55,884 research outputs found
On the equivalence problem for E-pattern languages over small alphabets
We contribute new facets to the discussion on the equivalence
problem for E-pattern languages (also referred to as extended or
erasing pattern languages). This fundamental open question asks for the
existence of a computable function that, given any pair of patterns, decides
whether or not they generate the same language. Our main result
disproves Ohlebusch and Ukkonen’s conjecture (Theoretical Computer
Science 186, 1997) on the equivalence problem; the respective argumentation,
that largely deals with the nondeterminism of pattern languages,
is restricted to terminal alphabets with at most four distinct letters
Bad news on decision problems for patterns
We study the inclusion problem for pattern languages, which
is shown to be undecidable by Jiang et al. (J. Comput. System Sci. 50,
1995). More precisely, Jiang et al. demonstrate that there is no effective
procedure deciding the inclusion for the class of all pattern languages
over all alphabets. Most applications of pattern languages, however, consider
classes over fixed alphabets, and therefore it is practically more
relevant to ask for the existence of alphabet-specific decision procedures.
Our first main result states that, for all but very particular cases, this
version of the inclusion problem is also undecidable. The second main
part of our paper disproves the prevalent conjecture on the inclusion
of so-called similar E-pattern languages, and it explains the devastating
consequences of this result for the intensive previous research on the
most prominent open decision problem for pattern languages, namely
the equivalence problem for general E-pattern languages
Bad news on decision problems for patterns
We study the inclusion problem for pattern languages, which - due to Jiang
et al. (Journal of Computer and System Sciences 50, 1995) - is known to be
undecidable. More precisely, Jiang et al. demonstrate that there is no effective
procedure deciding the inclusion for the class of all pattern languages over all
alphabets. Most applications of pattern languages, however, consider classes
over fixed alphabets, and therefore it is practically more relevant to ask for the
existence of alphabet-specific decision procedures. Our first main result states
that, for all but very particular cases, this version of the inclusion problem is also
undecidable. The second main part of our paper disproves the prevalent conjecture
on the inclusion of so-called similar E-pattern languages, and it explains
the devastating consequences of this result for the intensive previous research
on the most prominent open decision problem for pattern languages, namely
the equivalence problem for general E-pattern languages
A Characterization for Decidable Separability by Piecewise Testable Languages
The separability problem for word languages of a class by
languages of a class asks, for two given languages and
from , whether there exists a language from that
includes and excludes , that is, and . In this work, we assume some mild closure properties for
and study for which such classes separability by a piecewise
testable language (PTL) is decidable. We characterize these classes in terms of
decidability of (two variants of) an unboundedness problem. From this, we
deduce that separability by PTL is decidable for a number of language classes,
such as the context-free languages and languages of labeled vector addition
systems. Furthermore, it follows that separability by PTL is decidable if and
only if one can compute for any language of the class its downward closure wrt.
the scattered substring ordering (i.e., if the set of scattered substrings of
any language of the class is effectively regular).
The obtained decidability results contrast some undecidability results. In
fact, for all (non-regular) language classes that we present as examples with
decidable separability, it is undecidable whether a given language is a PTL
itself.
Our characterization involves a result of independent interest, which states
that for any kind of languages and , non-separability by PTL is
equivalent to the existence of common patterns in and
Decision Problems for Deterministic Pushdown Automata on Infinite Words
The article surveys some decidability results for DPDAs on infinite words
(omega-DPDA). We summarize some recent results on the decidability of the
regularity and the equivalence problem for the class of weak omega-DPDAs.
Furthermore, we present some new results on the parity index problem for
omega-DPDAs. For the specification of a parity condition, the states of the
omega-DPDA are assigned priorities (natural numbers), and a run is accepting if
the highest priority that appears infinitely often during a run is even. The
basic simplification question asks whether one can determine the minimal number
of priorities that are needed to accept the language of a given omega-DPDA. We
provide some decidability results on variations of this question for some
classes of omega-DPDAs.Comment: In Proceedings AFL 2014, arXiv:1405.527
Regular Languages meet Prefix Sorting
Indexing strings via prefix (or suffix) sorting is, arguably, one of the most
successful algorithmic techniques developed in the last decades. Can indexing
be extended to languages? The main contribution of this paper is to initiate
the study of the sub-class of regular languages accepted by an automaton whose
states can be prefix-sorted. Starting from the recent notion of Wheeler graph
[Gagie et al., TCS 2017]-which extends naturally the concept of prefix sorting
to labeled graphs-we investigate the properties of Wheeler languages, that is,
regular languages admitting an accepting Wheeler finite automaton.
Interestingly, we characterize this family as the natural extension of regular
languages endowed with the co-lexicographic ordering: when sorted, the strings
belonging to a Wheeler language are partitioned into a finite number of
co-lexicographic intervals, each formed by elements from a single Myhill-Nerode
equivalence class. Moreover: (i) We show that every Wheeler NFA (WNFA) with
states admits an equivalent Wheeler DFA (WDFA) with at most
states that can be computed in time. This is in sharp contrast with
general NFAs. (ii) We describe a quadratic algorithm to prefix-sort a proper
superset of the WDFAs, a -time online algorithm to sort acyclic
WDFAs, and an optimal linear-time offline algorithm to sort general WDFAs. By
contribution (i), our algorithms can also be used to index any WNFA at the
moderate price of doubling the automaton's size. (iii) We provide a
minimization theorem that characterizes the smallest WDFA recognizing the same
language of any input WDFA. The corresponding constructive algorithm runs in
optimal linear time in the acyclic case, and in time in the
general case. (iv) We show how to compute the smallest WDFA equivalent to any
acyclic DFA in nearly-optimal time.Comment: added minimization theorems; uploaded submitted version; New version
with new results (W-MH theorem, linear determinization), added author:
Giovanna D'Agostin
Trustworthy Refactoring via Decomposition and Schemes: A Complex Case Study
Widely used complex code refactoring tools lack a solid reasoning about the
correctness of the transformations they implement, whilst interest in proven
correct refactoring is ever increasing as only formal verification can provide
true confidence in applying tool-automated refactoring to industrial-scale
code. By using our strategic rewriting based refactoring specification
language, we present the decomposition of a complex transformation into smaller
steps that can be expressed as instances of refactoring schemes, then we
demonstrate the semi-automatic formal verification of the components based on a
theoretical understanding of the semantics of the programming language. The
extensible and verifiable refactoring definitions can be executed in our
interpreter built on top of a static analyser framework.Comment: In Proceedings VPT 2017, arXiv:1708.0688
Deciding definability in FO2(<h,<v) on trees
We provide a decidable characterization of regular forest languages definable
in FO2(<h,<v). By FO2(<h,<v) we refer to the two variable fragment of first
order logic built from the descendant relation and the following sibling
relation. In terms of expressive power it corresponds to a fragment of the
navigational core of XPath that contains modalities for going up to some
ancestor, down to some descendant, left to some preceding sibling, and right to
some following sibling. We also show that our techniques can be applied to
other two variable first-order logics having exactly the same vertical
modalities as FO2(<h,<v) but having different horizontal modalities
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