Certified Context-Free Parsing: A formalisation of Valiant's Algorithm in Agda

Valiant (1975) has developed an algorithm for recognition of context free languages. As of today, it remains the algorithm with the best asymptotic complexity for this purpose. In this paper, we present an algebraic specification, implementation, and proof of correctness of a generalisation of Valiant's algorithm. The generalisation can be used for recognition, parsing or generic calculation of the transitive closure of upper triangular matrices. The proof is certified by the Agda proof assistant. The certification is representative of state-of-the-art methods for specification and proofs in proof assistants based on type-theory. As such, this paper can be read as a tutorial for the Agda system

Hamiltonian Coupling of Electromagnetic Field and Matter

Reversible part of evolution equations of physical systems is often generated by a Poisson bracket. We discuss geometric means of construction of Poisson brackets and their mutual coupling (direct, semidirect and matched-pair products) as well as projections of Poisson brackets to less detailed Poisson brackets. This way the Hamiltonian coupling between transport of mixtures and electrodynamics is elucidated

Relative Expressive Power of Navigational Querying on Graphs

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; 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 some nontrivial separations and a few surprising collapses.Comment: An extended abstract announcing the results of this paper was presented at the 14th International Conference on Database Theory, Uppsala, Sweden, March 201

Pseudofinite structures and simplicity

We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity of the underlying theory, and that a drop in pseudofinite dimension is equivalent to forking. Under a suitable assumption, a measure-theoretic condition is shown to be equivalent to local stability. Many examples are explored, including vector spaces over finite fields viewed as 2-sorted finite structures, and homocyclic groups. Connections are made to products of sets in finite groups, in particular to word maps, and a generalization of Tao's algebraic regularity lemma is noted

To found or not to found: that is the question

Aim of this paper is to confute two views, the first about Schr\"oder's presumptive foundationalism, according to he founded mathematics on the calculus of relatives; the second one mantaining that Schr\"oder only in his last years (from 1890 onwards) focused on an universal and symbolic language (by him called pasigraphy). We will argue that, on the one hand Schr\"oder considered the problem of founding mathematics already solved by Dedekind, limiting himself in a mere translation of the Chain Theory in the language of the relatives. On the other hand, we will show that Schr\"oder's pasigraphy was connaturate to himself and that it roots in his very childhood and in his love for foreign languages.Comment: Next to be published in Logic and Logical Philosoph