229 research outputs found
Heinrich Behmann's 1921 lecture on the decision problem and the algebra of logic
Heinrich Behmann (1891-1970) obtained his Habilitation under David Hilbert in
G\"ottingen in 1921 with a thesis on the decision problem. In his thesis, he
solved-independently of L\"owenheim and Skolem's earlier work-the decision
problem for monadic second-order logic in a framework that combined elements of
the algebra of logic and the newer axiomatic approach to logic then being
developed in G\"ottingen. In a talk given in 1921, he outlined this solution,
but also presented important programmatic remarks on the significance of the
decision problem and of decision procedures more generally. The text of this
talk as well as a partial English translation are included
Existential Second-Order Logic Over Graphs: A Complete Complexity-Theoretic Classification
Descriptive complexity theory aims at inferring a problem's computational
complexity from the syntactic complexity of its description. A cornerstone of
this theory is Fagin's Theorem, by which a graph property is expressible in
existential second-order logic (ESO logic) if, and only if, it is in NP. A
natural question, from the theory's point of view, is which syntactic fragments
of ESO logic also still characterize NP. Research on this question has
culminated in a dichotomy result by Gottlob, Kolatis, and Schwentick: for each
possible quantifier prefix of an ESO formula, the resulting prefix class either
contains an NP-complete problem or is contained in P. However, the exact
complexity of the prefix classes inside P remained elusive. In the present
paper, we clear up the picture by showing that for each prefix class of ESO
logic, its reduction closure under first-order reductions is either FO, L, NL,
or NP. For undirected, self-loop-free graphs two containment results are
especially challenging to prove: containment in L for the prefix and containment in FO for the prefix
for monadic . The complex argument by
Gottlob, Kolatis, and Schwentick concerning polynomial time needs to be
carefully reexamined and either combined with the logspace version of
Courcelle's Theorem or directly improved to first-order computations. A
different challenge is posed by formulas with the prefix : We show that they express special constraint satisfaction problems
that lie in L.Comment: Technical report version of a STACS 2015 pape
Synchronizing Data Words for Register Automata
Register automata (RAs) are finite automata extended with a finite set of
registers to store and compare data from an infinite domain. We study the
concept of synchronizing data words in RAs: does there exist a data word that
sends all states of the RA to a single state?
For deterministic RAs with k registers (k-DRAs), we prove that inputting data
words with 2k+1 distinct data from the infinite data domain is sufficient to
synchronize. We show that the synchronization problem for DRAs is in general
PSPACE-complete, and it is NLOGSPACE-complete for 1-DRAs. For nondeterministic
RAs (NRAs), we show that Ackermann(n) distinct data (where n is the size of the
RA) might be necessary to synchronize. The synchronization problem for NRAs is
in general undecidable, however, we establish Ackermann-completeness of the
problem for 1-NRAs.
Another main result is the NEXPTIME-completeness of the length-bounded
synchronization problem for NRAs, where a bound on the length of the
synchronizing data word, written in binary, is given. A variant of this last
construction allows to prove that the length-bounded universality problem for
NRAs is co-NEXPTIME-complete
Ordered Navigation on Multi-attributed Data Words
We study temporal logics and automata on multi-attributed data words.
Recently, BD-LTL was introduced as a temporal logic on data words extending LTL
by navigation along positions of single data values. As allowing for navigation
wrt. tuples of data values renders the logic undecidable, we introduce ND-LTL,
an extension of BD-LTL by a restricted form of tuple-navigation. While complete
ND-LTL is still undecidable, the two natural fragments allowing for either
future or past navigation along data values are shown to be Ackermann-hard, yet
decidability is obtained by reduction to nested multi-counter systems. To this
end, we introduce and study nested variants of data automata as an intermediate
model simplifying the constructions. To complement these results we show that
imposing the same restrictions on BD-LTL yields two 2ExpSpace-complete
fragments while satisfiability for the full logic is known to be as hard as
reachability in Petri nets
Separateness of Variables -- A Novel Perspective on Decidable First-Order Fragments
The classical decision problem, as it is understood today, is the quest for a
delineation between the decidable and the undecidable parts of first-order
logic based on elegant syntactic criteria. In this paper, we treat the concept
of separateness of variables and explore its applicability to the classical
decision problem. Two disjoint sets of first-order variables are separated in a
given formula if variables from the two sets never co-occur in any atom of that
formula. This simple notion facilitates extending many well-known decidable
first-order fragments significantly and in a way that preserves decidability.
We will demonstrate that for several prefix fragments, several guarded
fragments, the two-variable fragment, and for the fluted fragment. Altogether,
we will investigate nine such extensions more closely. Interestingly, each of
them contains the relational monadic first-order fragment without equality.
Although the extensions exhibit the same expressive power as the respective
originals, certain logical properties can be expressed much more succinctly. In
three cases the succinctness gap cannot be bounded using any elementary
function.Comment: 44 pages, 1 figur
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