arXiv subjects

David Corfield

Publications and source records attributed to David Corfield.

2 recordsLinked to original sources

arXiv2026

Categorical Models of Amortized Cost: An Adjoint Relationship between Cost and Potential

Various type systems have been developed to track the cost $\kappa$ of a computation using a cost-tracking monad $M\ \kappa \tau$. On its own, this only tracks the worst-case cost of a computation. If we also want to track amortized cost, then we can add a type $[\kappa]\tau$ which stores potential $\kappa$ with a type $\tau$, together with operations for storing and releasing potential. In this work, we build on one such system, $\lambda$-amor: $\lambda$-amor allows to track cost and potential in the type system and subsumes effect and coeffect-based systems, call-by-value and call-by-name based languages. In this paper, we identify the abstract properties that denotational models of type theories for cost and potential have to satisfy: Cost and potential must be modelled by an adjoint pair of graded functors, where the functor modelling cost forms both a graded monad and a compatible graded comonad. We present three concrete instances of this general abstract scheme: (1) A simple set-theoretic model that ignores the cost tracked by the type system, (2) the Kripke logical relations model in the original $\lambda$-amor paper (which we show can be turned into an instance of the adjoint model), and (3) a novel model based on copresheaves on a monoidal category of costs, where we model pairs and functions by Day convolution and its right-adjoint.

cs.PL↗

arXiv2021

Fundamental weight systems are quantum states

Weight systems on chord diagrams play a central role in knot theory and Chern-Simons theory; and more recently in stringy quantum gravity. We highlight that the noncommutative algebra of horizontal chord diagrams is canonically a star-algebra, and ask which weight systems are positive with respect to this structure; hence we ask: Which weight systems are quantum states, if horizontal chord diagrams are quantum observables? We observe that the fundamental gl(n)-weight systems on horizontal chord diagrams with N strands may be identified with the Cayley distance kernel at inverse temperature beta=ln(n) on the symmetric group on N elements. In contrast to related kernels like the Mallows kernel, the positivity of the Cayley distance kernel had remained open. We characterize its phases of indefinite, semi-definite and definite positivity, in dependence of the inverse temperature beta; and we prove that the Cayley distance kernel is positive (semi-)definite at beta=ln(n) for all n=1,2,3,... In particular, this proves that all fundamental gl(n)-weight systems are quantum states, and hence so are all their convex combinations. We close with briefly recalling how, under our "Hypothesis H", this result impacts on the identification of bound states of multiple M5-branes.

math.GT↗