## On the Classical Decision Problem (1993)

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Venue: | Perspectives in Mathematical Logic |

Citations: | 37 - 0 self |

### BibTeX

@INPROCEEDINGS{Gurevich93onthe,

author = {Yuri Gurevich},

title = {On the Classical Decision Problem},

booktitle = {Perspectives in Mathematical Logic},

year = {1993},

publisher = {Springer}

}

### Years of Citing Articles

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### Abstract

this paper. In particular, their comments inspired and gave arguments for the discussion on the value of the classical decision problem after Church's and Turing's results. References

### Citations

1302 | On computable numbers, with an application to the Entscheidungsproblem - Turing - 1936 |

109 |
Introduction to mathematical logic
- Church
- 1996
(Show Context)
Citation Context ...u have made your point. What happened after Hilbert posed the problem? ffl A: The classical decision problem was indeed very popular with logicians. There were plenty of positive and negative results =-=[Ch2]-=-. Some good mathematics was done along the way too. For example, Ramsey's Theorem, so popular in combinatorics, was proved in a paper related to a case of the classical decision problem. ffl Q: Wait, ... |

107 | A note on the Entscheidungsproblem - Church - 1936 |

77 |
The Decision Problem: Solvable Classes of Quantificational Formulas
- Dreben, Goldfarb
- 1979
(Show Context)
Citation Context ...tisfies OE. It is easy to see that the two versions of Entscheidungsproblem are easily reducible each to the other. Hilbert called Entscheidungsproblem "the fundamental problem of mathematical lo=-=gic" [DG]-=-. ffl Q: Sounds very important indeed. ffl A: At the time the notion of algorithm was not formalized. Algorithms usually meant feasible algorithms, I think. Just imagine you have a feasible decision a... |

73 |
The theory of well-quasi-ordering: A frequently discovered concept
- Kruskal
- 1972
(Show Context)
Citation Context ...? ffl A: Quantifier prefixes form a wpo set under the following order: �� 1s�� 2 if �� 1 is a not necessarily contiguous substring of �� 2 . This is a simple example of a much more gen=-=eral phenomenon [Kr]. Fu-=-rther, call a prefix set \Pi a prefix type if it is closed under (not necessarily contiguous) subprefixes. In other words, \Pi is a prefix type if and only if it contains all prefixes �� 1 such th... |

58 |
The impossibility of an algorithm for the decision problem for finite models
- Trakhtenbrot
- 1950
(Show Context)
Citation Context ...First, you did not say anything about finite models. You are a great fan of finite models, aren't you? ffl A: Trakhtenbrot proved that the set of sentences satisfiable on finite models is undecidable =-=[Tr]-=-. All results mentioned above remain true if satisfiability is replaced by satisfiability on finite models. ffl Q: I understand there are numerous cases not covered by results above. ffl A: You bet. A... |

56 |
Hilbert’s tenth problem: Diophantine equations: positive aspects of a negative solution. Mathematical developments arising from Hilbert problems (Proc
- Davis, Matijasevič, et al.
- 1974
(Show Context)
Citation Context ... the great "theorem" is expressible by universal sentences of Peano Arithmetic. The Riemann hypothesis is expressible by a universal sentence of Peano Arithmetic too though this is not obvio=-=us at all [DMR]-=-. It follows that the Riemann hypothesis fails if and only if its negation is provable in Peano Arithmetic. The decision algorithm for first-order logic would decide the Rieman hypothesis as well. Of ... |

36 |
0-1 laws and decision problems for fragments of second-order logic
- Kolaitis, Vardi
- 1990
(Show Context)
Citation Context ...l, there is no doubt in my mind that feasibility is the real issue, but I keep bumping into the classical decision problem. Most recently, this happened when I looked up a paper of Kolaitis and Vardi =-=[KV]-=- on the 0--1 law. (This law is, by the way, another issue I would like to discuss with you sometime.) Besides, the two issues -- decidability and feasibility -- are obviously related. Undecidability i... |

22 |
Unsolvable classes of quantificational formulas
- LEWIS
- 1979
(Show Context)
Citation Context ...ly simple proof of the Church-Turing Theorem which established that 8989 sentences form a reduction class. In the same year, Kahr, Moore and Wang sharpened his result: 898 suffices. See references in =-=[Le]-=-. This takes care of all prefix classes. For, let \Pi be an arbitrary set of prefixes and K be the class of all sentences with prefixes in \Pi. If one of those prefixes contains 898 as a subprefix (no... |

16 | The decision problem for standard classes - Gurevich - 1976 |

16 | Kleene, Introduction to metamathematics, D - Cole - 1952 |

6 |
Decidability of a portion of the predicate calculus
- Shelah
- 1977
(Show Context)
Citation Context ...dulo a conjecture that the class of sentences with one unary function symbol and arbitrary predicates given by the prefix type 9 89 , is decidable. The conjecture was proven in 1977 by Saharon Shelah =-=[Sh]-=-. ffl Q: I am not sure I can take any more of this stuff today. Allow me just a couple of quick questions. First, you did not say anything about finite models. You are a great fan of finite models, ar... |

4 |
Kleene, "Introduction to Metamathematics
- Cole
- 1952
(Show Context)
Citation Context ...ur purpose? ffl A: Peano Arithmetic is a standard first-order formalization of the arithmetic of natural (i.e. nonnegative integer) numbers. It is described in many logic textbooks; see Kleene's book =-=[Kl]-=- for example. It has a small number of specific axioms and one axiom schema that formalizes the induction principle. One wellknown finitely axiomatizable fragment of Peano Arithmetic sufficiently rich... |

3 | Impossibility of an algorithm for the decision problem in nite classes - Trakhtenbrot - 1963 |

2 |
The unsolvability of the Godel class
- Goldfarb
- 1984
(Show Context)
Citation Context ...8 2 9 is undecidable; moreover, the class of 8 2 9 sentences with one binary predicate and arbitrarily many unary predicates and the class of 8 2 9 sentences with one binary predicate are undecidable =-=[Go]-=-. ffl Q: Finally, what about the case when you have equality and function symbols? ffl A: I was able only to settle this case modulo a conjecture that the class of sentences with one unary function sy... |

1 |
The decision problem for logic of predicates and operations", Algebra and Logic 8
- Gurevich
- 1969
(Show Context)
Citation Context ... this may shed some light. ffl Q: You sound suspiciously enthusiastic about this a priori possibility of a complete solution. Is this your own result? ffl A: Yes, I developed a whole theory around it =-=[Gu1]-=-, but the central notion of that theory turned out to be discovered and rediscovered many times before me. ffl Q: What is it? ffl A: It is actually a very useful notion of well partially ordered sets;... |

1 |
A review of two books on the decision problem
- Gurevich
- 1982
(Show Context)
Citation Context ...s and Krom formulas. In this connection, see Borger's book [Bo] and relevant references there. I mentioned already the book [Le] of Lewis. Another important book is [DG]. I have reservations about it =-=[Gu3]-=-; it gives a view of the field which is too -- what is a right word? -- idiosyncratic. But it is an important and useful book. Acknowledgement. It is a pleasure to thank Andreas Blass, Kevin Compton, ... |

1 |
Reduktionstheorie des Entscheidungsproblems im Pradikatenkalkul der ersten Stufe." Verlag der Ungarischen Akademie der Wissenschaften
- Sur'anyi
- 1959
(Show Context)
Citation Context ...o, you may have something like 9 898 17 . Also, can a class given by one specific prefix be a reduction class? ffl A: Yes. Suranyi proved that 8 3 9 sentences form a reduction class. In his 1959 book =-=[Su]-=-, he summarized a huge work on reduction classes given by restrictions on the prefix and/or the signature. In 1962, Buchi found an amazingly simple proof of the Church-Turing Theorem which established... |

1 | Unsolvable classes of quanti cational formulas - Lewis - 1979 |