Papers for
programming language theorists
Papers whose findings have a practical use for this group, as judged from the abstract. Open a paper to read what it means in practice.
Graphoid automata and frobenius props unify graph evaluations by matrices
Graphoid Automata, Frobenius PROPs, and Semiring Evaluation
Abstract: Graphoid automata evaluate graphs by composing relations between words of states. We examine the equations that make this evaluation independent of the chosen graph expression, and what changes when relations are replaced by matrices over a commutative semiring. We prove that the classical graphoid equations are equivalent to the special commutative Frobenius equations together with the required symmetry laws. The known cospan and hypergraph presentations then describe equality of structural and labelled expressions, with the bone law deleting precisely the isolated internal vertices. We use established partition representations to compare this equality with equality of evaluated matrices. For any nonempty finite state set $Q$ and commutative semiring $K$, we give a uniform criterion for two structural expressions to have the same matrix. The criterion determines which isolated-vertex counts are distinguished and when a vanishing scalar makes matrices from different boundary partitions equal. For the canonical structure, the bone law holds precisely when $|Q|\cdot1_K=1_K$. Preserving all relational compositions is a separate requirement, equivalent to additive idempotency for $|Q|\geq2$. We then examine the characteristic matrices of the known relational structures on $Q$ given by disjoint unions of Abelian groups. Their Frobenius equations hold over every commutative semiring, whereas specialness depends on the order of each group and the bone law depends on the number of groups. These conditions specify which equations remain valid when weights are assigned to a graphoid automaton.
Concurrent game strategies characterized as street fibrations with symmetry
Concurrent Strategies as Street Fibrations
Abstract: Concurrent games and strategies based on event structures are an established and expressive model of interactive computation. In this paper we revisit the problem of equipping this model with symmetry. Symmetry is essential for many applications, but it is also a source of mathematical complexity, not least because strategies with symmetry can be compared up to several distinct notions of equivalence. Our first contribution is a complete characterization of two classes of strategies for which composition admits an identity. For these two classes the identity laws hold, respectively, up to a "weak" and a "strong" notion of equivalence. The main result is that the weak class of strategies corresponds precisely to those inducing Street fibrations over the game. We also show how the established "thin" approach to symmetry also fits in this general framework. A second contribution is a formal connection between games and setoid-valued profunctors. This follows a long tradition of relating games and relational models, but here additionally includes a very general model of symmetries. Our characterization of strategies as Street fibrations makes the connection to profunctors significantly easier to establish.