## Honest Universality (2012)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Dershowitz12honestuniversality,

author = {Nachum Dershowitz and Evgenia Falkovich},

title = {Honest Universality},

year = {2012}

}

### OpenURL

### Abstract

We extend the notion of universality of a function, due to Turing, to arbitrary (countable) effective domains, taking care to disallow any cheating on the part of the representations used. universal function, representation, encoding, effectiveness, com-Keywords: putability 1

### Citations

1175 |
On computable numbers, with an application to the Entscheidungs problem
- Turing
- 1936
(Show Context)
Citation Context ...representation, encoding, effectiveness, comKeywords: putability 1 Introduction Alan Turing, in his groundbreaking 1936 paper on the undecidability of the Halting Problem and the Entscheidungsproblem =-=[1]-=-, also invented the notion of a universal machine. He explained the idea as follows: The universal computing machine. It is possible to invent a single machine which can be used to compute any computa... |

838 | Theory of Recursive Functions and Effective Computability - ROGERS - 1967 |

692 |
Introduction to Metamathematics. D
- Kleene
- 1952
(Show Context)
Citation Context ...ctively within the model of computation itself. Some directions for further thought are the following: 81. Extend the notion of universal evaluation to partial evaluations as in Kleene’s smn Theorem =-=[14]-=- and the Futamura projections [15]. 2. Consider steppers and interpreters, which take programs and simulate their step-by-step behavior, in a controlled fashion. Abstract state machines provide a mean... |

174 | Partial evaluation of computation process – an approach to a compiler-compiler
- Futamura
- 1971
(Show Context)
Citation Context ...er effective domain. Some directions for further thought are the following: 1. Extend the notion of universal evaluation to partial evaluations as in Kleene’s smn Theorem and the Futamura projections =-=[12]-=-. 2. Consider steppers and interpreters, which take programs and simulate their step-by-step behavior, in a controlled fashion. Abstract state machines provide a means for capturing the step-by-step b... |

114 | Sequential abstract state machines capture sequential algorithms
- Gurevich
(Show Context)
Citation Context ... we need not dwell on programming details. The precise definition of effectiveness over arbitrary (countable) domains given in [4] is based on the generic abstract-state machine formalism of Gurevich =-=[5]-=-. States can be any (isomorphism-closed) class of logical structures over some fixed finite vocabulary. Furthermore, transitions must be effectively describable. This is formalized by saying that ther... |

65 |
Computable algebras, general theory and theory of computable fields
- Rabin
- 1960
(Show Context)
Citation Context ...ovably from the formalization of the earlier-mentioned characterization. Two other natural characterizations of effectiveness, based on Turing machines [8] and recursive functions (along the lines of =-=[9]-=- and others), are shown to be equivalent to this one in [10]. A successor function s : C → C is a unary function that enumerates its domain: C = {sn(e): n ∈ N} for some e ∈ C. For any finitely-generat... |

23 | A natural axiomatization of computability and proof of Church’s Thesis
- Dershowitz, Gurevich
(Show Context)
Citation Context ...re is a fixed finite set of terms that “determines behavior” of the algorithm in a sense made precise in [5, 6]. (Transitions must also preserve the domain of states and commute with isomorphism.) In =-=[7, 4]-=-, it has been shown how the Church-Turing thesis, namely, that no effective model can do more than a Turing machine can, follows provably from the formalization of the above characterization. Two othe... |

17 |
Markov functions
- ROGERS, W
- 1981
(Show Context)
Citation Context ...hich the Turing-computable partial functions are the effective ones, and (2) the natural numbers—for which the effective functions are identified with the partial recursive ones. Davis [2] and Rogers =-=[3]-=- have proposed general definitions of universality for Turing machines and for partial recursive functions, respectively. For other (countable) domains, we will adopt the analogous notion of effective... |

17 |
The definition of universal Turing machine
- Davis
- 1957
(Show Context)
Citation Context ...alphabets—for which the Turing-computable partial functions are the effective ones, and (2) the natural numbers—for which the effective functions are identified with the partial recursive ones. Davis =-=[2]-=- and Rogers [3] have proposed general definitions of universality for Turing machines and for partial recursive functions, respectively. For other (countable) domains, we will adopt the analogous noti... |

12 | The Church-Turing thesis over arbitrary domains
- Boker, Dershowitz
(Show Context)
Citation Context ...r Turing machines and for partial recursive functions, respectively. For other (countable) domains, we will adopt the analogous notion of effectiveness of a model of computation that was developed in =-=[4]-=-. Armed with the appropriate concept of generic effectiveness (Section 3), we extend the notion of universality to arbitrary sets of functions over arbitrary domains (Section 5), 1while addressing of... |

7 | Exact exploration and hanging algorithms
- Blass, Dershowitz, et al.
(Show Context)
Citation Context ...Furthermore, transitions must be effectively describable. This is formalized by saying that there is a fixed finite set of terms that “determines behavior” of the algorithm in a sense made precise in =-=[5, 6]-=-. (Transitions must also preserve the domain of states and commute with isomorphism.) In [7, 4], it has been shown how the Church-Turing thesis, namely, that no effective model can do more than a Turi... |

7 |
Acceptable notation. Notre Dame
- Shapiro
- 1982
(Show Context)
Citation Context ... of Thm. 4.7] that ρ is necessarily effectively computable (over C ∪ D) if there is an effective function s ∈ H that simulates a successor function s ∈ G. (The significance of successor was noted in =-=[12]-=-.) Even when σ = ρ, it turns out that both must be effectively computable, provided that—in addition—the identity function is simulated effectively. Proposition 3. If there are effective simulations ... |

6 |
The computable kernel of Abstract State Machines
- Reisig
- 2008
(Show Context)
Citation Context ...an do more than a Turing machine can, follows provably from the formalization of the earlier-mentioned characterization. Two other natural characterizations of effectiveness, based on Turing machines =-=[8]-=- and recursive functions (along the lines of [9] and others), are shown to be equivalent to this one in [10]. A successor function s : C → C is a unary function that enumerates its domain: C = {sn(e):... |

5 | Nachum Dershowitz. Comparing computational power - Boker - 2006 |

4 | Three paths to effectiveness
- Boker, Dershowitz
- 2010
(Show Context)
Citation Context ...from the formalization of the above characterization. Two other natural characterizations of effectiveness, based on Turing machines and recursive functions, are shown to be equivalent to this one in =-=[8]-=-. 3Since the whole domain is reachable, one can effectively enumerate all terms (or just all constructor terms) and thereby enumerate all domain values. Therefore, by standard search methods, any eff... |

4 | The linear time hierarchy theorems for RAMs and abstract state machines - Blass, Gurevich - 1997 |

2 |
Acceptable notation. Notre Dame Journal of Formal Logic
- Shapiro
- 1982
(Show Context)
Citation Context ...n effective function ̂s ∈ H that simulates a successor function s ∈ G, that is, a function that enumerates its domain: C = {s i (e): i ∈ N} for some e ∈ C. (The significance of successor was noted in =-=[10]-=-.) The analogous requirement for non-numerical domains is that its constructor functions be simulated. Even when σ ̸= ρ, it turns out that both must be effectively computable, provided that—in additio... |

2 | Universality of Wolfram’s 2,3 Turing Machine - Smith - 2007 |