Statistical properties of simple types

  • Authors:

  • M. Moczurad;J. Tyszkiewicz;M. Zaionc

  • Affiliations:

  • Computer Science Department, Jagiellonian University, Nawojki 11, 30-072 Krakow, Poland. Email madry@softlab.ii.uj.edu.pl/ zaionc@ii.uj.edu.pl;Institute of Informatics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland, Email jty@mimuw.edu.pl on leave at School CSE, UNSW, 2052 Sydney, Australia;Computer Science Department, Jagiellonian University, Nawojki 11, 30-072 Krakow, Poland. Email madry@softlab.ii.uj.edu.pl/ zaionc@ii.uj.edu.pl

  • Venue:
  • Mathematical Structures in Computer Science
  • Year:
  • 2000

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Abstract

We consider types and typed lambda calculus over a finite number of ground types. We are going to investigate the size of the fraction of inhabited types of the given length n against the number of all types of length n. The plan of this paper is to find the limit of that fraction when n → ∞. The answer to this question is equivalent to finding the ‘density’ of inhabited types in the set of all types, or the so-called asymptotic probability of finding an inhabited type in the set of all types. Under the Curry–Howard isomorphism this means finding the density or asymptotic probability of provable intuitionistic propositional formulas in the set of all formulas. For types with one ground type (formulas with one propositional variable), we prove that the limit exists and is equal to 1/2 + √5/10, which is approximately 72.36%. This means that a long random type (formula) has this probability of being inhabited (tautology). We also prove that for every finite number k of ground-type variables, the density of inhabited types is always positive and lies between (4k + 1)/(2k + 1)2 and (3k + 1)/(k + 1)2. Therefore we can easily see that the density is decreasing to 0 with k going to infinity. From the lower and upper bounds presented we can deduce that at least 1/3 of classical tautologies are intuitionistic.