The equational theory of the natural join and inner union is decidable

The natural join and the inner union operations combine relations of a database. Tropashko and Spight [24] realized that these two operations are the meet and join operations in a class of lattices, known by now as the relational lattices. They proposed then lattice theory as an algebraic approach to the theory of databases, alternative to the relational algebra. Previous works [17, 22] proved that the quasiequational theory of these lattices-that is, the set of definite Horn sentences valid in all the relational lattices-is undecidable, even when the signature is restricted to the pure lattice signature. We prove here that the equational theory of relational lattices is decidable. That, is we provide an algorithm to decide if two lattice theoretic terms t, s are made equal under all intepretations in some relational lattice. We achieve this goal by showing that if an inclusion t $\le$ s fails in any of these lattices, then it fails in a relational lattice whose size is bound by a triple exponential function of the sizes of t and s.Comment: arXiv admin note: text overlap with arXiv:1607.0298

Datalog as a parallel general purpose programming language

The increasing available parallelism of computers demands new programming languages that make parallel programming dramatically easier and less error prone. It is proposed that datalog with negation and timestamps is a suitable basis for a general purpose programming language for sequential, parallel and distributed computers. This paper develops a fully incremental bottom-up interpreter for datalog that supports a wide range of execution strategies, with trade-offs affecting efficiency, parallelism and control of resource usage. Examples show how the language can accept real-time external inputs and outputs, and mimic assignment, all without departing from its pure logical semantics

Specifying and Executing Optimizations for Parallel Programs

Compiler optimizations, usually expressed as rewrites on program graphs, are a core part of all modern compilers. However, even production compilers have bugs, and these bugs are difficult to detect and resolve. The problem only becomes more complex when compiling parallel programs; from the choice of graph representation to the possibility of race conditions, optimization designers have a range of factors to consider that do not appear when dealing with single-threaded programs. In this paper we present PTRANS, a domain-specific language for formal specification of compiler transformations, and describe its executable semantics. The fundamental approach of PTRANS is to describe program transformations as rewrites on control flow graphs with temporal logic side conditions. The syntax of PTRANS allows cleaner, more comprehensible specification of program optimizations; its executable semantics allows these specifications to act as prototypes for the optimizations themselves, so that candidate optimizations can be tested and refined before going on to include them in a compiler. We demonstrate the use of PTRANS to state, test, and refine the specification of a redundant store elimination optimization on parallel programs.Comment: In Proceedings GRAPHITE 2014, arXiv:1407.767

A Faster Parameterized Algorithm for Treedepth

The width measure \emph{treedepth}, also known as vertex ranking, centered coloring and elimination tree height, is a well-established notion which has recently seen a resurgence of interest. We present an algorithm which---given as input an $n$-vertex graph, a tree decomposition of the graph of width $w$, and an integer $t$---decides Treedepth, i.e. whether the treedepth of the graph is at most $t$, in time $2^{O(wt)} \cdot n$. If necessary, a witness structure for the treedepth can be constructed in the same running time. In conjunction with previous results we provide a simple algorithm and a fast algorithm which decide treedepth in time $2^{2^{O(t)}} \cdot n$ and $2^{O(t^2)} \cdot n$, respectively, which do not require a tree decomposition as part of their input. The former answers an open question posed by Ossona de Mendez and Nesetril as to whether deciding Treedepth admits an algorithm with a linear running time (for every fixed $t$) that does not rely on Courcelle's Theorem or other heavy machinery. For chordal graphs we can prove a running time of $2^{O(t \log t)}\cdot n$ for the same algorithm.Comment: An extended abstract was published in ICALP 2014, Track