333 research outputs found
The Church Problem for Countable Ordinals
A fundamental theorem of Buchi and Landweber shows that the Church synthesis
problem is computable. Buchi and Landweber reduced the Church Problem to
problems about ω-games and used the determinacy of such games as one of
the main tools to show its computability. We consider a natural generalization
of the Church problem to countable ordinals and investigate games of arbitrary
countable length. We prove that determinacy and decidability parts of the
Bu}chi and Landweber theorem hold for all countable ordinals and that its full
extension holds for all ordinals < \omega\^\omega
A Proof of Kamp's theorem
We provide a simple proof of Kamp's theorem
A Proof of Stavi's Theorem
Kamp's theorem established the expressive equivalence of the temporal logic
with Until and Since and the First-Order Monadic Logic of Order (FOMLO) over
the Dedekind-complete time flows. However, this temporal logic is not
expressively complete for FOMLO over the rationals. Stavi introduced two
additional modalities and proved that the temporal logic with Until, Since and
Stavi's modalities is expressively equivalent to FOMLO over all linear orders.
We present a simple proof of Stavi's theorem.Comment: arXiv admin note: text overlap with arXiv:1401.258
The Church Synthesis Problem with Parameters
For a two-variable formula ψ(X,Y) of Monadic Logic of Order (MLO) the
Church Synthesis Problem concerns the existence and construction of an operator
Y=F(X) such that ψ(X,F(X)) is universally valid over Nat.
B\"{u}chi and Landweber proved that the Church synthesis problem is
decidable; moreover, they showed that if there is an operator F that solves the
Church Synthesis Problem, then it can also be solved by an operator defined by
a finite state automaton or equivalently by an MLO formula. We investigate a
parameterized version of the Church synthesis problem. In this version ψ
might contain as a parameter a unary predicate P. We show that the Church
synthesis problem for P is computable if and only if the monadic theory of
is decidable. We prove that the B\"{u}chi-Landweber theorem can be
extended only to ultimately periodic parameters. However, the MLO-definability
part of the B\"{u}chi-Landweber theorem holds for the parameterized version of
the Church synthesis problem
Synthesis of Finite-state and Definable Winning Strategies
Church\u27s Problem asks for the construction of a procedure which,
given a logical specification on sequence pairs, realizes
for any input sequence an output sequence such that
satisfies . McNaughton reduced Church\u27s Problem to a problem about two-player-games.
B"uchi and Landweber gave a solution for
Monadic Second-Order Logic of Order () specifications in terms of finite-state strategies.
We consider two natural generalizations of the Church problem to
countable ordinals: the first deals with finite-state strategies;
the second deals with -definable strategies. We investigate
games of arbitrary countable length and prove the computability of
these generalizations of Church\u27s problem
ParseNet: Looking Wider to See Better
We present a technique for adding global context to deep convolutional
networks for semantic segmentation. The approach is simple, using the average
feature for a layer to augment the features at each location. In addition, we
study several idiosyncrasies of training, significantly increasing the
performance of baseline networks (e.g. from FCN). When we add our proposed
global feature, and a technique for learning normalization parameters, accuracy
increases consistently even over our improved versions of the baselines. Our
proposed approach, ParseNet, achieves state-of-the-art performance on SiftFlow
and PASCAL-Context with small additional computational cost over baselines, and
near current state-of-the-art performance on PASCAL VOC 2012 semantic
segmentation with a simple approach. Code is available at
https://github.com/weiliu89/caffe/tree/fcn .Comment: ICLR 2016 submissio
Degrees of Ambiguity for Parity Tree Automata
An automaton is unambiguous if for every input it has at most one accepting computation. An automaton is finitely (respectively, countably) ambiguous if for every input it has at most finitely (respectively, countably) many accepting computations. An automaton is boundedly ambiguous if there is k ? ?, such that for every input it has at most k accepting computations. We consider Parity Tree Automata (PTA) and prove that the problem whether a PTA is not unambiguous (respectively, is not boundedly ambiguous, not finitely ambiguous) is co-NP complete, and the problem whether a PTA is not countably ambiguous is co-NP hard
A Logic of Reachable Patterns in Linked Data-Structures
We define a new decidable logic for expressing and checking invariants of
programs that manipulate dynamically-allocated objects via pointers and
destructive pointer updates. The main feature of this logic is the ability to
limit the neighborhood of a node that is reachable via a regular expression
from a designated node. The logic is closed under boolean operations
(entailment, negation) and has a finite model property. The key technical
result is the proof of decidability. We show how to express precondition,
postconditions, and loop invariants for some interesting programs. It is also
possible to express properties such as disjointness of data-structures, and
low-level heap mutations. Moreover, our logic can express properties of
arbitrary data-structures and of an arbitrary number of pointer fields. The
latter provides a way to naturally specify postconditions that relate the
fields on entry to a procedure to the fields on exit. Therefore, it is possible
to use the logic to automatically prove partial correctness of programs
performing low-level heap mutations
Ambiguity Hierarchy of Regular Infinite Tree Languages
An automaton is unambiguous if for every input it has at most one accepting
computation. An automaton is k-ambiguous (for k>0) if for every input it has at
most k accepting computations. An automaton is boundedly ambiguous if there is
k, such that for every input it has at most k accepting computations. An
automaton is finitely (respectively, countably) ambiguous if for every input it
has at most finitely (respectively, countably) many accepting computations.
The degree of ambiguity of a regular language is defined in a natural way. A
language is k-ambiguous (respectively, boundedly, finitely, countably
ambiguous) if it is accepted by a k-ambiguous (respectively, boundedly,
finitely, countably ambiguous) automaton. Over finite words, every regular
language is accepted by a deterministic automaton. Over finite trees, every
regular language is accepted by an unambiguous automaton. Over -words
every regular language is accepted by an unambiguous B\"uchi automaton and by a
deterministic parity automaton. Over infinite trees, Carayol et al. showed that
there are ambiguous languages.
We show that over infinite trees there is a hierarchy of degrees of
ambiguity: For every k>1 there are k-ambiguous languages which are not k-1
ambiguous; and there are finitely (respectively countably, uncountably)
ambiguous languages which are not boundedly (respectively finitely, countably)
ambiguous.Comment: Revised according to the reviewers comment
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