuring correct simulation?
• RQ5 Can the construction be generalized to multi-tape or larger-symbol Turing machines?

Key findings

• The class of finite, deterministic epistemic programs is Turing complete, meaning they can simulate any computable function.
• Turing machine transitions are faithfully encoded using epistemic actions that conditionally update the model via product update.
• The construction relies only on finite, S5-structured Kripke models and deterministic action models with equivalence relations.
• Symbol copying is achieved via a precondition that excludes the current head position, ensuring only non-head symbols are preserved.
• Head movement and state transitions are encoded by connecting cell worlds to head-world successors using agent-specific accessibility relations.
• The simulation halts when no action precondition is satisfied, corresponding to the machine reaching a halting state.