119 research outputs found
Compressed Membership for NFA (DFA) with Compressed Labels is in NP (P)
In this paper, a compressed membership problem for finite automata, both
deterministic and non-deterministic, with compressed transition labels is
studied. The compression is represented by straight-line programs (SLPs), i.e.
context-free grammars generating exactly one string. A novel technique of
dealing with SLPs is introduced: the SLPs are recompressed, so that substrings
of the input text are encoded in SLPs labelling the transitions of the NFA
(DFA) in the same way, as in the SLP representing the input text. To this end,
the SLPs are locally decompressed and then recompressed in a uniform way.
Furthermore, such recompression induces only small changes in the automaton, in
particular, the size of the automaton remains polynomial.
Using this technique it is shown that the compressed membership for NFA with
compressed labels is in NP, thus confirming the conjecture of Plandowski and
Rytter and extending the partial result of Lohrey and Mathissen; as it is
already known, that this problem is NP-hard, we settle its exact computational
complexity. Moreover, the same technique applied to the compressed membership
for DFA with compressed labels yields that this problem is in P; for this
problem, only trivial upper-bound PSPACE was known
Tree-Automatic Well-Founded Trees
We investigate tree-automatic well-founded trees. Using Delhomme's
decomposition technique for tree-automatic structures, we show that the
(ordinal) rank of a tree-automatic well-founded tree is strictly below
omega^omega. Moreover, we make a step towards proving that the ranks of
tree-automatic well-founded partial orders are bounded by omega^omega^omega: we
prove this bound for what we call upwards linear partial orders. As an
application of our result, we show that the isomorphism problem for
tree-automatic well-founded trees is complete for level Delta^0_{omega^omega}
of the hyperarithmetical hierarchy with respect to Turing-reductions.Comment: Will appear in Logical Methods of Computer Scienc
A really simple approximation of smallest grammar
In this paper we present a really simple linear-time algorithm constructing a
context-free grammar of size O(g log (N/g)) for the input string, where N is
the size of the input string and g the size of the optimal grammar generating
this string. The algorithm works for arbitrary size alphabets, but the running
time is linear assuming that the alphabet Sigma of the input string can be
identified with numbers from 1,ldots, N^c for some constant c. Algorithms with
such an approximation guarantee and running time are known, however all of them
were non-trivial and their analyses were involved. The here presented algorithm
computes the LZ77 factorisation and transforms it in phases to a grammar. In
each phase it maintains an LZ77-like factorisation of the word with at most l
factors as well as additional O(l) letters, where l was the size of the
original LZ77 factorisation. In one phase in a greedy way (by a left-to-right
sweep and a help of the factorisation) we choose a set of pairs of consecutive
letters to be replaced with new symbols, i.e. nonterminals of the constructed
grammar. We choose at least 2/3 of the letters in the word and there are O(l)
many different pairs among them. Hence there are O(log N) phases, each of them
introduces O(l) nonterminals to a grammar. A more precise analysis yields a
bound O(l log(N/l)). As l \leq g, this yields the desired bound O(g log(N/g)).Comment: Accepted for CPM 201
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
Deciding Equivalence of Linear Tree-to-Word Transducers in Polynomial Time
We show that the equivalence of deterministic linear top-down tree-to-word
transducers is decidable in polynomial time. Linear tree-to-word transducers
are non-copying but not necessarily order-preserving and can be used to express
XML and other document transformations. The result is based on a partial normal
form that provides a basic characterization of the languages produced by linear
tree-to-word transducers.Comment: short version of this paper will be published in the proceedings of
the 20th Conference on Developments in Language Theory (DLT 2016), Montreal,
Canad
A Risk Assessment of Intangible Asset Valuation: The Post-Hoc Association between Goodwill Impairments and Risk Hazards in Mergers and Acquisitions
We tested whether the change from SFAS 141 to SFAS 141r/ASC 805 had any effect on restatements due to goodwill impairment. Our findings suggest that the implementation of SFAS 141r increased the likelihood of a financial restatement by 2.5 times. Board of Director and Audit Committee involvement in the goodwill impairment decision reduced the likelihood of a restatement occurring. Service industry companies were 3.2 times more likely to restate their assets due to goodwill impairment. Companies who were audited by a Big4 firm reduced the odds of restatement by 47%
Processing Succinct Matrices and Vectors
We study the complexity of algorithmic problems for matrices that are
represented by multi-terminal decision diagrams (MTDD). These are a variant of
ordered decision diagrams, where the terminal nodes are labeled with arbitrary
elements of a semiring (instead of 0 and 1). A simple example shows that the
product of two MTDD-represented matrices cannot be represented by an MTDD of
polynomial size. To overcome this deficiency, we extended MTDDs to MTDD_+ by
allowing componentwise symbolic addition of variables (of the same dimension)
in rules. It is shown that accessing an entry, equality checking, matrix
multiplication, and other basic matrix operations can be solved in polynomial
time for MTDD_+-represented matrices. On the other hand, testing whether the
determinant of a MTDD-represented matrix vanishes PSPACE$-complete, and the
same problem is NP-complete for MTDD_+-represented diagonal matrices. Computing
a specific entry in a product of MTDD-represented matrices is #P-complete.Comment: An extended abstract of this paper will appear in the Proceedings of
CSR 201
One-variable word equations in linear time
In this paper we consider word equations with one variable (and arbitrary
many appearances of it). A recent technique of recompression, which is
applicable to general word equations, is shown to be suitable also in this
case. While in general case it is non-deterministic, it determinises in case of
one variable and the obtained running time is O(n + #_X log n), where #_X is
the number of appearances of the variable in the equation. This matches the
previously-best algorithm due to D\k{a}browski and Plandowski. Then, using a
couple of heuristics as well as more detailed time analysis the running time is
lowered to O(n) in RAM model. Unfortunately no new properties of solutions are
shown.Comment: submitted to a journal, general overhaul over the previous versio
Silent Transitions in Automata with Storage
We consider the computational power of silent transitions in one-way automata
with storage. Specifically, we ask which storage mechanisms admit a
transformation of a given automaton into one that accepts the same language and
reads at least one input symbol in each step.
We study this question using the model of valence automata. Here, a finite
automaton is equipped with a storage mechanism that is given by a monoid.
This work presents generalizations of known results on silent transitions.
For two classes of monoids, it provides characterizations of those monoids that
allow the removal of \lambda-transitions. Both classes are defined by graph
products of copies of the bicyclic monoid and the group of integers. The first
class contains pushdown storages as well as the blind counters while the second
class contains the blind and the partially blind counters.Comment: 32 pages, submitte
Compressed Tree Canonization
Straight-line (linear) context-free tree (SLT) grammars have been used to
compactly represent ordered trees. It is well known that equivalence of SLT
grammars is decidable in polynomial time. Here we extend this result and show
that isomorphism of unordered trees given as SLT grammars is decidable in
polynomial time. The proof constructs a compressed version of the canonical
form of the tree represented by the input SLT grammar. The result is
generalized to unrooted trees by "re-rooting" the compressed trees in
polynomial time. We further show that bisimulation equivalence of unrooted
unordered trees represented by SLT grammars is decidable in polynomial time.
For non-linear SLT grammars which can have double-exponential compression
ratios, we prove that unordered isomorphism is PSPACE-hard and in EXPTIME. The
same complexity bounds are shown for bisimulation equivalence
- …