Equivalence of the Frame and Halting Problems
Abstract
:1. Introduction
2. Results
2.1. Notation and Operational Set-Up
2.2. The FP as an Operational Problem
2.3. Equivalence of the FP and HP
- FP →
- HP: Let a = the algorithm that generates the next state of and let i be some instantaneous state of . Let A’s state be , and the states of and be and respectively. Consistent with the above, we assume A knows but not , and knows one component of a, namely the algorithm implemented by A. Now assume the FP is undecidable: A cannot deduce the state of at from knowledge of and at t. In this case, A cannot deduce either i or a. In this case A cannot build an oracle 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 exists. In this case, A cannot deduce, even if given all of i at the current step t, that A’s next state (at ) is not a halting state. Hence A cannot deduce even its own next state at , let alone the full state . 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 of A on E at t affect the entanglement entropy of E at ?
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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Dietrich, E.; Fields, C. Equivalence of the Frame and Halting Problems. Algorithms 2020, 13, 175. https://doi.org/10.3390/a13070175
Dietrich E, Fields C. Equivalence of the Frame and Halting Problems. Algorithms. 2020; 13(7):175. https://doi.org/10.3390/a13070175
Chicago/Turabian StyleDietrich, Eric, and Chris Fields. 2020. "Equivalence of the Frame and Halting Problems" Algorithms 13, no. 7: 175. https://doi.org/10.3390/a13070175
APA StyleDietrich, E., & Fields, C. (2020). Equivalence of the Frame and Halting Problems. Algorithms, 13(7), 175. https://doi.org/10.3390/a13070175