# Equivalence of the Frame and Halting Problems

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Notation and Operational Set-Up

**Lemma**

**1**

**Proof.**

#### 2.2. The FP as an Operational Problem

#### 2.3. Equivalence of the FP and HP

**Theorem**

**1.**

**Proof:**

**FP**→**HP:**Let a = the algorithm that generates the next state of $A\times E$ and let i be some instantaneous state of $A\times E$. Let A’s state be ${i}_{A}$, and the states of ${E}_{A}$ and ${E}_{U}$ be ${i}_{EA}$ and ${i}_{EU}$ respectively. Consistent with the above, we assume A knows ${i}_{EA}$ but not ${i}_{EU}$, and knows one component of a, namely the algorithm ${a}_{A}$ implemented by A. Now assume the FP is undecidable: A cannot deduce the state of ${E}_{A}\bigsqcup {E}_{U}$ at $t+\Delta t$ from knowledge of ${a}_{A}$ and ${i}_{EA}$ at t. In this case, A cannot deduce either i or a. In this case A cannot build an oracle $f(a,i)$ that decides whether a halts on i and cannot recognize such an oracle if it exists a priori. Hence the HP is also undecidable by A. Since A is a generic finite agent, the HP is undecidable by any such agent.**HP**→**FP:**Assume that the HP is undecidable (as shown in Reference [20]) and hence that no oracle $f(a,i)$ exists. In this case, A cannot deduce, even if given all of i at the current step t, that A’s next state (at $t+\Delta t$) is not a halting state. Hence A cannot deduce even its own next state ${i}_{A}$ at $t+\Delta t$, let alone the full state $i(t+\Delta t)$. Hence the FP is undecidable. □

## 3. Discussion

#### 3.1. The System Identification Problem

- Given a system in the form of a black box (BB) allowing finite input-output interactions, deduce a complete specification of the system’s machine table (i.e., algorithm or internal dynamics).
- Given a complete specification of a machine table (i.e., algorithm or internal dynamics), recognize any BB having that description.

#### 3.2. The Symbol-Grounding Problem

#### 3.3. Undecidability of the QFP

QFP:Given a quantum agent A interacting with a quantum environment E, how does an action $a\left(t\right)$ of A on E at t affect the entanglement entropy of E at $t+\Delta t$?

**Theorem**

**2.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ACT-R | Adaptive Control of Thought-Rational |

AI | Artificial intelligence |

BB | Black Box |

CLARION | Connectionist Learning with Adaptive Rule Induction On-line |

EPR | Einstein-Podolsky-Rosen |

FAPP | For all practical purposes |

FP | Frame problem |

HP | Halting problem |

I/O | Input/Output |

LIDA | Learning Intelligent Distribution Agent |

MIP* | Multiprover Interactive Proof* |

QFP | Quantum Frame problem |

RE | Recursively enumerable |

VM | Virtual machine |

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**Figure 1.**General set-up for the Frame Problem (FP): a finite agent A interacts with an environment E. The agent has perception and action modules and a problem solver. The problem solver has access to two generative models, one of the environment and the other a “self” model of its own problem-solving capabilities. The environment is divided into a part ${E}_{A}$ to which A has observational access and a remainder ${E}_{U}$ to which A does not have observational access.

**Figure 2.**An interactive proof system with two otherwise noninteracting provers (A and B) that share an entangled state $\psi $ (red rectangle). The verifier interacts separately with the two provers. This is the set-up used in Reference [26] to prove Multiprover Interactive Proof* (MIP*) = RE, and hence that the Halting Problem (HP) can be solved by interactive proof with entanglement as a resource.

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Dietrich, E.; Fields, C. Equivalence of the Frame and Halting Problems. *Algorithms* **2020**, *13*, 175.
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Dietrich E, Fields C. Equivalence of the Frame and Halting Problems. *Algorithms*. 2020; 13(7):175.
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Dietrich, Eric, and Chris Fields. 2020. "Equivalence of the Frame and Halting Problems" *Algorithms* 13, no. 7: 175.
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