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 &#969;-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 &psi;(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 &psi;(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 &psi; 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 $varphi$ on sequence pairs, realizes for any input sequence $I$ an output sequence $O$ such that $(I,O)$ satisfies $varphi$. McNaughton reduced Church\u27s Problem to a problem about two-player$omega$-games. B"uchi and Landweber gave a solution for Monadic Second-Order Logic of Order ($MLO$) 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 $MLO$-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 $\omega$-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