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Arhinia, or absence of the nose, is a rare malformation of unknown etiology that is often accompanied by ocular and reproductive defects. Sequencing of 40 people with arhinia revealed that 84% of probands harbor a missense mutation localized to a constrained region of SMCHD1 encompassing the ATPase domain. SMCHD1 mutations cause facioscapulohumeral muscular dystrophy type 2 (FSHD2) via a trans-acting loss-of-function epigenetic mechanism. We discovered shared mutations and comparable DNA hypomethylation patterning between these distinct disorders. CRISPR/Cas9-mediated alteration of smchd1 in zebrafish yielded arhinia-relevant phenotypes. Transcriptome and protein analyses in arhinia probands and controls showed no differences in SMCHD1 mRNA or protein abundance but revealed regulatory changes in genes and pathways associated with craniofacial patterning. Mutations in SMCHD1 thus contribute to distinct phenotypic spectra, from craniofacial malformation and reproductive disorders to muscular dystrophy, which we speculate to be consistent with oligogenic mechanisms resulting in pleiotropic outcomes.

Abstract. This paper takes a fresh look at the topic of trace semantics in the theory of coalgebras. The first development of coalgebraic trace semantics used final coalgebras in Kleisli categories, stemming from an initial algebra in the underlying category. This approach requires some non-trivial assumptions, like dcpo enrichment, which do not always hold, even in cases where one can reasonably speak of traces (like for weighted automata). More recently, it has been noticed that trace semantics can also arise by first performing a determinization construction. In this paper, we develop a systematic approach, in which the two approaches correspond to different orders of composing a functor and a monad, and accordingly, to different distributive laws. The relevant final coalgebra that gives rise to trace semantics does not live in a Kleisli category, but more generally, in a category of Eilenberg-Moore algebras. In order to exploit its finality, we identify an extension operation, that changes the state space of a coalgebra into a free algebra, which abstractly captures determinization of automata. Notably, we show that the two different views on trace semantics are equivalent, in the examples where both approaches are applicable.

Guarded Kleene Algebra with Tests (GKAT) is a variation on Kleene Algebra with Tests (KAT) that arises by restricting the union (+) and iteration ( * ) operations from KAT to predicate-guarded versions. We develop the (co)algebraic theory of GKAT and show how it can be efficiently used to reason about imperative programs. In contrast to KAT, whose equational theory is PSPACE-complete, we show that the equational theory of GKAT is (almost) linear time. We also provide a full Kleene theorem and prove completeness for an analogue of Salomaa's axiomatization of Kleene Algebra. INTRODUCTIONComputer scientists have long explored the connections between families of programming languages and abstract machines. This dual perspective has furnished deep theoretical insights as well as practical tools. As an example, Kleene's classic result establishing the equivalence of regular expressions and finite automata [22] inspired decades of work across a variety of areas including programming language design, mathematical semantics, and formal verification.Kleene Algebra with Tests (KAT) [25], which combines Kleene Algebra (KA) with Boolean Algebra (BA), is a modern example of this approach. Viewed from the program-centric perspective, a KAT models the fundamental constructs that arise in programs: sequencing, branching, iteration, non-determinism, etc. The equational theory of KAT enables algebraic reasoning and can be finitely axiomatized [30]. Viewed from the machine-centric perspective, a KAT describes a kind of automaton that generates a regular language of traces. This shift in perspective admits techniques from the theory of coalgebras for reasoning about program behavior. In particular, there are efficient algorithms for checking bisimulation, which can be optimized using properties of bisimulations [8,19] or symbolic automata representations [40].KAT has been used to model computation across a wide variety of areas including program transformations [2, 26], concurrency control [10], compiler optimizations [29], cache control [5,9], and more [9]. A prominent recent application is NetKAT [1], a language for reasoning about the packetforwarding behavior of software-defined networks. NetKAT has a sound and complete equational theory, and a coalgebraic decision procedure that can be used to automatically verify many important networking properties including reachability, loop-freedom, and isolation [15]. However, while NetKAT's implementation scales well in practice, the complexity of deciding equivalence for NetKAT is PSPACE-complete in the worst case [1].A natural question to ask is whether there is an efficient fragment of KAT that is reasonably expressive, while retaining a solid foundation. We answer this question positively in this paper with a comprehensive study of Guarded Kleene Algebra with Tests (GKAT), the guarded fragment of KAT. The language is a propositional abstraction of imperative while programs, which have been well-studied in the literature. We establish the fundamental properties of GKAT and develop its ...