43,860 research outputs found
Computational Complexity of Synchronization under Regular Commutative Constraints
Here we study the computational complexity of the constrained synchronization
problem for the class of regular commutative constraint languages. Utilizing a
vector representation of regular commutative constraint languages, we give a
full classification of the computational complexity of the constraint
synchronization problem. Depending on the constraint language, our problem
becomes PSPACE-complete, NP-complete or polynomial time solvable. In addition,
we derive a polynomial time decision procedure for the complexity of the
constraint synchronization problem, given some constraint automaton accepting a
commutative language as input.Comment: Published in COCOON 2020 (The 26th International Computing and
Combinatorics Conference); 2nd version is update of the published version and
1st version; both contain a minor error, the assumption of maximality in the
NP-c and PSPACE-c results (propositions 5 & 6) is missing, and of
incomparability of the vectors in main theorem; fixed in this version. See
(new) discussion after main theore
Mean-payoff Automaton Expressions
Quantitative languages are an extension of boolean languages that assign to
each word a real number. Mean-payoff automata are finite automata with
numerical weights on transitions that assign to each infinite path the long-run
average of the transition weights. When the mode of branching of the automaton
is deterministic, nondeterministic, or alternating, the corresponding class of
quantitative languages is not robust as it is not closed under the pointwise
operations of max, min, sum, and numerical complement. Nondeterministic and
alternating mean-payoff automata are not decidable either, as the quantitative
generalization of the problems of universality and language inclusion is
undecidable.
We introduce a new class of quantitative languages, defined by mean-payoff
automaton expressions, which is robust and decidable: it is closed under the
four pointwise operations, and we show that all decision problems are decidable
for this class. Mean-payoff automaton expressions subsume deterministic
mean-payoff automata, and we show that they have expressive power incomparable
to nondeterministic and alternating mean-payoff automata. We also present for
the first time an algorithm to compute distance between two quantitative
languages, and in our case the quantitative languages are given as mean-payoff
automaton expressions
What's Decidable About Sequences?
We present a first-order theory of sequences with integer elements,
Presburger arithmetic, and regular constraints, which can model significant
properties of data structures such as arrays and lists. We give a decision
procedure for the quantifier-free fragment, based on an encoding into the
first-order theory of concatenation; the procedure has PSPACE complexity. The
quantifier-free fragment of the theory of sequences can express properties such
as sortedness and injectivity, as well as Boolean combinations of periodic and
arithmetic facts relating the elements of the sequence and their positions
(e.g., "for all even i's, the element at position i has value i+3 or 2i"). The
resulting expressive power is orthogonal to that of the most expressive
decidable logics for arrays. Some examples demonstrate that the fragment is
also suitable to reason about sequence-manipulating programs within the
standard framework of axiomatic semantics.Comment: Fixed a few lapses in the Mergesort exampl
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