One-Counter Automata with Counter Observability

In a one-counter automaton (OCA), one can produce a letter from some finite alphabet, increment and decrement the counter by one, or compare it with constants up to some threshold. It is well-known that universality and language inclusion for OCAs are undecidable. In this paper, we consider OCAs with counter observability: Whenever the automaton produces a letter, it outputs the current counter value along with it. Hence, its language is now a set of words over an infinite alphabet. We show that universality and inclusion for that model are PSPACE-complete, thus no harder than the corresponding problems for finite automata. In fact, by establishing a link with visibly one-counter automata, we show that OCAs with counter observability are effectively determinizable and closed under all boolean operations. Moreover, it turns out that they are expressively equivalent to strong automata, in which transitions are guarded by MSO formulas over the natural numbers with successor

Incompleteness Theorems, Large Cardinals, and Automata over Finite Words

International audienceWe prove that one can construct various kinds of automata over finite words for which some elementary properties are actually independent from strong set theories like Tn =:ZFC + "There exist (at least) n inaccessible cardinals", for integers n ≥ 0. In particular, we prove independence results for languages of finite words generated by context-free grammars, or accepted by 2-tape or 1-counter automata. Moreover we get some independence results for weighted automata and for some related finitely generated subsemigroups of the set Z ^{3×3} of 3-3 matrices with integer entries. Some of these latter results are independence results from the Peano axiomatic system PA

One-Counter Automata with Counter Observability

International audienceIn a one-counter automaton (OCA), one can produce a letter from some finite alphabet, increment and decrement the counter by one, or compare it with constants up to some threshold. It is well-known that universality and language inclusion for OCAs are undecidable. In this paper, we consider OCAs with counter observability: Whenever the automaton produces a letter, it outputs the current counter value along with it. Hence, its language is now a set of words over an infinite alphabet. We show that universality and inclusion for that model are PSPACE-complete, thus no harder than the corresponding problems for finite automata. In fact, by establishing a link with visibly one-counter automata, we show that OCAs with counter observability are e ectively determinizable and closed under all boolean operations. Moreover, it turns out that they are expressively equivalent to strong automata, in which transitions are guarded by MSO formulas over the natural numbers with successor