## Computational Processes and Incompleteness (906)

### BibTeX

@MISC{Sutner906computationalprocesses,

author = {Klaus Sutner},

title = {Computational Processes and Incompleteness},

year = {906}

}

### OpenURL

### Abstract

We introduce a formal definition of Wolfram’s notion of computational process based on cellular automata, a physics-like model of computation. There is a natural classification of these processes into decidable, intermediate and complete. It is shown that in the context of standard finite injury priority arguments one cannot establish the existence of an intermediate computational process. 1 Computational Processes Degrees of unsolvability were introduced in two important papers by Post [21] and Kleene and Post [12]. The object of these papers was the study of the complexity of decision problems and in particular their relative complexity: how does a solution to one problem contribute to the solution of another, a notion that can be formalized in terms of Turing reducibility and Turing degrees. Post was particularly interested in the degrees of recursively enumerable (r.e.) degrees. The Turing degrees of r.e. sets together with Turing reducibility form a partial order and in fact an upper semi-lattice R. It is easy to see that R has least element /0, the degree of decidable sets, and a largest element /0 ′ , the degree of the halting set. Post asked whether there are any other r.e. degrees and embarked on a program to establish the existence of such an intermediate degree by constructing a suitable r.e. set. Post’s efforts produced a number of interesting ideas such as simple, hypersimple and hyperhypersimple sets but failed to produce

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Citation Context ...er directions and open problems in the last section. To keep this paper reasonably short we will refrain from introducing standard concepts from recursion theory and refer the reader to texts such as =-=[22, 26, 4]-=-. 2 Computational Processes and Cellular Automata It stands to reason that any definition of a computational process in Wolfram’s sense should capture some aspect of a physical computation rather than... |

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Citation Context ...ly, the technique was invented independently and nearly simultaneously by Friedberg and Muchnik, see [7, 19]. Our structural understanding of R has grown significantly over the last half century, see =-=[26]-=- for a slightly dated but excellent overview or [1] for a more recent account. For example, it is known that every countable partial order can be embedded into R which fact leads to the decidability o... |

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Citation Context ...uitable to model such devices is sufficiently simple. Third, Kucera gave a priority-free solution to Post’s Problem, see [13]. Their argument relies on the low basis theorem due to Jockusch and Soare =-=[10]-=- whose proof appears to require universality. Moreover, it is not entirely clear how to rephrase the entirety of Kucera’s argument as a computational process. Lastly, one may question whether our defi... |

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(Show Context)
Citation Context ...jury priority arguments one cannot establish the existence of an intermediate computational process. 1 Computational Processes Degrees of unsolvability were introduced in two important papers by Post =-=[21]-=- and Kleene and Post [12]. The object of these papers was the study of the complexity of decision problems and in particular their relative complexity: how does a solution to one problem contribute to... |

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Citation Context ...s is dense: whenever A <T B for two r.e. sets A and B there is a third r.e. set such that A <T C <T B, [24]. Overall, the first order theory of R is highly undecidable [8]. More recently, in his book =-=[35]-=-, Wolfram suggests that “. . . all processes, whether they are produced by human effort or occur spontaneously in nature, can be viewed as computations.” This assertion is not particularly controversi... |

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Citation Context ...ations of cellular automata are more or less based on attempts to transport concepts from classical dynamics into the realm of cellular automata, perhaps augmented by ideas from recursion theory, see =-=[33, 35, 16, 17, 14, 15]-=-. As one might expect, from a computational perspective these are all fraught with undecidability. For example, it is Π0 2-complete to determine whether all orbits end in a fixed point, it is Σ0 3-com... |

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Citation Context ...ized structures. It is not inconceivable that, given proper initial conditions, this type of behavior might be exploited to perform computations. This turns out to be indeed the case as shown by Cook =-=[3]-=-; the proof is quite difficult, however. In fact, it would be rather challenging to present this argument in a purely non-geometric way suitable by verification through a proof-checker. In order to mo... |

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Citation Context ...ry in the abstract theory of computation. When it comes to selecting a physics-like model of computation the examples in [35], as well as previous work by Wolfram, suggest cellular automata, see also =-=[9]-=-. Technically these are continuous shift-invariant maps on Cantor spaces of the form ΣZd where Σ is some finite alphabet. For simplicity we will only consider the one-dimensional case d = 1 here. Thus... |

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Citation Context ...a priority argument that has since become one of the hallmarks of computability theory. Interestingly, the technique was invented independently and nearly simultaneously by Friedberg and Muchnik, see =-=[7, 19]-=-. Our structural understanding of R has grown significantly over the last half century, see [26] for a slightly dated but excellent overview or [1] for a more recent account. For example, it is known ... |

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Citation Context ...ations of cellular automata are more or less based on attempts to transport concepts from classical dynamics into the realm of cellular automata, perhaps augmented by ideas from recursion theory, see =-=[33, 35, 16, 17, 14, 15]-=-. As one might expect, from a computational perspective these are all fraught with undecidability. For example, it is Π0 2-complete to determine whether all orbits end in a fixed point, it is Σ0 3-com... |

46 |
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Citation Context ... 3.1 The simplified Friedberg-Muchnik priority argument does not yield an intermediate computational process. There are variants of this construction that produce stronger results. For example, Sacks =-=[23]-=- has shown how to modify the construction so that A can be any given undecidable r.e. set and we obtain S such that A ̸≤T S. Thus S avoids a whole upper cone rather than just complete sets. Alas, as a... |

39 |
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Citation Context ...er directions and open problems in the last section. To keep this paper reasonably short we will refrain from introducing standard concepts from recursion theory and refer the reader to texts such as =-=[22, 26, 4]-=-. 2 Computational Processes and Cellular Automata It stands to reason that any definition of a computational process in Wolfram’s sense should capture some aspect of a physical computation rather than... |

37 |
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(Show Context)
Citation Context ...ne cannot establish the existence of an intermediate computational process. 1 Computational Processes Degrees of unsolvability were introduced in two important papers by Post [21] and Kleene and Post =-=[12]-=-. The object of these papers was the study of the complexity of decision problems and in particular their relative complexity: how does a solution to one problem contribute to the solution of another,... |

36 |
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Citation Context ... the Σ1 theory of R. According to a theorem by Sacks, the partial order of the r.e. Turing degrees is dense: whenever A <T B for two r.e. sets A and B there is a third r.e. set such that A <T C <T B, =-=[24]-=-. Overall, the first order theory of R is highly undecidable [8]. More recently, in his book [35], Wolfram suggests that “. . . all processes, whether they are produced by human effort or occur sponta... |

32 |
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Citation Context ...a priority argument that has since become one of the hallmarks of computability theory. Interestingly, the technique was invented independently and nearly simultaneously by Friedberg and Muchnik, see =-=[7, 19]-=-. Our structural understanding of R has grown significantly over the last half century, see [26] for a slightly dated but excellent overview or [1] for a more recent account. For example, it is known ... |

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(Show Context)
Citation Context ...ations of cellular automata are more or less based on attempts to transport concepts from classical dynamics into the realm of cellular automata, perhaps augmented by ideas from recursion theory, see =-=[33, 35, 16, 17, 14, 15]-=-. As one might expect, from a computational perspective these are all fraught with undecidability. For example, it is Π0 2-complete to determine whether all orbits end in a fixed point, it is Σ0 3-com... |

25 |
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(Show Context)
Citation Context ...bits end in a fixed point, it is Σ0 3-complete to determine whether all orbits are decidable, and it is Σ0 4-complete to determine whether a given cellular automaton is computationally universal, see =-=[29, 30]-=-. It was suggested in [32] to turn this difficulty into a tool: one can use the complexity of Reachability to classify cellular automaton in a much more fine-grained manner than usual.230 Computation... |

21 |
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Citation Context ... for all z But then e ∈ S ′ ⇐⇒ { f(e)} S (w(e)) ≃ 0 ⇐⇒ w(e) ∈ A so that S ′ ≤T S ⊕ A. Similar arguments seem to apply to all priority constructions. In fact, it was suggested by Jockusch and Soare in =-=[11]-=- that priority constructions obey a kind of “maximum degree principle” in the sense that the construction of an r.e. set S with weak negative requirements automatically produces a complete set. If the... |

20 |
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Citation Context ...bits end in a fixed point, it is Σ0 3-complete to determine whether all orbits are decidable, and it is Σ0 4-complete to determine whether a given cellular automaton is computationally universal, see =-=[29, 30]-=-. It was suggested in [32] to turn this difficulty into a tool: one can use the complexity of Reachability to classify cellular automaton in a much more fine-grained manner than usual.230 Computation... |

18 |
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Citation Context ...be taken to make sure that the one-step relation in a computational processes suitable to model such devices is sufficiently simple. Third, Kucera gave a priority-free solution to Post’s Problem, see =-=[13]-=-. Their argument relies on the low basis theorem due to Jockusch and Soare [10] whose proof appears to require universality. Moreover, it is not entirely clear how to rephrase the entirety of Kucera’s... |

17 | Cellular automata and intermediate degrees
- Sutner
(Show Context)
Citation Context ...is Σ0 3-complete to determine whether all orbits are decidable, and it is Σ0 4-complete to determine whether a given cellular automaton is computationally universal, see [29, 30]. It was suggested in =-=[32]-=- to turn this difficulty into a tool: one can use the complexity of Reachability to classify cellular automaton in a much more fine-grained manner than usual.230 Computational Processes and Incomplet... |

16 |
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(Show Context)
Citation Context ... order of the r.e. Turing degrees is dense: whenever A <T B for two r.e. sets A and B there is a third r.e. set such that A <T C <T B, [24]. Overall, the first order theory of R is highly undecidable =-=[8]-=-. More recently, in his book [35], Wolfram suggests that “. . . all processes, whether they are produced by human effort or occur spontaneously in nature, can be viewed as computations.” This assertio... |

11 |
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Citation Context ...licated construction of an intermediate degree known. In this section we will discuss what appears to be the most basic construction. Our description here will be rather terse; we refer the reader to =-=[26, 20]-=- for background. As already mentioned, the classical Friedberg-Muchnik construction establishes the existence of two Turing incomparable r.e. sets. Here we use a method that is slightly less ambitious... |

11 |
The Friedberg-Muchnik theorem re-examined
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(Show Context)
Citation Context ... incomparable with respect to Turing reductions. Hence, each individual set so constructed has intermediate degree. However, it was shown by Soare that the disjoint sum A ⊕ B is in fact complete, see =-=[25]-=-. It is thus hard to see how this priority argument could be construed as not being a complete process. The difficulty in identifying intermediate processes is closely related to another often observe... |

10 | Computability theory and differential geometry
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(Show Context)
Citation Context ...tion of theorems has degree exactly d, see [6]. For a more recent and more complicated application of degree theory to differential geometry see Soare’s contribution to the proof of Gromov’s theorem, =-=[27]-=-. There appears to be a fairly strong connection between the lack of natural intermediate degrees and PCE. Indeed, any of Davis’ previously studied and named decision problems could presumably be tran... |

10 | Cellular automata and intermediate reachability problems
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(Show Context)
Citation Context ... other computational model the automaton may emulate). It requires a bit of care to make sure that the other orbits do not violate the degree condition. At any rate, we have the following result, see =-=[31, 32]-=-. Theorem 2.2 Degree Theorem For every r.e. degree d there is a one-dimensional cellular automaton whose Reachability Problem has degree precisely d. In fact, the cellular automaton can be chosen to b... |

8 |
private communication
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Citation Context ...e assertion. However, most recursion theorists would agree that PCE coexists uneasily with the well-established theory of the r.e. degrees. Wolfram’s response to such criticism can be summarized thus =-=[34]-=-: any of the standard T. Neary, D. Woods, A.K. Seda and N. Murphy (Eds.): The Complexity of Simple Programs 2008. EPTCS 1, 2009, pp. 226–234, doi:10.4204/EPTCS.1.22 c○ K. SutnerK. Sutner 227 construc... |

7 |
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Citation Context ... that derivability in a formal theory fully reflects the structure of the r.e. degrees: for every r.e. degree d there is an axiomatizable theory whose collection of theorems has degree exactly d, see =-=[6]-=-. For a more recent and more complicated application of degree theory to differential geometry see Soare’s contribution to the proof of Gromov’s theorem, [27]. There appears to be a fairly strong conn... |

7 |
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Citation Context |

6 | Model Checking One-Dimensional Cellular Automata
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(Show Context)
Citation Context ...idable. We note in passing that this result also holds over the full space Σ Z of infinite configurations, though the machinery required to establish decidability is quite a bit more complicated, see =-=[28]-=-. Dimensionality is crucial here, no similar result holds in dimensions two or higher. At any rate, we can now propose a formal definition of a computational process. For our purposes, a computational... |

4 |
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Citation Context ...rly simultaneously by Friedberg and Muchnik, see [7, 19]. Our structural understanding of R has grown significantly over the last half century, see [26] for a slightly dated but excellent overview or =-=[1]-=- for a more recent account. For example, it is known that every countable partial order can be embedded into R which fact leads to the decidability of the Σ1 theory of R. According to a theorem by Sac... |

4 |
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Citation Context ...ith intermediate degrees: all existence proofs of intermediate degrees are artificial in the sense that the constructed r.e. sets are entirely ad hoc. This was perhaps stated most clearly by M. Davis =-=[5]-=-: “But one can be quite precise in stating that no one has produced an intermediate r.e. degree about which it can be said that it is the degree of a decision problem that had been previously studied ... |

3 |
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Citation Context ... the great complexity in the structure of the c.e. degrees arises solely from studying unnatural problem.” Note that results from degree theory have been transfered to other areas. For example, Boone =-=[2]-=- has shown how to construct a finitely presented group whose word problem has a given, arbitrary r.e. degree. Of course, the translation itself is entirely unsuspicious, it is only the instantiation r... |

3 |
Topological and Symbolic Dynamics. Number 11
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(Show Context)
Citation Context |

3 |
Two-state, reversible, universal cellular automata in three dimensions
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(Show Context)
Citation Context ...mputation. For example, one could consider higher-dimensional cellular automata that are reversible and in addition obey certain conservation laws, much in the spirit of Fredkin’s recent work on SALT =-=[18]-=-. The motivation for Fredkin is the construction of physically feasible, three-dimensional systems that dissipate as little energy as possible. It is conceivable that a narrow class of such systems co... |