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Tarski’s Influence on Computer Science

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The Lvov-Warsaw School. Past and Present

Part of the book series: Studies in Universal Logic ((SUL))

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Abstract

Alfred Tarski’s influence on computer science was indirect but significant in a number of directions and was in certain respects fundamental. Here surveyed is Tarski’s work on the decision procedure for algebra and geometry, the method of elimination of quantifiers, the semantics of formal languages, model-theoretic preservation theorems, and algebraic logic; various connections of each with computer science are taken up.

The author “Solomon Feferman” is deceased.Reprinted from Logical Methods in Computer Science, vol. 2 (3:6), pp. 1–13, https://doi.org/10.2168/LMCS-2(3:6)2006. This reprint for Studies in Universal Logic style in LATE X is prepared by Zbigniew Bonikowski.

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Notes

  1. 1.

    I want to thank the organizers of LICS 2005 for inviting me to give this lecture and for suggesting the topic of Tarski’s influence on computer science, a timely suggestion for several reasons. I appreciate the assistance of Deian Tabakov and Shawn Standefer in preparing the LATE X version of this text. Except for the addition of references, footnotes, corrections of a few points and stylistic changes, the text is essentially as delivered. Subsequent to the lecture I received interesting comments from several colleagues that would have led me to expand on some of the topics as well as the list of references, had I had the time to do so.

  2. 2.

    For those who may not know what the “big shiny Oracle towers” are, the reference is to the headquarters of Oracle Corporation on the Redwood Shores area of the San Francisco Peninsula. A duly shiny photograph of a few of these towers may be found at http://en.wikipedia.org/wiki/Image:OracleCorporationHQ.png.

  3. 3.

    I am most indebted in this respect to Phokion Kolaitis. Besides him I have also received useful comments from Michael Beeson, Bruno Buchberger, George Collins, John Etchemendy, Donald Knuth, Janos Makowsky, Victor Marek, Ursula Martin, John Mitchell, Vaughan Pratt, Natarajan Shankar, and Adam Strzebonski. And finally, I would like to thank the two anonymous referees for a number of helpful corrections.

  4. 4.

    Collins reports [7, p. 86] a communication from Leonard Monk in 1974 stating that he and Bob Solovay had obtained a triply exponential upper bound decision procedure for real algebra, though not a quantifier elimination procedure. Fischer and Rabin say (op. cit., p. 124) that Solovay found a doubly exponential upper bound, based on Monk’s work.

  5. 5.

    This includes a reprint of Tarski’s “A decision method for elementary algebra and geometry” [39].

  6. 6.

    Just minutes before my lecture, I learned from Prakash Panangaden that John Canny (U.C. Berkeley School of Engineering) proved [6] that the existential theory of the reals is in PSPACE.

  7. 7.

    A sad coda to this story is that Presburger, a Jew, perished in the Holocaust in 1943.

  8. 8.

    Shankar [37] takes as an epigram a quote from Davis [12] re his experiment with Presburger Arithmetic: “Its great triumph was to prove that the sum of two even numbers is even.” A second epigram from the same source, quoting Hao Wang, is that: “The most interesting lesson from these results is perhaps that even in a fairly rich domain, the theorems actually proved are mostly ones which call on a very small portion of the available resources …”

  9. 9.

    By the elementary theory of a structure, Tarski means the set of its first-order truths.

  10. 10.

    It was not until 1957, in a paper with Robert L. Vaught [45], that Tarski explicitly presented these notions as those of satisfaction and truth in a structure. See the discussion by Hodges [26] and Feferman [15] of the relationship of that to the “Wahrheitsbegriff” paper.

  11. 11.

    Tarski proved that every monotonic function over a complete lattice has a complete lattice of fixed points, and hence a least fixed point. This is a generalization of a much earlier joint result of Knaster and Tarski and so is sometimes referred to as the Knaster-Tarski theorem. A related result used in applications is that every continuous function on a complete lattice has a least fixed point; credit for it is unclear, and thus it is considered a “folk theorem”. The history of these and other fixed point theorems relevant to computer science is surveyed in [29].

  12. 12.

    Scott informed me that his use of lattice fixed points was initiated in the fall of 1969 in work with Christopher Strachey and exposed in many lectures in Oxford while on leave there. For further developments and a large bibliography see [21].

  13. 13.

    Lyndon’s famous positivity theorem implies OHPT. Ajtai and Gurevich, and then Stolboushkin in a simpler way, proved failure in the finite of the positivity theorem, but their constructions did not prove failure in the finite of OHPT.

  14. 14.

    According to Rossman, the implication was known to hold prior to his discovery.

  15. 15.

    According to Van den Bussche (personal communication) the first people from the database community to recognize the connection between Codd’s relational algebra and Tarski’s cylindric algebras were Witold Lipski and his student Tomasz Imielinski, in a talk given at the very first edition of PODS (the ACM Symposium on Principles of Database Systems), held in Los Angeles, March 29–31, 1982. Their work was later published in Imielinski and Lipski [27].

  16. 16.

    Some other applications to computer science — not discussed here — of Tarski’s work on relation algebra are indicated on p. 339 and its footnote 4 of that interlude.

  17. 17.

    Cf. the papers [18, 19] dealing with expressibility/inexpressibility in Tarski’s algebraic framework.

  18. 18.

    Van den Bussche’s article concludes with a survey of some interesting connections to constraint databases and geometric databases.

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, CA, USA

    Solomon Feferman

  2. Department of Philosophy, Stanford University, Stanford, CA, USA

    Solomon Feferman

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  1. Department of Fundamental Mathematics, UNED, Madrid, Spain

    Ángel Garrido

  2. Cardinal Stefan Wyszyński University in Warsaw, Warsaw, Poland

    Urszula Wybraniec-Skardowska

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Feferman, S. (2018). Tarski’s Influence on Computer Science. In: Garrido, Á., Wybraniec-Skardowska, U. (eds) The Lvov-Warsaw School. Past and Present. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-65430-0_29

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