41 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
Decision problems for Clark-congruential languages
A common question when studying a class of context-free grammars is whether
equivalence is decidable within this class. We answer this question positively
for the class of Clark-congruential grammars, which are of interest to
grammatical inference. We also consider the problem of checking whether a given
CFG is Clark-congruential, and show that it is decidable given that the CFG is
a DCFG.Comment: Version 2 incorporates revisions prompted by the comments of
anonymous referees at ICGI and LearnAu
Equivalence of Deterministic One-Counter Automata is NL-complete
We prove that language equivalence of deterministic one-counter automata is
NL-complete. This improves the superpolynomial time complexity upper bound
shown by Valiant and Paterson in 1975. Our main contribution is to prove that
two deterministic one-counter automata are inequivalent if and only if they can
be distinguished by a word of length polynomial in the size of the two input
automata
Bisimulation Equivalence of First-Order Grammars is ACKERMANN-Complete
Checking whether two pushdown automata with restricted silent actions are
weakly bisimilar was shown decidable by S\'enizergues (1998, 2005). We provide
the first known complexity upper bound for this famous problem, in the
equivalent setting of first-order grammars. This ACKERMANN upper bound is
optimal, and we also show that strong bisimilarity is primitive-recursive when
the number of states of the automata is fixed
Bisimilarity of Pushdown Systems is Nonelementary
Given two pushdown systems, the bisimilarity problem asks whether they are
bisimilar. While this problem is known to be decidable our main result states
that it is nonelementary, improving EXPTIME-hardness, which was the previously
best known lower bound for this problem. Our lower bound result holds for
normed pushdown systems as well
Deciding Semantic Finiteness of Pushdown Processes and First-Order Grammars w.r.t. Bisimulation Equivalence
The problem if a given configuration of a pushdown automaton (PDA) is
bisimilar with some (unspecified) finite-state process is shown to be
decidable. The decidability is proven in the framework of first-order grammars, which are given by finite sets of labelled rules that rewrite roots of first-order terms. The framework is equivalent to PDA where also deterministic popping epsilon-steps are allowed, i.e. to the model for which Senizergues showed an involved procedure deciding bisimilarity (FOCS 1998). Such a procedure is here used as a black-box part of the algorithm. For deterministic PDA the regularity problem was shown decidable by Valiant (JACM 1975) but the decidability question for nondeterministic PDA, answered positively here, had been open (as indicated, e.g., by Broadbent and Goeller, FSTTCS 2012)
Fragments of ML Decidable by Nested Data Class Memory Automata
The call-by-value language RML may be viewed as a canonical restriction of
Standard ML to ground-type references, augmented by a "bad variable" construct
in the sense of Reynolds. We consider the fragment of (finitary) RML terms of
order at most 1 with free variables of order at most 2, and identify two
subfragments of this for which we show observational equivalence to be
decidable. The first subfragment consists of those terms in which the
P-pointers in the game semantic representation are determined by the underlying
sequence of moves. The second subfragment consists of terms in which the
O-pointers of moves corresponding to free variables in the game semantic
representation are determined by the underlying moves. These results are shown
using a reduction to a form of automata over data words in which the data
values have a tree-structure, reflecting the tree-structure of the threads in
the game semantic plays. In addition we show that observational equivalence is
undecidable at every third- or higher-order type, every second-order type which
takes at least two first-order arguments, and every second-order type (of arity
greater than one) that has a first-order argument which is not the final
argument