BCK逻辑的BCI片段有多大
How big is BCI fragment of BCK logic
摘要：
我们研究BCI和BCK逻辑的定量性质。本文的第一部分比较了BCI与BCK逻辑中可证明公式的数量。我们考虑建立在蕴涵和一组固定的$k$变量的公式。我们研究了在BCI逻辑中可证明的具有给定长度的$n$的公式的数量与在更丰富的BCK逻辑中可证明的具有长度的$n$的公式的数量之间的比例。当公式的长度$n$趋于无穷大时，我们研究了这个分数的渐近行为。这个极限给出了随机选择的BCK公式在BCI中也是可证明的概率度量。我们证明，这个概率趋于零的变量的数量趋于无穷大。第二部分的文件是专门的lambda项表示BCI和BCK逻辑的证明的数量。我们建立了一个比例之间的数量相同长度的证明$n$，我们调查这个比例的渐近行为时，长度的证明趋于无穷大。我们证明了概率为0的随机选择的BCK证明也是BCI公式的证明。
英文摘要：
We investigate quantitative properties of BCI and BCK logics. The first part of the paper 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 behavior 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 paper 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 behavior 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.
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