719 research outputs found
Unary Pushdown Automata and Straight-Line Programs
We consider decision problems for deterministic pushdown automata over a
unary alphabet (udpda, for short). Udpda are a simple computation model that
accept exactly the unary regular languages, but can be exponentially more
succinct than finite-state automata. We complete the complexity landscape for
udpda by showing that emptiness (and thus universality) is P-hard, equivalence
and compressed membership problems are P-complete, and inclusion is
coNP-complete. Our upper bounds are based on a translation theorem between
udpda and straight-line programs over the binary alphabet (SLPs). We show that
the characteristic sequence of any udpda can be represented as a pair of
SLPs---one for the prefix, one for the lasso---that have size linear in the
size of the udpda and can be computed in polynomial time. Hence, decision
problems on udpda are reduced to decision problems on SLPs. Conversely, any SLP
can be converted in logarithmic space into a udpda, and this forms the basis
for our lower bound proofs. We show coNP-hardness of the ordered matching
problem for SLPs, from which we derive coNP-hardness for inclusion. In
addition, we complete the complexity landscape for unary nondeterministic
pushdown automata by showing that the universality problem is -hard, using a new class of integer expressions. Our techniques have
applications beyond udpda. We show that our results imply -completeness for a natural fragment of Presburger arithmetic and coNP lower
bounds for compressed matching problems with one-character wildcards
Multi-Head Finite Automata: Characterizations, Concepts and Open Problems
Multi-head finite automata were introduced in (Rabin, 1964) and (Rosenberg,
1966). Since that time, a vast literature on computational and descriptional
complexity issues on multi-head finite automata documenting the importance of
these devices has been developed. Although multi-head finite automata are a
simple concept, their computational behavior can be already very complex and
leads to undecidable or even non-semi-decidable problems on these devices such
as, for example, emptiness, finiteness, universality, equivalence, etc. These
strong negative results trigger the study of subclasses and alternative
characterizations of multi-head finite automata for a better understanding of
the nature of non-recursive trade-offs and, thus, the borderline between
decidable and undecidable problems. In the present paper, we tour a fragment of
this literature
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
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