12,374 research outputs found
Boolean Dependence Logic and Partially-Ordered Connectives
We introduce a new variant of dependence logic called Boolean dependence
logic. In Boolean dependence logic dependence atoms are of the type
=(x_1,...,x_n,\alpha), where \alpha is a Boolean variable. Intuitively, with
Boolean dependence atoms one can express quantification of relations, while
standard dependence atoms express quantification over functions.
We compare the expressive power of Boolean dependence logic to dependence
logic and first-order logic enriched by partially-ordered connectives. We show
that the expressive power of Boolean dependence logic and dependence logic
coincide. We define natural syntactic fragments of Boolean dependence logic and
show that they coincide with the corresponding fragments of first-order logic
enriched by partially-ordered connectives with respect to expressive power. We
then show that the fragments form a strict hierarchy.Comment: 41 page
A Fragment of Dependence Logic Capturing Polynomial Time
In this paper we study the expressive power of Horn-formulae in dependence
logic and show that they can express NP-complete problems. Therefore we define
an even smaller fragment D-Horn* and show that over finite successor structures
it captures the complexity class P of all sets decidable in polynomial time.
Furthermore we study the question which of our results can ge generalized to
the case of open formulae of D-Horn* and so-called downwards monotone
polynomial time properties of teams
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
On Generalizing Decidable Standard Prefix Classes of First-Order Logic
Recently, the separated fragment (SF) of first-order logic has been
introduced. Its defining principle is that universally and existentially
quantified variables may not occur together in atoms. SF properly generalizes
both the Bernays-Sch\"onfinkel-Ramsey (BSR) fragment and the relational monadic
fragment. In this paper the restrictions on variable occurrences in SF
sentences are relaxed such that universally and existentially quantified
variables may occur together in the same atom under certain conditions. Still,
satisfiability can be decided. This result is established in two ways: firstly,
by an effective equivalence-preserving translation into the BSR fragment, and,
secondly, by a model-theoretic argument.
Slight modifications to the described concepts facilitate the definition of
other decidable classes of first-order sentences. The paper presents a second
fragment which is novel, has a decidable satisfiability problem, and properly
contains the Ackermann fragment and---once more---the relational monadic
fragment. The definition is again characterized by restrictions on the
occurrences of variables in atoms. More precisely, after certain
transformations, Skolemization yields only unary functions and constants, and
every atom contains at most one universally quantified variable. An effective
satisfiability-preserving translation into the monadic fragment is devised and
employed to prove decidability of the associated satisfiability problem.Comment: 34 page
Model Checking CTL is Almost Always Inherently Sequential
The model checking problem for CTL is known to be P-complete (Clarke,
Emerson, and Sistla (1986), see Schnoebelen (2002)). We consider fragments of
CTL obtained by restricting the use of temporal modalities or the use of
negations---restrictions already studied for LTL by Sistla and Clarke (1985)
and Markey (2004). For all these fragments, except for the trivial case without
any temporal operator, we systematically prove model checking to be either
inherently sequential (P-complete) or very efficiently parallelizable
(LOGCFL-complete). For most fragments, however, model checking for CTL is
already P-complete. Hence our results indicate that, in cases where the
combined complexity is of relevance, approaching CTL model checking by
parallelism cannot be expected to result in any significant speedup. We also
completely determine the complexity of the model checking problem for all
fragments of the extensions ECTL, CTL+, and ECTL+
Generating Functions For Kernels of Digraphs (Enumeration & Asymptotics for Nim Games)
In this article, we study directed graphs (digraphs) with a coloring
constraint due to Von Neumann and related to Nim-type games. This is equivalent
to the notion of kernels of digraphs, which appears in numerous fields of
research such as game theory, complexity theory, artificial intelligence
(default logic, argumentation in multi-agent systems), 0-1 laws in monadic
second order logic, combinatorics (perfect graphs)... Kernels of digraphs lead
to numerous difficult questions (in the sense of NP-completeness,
#P-completeness). However, we show here that it is possible to use a generating
function approach to get new informations: we use technique of symbolic and
analytic combinatorics (generating functions and their singularities) in order
to get exact and asymptotic results, e.g. for the existence of a kernel in a
circuit or in a unicircuit digraph. This is a first step toward a
generatingfunctionology treatment of kernels, while using, e.g., an approach "a
la Wright". Our method could be applied to more general "local coloring
constraints" in decomposable combinatorial structures.Comment: Presented (as a poster) to the conference Formal Power Series and
Algebraic Combinatorics (Vancouver, 2004), electronic proceeding
Quantified CTL: Expressiveness and Complexity
While it was defined long ago, the extension of CTL with quantification over
atomic propositions has never been studied extensively. Considering two
different semantics (depending whether propositional quantification refers to
the Kripke structure or to its unwinding tree), we study its expressiveness
(showing in particular that QCTL coincides with Monadic Second-Order Logic for
both semantics) and characterise the complexity of its model-checking and
satisfiability problems, depending on the number of nested propositional
quantifiers (showing that the structure semantics populates the polynomial
hierarchy while the tree semantics populates the exponential hierarchy)
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