2,351 research outputs found
(Un)Decidability Results for Word Equations with Length and Regular Expression Constraints
We prove several decidability and undecidability results for the
satisfiability and validity problems for languages that can express solutions
to word equations with length constraints. The atomic formulas over this
language are equality over string terms (word equations), linear inequality
over the length function (length constraints), and membership in regular sets.
These questions are important in logic, program analysis, and formal
verification. Variants of these questions have been studied for many decades by
mathematicians. More recently, practical satisfiability procedures (aka SMT
solvers) for these formulas have become increasingly important in the context
of security analysis for string-manipulating programs such as web applications.
We prove three main theorems. First, we give a new proof of undecidability
for the validity problem for the set of sentences written as a forall-exists
quantifier alternation applied to positive word equations. A corollary of this
undecidability result is that this set is undecidable even with sentences with
at most two occurrences of a string variable. Second, we consider Boolean
combinations of quantifier-free formulas constructed out of word equations and
length constraints. We show that if word equations can be converted to a solved
form, a form relevant in practice, then the satisfiability problem for Boolean
combinations of word equations and length constraints is decidable. Third, we
show that the satisfiability problem for quantifier-free formulas over word
equations in regular solved form, length constraints, and the membership
predicate over regular expressions is also decidable.Comment: Invited Paper at ADDCT Workshop 2013 (co-located with CADE 2013
Decidability in the logic of subsequences and supersequences
We consider first-order logics of sequences ordered by the subsequence
ordering, aka sequence embedding. We show that the \Sigma_2 theory is
undecidable, answering a question left open by Kuske. Regarding fragments with
a bounded number of variables, we show that the FO2 theory is decidable while
the FO3 theory is undecidable
Expansions of MSO by cardinality relations
We study expansions of the Weak Monadic Second Order theory of (N,<) by
cardinality relations, which are predicates R(X1,...,Xn) whose truth value
depends only on the cardinality of the sets X1, ...,Xn. We first provide a
(definable) criterion for definability of a cardinality relation in (N,<), and
use it to prove that for every cardinality relation R which is not definable in
(N,<), there exists a unary cardinality relation which is definable in (N,<,R)
and not in (N,<). These results resemble Muchnik and Michaux-Villemaire
theorems for Presburger Arithmetic. We prove then that + and x are definable in
(N,<,R) for every cardinality relation R which is not definable in (N,<). This
implies undecidability of the WMSO theory of (N,<,R). We also consider the
related satisfiability problem for the class of finite orderings, namely the
question whether an MSO sentence in the language {<,R} admits a finite model M
where < is interpreted as a linear ordering, and R as the restriction of some
(fixed) cardinality relation to the domain of M. We prove that this problem is
undecidable for every cardinality relation R which is not definable in (N,<).Comment: to appear in LMC
Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0
Let be a one-variable function field over a field of constants of
characteristic 0. Let be a holomorphy subring of , not equal to . We
prove the following undecidability results for : If is recursive, then
Hilbert's Tenth Problem is undecidable in . In general, there exist
such that there is no algorithm to tell whether a
polynomial equation with coefficients in \Q(x_1,...,x_n) has solutions in
.Comment: This version contains minor revisions and will appear in Annales de l
Institut Fourie
The Decidability Frontier for Probabilistic Automata on Infinite Words
We consider probabilistic automata on infinite words with acceptance defined
by safety, reachability, B\"uchi, coB\"uchi, and limit-average conditions. We
consider quantitative and qualitative decision problems. We present extensions
and adaptations of proofs for probabilistic finite automata and present a
complete characterization of the decidability and undecidability frontier of
the quantitative and qualitative decision problems for probabilistic automata
on infinite words
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
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