Asymptotically almost all \lambda-terms are strongly normalizing

We present quantitative analysis of various (syntactic and behavioral) properties of random \lambda-terms. Our main results are that asymptotically all the terms are strongly normalizing and that any fixed closed term almost never appears in a random term. Surprisingly, in combinatory logic (the translation of the \lambda-calculus into combinators), the result is exactly opposite. We show that almost all terms are not strongly normalizing. This is due to the fact that any fixed combinator almost always appears in a random combinator.

[1]  Danièle Gardy.  Random Boolean expressions , 2005 .

[2]  Marek Zaionc.  On the Asymptotic Density of Tautologies in Logic of Implication and Negation , 2005, Reports Math. Log..

[3]  Danièle Gardy,et al.  Classical and Intuitionistic Logic Are Asymptotically Identical , 2007, CSL.

[4]  Antoine Genitrini,et al.  Intuitionistic vs. Classical Tautologies, Quantitative Comparison , 2007, TYPES.

[5]  Jerzy Tyszkiewicz,et al.  Statistical properties of simple types , 2000, Mathematical Structures in Computer Science.

[6]  Philippe Flajolet,et al.  Analytic Combinatorics , 2009 .

[7]  Guillaume Theyssier,et al.  ON LOCAL SYMMETRIES AND UNIVERSALITY IN CELLULAR , 2009 .

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

[9]  René David,et al.  Normalization without reducibility , 2001, Ann. Pure Appl. Log..

[10]  Marek Zaionc,et al.  Probability distribution for simple tautologies , 2006, Theor. Comput. Sci..

[11]  Zofia Kostrzycka,et al.  Statistics of Intuitionistic versus Classical Logics , 2004, Stud Logica.

[12]  Philippe Flajolet,et al.  And/Or Trees Revisited , 2004, Comb. Probab. Comput..

[13]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[14]  Joel David Hamkins,et al.  The Halting Problem Is Decidable on a Set of Asymptotic Probability One , 2006, Notre Dame J. Formal Log..

[15]  Alexander N. Rybalov.  On the strongly generic undecidability of the Halting Problem , 2007, Theor. Comput. Sci..

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

[17]  Guillaume Theyssier,et al.  On Local Symmetries And Universality In Cellular Autmata , 2009, STACS.

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

[19]  Danièle Gardy,et al.  And/or tree probabilities of Boolean functions , 2005 .

[20]  Hanno Lefmann,et al.  Some typical properties of large AND/OR Boolean formulas , 1997, Random Struct. Algorithms.

[21]  Karim Nour,et al.  A short proof of the strong normalization of the simply typed $\lambda\mu$-calculus , 2003, LICS 2003.

[22]  Hanno Lefmann,et al.  Some typical properties of large AND/OR Boolean formulas , 1997 .

[23]  Laurent Regnier,et al.  Une équivalence sur les lambda-termes , 1994, Theor. Comput. Sci..

[24]  R. Smullyan.  To mock a mockingbird and other logic puzzles : including an amazing adventure in combinatory logic , 1985 .

[25]  Antoine Genitrini,et al.  Quantitative Comparison of Intuitionistic and Classical Logics - Full Propositional System , 2009, LFCS.