132 research outputs found
Complete Axiomatizations of Fragments of Monadic Second-Order Logic on Finite Trees
We consider a specific class of tree structures that can represent basic
structures in linguistics and computer science such as XML documents, parse
trees, and treebanks, namely, finite node-labeled sibling-ordered trees. We
present axiomatizations of the monadic second-order logic (MSO), monadic
transitive closure logic (FO(TC1)) and monadic least fixed-point logic
(FO(LFP1)) theories of this class of structures. These logics can express
important properties such as reachability. Using model-theoretic techniques, we
show by a uniform argument that these axiomatizations are complete, i.e., each
formula that is valid on all finite trees is provable using our axioms. As a
backdrop to our positive results, on arbitrary structures, the logics that we
study are known to be non-recursively axiomatizable
Advances and applications of automata on words and trees : abstracts collection
From 12.12.2010 to 17.12.2010, the Dagstuhl Seminar 10501 "Advances and Applications of Automata on Words and Trees" was held in Schloss Dagstuhl - Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available
10501 Abstracts Collection -- Advances and Applications of Automata on Words and Trees
From 12.12.2010 to 17.12.2010, the Dagstuhl Seminar 10501
``Advances and Applications of Automata on Words and Trees\u27\u27 was held
in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Undecidable First-Order Theories of Affine Geometries
Tarski initiated a logic-based approach to formal geometry that studies
first-order structures with a ternary betweenness relation (\beta) and a
quaternary equidistance relation (\equiv). Tarski established, inter alia, that
the first-order (FO) theory of (R^2,\beta,\equiv) is decidable. Aiello and van
Benthem (2002) conjectured that the FO-theory of expansions of (R^2,\beta) with
unary predicates is decidable. We refute this conjecture by showing that for
all n>1, the FO-theory of monadic expansions of (R^2,\beta) is \Pi^1_1-hard and
therefore not even arithmetical. We also define a natural and comprehensive
class C of geometric structures (T,\beta), where T is a subset of R^2, and show
that for each structure (T,\beta) in C, the FO-theory of the class of monadic
expansions of (T,\beta) is undecidable. We then consider classes of expansions
of structures (T,\beta) with restricted unary predicates, for example finite
predicates, and establish a variety of related undecidability results. In
addition to decidability questions, we briefly study the expressivity of
universal MSO and weak universal MSO over expansions of (R^n,\beta). While the
logics are incomparable in general, over expansions of (R^n,\beta), formulae of
weak universal MSO translate into equivalent formulae of universal MSO.
This is an extended version of a publication in the proceedings of the 21st
EACSL Annual Conferences on Computer Science Logic (CSL 2012).Comment: 21 pages, 3 figure
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
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