2,167 research outputs found
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
Deletion codes in the high-noise and high-rate regimes
The noise model of deletions poses significant challenges in coding theory,
with basic questions like the capacity of the binary deletion channel still
being open. In this paper, we study the harder model of worst-case deletions,
with a focus on constructing efficiently decodable codes for the two extreme
regimes of high-noise and high-rate. Specifically, we construct polynomial-time
decodable codes with the following trade-offs (for any eps > 0):
(1) Codes that can correct a fraction 1-eps of deletions with rate poly(eps)
over an alphabet of size poly(1/eps);
(2) Binary codes of rate 1-O~(sqrt(eps)) that can correct a fraction eps of
deletions; and
(3) Binary codes that can be list decoded from a fraction (1/2-eps) of
deletions with rate poly(eps)
Our work is the first to achieve the qualitative goals of correcting a
deletion fraction approaching 1 over bounded alphabets, and correcting a
constant fraction of bit deletions with rate aproaching 1. The above results
bring our understanding of deletion code constructions in these regimes to a
similar level as worst-case errors
(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
Ordering constraints on trees
We survey recent results about ordering constraints on trees and discuss their applications. Our main interest lies in the family of recursive path orderings which enjoy the properties of being total, well-founded and compatible with the tree constructors. The paper includes some new results, in particular the undecidability of the theory of lexicographic path orderings in case of a non-unary signature
The weak pigeonhole principle for function classes in S^1_2
It is well known that S^1_2 cannot prove the injective weak pigeonhole
principle for polynomial time functions unless RSA is insecure. In this note we
investigate the provability of the surjective (dual) weak pigeonhole principle
in S^1_2 for provably weaker function classes.Comment: 11 page
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