Mathematical logic; particularly lambda-calculus, combinatory logic and type-theories, with a current bias towards historical aspects.
Lambda-calculus and combinatory logic are formal systems, to some extent rivals, used in the construction and study of programming languages which are higher-order (i.e. in which programs may change other programs). These two systems were invented in the 1920s by mathematicians for use in higher-order logic, and came to be applied in programming theory from the 1970s onward, when that theory expanded to cover higher-order computations.
In a type-theory, types are labels which may be attached to certain programs to show what other programs they can change. A type-system is a particular set of rules for attaching types; the rules themselves are usually reasonably simple, but such questions as what programs are typable, what set of types a program may receive, and whether a typable computation can continue for ever, are not always easy to answer and have occupied many researchers.
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Basic Simple Type Theory; Cambridge University Press, 1997, 2008. MathSciNet MR1466699. An Errata-list is available; see Files for Downloading above.
(Co-author J. P. Seldin) Lambda-calculus and Combinators, an Introduction; Cambridge University Press, 2008. (Earlier edition was 1986 with title Introduction to Combinators and Lambda-calculus.) An Errata-list is available, also a supplementary chapter on Goedel's consistency-proof for arithmetic; see Files for Downloading above.
(Co-authors B. Lercher, J. P. Seldin) Introduction to Combinatory Logic; Cambridge University Press, 1972, No.7 in series "London Mathematical Society Lecture Notes". ( MathSciNet MR0335242.)
(Co-authors H. B. Curry, J. P. Seldin) Combinatory Logic, Volume 2; North-Holland Co., Amsterdam, 1972, No.65 in series "Studies in Logic".
(Edited; co-editor J. P. Seldin) To H. B. Curry, Essays on Combinatory Logic, Lambda Calculus and Formalism Academic Press, London and New York, 1980. ( MathSciNet MR0592792.)
(Edited; co-editor P. de Groote) Typed Lambda Calculi and Applications, Proceedings of the Third International Conference TLCA '97, Nancy, France; Springer-Verlag, Berlin and Heidelberg, 1997. No.1210 in series "Lecture Notes in Computer Science".
(Co-author Mariangiola Dezani-Ciancaglini) Lambda Calculus. In Wiley Encyclopedia of Computer Science and Engineering, Volume 3, pp. 1701--1708, edited by Benjamin Wah; published by Wiley, 2008.
Curry's last problem: imitating lambda-beta-reduction in combinatory logic, Proc. 32nd MLG Meeting, Shizuoka, Japan, November 26--28, 1998, pp. 20--22. PDF of preprint.
M. H. Newman's typability algorithm for lambda-calculus, Journal of Logic and Computation 18(2) (April 2008), 229--238. PDF of preprint.
(Co-author F. Cardone) Lambda-calculus and combinators in the 20th century. In Handbook of the History of Logic, Volume 5: Logic from Russell to Church, edited by D. M. Gabbay and J. Woods, published by Elsevier (North-Holland Co.), Amsterdam, 2009, pp. 723--817. PDFs of preprint and Errata list: see Files for Downloading above.
(Co-author N. Çağman) Combinatory Weak Reduction in Lambda Calculus, Theoretical Computer Science 198 (1998), 239--247. ( MathSciNet MR1616965.) PDF available here.
(Co-author M. W. Bunder) Two beta-equal lambda-I-terms with no types in common, Theoretical Computer Science 155 (1996), 265--266. ( MathSciNet MR1379073.) PDF available here.
(Co-authors P. Trigg, M. W. Bunder) Combinatory abstraction using B, B' and friends, Theoretical Computer Science 135 (1994), 405--422. ( MathSciNet MR1311210.) PDF available here.
BCK and BCI logics, condensed detachment and the 2-property, Notre Dame Journal of Formal Logic 34 (1993), 231--250. ( MathSciNet MR1231287.) PDF available here.
Types with intersection: an introduction, Formal Aspects of Computing 4 (1992), 470--486. PDF: see SpringerLink.
(Co-author Mariangiola Dezani-Ciancaglini) Intersection types for combinatory logic, Theoretical Computer Science 100 (1992), 303--324. ( MathSciNet MR1173628.) PDF: see Science Direct.
(Co-author D. Meredith) Principal type-schemes and condensed detachment, Journal of Symbolic Logic 55 (1990), 90--105. ( MathSciNet MR1043546.) PDF: see Project Euclid.
(Co-authors M. W. Bunder, J. P. Seldin) On adding (ξ) to weak equality in combinatory logic, Journal of Symbolic Logic 54 (1989), 590--607. ( MathSciNet MR0997891.) PDF: see Project Euclid.
BCK-combinators and linear λ-terms have types, Theoretical Computer Science 64 (1989), 97--105. ( MathSciNet MR0993977.) PDF: see Science Direct.
Combinators and lambda-calculus, a short outline. In Combinators and Functional Programming Languages, edited by G. Cousineau, P.-L. Curien and B. Robinet, no.242 in series "Lecture Notes in Computer Science", Springer-Verlag, Berlin, 1985. Pp. 104--122. PDF here.
Coppo-Dezani types do not correspond to propositional logic, Theoretical Computer Science 28 (1984), 235--236. PDF: see Science Direct.
Curry's type-rules are complete with respect to the F-semantics too, Theoretical Computer Science 22 (1983), 127--133. PDF: see Science Direct.
The completeness theorem for typing lambda terms, Theoretical Computer Science 22 (1983), 1--17. ( MathSciNet MR0693047.) PDF: see Science Direct.
The simple semantics for Coppo-Dezani-Sallé types. In International Symposium on Programming, 5th Colloquium, Proceedings 1982, edited by M. Dezani and U. Montanari, no.137 in series "Lecture Notes in Computer Science", Springer-Verlag, Berlin, 1982. Pp. 212--226. ( MathSciNet MR0807180.) PDF: see SpringerLink.
(Co-author G. Longo) Lambda-calculus models and extensionality, Zeitschrift fur Mathematische Logik 26 (1980), 289--310. ( MathSciNet MR0582407.) PDF: see Wiley Online Library.
The discrimination theorem holds for combinatory weak reduction, Theoretical Computer Science 8 (1979), 393--394. PDF: see Science Direct.
Standard and normal reductions, Transactions of the American Mathematical Society 241 (1978), 253--271. ( MathSciNet MR0492300.) PDF: see Jstor.
Reductions of residuals are finite, Transactions of the American Mathematical Society 240 (1978), 345--361. ( MathSciNet MR0489603.) PDF: see Jstor.
The equivalence of complete reductions, Transactions of the American Mathematical Society 229 (1977), 227--248. ( MathSciNet MR0444445.) PDF: see Jstor.
(Co-author G. Mitschke) Some remarks about the connections between combinatory logic and axiomatic recursion theory, Archiv fur Mathematische Logik 18 (1977), 99--103. ( MathSciNet MR0469717.) PDF: see SpringerLink.
Combinatory reductions and lambda reductions compared, Zeitschrift fur Mathematische Logik 23 (1977), 169--180. ( MathSciNet MR0485229.) PDF: see Wiley Online Library. .
An abstract Church-Rosser theorem, II: applications, Journal of Symbolic Logic 39 (1974), 1--21. ( MathSciNet MR0347558.) PDF: see Project Euclid.
(Co-author B. Lercher) A short proof of Curry's normal form theorem, Proceedings of the American Mathematical Society 24 (1970), 808--810. ( MathSciNet MR0255405.) PDF: see Jstor.
The principal type-scheme of an object in combinatory logic, Transactions of the American Mathematical Society 146 (1969), 29--60. ( MathSciNet MR0253905.) PDF: see Jstor.
An abstract form of the Church-Rosser theorem, I, Journal of Symbolic Logic 34 (1969), 545--560. ( MathSciNet MR0302434.) PDF: see Project Euclid.
Axioms for strong reduction in combinatory logic, Journal of Symbolic Logic 32 (1967), 224--236. ( MathSciNet MR0214470.) PDF: see Project Euclid.