Finite Tree Automata with Cost Functions

Cost functions for tree automata are mappings from transitions to (tuples o0 polynomials over some semiring. We consider four semirings, namely N the semiring of nonnegative integers, A the "arctical semiring", T the tropical semiring and F the semiring of finite subsets of nonnegative integers. We show: for semirings N and A it is decidable in polynomial time whether or not the costs of accepting computations is bounded; for F it is decidable in polynomial time whether or not tile cardinality of occurring cost sets is bounded. In all three cases we derive explicit upper bounds. For semiring T we are able to derive similar results at least in case of polynomials of degree at most 1. For N and A we extend our results to multi-dimensional cost functions. 0. I n t roduc t ion Finite tree automata are finite state devices which operate on labeled ordered trees. Cost functions c for tree automata map transitions to (tuples of) polynomials over some semiring R such that every computation obtains a cost c(~)) in R (resp. R d in the mult i -dimensional case). A pair of a finite tree automaton and a cost function is called cost automaton. There are two reasons why we are interested in finite tree automata with cost functions. First, finite tree automata are an important tool o f ' compi l e r generating systems like O P T R A N where they are applied for generating code selectors f rom descriptions o f target machine assembly languages [GieSch88, WeiWi88]. A generated tree automaton A is meant to traverse the abstract syntax tree o f an input program. The different accepting computat ions o f A correspond to different possibilities of target machine code generation. F rom these the "cheapest" one is selected according to some suitable cost measure. In fact, these measures are of a type similar to the cost functions we consider here. The second reason why we are interested in cost automata is o f a more theoretical nature. In [CouMo90] Courcelle and Mosbah investigated MS (i.e. Monadic Second-Order) evaluations