Ordered logics and type systems have been used in a variety of applications including computational linguistics, memory allocation, stream processing, logical frameworks, parametricity, and enforcing security protocols. In most formulations, ordered types are also linear, requiring each resource to be used exactly once. Prior work by Kanovich et al. has investigated calculi that relax this constraint through subexponentials within a linear ordered logic. We generalize their work by using adjoint modalities to combine logics with varying fine-grained structural properties, including weakening, left contraction, right contraction, left mobility, and right mobility. We show that the resulting sequent calculus admits cut elimination. We further provide a natural deduction formulation in which structural rules are implicit, and show that proof checking for this system is decidable. This makes it a suitable foundation for an expressive adjoint programming language or logical framework.

Adjoint logic is a general approach to combining multiple logics with different structural properties, including linear, affine, strict, and (ordinary) intuitionistic logics, where each proposition has an intrinsic mode of truth. It has been defined in the form of a sequent calculus because the central concept of independence is most clearly understood in this form, and because it permits a proof of cut elimination following standard techniques. In this paper we present a natural deduction formulation of adjoint logic and show how it is related to the sequent calculus. As a consequence, every provable proposition has a verification (sometimes called a long normal form). We also give a computational interpretation of adjoint logic in the form of a functional language and prove properties of computations that derive from the structure of modes, including freedom from garbage (for modes without weakening and contraction), strictness (for modes disallowing weakening), and erasure (based on a preorder between modes). Finally, we present a surprisingly subtle algorithm for type checking.