Extending partial combinatory algebras

We give a negative answer to the question of whether every partial combinatory algebra can be completed. The explicit counterexample will be an intricately constructed term model. The construction and the proof that it works depend heavily on syntactic techniques. In particular, it provides a nice example of reasoning with elementary diagrams and descendants. We also include a domain-theoretic proof of the existence of an incompletable partial combinatory algebra.

[1]  Eric G. Wagner.  Uniformly reflexive structures: An axiomatic approach to computability , 1969, Inf. Sci..

[2]  Jan Willem Klop,et al.  Combinatory reduction systems , 1980 .

[3]  Henk Barendregt,et al.  The Lambda Calculus: Its Syntax and Semantics , 1985 .

[4]  Hendrik Pieter Barendregt,et al.  Normed uniformly reflexive structures , 1975, Lambda-Calculus and Computer Science Theory.

[5]  Cyrus O. Harbourt,et al.  Looking back , 2007, IEEE Potentials.

[6]  Harold T. Hodes,et al.  The | lambda-Calculus. , 1988 .

[7]  John C. Mitchell,et al.  Foundations for programming languages , 1996, Foundation of computing series.

[8]  H. R. Strong.  Algebraically generalized recursive function theory , 1968 .

[9]  Andrea Asperti,et al.  Effective Applicative Structures , 1995, Category Theory and Computer Science.

[10]  Tobias Nipkow,et al.  Higher-Order Algebra, Logic, and Term Rewriting , 1993, Lecture Notes in Computer Science.

[11]  Haskell B. Curry,et al.  Combinatory Logic, Volume I , 1959 .

[12]  Jan Willem Klop,et al.  Collapsing Partial Combinatory Algebras , 1995, HOA.

[13]  Jean-Jacques Lévy,et al.  An algebraic interpretation of the lambda beta - calculus and a labeled lambda - calculus , 1975, Lambda-Calculus and Computer Science Theory.

[14]  Ingemarie Bethke,et al.  On the existence of extensional partial combinatory algebras , 1987, Journal of Symbolic Logic.

[15]  C. Pollard,et al.  Center for the Study of Language and Information , 2022 .

[16]  Andrea Asperti,et al.  A Sufficient Condition for Completability of Partial Combinatory Algebras , 1997, J. Symb. Log..

[17]  Roger Hindley,et al.  Some remarks about the connections between Combinatory Logic and axiomatic recursion theory , 1977, Arch. Math. Log..

[18]  Jan Willem Klop,et al.  Completing partial combinatory algebras with unique head-normal forms , 1996, Proceedings 11th Annual IEEE Symposium on Logic in Computer Science.

[19]  Eric G. Wagner.  Uniformly reflexive structures: On the nature of gödelizations and relative computability , 1969 .

[20]  Tobias Nipkow,et al.  Term rewriting and all that , 1998 .