Computing and Software 701
Logic and Discrete Mathematics in Software Engineering
Fall 2004

Instructor: William M. Farmer

References

01 What is Mathematics?

  1. D. M. Bressoud, Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture, Cambridge University Press, 1999.
  2. I. Lakatos, Proofs and Refutations, Cambridge University Press, 1976.
  3. Biographical Sketch of I. Lakatos.

02 What is Logic?

03 Propositional Logic

04 Numbers, Sets, Functions, and Relations

  1. P. R. Halmos, Abstract Naive Set Theory, Springer-Verlag, 1987.

05 Mathematical Models

  1. J. Hopcraft, R. Motwani, and J. Ullman, Introduction to Automata Theory, Languages, and Computation, Second Edition, Addison Wesley, 2001.

06 First-Order Logic

  1. C. C. Chang and H. J. Keisler, Model Theory, North Holland, 1990.
  2. H. B. Enderton, A Mathematical Introduction to Logic, Academic Press, 1972.
  3. R. M. Smullyan, First-Order Logic, Dover publications, 1995.

07 Equational Logic and Algebraic Reasoning

  1. V. Sperschneider and G. Antoniou, Logic: A Foundation for Computer Science, Addison Wesley, 1991.
  2. F. Baader and T. Nipkow, Term Rewriting and All That, Springer, 1999.

08 Recursion and Induction

  1. W. M. Farmer, "A scheme for defining partial higher-order functions by recursion", In: A. Butterfield, ed., 3rd Irish Workshop on Formal Methods, 13 pp., electronic Workshops in Computing, Springer-Verlag, 1999. PostScript PDF

09 Simple Type Theory

  1. P. B. Andrews, An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof, Kluwer, 2002.
  2. A. Church, "A formulation of the simple theory of types", Journal of Symbolic Logic, 5:56-68, 1940.
  3. W. M. Farmer, "A basic extended simple type theory", SQRL Report No. 14, 12 pp., McMaster University, 2004. PostScript PDF
  4. W. M. Farmer, "The seven virtues of simple type theory", SQRL Report No. 18, 27 pp., McMaster University, 2004. PostScript PDF

10 A Compendium of Mathematical Structures

  1. D. Dummit, R. Foote, and B. Holland, Abstract Algebra, Second Edition, Wiley, 1999.
  2. MathWorld
  3. The Mathematics Section of the Wikipedia
  4. E. Mendelson, Schaum's Outline of Boolean Algebra and Switching Circuits, McGraw-Hill, 1970.

11 Simple Type Theory with Undefinedness

  1. W. M. Farmer, "A partial functions version of Church's simple theory of types", Journal of Symbolic Logic, 55:1269-1291, 1990.
  2. W. M. Farmer, "Reasoning about partial functions with the aid of a computer", Erkenntnis, 43:279-294, 1995.
  3. W. M. Farmer, "STMM: A Set Theory for Mechanized Mathematics", Journal of Automated Reasoning, 26:269-289, 2001. PostScript PDF
  4. W. M. Farmer, "Formalizing undefinedness arising in calculus", in: D. Basin and M. Rusinowitch, eds., Automated Reasoning, LNCS, 3097:475-489, 2004.

12 Axiomatic Method

  1. J. Barwise and J. Seligman, Information Flow : The Logic of Distributed Systems, Cambridge University Press, 1997.
  2. N. Bourbaki, Eléments de mathématique, mid 1900s.
  3. R. Burstall and J. Goguen, "Putting theories together to make specifications", Proceedings of the Fifth International Joint Conference on Artificial Intelligence, pp. 1045-1058, 1977.
  4. Euclid, The Elements, 300 BC.
  5. W. M. Farmer, "Theory interpretation in simple type theory", in: J. Heering et al., eds., Higher-Order Algebra, Logic, and Term Rewriting, LNCS, 816:96-123, 1994. PostScript PDF
  6. W. M. Farmer, J. D. Guttman, and F. J. Thayer, "Little theories", in: D. Kapur, ed., Automated Deduction--CADE-11, LNCS, 607:567-581, 1992. PostScript PDF
  7. R. Nakajima and T. Yuasa (eds.), The IOTA Programming System, LNCS, Vol. 160, 1982.
  8. Noneuclidean Geometry.
  9. D. Sannella and A. Tarlecki, "Towards formal development of programs from algebraic specifications: Implementations revisited", Acta Informatica, 25:233-281, 1988.
  10. D. Smith and M. Lowry, "Algorithmic theories and design tactics", Science of Computer Programming, 14:305-321, 1990
  11. W. Turski and T. Maibaum, The Specification of Computer Programs, Addison-Wesley, 1987.
  12. A. N. Whitehead and B. Russell, Principia Mathematica, Cambridge University Press, 1910-1913.