199 research outputs found
Parikh Image of Pushdown Automata
We compare pushdown automata (PDAs for short) against other representations.
First, we show that there is a family of PDAs over a unary alphabet with
states and stack symbols that accepts one single long word for
which every equivalent context-free grammar needs
variables. This family shows that the classical algorithm for converting a PDA
to an equivalent context-free grammar is optimal even when the alphabet is
unary. Moreover, we observe that language equivalence and Parikh equivalence,
which ignores the ordering between symbols, coincide for this family. We
conclude that, when assuming this weaker equivalence, the conversion algorithm
is also optimal. Second, Parikh's theorem motivates the comparison of PDAs
against finite state automata. In particular, the same family of unary PDAs
gives a lower bound on the number of states of every Parikh-equivalent finite
state automaton. Finally, we look into the case of unary deterministic PDAs. We
show a new construction converting a unary deterministic PDA into an equivalent
context-free grammar that achieves best known bounds.Comment: 17 pages, 2 figure
On Probabilistic Parallel Programs with Process Creation and Synchronisation
We initiate the study of probabilistic parallel programs with dynamic process
creation and synchronisation. To this end, we introduce probabilistic
split-join systems (pSJSs), a model for parallel programs, generalising both
probabilistic pushdown systems (a model for sequential probabilistic procedural
programs which is equivalent to recursive Markov chains) and stochastic
branching processes (a classical mathematical model with applications in
various areas such as biology, physics, and language processing). Our pSJS
model allows for a possibly recursive spawning of parallel processes; the
spawned processes can synchronise and return values. We study the basic
performance measures of pSJSs, especially the distribution and expectation of
space, work and time. Our results extend and improve previously known results
on the subsumed models. We also show how to do performance analysis in
practice, and present two case studies illustrating the modelling power of
pSJSs.Comment: This is a technical report accompanying a TACAS'11 pape
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
Synchronous Subsequentiality and Approximations to Undecidable Problems
We introduce the class of synchronous subsequential relations, a subclass of
the synchronous relations which embodies some properties of subsequential
relations. If we take relations of this class as forming the possible
transitions of an infinite automaton, then most decision problems (apart from
membership) still remain undecidable (as they are for synchronous and
subsequential rational relations), but on the positive side, they can be
approximated in a meaningful way we make precise in this paper. This might make
the class useful for some applications, and might serve to establish an
intermediate position in the trade-off between issues of expressivity and
(un)decidability.Comment: In Proceedings GandALF 2015, arXiv:1509.0685
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
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