## From Hilbert’s Program to a Logic Tool Box

### BibTeX

@MISC{Makowsky_fromhilbert’s,

author = {J. A. Makowsky},

title = {From Hilbert’s Program to a Logic Tool Box},

year = {}

}

### OpenURL

### Abstract

www.cs.technion.ac.il/∼janos Abstract. In this paper I discuss what, according to my long experience, every computer scientists should know from logic. We concentrate on issues of modeling, interpretability and levels of abstraction. We discuss what the minimal toolbox of logic tools should look like for a computer scientist who is involved in designing and analyzing reliable systems. We shall conclude that many classical topics dear to logicians are less important than usually presented, and that less known ideas from logic may be more useful for the working computer scientist. For Witek Marek, first mentor, then colleague and true friend, on the occasion of his 65th birthday.

### Citations

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(Show Context)
Citation Context ...omputer Science curriculum. Be that as it may, Computability Theory has emancipated itself from Logic and in Computer Science the two are often taught independently. A good exception is the monograph =-=[Pap94]-=-. 9Proof Theory Proof theory evolved around the question of what type of consistency proofs are at all possible. As a spin-off the field of deduction-based automated reasoning and automated theorem p... |

1534 |
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(Show Context)
Citation Context ...classical logics. These logics can also be modeled using set-theoretic tools, and indeed they are. When they find applications to Computer Science, as temporal logic [MP95] or the logic of knowledge, =-=[FHMV95]-=-, they also find their way into more advanced courses. I have also omitted a discussion of the role of logic research in the development of more and more sophisticated programming languages and progra... |

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272 |
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Citation Context ...Boris Trakhtenbrot showed that Logic restricted to finite interpretations is not axiomatizable. The most comprehensive account of solvable and unsolvable cases of the Decision Problem can be found in =-=[BGG97]-=-. Act V: Clarifications and Repairs Out of the ashes rise the four classical sub-disciplines of mathematical logic: 6Set Theory arises from work by E. Zermelo, D. Mirimanoff, J. von Neumann, A. Fränk... |

203 | On full abstraction for PCF - Hyland, Ong |

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(Show Context)
Citation Context ...e go ultimately back to Cantor’s original work and are very basic. The Recursion Theorem and Fixed Point Theorems should probably be taught in a more advanced course, using, say, Y. Moschovakis’ book =-=[Mos94]-=-. Recursion Theory vs Computability Theory Recursion Theory got its name for a good reason: The computable functions over the natural numbers were defined recursively, and early Recursion Theory consi... |

154 | Logic and the challenge of computer science
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(Show Context)
Citation Context ... found in [Mak04]. For other failures of classical theorems of First Order Logic when restricted to the finite case, such as the compactness theorem and various preservation theorems, see Y. Gurevich =-=[Gur88]-=- and B. Rossman [Ros08]. One should add here that the combination of the Compactness Theorem with the Ehrenfeucht-Fraïssé games leads to Lindström’s characterization of First Order Logic. The attempts... |

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(Show Context)
Citation Context ...926). However, J. von Neumann, in his thesis, credits D. Mirimanoff for some of his ideas. J. von Neumann’s approach was widely adopted, cf. [Lev79,Jec78,Mos94], or for the less sophisticated reader, =-=[Hal60]-=- and [Dev93]. 13differ from each other? Syntactically, Kuratowski’s pair and its simplified version are shorter than Wiener’s version, and are simpler in the sense that they appear earlier in the cum... |

86 | Temporal Verification of Reactive Systems
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(Show Context)
Citation Context ...n the standard classification non-classical logics. These logics can also be modeled using set-theoretic tools, and indeed they are. When they find applications to Computer Science, as temporal logic =-=[MP95]-=- or the logic of knowledge, [FHMV95], they also find their way into more advanced courses. I have also omitted a discussion of the role of logic research in the development of more and more sophistica... |

79 | Grundzüge der Theoretischen Logik - Hilbert, Ackermann - 1928 |

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46 |
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(Show Context)
Citation Context ...renfeucht-Fraïssé games and the Feferman-Vaught Theorem and its variations. The former is omnipresent in Finite Model Theory, cf. [EF95,Lib04] and the survey of the uses of the latter can be found in =-=[Mak04]-=-. For other failures of classical theorems of First Order Logic when restricted to the finite case, such as the compactness theorem and various preservation theorems, see Y. Gurevich [Gur88] and B. Ro... |

39 | Naturalism in Mathematics - Maddy - 1997 |

31 |
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Citation Context ...in his Foundations of Analysis first published in German in 1930, and in English in 1951, with many reprints, the latest in 1999 with a German-English vocabulary by the American Mathematical Society, =-=[Lan99]-=-. In this book he explicitly constructs the real and complex numbers from the standard model of Peano arithmetic. Landau’s style is very dry and concise, and the text was written for mature mathematic... |

29 |
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(Show Context)
Citation Context ...r, J. von Neumann, in his thesis, credits D. Mirimanoff for some of his ideas. J. von Neumann’s approach was widely adopted, cf. [Lev79,Jec78,Mos94], or for the less sophisticated reader, [Hal60] and =-=[Dev93]-=-. 13differ from each other? Syntactically, Kuratowski’s pair and its simplified version are shorter than Wiener’s version, and are simpler in the sense that they appear earlier in the cumulative hier... |

19 |
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(Show Context)
Citation Context ...ems. A thorough discussion of the pros and cons of Neo-Logicism can be found in J. Burgess [Bur05]. A thorough discussion of abstraction principles similar to Hume’s Principle can be found in K. Fine =-=[Fin02]-=-. So much for the ”big crisis”. However, the set theory needed for the foundations of Computer Science can be derived from logical principles without running into new foundational problems. In the seq... |

19 | Finite Model Theory and its applications - Gädel, Kolaitis, et al. - 2007 |

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Citation Context ...he modern Logical Formalisms, including the correct binding rules for quantification, and – set out to lay the Foundations of Mathematics with his Die Grundgesetze der Arithmetik, Volume1 (1893), see =-=[Bur05]-=-. 4– The book was not well received. Only G. Peano, author of The principles of arithmetic, presented by a new method (1889), [Ken73] wrote a positive review of it. Act II: Paradise lost – On 16 June... |

13 | Sequentiality vs. concurrency in games and logic - Abramsky - 1994 |

13 |
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(Show Context)
Citation Context ...athematics and Computer Science: Epistemology and degrees of constructivism. A delightful and insightful presentation and discussion of these matters from a contemporary point of view can be found in =-=[Sha00]-=-. Although I tend to be a Platonist, viewing mathematical concepts as real, I am aware of the difficulties inherent in this position, cf. [Mad90,Mad98]. I am also aware of the social and cultural mech... |

12 | Model theory and computer science: An appetizer - Makowsky - 1992 |

12 | Homomorphism preservation theorems
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(Show Context)
Citation Context ...other failures of classical theorems of First Order Logic when restricted to the finite case, such as the compactness theorem and various preservation theorems, see Y. Gurevich [Gur88] and B. Rossman =-=[Ros08]-=-. One should add here that the combination of the Compactness Theorem with the Ehrenfeucht-Fraïssé games leads to Lindström’s characterization of First Order Logic. The attempts to develop an Abstract... |

11 | The Consistency of Frege’s Foundation of Arithmetic - Boolos - 1987 |

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10 |
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(Show Context)
Citation Context ...s as real, I am aware of the difficulties inherent in this position, cf. [Mad90,Mad98]. I am also aware of the social and cultural mechanisms at work which strongly influence how science evolves, cf. =-=[Wil81]-=-. However, I strongly object to the arguments which take the social and cultural mechanisms at work as a justification for the erroneous claim that scientific truth is purely social and cultural. 25F... |

9 |
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(Show Context)
Citation Context ... they thought was worth teaching in their field, I would like to write such a book on Logic for the Computer Scientist. Two of the last great books in this tradition were written by Laurent Schwartz (=-=[Sch67]-=-) and was used as a course for the elitist École Polytechnique, and by R. Godement ([God66]) which was used for the equally elitist École Normal Supérieure. My 24French colleagues now tell me, that e... |

6 |
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(Show Context)
Citation Context ...onal programming do belong in a basic course on Programming Languages. However, only little of this rich material is suitable for the student I have in mind. A comprehensive survey of Proof Theory is =-=[Bus98]-=-. It remains to be seen to what extent Girard’s ambitious program of using Linear Logic (a resource conscious version of natural deduction) is valuable for the sophisticated practitioner. The current ... |

6 |
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(Show Context)
Citation Context ...del of Peano arithmetic. Landau’s style is very dry and concise, and the text was written for mature mathematicians. A more pedagogical version of the same constructions can be found in S. Feferman’s =-=[Fef64]-=-, which I also would love to see reprinted. One can view such a foundation of Analysis as a pragmatic version of Frege’s program. Roughly, we proceed as follows: – One starts with a cumulative hierarc... |

6 |
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(Show Context)
Citation Context ...gh a strict separation of my interests is impossible, my research roughly can be divided as follows: My early work (1970-85) was mainly in mathematical logic, in particular in classical model theory (=-=[Mak74]-=-), abstract model theory and generalized quantifiers ([BF85,Mak85b,MM85,Mak85a]). In the period of 1980-1995 my work shifted to applications of logic to semantics of programming languages, logic progr... |

5 |
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(Show Context)
Citation Context ...their technical abilities and limitations, and who can take full responsibility for the safe functioning of their creations. Roger Godement expressed this well in his preface to his ”Cours d’Algèbre” =-=[God66]-=- 1 . Even in teaching mathematics we can at least attempt to teach the students the flavor of freedom and critical thought, and to get them used to the idea of being treated as humans empowered with t... |

5 |
Problem #1: Ein ungelostes Problem in der symbolischen
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- 1952
(Show Context)
Citation Context ...nd geometric theory, linking model theory with algebraic geometry and number theory. Although finite model theory has its early origin in the study of the “spectrum problem” initiated by H. Scholz in =-=[Sch52]-=-, it was through Automata Theory, Database Theory and Complexity Theory that it evolved into its own discipline with a legitimate place in advanced Computer Science education. . . . and for long this ... |

4 | Abstract embedding relations - Makowsky - 1985 |

3 | NCE graph grammars and clique-width - Glikson, Makowsky - 2003 |

2 | Why sets
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- 2008
(Show Context)
Citation Context ...d by a large social consensus, that the language of sets is the most used and most accepted way of modeling mathematical objects. A very convincing discussion, why sets are used that way, is given in =-=[BG08]-=-. We are used to modeling automata of all sorts including Turing machines as tuples of sets, functions and relations. We do the same when we discuss behavior of hardware and software, when we prove pr... |

2 | Grundzuge der theoretischen Logik, 3rd edition - Hilbert, Ackermann - 1949 |

2 |
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(Show Context)
Citation Context ...ics with his Die Grundgesetze der Arithmetik, Volume1 (1893), see [Bur05]. 4– The book was not well received. Only G. Peano, author of The principles of arithmetic, presented by a new method (1889), =-=[Ken73]-=- wrote a positive review of it. Act II: Paradise lost – On 16 June 1902, Bertrand Russell pointed out, with great modesty, that the Russell paradox gives a contradiction in Frege’s system of axioms. –... |

2 | Abstract equivalence relations - Makowsky, Mundici - 1985 |

1 | 26 G. Boolos. On the proof of Frege’s theorem - Logic, Logic - 1998 |