Boolean Algebras for Lambda Calculus

In this paper we show that the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras. In every combinatory algebra there is a Boolean algebra of central elements (playing the role of idempotent elements in rings), whose operations are defined by suitable combinators. Central elements are used to represent any combinatory algebra as a Boolean product of directly indecomposable combinatory algebras (i.e., algebras which cannot be decomposed as the Cartesian product of two other nontrivial algebras). Central elements are also used to provide applications of the representation theorem to lambda calculus. We show that the indecomposable semantics (i.e., the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics, and the term models of all semisensible lambda theories. In one of the main results of the paper we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible: for every recursively enumerable lambda theory Tscr, there is a continuum of lambda theories including Tscr which are omitted by the indecomposable semantics

[1]  Samson Abramsky,et al.  Domain Theory in Logical Form , 1991, LICS.

[2]  D. Vaggione.  Varieties in Which the Pierce Stalks Are Directly Indecomposable , 1996 .

[3]  R. McKenzie,et al.  Algebras, Lattices, Varieties , 1988 .

[4]  M. Schönfinkel.  Über die Bausteine der mathematischen Logik , 1924 .

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

[6]  Antonino Salibra,et al.  Topological incompleteness and order incompleteness of the lambda calculus , 2003, TOCL.

[7]  Antonino Salibra,et al.  The Lattice of Lambda Theories , 2004, J. Log. Comput..

[8]  Mariangiola Dezani-Ciancaglini,et al.  A filter lambda model and the completeness of type assignment , 1983, Journal of Symbolic Logic.

[9]  Mariangiola Dezani-Ciancaglini,et al.  Filter Models and Easy Terms , 2001, ICTCS.

[10]  Stanley Burris,et al.  A course in universal algebra , 1981, Graduate texts in mathematics.

[11]  J. Heijenoort.  From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 , 1967 .

[12]  R. Pierce.  Modules over Commutative Regular Rings , 1967 .

[13]  Chantal Berline,et al.  Graph models of $\lambda$-calculus at work, and variations , 2006, Mathematical Structures in Computer Science.

[14]  S. D. Comer,et al.  Representations by algebras of sections over Boolean spaces. , 1971 .

[15]  Antonino Salibra,et al.  A continuum of theories of lambda calculus without semantics , 2001, Proceedings 16th Annual IEEE Symposium on Logic in Computer Science.

[16]  Antonio Bucciarelli,et al.  Sequentiality and strong stability , 1991, [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science.

[17]  Gérard Berry,et al.  Stable Models of Typed lambda-Calculi , 1978, ICALP.

[18]  Furio Honsell,et al.  An Approximation Theorem for Topological Lambda Models and the Topological Incompleteness of Lambda Calculus , 1992, J. Comput. Syst. Sci..

[19]  G. Plotkin.  Set-theoretical and Other Elementary Models of the -calculus Part 1: a Set-theoretical Deenition of Applica- Tion 1 Introduction , 2007 .

[20]  Antonio Bucciarelli,et al.  The sensible graph theories of lambda calculus , 2004, LICS 2004.

[21]  Rainer Kerth.  On the construction of stable models of untyped lambda-calculus , 2001, Theor. Comput. Sci..

[22]  S. Shelah,et al.  Annals of Pure and Applied Logic , 1991 .

[23]  Gordon D. Plotkin,et al.  Set-Theoretical and Other Elementary Models of the lambda-Calculus , 1993, Theor. Comput. Sci..

[24]  Antonino Salibra,et al.  Easiness in graph models , 2006, Theor. Comput. Sci..

[25]  Antonio Bucciarelli,et al.  The sensible graph theories of lambda calculus , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[26]  Olivier Bastonero,et al.  Strong Stability and the Incompleteness of Stable Models for lambda-Calculus , 1999, Ann. Pure Appl. Log..

[27]  Chantal Berline,et al.  From computation to foundations via functions and application: The -calculus and its webbed models , 2000, Theor. Comput. Sci..

[28]  Peter Selinger.  Order-incompleteness and finite lambda reduction models , 2003, Theor. Comput. Sci..

[29]  Rasmus Ejlers Møgelberg,et al.  Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science , 2007 .

[30]  Antonino Salibra.  On the algebraic models of lambda calculus , 2000, Theor. Comput. Sci..

[31]  William C. Frederick,et al.  A Combinatory Logic , 1995 .

[32]  B. M. Fulk.  MATH , 1992 .