## Algorithms: A quest for absolute definitions (2003)

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Venue: | Bulletin of the European Association for Theoretical Computer Science |

Citations: | 20 - 9 self |

### BibTeX

@ARTICLE{Blass03algorithms:a,

author = {Andreas Blass and Yuri Gurevich},

title = {Algorithms: A quest for absolute definitions},

journal = {Bulletin of the European Association for Theoretical Computer Science},

year = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the Church-Turing thesis and contrast Church's and Turing's approaches, and we finish with some recent investigations.

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Citation Context ...putable [10]. A few months later, independently of Church, Alan Turing published a powerful speculative proof of a similar conjecture: every computable real number is computable by the Turing machine =-=[54]-=-. Kurt G"odel found Church's thesis "thoroughly unsatisfactory" but later was convinced by Turing's argument. Later yet he worried about a possible flaw in Turing's argument. In Section 2 we recount b... |

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Citation Context ... as much time as used by common algorithms, in this case quadratic time. Unexpectedly, Karatsuba reduced the power to log2(3) [31]." (Readers interested in fast integer multiplication are referred to =-=[38]-=-.) It is not clear to us how Kolmogorov thought of the tape graph. One hypothesis is that edges reflect physical closeness. This hypothesis collides with the fact that our physical space is finite-dim... |

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Citation Context ...me interesting references. ffl Robin Gandy attempted to derive Turing's thesis from a number of "principles for mechanisms" [17]. Wilfried Sieg continues this line of research [52]. ffl David Deutsch =-=[15]-=- designed a universal quantum computer that is supposed to be able to simulate the behavior of any finite physical system. Gandy's approach is criticized in [16, pages 280-281]. Deutsch's approach and... |

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Citation Context ...SM B. In particular, B simulates A step for step. In Section 5 we outline the approach of [27]. In 1995, the second author published the definition of parallel and distributed abstract state machines =-=[25]-=-. The Foundations of Software Engineering group at Microsoft Research developed an industrial strength specification language AsmL that allows one to write and execute parallel and distributed abstrac... |

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Citation Context ...mbers to natural numbers should be identified with that of a recursive function. Church had in mind total functions [10]. Later Kleene improved on Church's thesis by extending it to partial functions =-=[32]-=-. The fascinating history of the thesis is recounted in [14]; see also [51]. Originally Church hypothesized that every effectively calculable function from natural numbers to natural numbers is defina... |

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Citation Context ...lgorithms. These issues are very briefly discussed in Section 4. In 1991, the second author published the definition of sequential abstract state machines (ASMs, called evolving algebras at the time) =-=[23]-=-. In 2000, he published a definition of sequential algorithms derived from first principles [27]. In the same paper he proved that every sequential algorithm A is behaviorally equivalent to some seque... |

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Citation Context ...ven for a particular input, can grow during the computation. "In 1970 the present author introduced a new machine model (cf. [47]) now called storage modification machine (SMM)," writes Sch"onhage in =-=[48]-=-, "and posed the intuitive thesis that this model possesses extreme flexibility and should therefore serve as a basis for an adequate notion of time complexity." In article [48], Sch"onhage gave "a co... |

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Citation Context ...probably will not greatly influence programming language theory. Here are some interesting references. ffl Robin Gandy attempted to derive Turing's thesis from a number of "principles for mechanisms" =-=[17]-=-. Wilfried Sieg continues this line of research [52]. ffl David Deutsch [15] designed a universal quantum computer that is supposed to be able to simulate the behavior of any finite physical system. G... |

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Citation Context ...machines [2]. In 2001, the present authors published a definition of parallel algorithms derived from first principles as well as a proof that every parallel algorithm is equivalent to a parallel ASM =-=[7]-=-. Section 6 is a quick discussion of parallel algorithms. The problem of defining distributed algorithms from first principles is open. In Section 7 we discuss a few related issues. Finally let us not... |

55 |
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Citation Context ...eved that quantum computers can be polynomial time simulated by Turing machines. For the believers in quantum computers, it is more natural to speak about probabilistic Turing machines. We quote from =-=[4]-=-. Just as the theory of computability has its foundations in the ChurchTuring thesis, computational complexity theory rests upon a modern strengthening of this thesis, which asserts that any "reasonab... |

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Citation Context ...rov machines and pointer machines enabled better complexity measures. Abstract state machines enable precise executable specifications of software systems though this story is only starting to unfold =-=[1, 2, 9]-=-. 2 The Church-Turing thesis 2.1 Church + Turing The celebrated Church-Turing thesis [10, 54] captured the notion of computable function. Every computable function from natural numbers to natural numb... |

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Citation Context ... P. At the 1991 Annual Meeting of the Association of Symbolic Logic, Steve Cook argued in favor of that thesis, and the second author argued against it. Some of the arguments can be found in [11] and =-=[24]-=- respectively. A related but different version of the polynomial time Turing thesis is that a problem is in P if it can be solved in polynomial time at all, by any means. The presumed reason is that a... |

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32 |
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Citation Context ... it is in P. At the 1991 Annual Meeting of the Association of Symbolic Logic, Steve Cook argued in favor of that thesis, and the second author argued against it. Some of the arguments can be found in =-=[11]-=- and [24] respectively. A related but different version of the polynomial time Turing thesis is that a problem is in P if it can be solved in polynomial time at all, by any means. The presumed reason ... |

30 |
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Citation Context ...fied order within any one step of the algorithm is at most d. There are several computation models for wide-step algorithms in the literature. The two most known models are Boolean circuits and PRAMs =-=[35]-=-. (PRAM stands for "Parallel Random Access Machines".) These two models and some other models of wide-step algorithms that occurred to us or to the referees are shown to satisfy the wide-step postulat... |

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Citation Context ...n build a binary tree over x and then move fast about x. Leonid Levin used a universal Kolmogorov machine to construct his algorithm for NP problems that is optimal 10sup to a multiplicative constant =-=[41, 22]-=-. The up-to-a-multiplicative-constant form is not believed to be achievable for the multitape Turing machine model popular in theoretical computer science. Similarly, the class of functions computable... |

25 | What is an algorithm
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- 2001
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Citation Context ...According to Yiannis Moschovakis, an algorithm is a "recursor", a monotone operator over partial functions whose least fixed point includes (as one component) the function that the algorithm computes =-=[44]-=-. He proposes a particular language for defining recursors. A definition may use various givens: functions or recursors. Moschovakis gives few examples and they are all small ones. The approach does n... |

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17 |
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- 1999
(Show Context)
Citation Context ...uage theory. Here are some interesting references. ffl Robin Gandy attempted to derive Turing's thesis from a number of "principles for mechanisms" [17]. Wilfried Sieg continues this line of research =-=[52]-=-. ffl David Deutsch [15] designed a universal quantum computer that is supposed to be able to simulate the behavior of any finite physical system. Gandy's approach is criticized in [16, pages 280-281]... |

16 |
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Citation Context ...nism is available, classical ruler-andcompass constructions can be regarded as nondeterministic ASMs operating on a suitable structure of geometric objects. A critical examination of [27] is found in =-=[46]-=-. 6 Formalization of parallel algorithms Encouraged by the success in capturing the notion of sequential algorithms in [27], we "attacked" parallel algorithms in [7]. The attack succeeded. We gave an ... |

15 |
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Citation Context ... system. Gandy's approach is criticized in [16, pages 280-281]. Deutsch's approach and quantum computers in general are criticized in [43, Section 2]. ffl Charles H. Bennett and Rolf Landauer pose in =-=[3]-=- important problems related to the fundamental physical limits of computation. ffl Marian Boykan Pour-El and Ian Richards [45] investigate the extent to which computability is preserved by fundamental... |

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12 | Step by recursive step: Church’s analysis of effective calculability
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- 1997
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Citation Context ...tion. Church had in mind total functions [10]. Later Kleene improved on Church's thesis by extending it to partial functions [32]. The fascinating history of the thesis is recounted in [14]; see also =-=[51]-=-. Originally Church hypothesized that every effectively calculable function from natural numbers to natural numbers is definable in his lambda calculus. G"odel didn't buy that. In 1935, Church wrote t... |

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8 |
On the concept of algorithm”, Uspekhi Mat. Nauk 8:4
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(Show Context)
Citation Context ... toward understanding algorithms, but it did not solve the problem what an algorithm is. Further progress in foundations of algorithms was achieved by Kolmogorov and his student Uspensky in the 1950s =-=[39, 40]-=-. The Kolmogorov machine with its 2sreconfigurable "tape" has a certain advantage over the Turing machine. The notion of pointer machine was an improvement of the notion of Kolmogorov machine. These i... |

8 |
Richards. Computability in analysis and physics
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Citation Context ...cized in [43, Section 2]. ffl Charles H. Bennett and Rolf Landauer pose in [3] important problems related to the fundamental physical limits of computation. ffl Marian Boykan Pour-El and Ian Richards =-=[45]-=- investigate the extent to which computability is preserved by fundamental constructions of analysis, such as those used in classical and quantum theories of physics. 13s4.2 Polynomial time Turing's t... |

7 |
webpage, http://www.eecs.umich.edu/gasm/, maintained by James K
- Michigan
(Show Context)
Citation Context ...rov machines and pointer machines enabled better complexity measures. Abstract state machines enable precise executable specifications of software systems though this story is only starting to unfold =-=[1, 2, 9]-=-. 2 The Church-Turing thesis 2.1 Church + Turing The celebrated Church-Turing thesis [10, 54] captured the notion of computable function. Every computable function from natural numbers to natural numb... |

7 |
and Saharon Shelah. Nearly linear time
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- 1989
(Show Context)
Citation Context ...mains the same if Kolmogorov machines are replaced e.g. by various random access computers in the literature; it is not believed, however, that the usual multitape Turing machines have the same power =-=[29]-=-. Kolmogorov machines allow one to do reasonable computations in reasonable time. This may have provoked Kolmogorov to ask new questions. "Kolmogorov ran a complexity seminar in the 50s or early 60s,"... |

6 | On kolmogorov machines and related issues
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(Show Context)
Citation Context ...n build a binary tree over x and then move fast about x. Leonid Levin used a universal Kolmogorov machine to construct his algorithm for NP problems that is optimal 10sup to a multiplicative constant =-=[41, 22]-=-. The up-to-a-multiplicative-constant form is not believed to be achievable for the multitape Turing machine model popular in theoretical computer science. Similarly, the class of functions computable... |

6 |
For every Sequential Algorithm there is an Equivalent Sequential Abstract State Machine
- Gurevich
(Show Context)
Citation Context ...ished the definition of sequential abstract state machines (ASMs, called evolving algebras at the time) [23]. In 2000, he published a definition of sequential algorithms derived from first principles =-=[27]-=-. In the same paper he proved that every sequential algorithm A is behaviorally equivalent to some sequential ASM B. In particular, B simulates A step for step. In Section 5 we outline the approach of... |

5 |
Kolmogorov Algorithms are Stronger than Turing
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Citation Context ...s an M 0 2 M 0 such that M r ! M 0 . Real time equivalence M r $ M 0 means M r ! M 0 and M 0 r ! M. \LambdasDima Grigoriev proved that Turing machines cannot simulate Kolmogorov machines in real time =-=[21]-=-. Sch"onhage introduced a precise language for programming his machines and complained that the Kolmogorov-Uspensky description of Kolmogorov machines is clumsy. For our purposes here, however, it is ... |

5 |
On the definition of an algorithm, Uspekhi Mat. Nauk 13
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- 1958
(Show Context)
Citation Context ... toward understanding algorithms, but it did not solve the problem what an algorithm is. Further progress in foundations of algorithms was achieved by Kolmogorov and his student Uspensky in the 1950s =-=[39, 40]-=-. The Kolmogorov machine with its 2sreconfigurable "tape" has a certain advantage over the Turing machine. The notion of pointer machine was an improvement of the notion of Kolmogorov machine. These i... |

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4 |
Mathematical Logic, Addison–Wesley 1967 T ⊢ φ ⇔ ψ. Monadic calculus does not allow the quantifier elimination. Indeed, if U is a unary predicate symbol, then the existential quantifier cannot be eliminated from the formula ∃xU(x
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(Show Context)
Citation Context ...sed under isomorphisms. Further, any isomorphism from a state X onto a state Y is also an isomorphism from o/A(X) onto o/A(Y ). The notion of first-order structure is well-known in mathematical logic =-=[49]-=-. We use the following conventions: ffl Every vocabulary contains the following logic symbols: the equality sign, the nullary relation symbols true and false, and the usual Boolean connectives. ffl Ev... |

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3 |
Background, reserve, and Gandy machines
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- 2000
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Citation Context ...e assigned values. In comparison to the case of sequential ASMs, there are two main new features in the syntax of BGS ASMs: ffl set-comprehension terms ft(x) : x 2 r : '(x)g, and ffl forall rules. In =-=[6]-=-, we introduced the notion of a background of an ASM. BGS ASMs have a set background. The specification language AsmL, mentioned in the introduction, has a rich background that includes a set backgrou... |

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3 |
Endre Tarjan. A class of algorithms which require nonlinear time to maintain disjoint sets
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- 1979
(Show Context)
Citation Context ...tendency is to make computers with several CPUs, central processing units, that run asynchronously.) Remark 3.3 In an influential 1979 article, Tarjan used the term "pointer machine" in a wider sense =-=[53]-=-. This wider notion of pointer machines has become better known in computer science than the older notion. 4 Related issues We mention a few issues touched upon in the talk that was the precursor of t... |