Abstract
We discuss connections, similarities, and differences between the concepts and issues arising in the study of abstract state machines and those arising in pure mathematics, particularly in set theory and logic. Among the topics from pure mathematics are the foundational role of set theory, permutation models of set theory without the axiom of choice, and interpretations (between theories or vocabularies) regarded as transformations acting on structures.
Preparation of this paper was partially supported by a grant from Microsoft Corporation. The opinions expressed here are, however, entirely my own.
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References
Andreas Blass and Yuri Gurevich, “Existential fixed-point logic”, in Computation Theory and Logic, ed. by E. Börger, Lecture Notes in Computer Science 270, Springer-Verlag (1987) 20–36.
Andreas Blass, Yuri Gurevich, and Saharon Shelah, “Choiceless polynomial time”, Ann. Pure Appl. Logic, 100 (1999) 141–187.
Andreas Blass, Yuri Gurevich, and Jan Van den Bussche, “Abstract state machines and computationally complete query languages”, this volume.
Nicolas Bourbaki, Elements of Mathematics. Theory of Sets, Hermann (1968).
Abraham Fraenkel, Der Begriff “definit” und die Unabhängigkeit des Aus-wahlaxioms, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse (1922) 253–257.
Murdoch J. Gabbay and Andrew M. Pitts, “A New Approach to Abstract Syn-tax Involving Binders”, in Proceedings 14th Annual IEEE Symposium on Logic in Computer Science, Trento, Italy, July 1999, IEEE Computer Society Press (1999) 214–224.
Yuri Gurevich, “Evolving Algebra 1993: Lipari Guide”, in Specification and Vali-dation Methods, ed. by E. Boerger, Oxford University Press, 1995, 9–36.
Yuri Gurevich, “Sequential abstract state machines capture sequential algorithms”, ACM Transactions on Computational Logic, to appear.
Thomas Jech, The Axiom of Choice, North-Holland (1973).
F. William Lawvere, “Variable quantities and variable structures in topoi”, in Algebra, Topology, and Category Theory (A Collection of Papers in Honor of Samuel Eilenberg), ed. by A. Heller and M. Tierney, Academic Press (1976) 101–131.
Eugene C. Luschei, The Logical Systems of Leśniewski, North-Holland (1962).
Anthony P. Morse, A Theory of Sets, Academic Press (1965).
Anand Pillay, Geometric Stability Theory, Oxford University Press (1996).
Saharon Shelah, Choiceless polynomial time logic: Inability to express, paper number 634, to appear.
Joseph Shoenfield, Mathematical Logic, Addison-Wesley (1967).
Edward E. Slaminka, “A Brouwer translation theorem for free homeomorphisms”, Trans. Amer. Math. Soc., 306 (1988) 277–291.
Alfred Tarski, Andrzej Mostowski, and Abraham Robinson, Undecidable Theories, North-Holland (1953).
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Blass, A. (2000). Abstract State Machines and Pure Mathematics. In: Gurevich, Y., Kutter, P.W., Odersky, M., Thiele, L. (eds) Abstract State Machines - Theory and Applications. ASM 2000. Lecture Notes in Computer Science, vol 1912. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44518-8_2
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DOI: https://doi.org/10.1007/3-540-44518-8_2
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