Not all properties are monitorable. This is a well-known fact, and it means there exist properties that cannot be fully verified at runtime. However, given a non-monitorable property, a monitor can still be synthesised, but it could end up in a state where no verdict will ever be concluded on the satisfaction (resp., violation) of the property. For this reason, non-monitorable properties are usually discarded. In this article, we carry out an in-depth analysis on monitorability, and how non-monitorable properties can still be partially verified. We present our theoretical results at a semantic level, without focusing on a specific formalism. Then, we show how our theory can be applied to achieve partial runtime verification of linear time properties.

There exists a broad family of multiparty sessions in which the progress of one session participant is not unconditional, but depends on the choices performed by other participants. These sessions fall outside the scope of currently available session type systems that guarantee progress. In this work we propose the first type system ensuring that well-typed multiparty sessions, including those exhibiting the aforementioned dependencies, fairly terminate. Fair termination is termination under a fairness assumption that disregards those interactions deemed unfair and therefore unrealistic. Fair termination, combined with the usual safety properties ensured within sessions, not only is desirable per se , but it entails livelock freedom and enables a compositional form of static analysis such that the well-typed composition of fairly terminating sessions results in a fairly terminating program.

Fair termination is the property of programs that may diverge "in principle" but that terminate "in practice", i.e. under suitable fairness assumptions concerning the resolution of non-deterministic choices. We study a conservative extension of $μ$MALL$^\infty$, the infinitary proof system of the multiplicative additive fragment of linear logic with least and greatest fixed points, such that cut elimination corresponds to fair termination. Proof terms are processes of $π$LIN, a variant of the linear $π$-calculus with (co)recursive types into which binary and (some) multiparty sessions can be encoded. As a result we obtain a behavioral type system for $π$LIN (and indirectly for session calculi through their encoding into $π$LIN) that ensures fair termination: although well-typed processes may engage in arbitrarily long interactions, they are fairly guaranteed to eventually perform all pending actions.

There exists a broad family of multiparty sessions in which the progress of one session participant is not unconditional, but depends on the choices performed by other participants. These sessions fall outside the scope of currently available session type systems that guarantee progress. In this work we propose the first type system ensuring that well-typed multiparty sessions, including those exhibiting the aforementioned dependencies, fairly terminate. Fair termination is termination under a fairness assumption that disregards those interactions deemed unfair and therefore unrealistic. Fair termination, combined with the usual safety properties ensured within sessions, not only is desirable per se, but it entails progress and enables a compositional form of static analysis such that the well-typed composition of fairly terminating sessions results in a fairly terminating program.

Many properties of communication protocols combine safety and liveness aspects. Characterizing such combined properties by means of a single inference system is difficult because of the fundamentally different techniques (coinduction and induction, respectively) usually involved in defining and proving them. In this paper we show that Generalized Inference Systems allow us to obtain sound and complete characterizations of (at least some of) these combined inductive/coinductive properties of binary session types. In particular, we illustrate the role of corules in characterizing fair termination (the property of protocols that can always eventually terminate), fair compliance (the property of interactions that can always be extended to reach client satisfaction) and fair subtyping, a liveness-preserving refinement relation for session types. The characterizations we obtain are simpler compared to the previously available ones and corules provide insight on the liveness properties being ensured or preserved. Moreover, we can conveniently appeal to the bounded coinduction principle to prove the completeness of the provided characterizations.

A binary session is a private communication channel that connects two processes, each adhering to a protocol description called session type . In this work, we study the first type system that ensures the fair termination of binary sessions. A session fairly terminates if all of the infinite executions admitted by its protocol are deemed unrealistic because they violate certain fairness assumptions . Fair termination entails the eventual completion of all pending input/output actions, including those that depend on the completion of an unbounded number of other actions in possibly different sessions. This form of lock freedom allows us to address a large family of natural communication patterns that fall outside the scope of existing type systems. Our type system is also the first to adopt fair subtyping , a liveness-preserving refinement of the standard subtyping relation for session types that so far has only been studied theoretically. Fair subtyping is surprisingly subtle not only to characterize concisely but also to use appropriately, to the point that the type system must carefully account for all usages of fair subtyping to avoid compromising its liveness-preserving properties.

Many properties of communication protocols stem from the combination of safety and liveness properties. Characterizing such combined properties by means of a single inference system is difficult because of the fundamentally different techniques (coinduction and induction, respectively) usually involved in defining and proving them. In this paper we show that Generalized Inference Systems allow for simple and insightful characterizations of (at least some of) these combined inductive/coinductive properties for dependent session types. In particular, we illustrate the role of corules in characterizing weak termination (the property of protocols that can always eventually terminate), fair compliance (the property of interactions that can always be extended to reach client satisfaction) and also fair subtyping, a liveness-preserving refinement relation for session types.

We provide an Agda library for inference systems, also supporting their recent generalization allowing flexible coinduction, that is, interpretations which are neither inductive, nor purely coinductive. A specific inference system can be obtained as an instance by writing a set of meta-rules, in an Agda format which closely resembles the usual one. In this way, the user gets for free the related properties, notably the inductive and coinductive intepretation and the corresponding proof principles. Moreover, a significant modularity is achieved. Indeed, rather than being defined from scratch and with a built-in interpretation, an inference system can also be obtained by composition operators, such as union and restriction to a smaller universe, and its semantics can be modularly chosen as well. In particular, flexible coinduction is obtained by composing in a certain way the interpretations of two inference systems. We illustrate the use of the library by several examples. The most significant one is a big-step semantics for the λ-calculus, where flexible coinduction allows to obtain a special result (∞) for all and only the diverging computations, and the proof of equivalence with small-step semantics is carried out by relying on the proof principles offered by the library.

Session types have consolidated as a formalism for the specification and static enforcement of communication protocols. Many different theories of dependent session types have been proposed, some enabling refined specifications on the content of messages, others allowing the structure of the protocols to depend on data exchanged in the protocol itself. In this work we continue a line of research studying the foundations of binary session types. In particular, we propose a variant of the linear π-calculus whose type structure encompasses virtually all dependent session types using just two type constructors: linear channel types and linear dependent pairs. We use Agda not only to formalize the metatheory of the calculus and obtain machine-checked proofs of type soundness, but also as host language in which we implement data-dependent protocols.