What is modal logic

Graph databases have received much attention as of late due to numerous applications in which data is naturally viewed as a graph; these include social networks, RDF and the Semantic Web, biological databases, and many others. There are many proposals for query languages for graph databases that mainly fall into two categories. One views graphs as a particular kind of relational data and uses traditional relational mechanisms for querying. The other concentrates on querying the topology of the graph. These approaches, however, lack the ability to combine data and topology, which would allow queries asking how data changes along paths and patterns enveloping it. In this article, we present a comprehensive study of languages that enable such combination of data and topology querying. These languages come in two flavors. The first follows the standard approach of path queries, which specify how labels of edges change along a path, but now we extend them with ways of specifying how both labels and data change. From the complexity point of view, the right type of formalisms are subclasses of register automata. These, however, are not well suited for querying. To overcome this, we develop several types of extended regular expressions to specify paths with data and study their querying power and complexity. The second approach adopts the popular XML language XPath and extends it from XML documents to graphs. Depending on the exact set of allowed features, we have a family of languages, and our study shows that it includes efficient and highly expressive formalisms for querying both the structure of the data and the data itself.

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Querying Graphs with Data

In this paper we present some ideas how to analyse data using logic tools. We define a logic language for expressing data analysis problems and we develop a deductive system for the language to use proof procedures to obtain solutions to these problems.

DAL: A Logic for Data Analysis.

P-type Mg doped CuCrO2 thin films have been deposited on fused silica substrates by Radio-Frequency (RF) magnetron sputtering. The as-deposited CuCrO2:Mg thin films have been annealed at different temperatures (from 450 to 650 ◦C) under primary vacuum to obtain the delafossite phase. The annealed samples exhibit 3R delafossite structure. Electrical conductivity σ and Seebeck coefficient S of all annealed films have been measured from 40 to 220 ◦C. The optimized properties have been obtained for CuCrO2:Mg thin film annealed at 550 ◦C. At a measurement temperature of 40 ◦C, this sample exhibited the highest electrical conductivity of 0.60 S·cm−1 with a Seebeck coefficient of +329 µV·K−1. The calculated power factor (PF = σS 2) was 6 µW·m−1·K−2 at 40 ◦C and due to the constant Seebeck coefficient and the increasing electrical conductivity with measurement temperature, it reached 38 µW·m−1·K−2 at 220 ◦C. Moreover, according to measurement of the Seebeck coefficient and electrical conductivity in temperature, we confirmed that CuCrO2:Mg exhibits hopping conduction and degenerates semiconductor behavior. Carrier concentration, Fermi level, and hole effective mass have been discussed.

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Thermoelectric and transport properties of Delafossite CuCrO2:Mg thin films prepared by RF magnetron sputtering

Motivated by both established and new applications, we study navigational query languages for graphs (binary relations). The simplest language has only the two operators union and composition, together with the identity relation. We make more powerful languages by adding any of the following operators: intersection; set difference; projection; coprojection; converse; transitive closure; and the diversity relation. All these operators map binary relations to binary relations. We compare the expressive power of all resulting languages. We do this not only for general path queries (queries where the result may be any binary relation) but also for boolean or yes/no queries (expressed by the nonemptiness of an expression). For both cases, we present the complete Hasse diagram of relative expressiveness. In particular, the Hasse diagram for boolean queries contains nontrivial separations and a few surprising collapses.

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Relative expressive power of navigational querying on graphs

We study satisfiability and infinite-state model checking in ICPDL, which extends Propositional Dynamic Logic (PDL) with intersection and converse operators on programs. The two main results of this paper are that (i) satisfiability is in 2EΧΡΤΙΜE, thus 2EΧΡΤΙΜE-complete by an existing lower bound, and (ii) infinite-state model checking of basic process algebras and pushdown systems is also 2EΧΡΤΙΜE-complete. Both upper bounds are obtained by polynomial time computable reductions to ω-regular tree satisfiability in ICPDL, a reasoning problem that we introduce specifically for this purpose. This problem is then reduced to the emptiness problem for alternating two-way automata on infinite trees. Our approach to (i) also provides a shorter and more elegant proof of Danecki's difficult result that satisfiability in IPDL is in 2EΧΡΤΙΜE. We prove the lower bound(s) for infinite-state model checking using an encoding of alternating Turing machines.

/pdf/pdl-with-intersection-and-converse-satisfiability-and-4j5xcfjozg.pdf

PDL with intersection and converse: satisfiability and infinite-state model checking

Equivalence of deterministic pushdown automata is a famous problem in theoretical computer science whose decidability has been shown by Senizergues. Our first result shows that decidability no longer holds when moving from finite words to infinite words. This solves an open problem that has recently been raised by Loding. In fact, we show that already the equivalence problem for deterministic Buchi one-counter automata is undecidable. Hence, the decidability border is rather tight when taking into account a recent result by Loding and Repke that equivalence of deterministic weak parity pushdown automata (a subclass of deterministic Buchi pushdown automata) is decidable.

Another known result on finite words is that the universality problem for vector addition systems is decidable. We show undecidability when moving to infinite words. In fact, we prove that already the universality problem for nondeterministic Buchi one-counter nets (or equivalently vector addition systems with one unbounded dimension) is undecidable.

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On Büchi One-Counter Automata

A standard way of building concurrent systems is by composing several individual processes by product operators. We show that even the simplest notion of product operators (i.e. asynchronous products) suces to increase the complexity of model checking simple logics like HennessyMilner (HM) logic and its extension with the reachability operator (EF-logic) from PSPACE to nonelementary. In particular, this nonelementary jump happens for EF-logic when we consider individual processes represented by pushdown systems (indeed, even with only one control state). Using this result, we prove nonelementary lower bounds on the size of formula decompositions provided by Feferman-Vaught (de)compositional methods for HM and EF logics, which reduce theories of asynchronous products to theories of the components. Finally, we show that the same nonelementary lower bounds also hold when we consider the relativization of such compositional methods to finite systems.

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Concurrency Makes Simple Theories Hard

Reachability on prefix-recognizable graphs

A pattern is encountered in a word if some infix of the word is the image of the pattern under some non-erasing morphism. A pattern p is unavoidable if, over every finite alphabet, every sufficiently long word encounters p. A theorem by Zimin and independently by Bean, Ehrenfeucht and McNulty states that a pattern over n distinct variables is unavoidable if, and only if, p itself is encountered in the n-th Zimin pattern. Given an alphabet size k, we study the minimal length f(n,k) such that every word of length f(n,k) encounters the n-th Zimin pattern. It is known that f is upper-bounded by a tower of exponentials. Our main result states that f(n,k) is lower-bounded by a tower of n-3 exponentials, even for k=2. To the best of our knowledge, this improves upon a previously best-known doubly-exponential lower bound. As a further result, we prove a doubly-exponential upper bound for encountering Zimin patterns in the abelian sense.

Stefan Göller

Papers

PDL with intersection and converse: satisfiability and infinite-state model checking

On Büchi One-Counter Automata

Concurrency Makes Simple Theories Hard

Reachability on prefix-recognizable graphs

On Long Words Avoiding Zimin Patterns.