13 research outputs found
Tightening the Complexity of Equivalence Problems for Commutative Grammars
We show that the language equivalence problem for regular and context-free
commutative grammars is coNEXP-complete. In addition, our lower bound
immediately yields further coNEXP-completeness results for equivalence problems
for communication-free Petri nets and reversal-bounded counter automata.
Moreover, we improve both lower and upper bounds for language equivalence for
exponent-sensitive commutative grammars.Comment: 21 page
The complexity of equivalence problems for commutative grammars
In this paper we investigate the computational complexity of the inequivalence problems for commutative grammars. We show that the inequivalence problems for type 0 and context-sensitive commutative grammars are undecidable whereas decidability in nondeterministic exponential-time holds for the classes of regular and context-free commutative grammars. For the latter the inequivalence problems are ÎŁp2-hard
Integer Vector Addition Systems with States
This paper studies reachability, coverability and inclusion problems for
Integer Vector Addition Systems with States (ZVASS) and extensions and
restrictions thereof. A ZVASS comprises a finite-state controller with a finite
number of counters ranging over the integers. Although it is folklore that
reachability in ZVASS is NP-complete, it turns out that despite their
naturalness, from a complexity point of view this class has received little
attention in the literature. We fill this gap by providing an in-depth analysis
of the computational complexity of the aforementioned decision problems. Most
interestingly, it turns out that while the addition of reset operations to
ordinary VASS leads to undecidability and Ackermann-hardness of reachability
and coverability, respectively, they can be added to ZVASS while retaining
NP-completness of both coverability and reachability.Comment: 17 pages, 2 figure
The Complexity of Downward Closure Comparisons
The downward closure of a language is the set of all (not necessarily contiguous) subwords of its members. It is well-known that the downward closure of every language is regular. Moreover, recent results show that downward closures are computable for quite powerful system models.
One advantage of abstracting a language by its downward closure is that then equivalence and inclusion become decidable. In this work, we study the complexity of these two problems. More precisely, we consider the following decision problems: Given languages K and L from classes C and D, respectively, does the downward closure of K include (equal) that of L?
These problems are investigated for finite automata, one-counter automata, context-free grammars, and reversal-bounded counter automata. For each combination, we prove a completeness result either for fixed or for arbitrary alphabets. Moreover, for Petri net languages, we show that both problems are Ackermann-hard and for higher-order pushdown automata of order k, we prove hardness for complements of nondeterministic k-fold exponential time
Complexity of Problems of Commutative Grammars
We consider commutative regular and context-free grammars, or, in other
words, Parikh images of regular and context-free languages. By using linear
algebra and a branching analog of the classic Euler theorem, we show that,
under an assumption that the terminal alphabet is fixed, the membership problem
for regular grammars (given v in binary and a regular commutative grammar G,
does G generate v?) is P, and that the equivalence problem for context free
grammars (do G_1 and G_2 generate the same language?) is in
The Complexity of Knapsack Problems in Wreath Products
We prove new complexity results for computational problems in certain wreath
products of groups and (as an application) for free solvable group. For a
finitely generated group we study the so-called power word problem (does a
given expression , where are
words over the group generators and are binary encoded
integers, evaluate to the group identity?) and knapsack problem (does a given
equation , where are words
over the group generators and are variables, has a solution in
the natural numbers). We prove that the power word problem for wreath products
of the form with nilpotent and iterated wreath products
of free abelian groups belongs to . As an application of the
latter, the power word problem for free solvable groups is in .
On the other hand we show that for wreath products , where
is a so called uniformly strongly efficiently non-solvable group (which
form a large subclass of non-solvable groups), the power word problem is
-hard. For the knapsack problem we show
-completeness for iterated wreath products of free abelian groups
and hence free solvable groups. Moreover, the knapsack problem for every wreath
product , where is uniformly efficiently non-solvable, is
-hard
A Type Checking Algorithm for Concurrent Object Protocols
Concurrent objects can be accessed and possibly modified concurrently by several running processes. It is notoriously difficult to make sure that such objects are consistent with – and are used according to – their intended protocol. In this paper we detail a type checking algorithm for concurrent objects protocols that provides automated support for this verification task. We model concurrent objects in the Objective Join Calculus and specify protocols using terms of a Commutative Kleene Algebra. The presented results are an essential first step towards the application of this static analysis technique to real-world programs