11 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
The fully compressed subgroup membership problem
Suppose that F is a free group and k is a natural number. We show that the fully compressed membership problem for k-generated subgroups of F is solvable in polynomial time. In order to do this, we adapt the theory of Stallings' foldings to handle edges with compressed labels. This partially answers a question of Markus Lohrey
Approximation of grammar-based compression via recompression
In this paper we present a 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, ..., N^c} for some constant c. Otherwise,
additional cost of O(n log|\Sigma|) is needed.
Algorithms with such approximation guarantees and running time are known, the
novelty of this paper is a particular simplicity of the algorithm as well as
the analysis of the algorithm, which uses a general technique of recompression
recently introduced by the author. Furthermore, contrary to the previous
results, this work does not use the LZ representation of the input string in
the construction, nor in the analysis.Comment: 22 pages, some many small improvements, to be submited to a journa
Subgroup Membership in GL(2,Z)
It is shown that the subgroup membership problem for a virtually free group can be decided in polynomial time where all group elements are represented by so-called power words, i.e., words of the form p_1^{z_1} p_2^{z_2} ? p_k^{z_k}. Here the p_i are explicit words over the generating set of the group and all z_i are binary encoded integers. As a corollary, it follows that the subgroup membership problem for the matrix group GL(2,?) can be decided in polynomial time when all matrix entries are given in binary notation
Context unification is in PSPACE
Contexts are terms with one `hole', i.e. a place in which we can substitute
an argument. In context unification we are given an equation over terms with
variables representing contexts and ask about the satisfiability of this
equation. Context unification is a natural subvariant of second-order
unification, which is undecidable, and a generalization of word equations,
which are decidable, at the same time. It is the unique problem between those
two whose decidability is uncertain (for already almost two decades). In this
paper we show that the context unification is in PSPACE. The result holds under
a (usual) assumption that the first-order signature is finite.
This result is obtained by an extension of the recompression technique,
recently developed by the author and used in particular to obtain a new PSPACE
algorithm for satisfiability of word equations, to context unification. The
recompression is based on performing simple compression rules (replacing pairs
of neighbouring function symbols), which are (conceptually) applied on the
solution of the context equation and modifying the equation in a way so that
such compression steps can be in fact performed directly on the equation,
without the knowledge of the actual solution.Comment: 27 pages, submitted, small notation changes and small improvements
over the previous tex
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
Recompression: a simple and powerful technique for word equations
In this paper we present an application of a simple technique of local
recompression, previously developed by the author in the context of compressed
membership problems and compressed pattern matching, to word equations. The
technique is based on local modification of variables (replacing X by aX or Xa)
and iterative replacement of pairs of letters appearing in the equation by a
`fresh' letter, which can be seen as a bottom-up compression of the solution of
the given word equation, to be more specific, building an SLP (Straight-Line
Programme) for the solution of the word equation.
Using this technique we give a new, independent and self-contained proofs of
most of the known results for word equations. To be more specific, the
presented (nondeterministic) algorithm runs in O(n log n) space and in time
polynomial in log N, where N is the size of the length-minimal solution of the
word equation. The presented algorithm can be easily generalised to a generator
of all solutions of the given word equation (without increasing the space
usage). Furthermore, a further analysis of the algorithm yields a doubly
exponential upper bound on the size of the length-minimal solution. The
presented algorithm does not use exponential bound on the exponent of
periodicity. Conversely, the analysis of the algorithm yields an independent
proof of the exponential bound on exponent of periodicity.
We believe that the presented algorithm, its idea and analysis are far
simpler than all previously applied. Furthermore, thanks to it we can obtain a
unified and simple approach to most of known results for word equations.
As a small additional result we show that for O(1) variables (with arbitrary
many appearances in the equation) word equations can be solved in linear space,
i.e. they are context-sensitive.Comment: Submitted to a journal. Since previous version the proofs were
simplified, overall presentation improve