Guarded Traced Categories

Abstract

Notions of guardedness serve to delineate the admissibility of cycles, e.g. in recursion, corecursion, iteration, or tracing. We introduce an abstract notion of guardedness structure on a symmetric monoidal category, along with a corresponding notion of guarded traces, which are defined only if the cycles they induce are guarded. We relate structural guardedness, determined by propagating guardedness along the operations of the category, to geometric guardedness phrased in terms of a diagrammatic language. In our setup, the Cartesian case (recursion) and the co-Cartesian case (iteration) become completely dual, and we show that in these cases, guarded tracedness is equivalent to presence of a guarded Conway operator, in analogy to an observation on total traces by Hasegawa and Hyland. Moreover, we relate guarded traces to unguarded categorical uniform fixpoint operators in the style of Simpson and Plotkin. Finally, we show that partial traces based on Hilbert-Schmidt operators in the category of Hilbert spaces are an instance of guarded traces.

A Metalanguage for Guarded Iteration

“…We have instantiated the notion of abstract guardedness [23,21] to a multivariable setting in the form of a metalanguage for guarded iteration, which incorporates both monad-based encapsulation of side-effects [35] and the fine-grain call-by-value paradigm [30]. As a side product, this has additionally resulted in a semantically justified unification of (guarded) iteration and exception handling, extending previous work by Geron and Levy [14].…”

“…In recent work, we have proposed a new axiomatic approach to unifying notions of guardedness [23,21], where the main idea is to provide an abstract notion of guardedness applicable to a wide range of (mutually irreducible) models, including, e.g., complete partial orders, complete metric spaces, and infinitedimensional Hilbert spaces, instead of designing a concrete model carrying a specific notion of guardedness. A salient feature of axiomatic guardedness is that it varies in a large spectrum starting from total guardedness (everything is guarded) and ending at vacuous guardedness (very roughly, guardedness in a variable means essentially non-occurrence of this variable in the defining expression) with proper examples as discussed above lying between these two extremes.…”

“…The developments in [21] are couched in terms of a special class of monoidal categories called guarded traced symmetric monoidal categories, equipped with a monoidal notion of guardedness and a monoidal notion of feedback allowing only such cyclic computations that are guarded in the corresponding sense. In the present work we explore a refinement of this notion by instantiating guarded traces to Kleisli categories of computational monads in sense of Moggi [35], with coproduct (inherited from the base category) as the monoidal structure.…”

“…Plan of the paper. In Section 2 we give the necessary technical preliminaries, and discuss and complement the semantic foundations for guarded iteration [23,21]. In Sections 3 and 4 we present our metalanguage for guarded iteration (without functional types) and its generic denotational semantics.…”

Differentiable Causal Computations via Delayed Trace

“…A common theme in recursively defined computations is that to have well-defined behaviour, a recursive computation must satisfy a guardedness condition [1], [26]. Goncharov and Schröder developed the theory of guarded traced categories to formalize this phenomena in [17]. The key idea is to restrict Joyal et al's trace operator [22] to a class of guarded morphisms, which are an abstractly given class of morphisms satisfying the guardedness condition.…”