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Relational type systems have been designed for several applications including information flow, differential privacy, and cost analysis. In order to achieve the best results, these systems often use relational refinements and relational effects to maximally exploit the similarity in the structure of the two programs being compared. Relational type systems are appealing for relational properties because they deliver simpler and more precise verification than what could be derived from typing the two programs separately. However, relational type systems do not yet achieve the practical appeal of their non-relational counterpart, in part because of the lack of a general foundations for implementing them. In this paper, we take a step in this direction by developing bidirectional relational type checking for systems with relational refinements and effects. Our approach achieves the benefits of bidirectional type checking, in a relational setting. In particular, it significantly reduces the need for typing annotations through the combination of type checking and type inference. In order to highlight the foundational nature of our approach, we develop bidirectional versions of several relational type systems which incrementally combine many different components needed for expressive relational analysis., Comment: 14 pages

Relational cost analysis aims at formally establishing bounds on the difference in the evaluation costs of two programs. As a particular case, one can also use relational cost analysis to establish bounds on the difference in the evaluation cost of the same program on two different inputs. One way to perform relational cost analysis is to use a relational type-and-effect system that supports reasoning about relations between two executions of two programs. Building on this basic idea, we present a type-and-effect system, called ARel, for reasoning about the relative cost of array-manipulating, higher-order functional-imperative programs. The key ingredient of our approach is a new lightweight type refinement discipline that we use to track relations (differences) between two arrays. This discipline combined with Hoare-style triples built into the types allows us to express and establish precise relative costs of several interesting programs which imperatively update their data., Comment: 14 pages