We introduce the class of P-finite automata. These are a generalisation of weighted automata, in which the weights of transitions can depend polynomially on the length of the input word. P-finite automata can also be viewed as simple tail-recursive programs in which the arguments of recursive calls can non-linearly refer to a variable that counts the number of recursive calls. The nomenclature is motivated by the fact that over a unary alphabet P-finite automata compute so-called P-finite sequences, that is, sequences that satisfy a linear recurrence with polynomial coefficients. Our main result shows that P-finite automata can be learned in polynomial time in Angluin's MAT exact learning model. This generalises the classical results that deterministic finite automata and weighted automata over a field are respectively polynomial-time learnable in the MAT model.

翻译：我们引入了P-有限自动机这一概念。该类自动机是加权自动机的推广,其中转移的权重可以多项式地依赖于输入词的长度。P-有限自动机也可被视为简单的尾递归程序,其中递归调用的参数可以非线性地引用一个记录递归调用次数的变量。这一命名的动机在于,在单字母表上,P-有限自动机计算所谓的P-有限序列,即满足具有多项式系数的线性递归关系的序列。我们的主要结果表明,在Angluin的MAT精确学习模型中,P-有限自动机可以在多项式时间内被学习。这一结果推广了经典结论,即确定性有限自动机以及域上加权自动机在MAT模型中分别是多项式时间可学习的。