Marek Zaionc

Statistical properties of simple types

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 1s2 + √5s10, 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)s(2k + 1)2 and (3k + 1)s(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 1s3 of classical tautologies are intuitionistic.

Asymptotically almost all \lambda-terms are strongly normalizing

We present a quantitative analysis of various (syntactic and behavioral) prop- erties of random �-terms. Our main results show that asymptotically, almost all 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 �-calculus into combi- nators), 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.

Statistics of Intuitionistic versus Classical Logics

For the given logical calculus we investigate the proportion of the number of true formulas of a certain length n to the number of all formulas of such length. We are especially interested in asymptotic behavior of this fraction when n tends to infinity. If the limit exists it is represented by a real number between 0 and 1 which we may call the density of truth for the investigated logic. In this paper we apply this approach to the intuitionistic logic of one variable with implication and negation. The result is obtained by reducing the problem to the same one of Dummetts intermediate linear logic of one variable (see [2]). Actually, this paper shows the exact density of intuitionistic logic and demonstrates that it covers a substantial part (more than 93%) of classical prepositional calculus. Despite using strictly mathematical means to solve all discussed problems, this paper in fact, may have a philosophical impact on understanding how much the phenomenon of truth is sporadic or frequent in random mathematics sentences.

Classical and intuitionistic logic are asymptotically identical

This paper considers logical formulas built on the single binary connector of implication and a finite number of variables. When the number of variables becomes large, we prove the following quantitative results: asymptotically, all classical tautologies are simple tautologies. It follows that asymptotically, all classical tautologies are intuitionistic.

Intuitionistic vs. classical tautologies, quantitative comparison

We consider propositional formulas built on implication. The size of a formula is the number of occurrences of variables in it. We assume that two formulas which differ only in the naming of variables are identical. For every n ∈ N, there is a finite number of different formulas of size n. For every n we consider the proportion between the number of intuitionistic tautologies of size n compared with the number of classical tautologies of size n. We prove that the limit of that fraction is 1 when n tends to infinity.

Probability distribution for simple tautologies

In this paper we investigate the size of the fraction of tautologies of the given length n against the number of all formulas of length n for implicational logic. We are specially interested in asymptotic behavior of this fraction. We demonstrate the relation between a number of premises of implicational formula and asymptotic probability of finding formula with this number of premises. Furthermore, we investigate the distribution of this asymptotic probabilities. Distribution for all formulas is contrasted with the same distribution for tautologies only. We prove those distributions to be so different that enable us to estimate likelihood of truth for a given long formula. Despite the fact that all discussed problems and methods in this paper are solved by mathematical means, the paper may have some philosophical impact on the understanding how much the phenomenon of truth is sporadic or frequent in random logical sentences.

λ-Definability on free algebras

Abstract Zaionc, M., λ-Definability on free algebras, Annals of Pure and Applied Logic 51 (1991) 279-300. A λ-language over a simple type structure is considered. There is a natural isomorphism which identifies free algebras with nonempty second-order types. If A is a free algebra determined by the signature SA = [α1,...,αn], then by a type τA we mean τ1,...,τn→0 where τi=0αi→0. It can be seen that closed terms of the type τA reflex constructions in the algebra A. Therefore any term of the type (τA)n→τA defines some n-ary mapping in algebra A. The problem is to characterize λ-definable mappings in any free algebra. It is proved that the set of λ-definable operations is the minimal set that contains constant functions and projections and is closed under composition and limited recursion. This result is a generalization of the result of Schwichtenberg (1975) and Statman (1979) which characterize the λ-definable functions over the natural number type (0→0)→(0→0), i.e algebra [1, 0], as well as of the result of Zaionc (1987) for λ-definable word operations over type (0→0)n→(0→0), i.e algebra [1,...,1,0], and of the results about λ-definable tree operations (Zaionc, 1988 and 1990), i.e in algebra [2, 0]. Some of the examples in Section 5 are based on a publication of Zaionc (1988).

How big is BCI fragment of BCK logic

We investigate quantitative properties of BCI and BCK logics. The first part of the article compares the number of formulas provable in BCI versus BCK logics. We consider formulas built on implication and a fixed set of k variables. We investigate the proportion between the number of such formulas of a given length n provable in BCI logic against the number of formulas of length n provable in richer BCK logic. We examine an asymptotic behaviour of this fraction when length n of formulas tends to infinity. This limit gives a probability measure that randomly chosen BCK formula is also provable in BCI. We prove that this probability tends to zero as the number of variables tends to infinity. The second part of the article is devoted to the number of lambda terms representing proofs of BCI and BCK logics. We build a proportion between number of such proofs of the same length n and we investigate asymptotic behaviour of this proportion when length of proofs tends to infinity. We demonstrate that with probability 0 a randomly chosen BCK proof is also a proof of a BCI formula.

A Natural Counting of Lambda Terms

We study the sequences of numbers corresponding to lambda terms of given sizes, where the size is this of lambda terms with de Bruijn indices in a very natural model where all the operators have size 1. For plain lambda terms, the sequence corresponds to two families of binary trees for which we exhibit bijections. We study also the distribution of normal forms, head normal forms and strongly normalizing terms. In particular we show that strongly normalizing terms are of density 0 among plain terms.

Counting proofs in propositional logic

We give a procedure for counting the number of different proofs of a formula in various sorts of propositional logic. This number is either an integer (that may be 0 if the formula is not provable) or infinite.