Graded monads refine traditional monads using effect annotations in order to describe quantitatively the computational effects that a program can generate. They have been successfully applied to a variety of formal systems for reasoning about effectful computations. However, existing categorical frameworks for graded monads do not support effects that may depend on program values, which we call dependent effects , thereby limiting their expressiveness. We address this limitation by introducing indexed graded monads , a categorical generalization of graded monads inspired by the fibrational “indexed” view and by classical categorical semantics of dependent type theories. We show how indexed graded monads provide semantics for a refinement type system with dependent effects. We also show how this type system can be instantiated with specific choices of parameters to obtain several formal systems for reasoning about specific program properties. These instances include, in particular, cost analysis, probability-bound reasoning, expectation-bound reasoning, and temporal safety verification.

We propose new supermartingale-based certificates for verifying almost sure satisfaction of ω-regular properties: (1) generalised Streett supermartingales (GSSMs) and their lexicographic extension (LexGSSMs), (2) distribution-valued Streett supermartingales (DVSSMs), and (3) progress-measure supermartingales (PMSMs) and their lexicographic extension (LexPMSMs). GSSMs, LexGSSMs, and DVSSMs are derived from least-fixed point characterisations of positive recurrence and null recurrence of Markov chains with respect to given Streett conditions; and PMSMs and LexPMSMs are probabilistic extensions of parity progress measures. We study the hierarchy among these certificates and existing certificates, namely Streett supermartingales, by comparing the classes of problems that can be verified by each type of certificates. Notably, we show that our certificates are strictly more powerful than Streett supermartingales. We also prove completeness of GSSMs for positive recurrence and of DVSSMs for null recurrence: DVSSMs are, in theory, the most powerful certificates in the sense that for any Markov chain that almost surely satisfies a given ω-regular property, there exists a DVSSM certifying it. We provide a sound and relatively complete algorithm for synthesising LexPMSMs, the second most powerful certificates in the hierarchy. We have implemented a prototype tool based on this algorithm, and our experiments show that our tool can successfully synthesise certificates for various examples including those that cannot be certified by existing supermartingales.

Many quantitative properties of probabilistic programs can be characterized as least fixed points, but verifying their lower bounds remains a challenging problem. We present a new approach to lower-bound verification that exploits and extends the connection between the uniqueness of fixed points and program termination. The core technical tool is a generalization of ranking supermartingales, which serves as witnesses of the uniqueness of fixed points. Our method provides a simple and unified reasoning principle applicable to a wide range of quantitative properties, including termination probability, the weakest preexpectation, expected runtime, higher moments of runtime, and conditional weakest preexpectation. We provide a template-based algorithm for automated verification of lower bounds and demonstrate the effectiveness of the proposed method via experiments.

We give a denotational account of logical relations for call-by-push-value (CBPV) in the fibrational style of Hermida, Jacobs, Katsumata and others. Fibrations—which axiomatise the usual notion of sets-with-relations—provide a clean framework for constructing new, logical relations-style, models. Such models can then be used to study properties such as effect simulation.Extending this picture to CBPV is challenging: the models incorporate both adjunctions and enrichment, making the appropriate notion of fibration unclear. We handle this using 2-category theory. We identify an appropriate 2-category, and define CBPV fibrations to be fibrations internal to this 2-category which strictly preserve the CBPV semantics.Next, we develop the theory so it parallels the classical setting. We give versions of the codomain and subobject fibrations, and show that new models can be constructed from old ones by pullback. The resulting framework enables the construction of new, logical relations-style, models for CBPV.Finally, we demonstrate the utility of our approach with particular examples. These include a generalisation of Katsumata’s ⊤⊤-lifting to CBPV models, an effect simulation result, and a relative full completeness result for CBPV without sum types.

Verification of higher-order probabilistic programs is a challenging problem. We present a verification method that supports several quantitative properties of higher-order probabilistic programs. Usually, extending verification methods to handle the quantitative aspects of probabilistic programs often entails extensive modifications to existing tools, reducing compatibility with advanced techniques developed for qualitative verification. In contrast, our approach necessitates only small amounts of modification, facilitating the reuse of existing techniques and implementations. On the theoretical side, we propose a dependent refinement type system for a generalised higher-order fixed point logic (HFL). Combined with continuation-passing style encodings of properties into HFL, our dependent refinement type system enables reasoning about several quantitative properties, including weakest pre-expectations, expected costs, moments of cost, and conditional weakest pre-expectations for higher-order probabilistic programs with continuous distributions and conditioning. The soundness of our approach is proved in a general setting using a framework of categorical semantics so that we don’t have to repeat similar proofs for each individual problem. On the empirical side, we implement a type checker for our dependent refinement type system that reduces the problem of type checking to constraint solving. We introduce admissible predicate variables and integrable predicate variables to constrained Horn clauses (CHC) so that we can soundly reason about the least fixed points and samplings from probability distributions. Our implementation demonstrates that existing CHC solvers developed for non-probabilistic programs can be extended to a solver for the extended CHC with only small efforts. We also demonstrate the ability of our type checker to verify various concrete examples.

Abstract Weakest precondition transformers are useful tools in program verification. One of their key properties is composability, that is, the weakest precondition predicate transformer (wppt for short) associated to program $f;\;g$ should be equal to the composition of the wppts associated to f and g . In this paper, we study the categorical structure behind wppts from a fibrational point of view. We characterize the wppts that satisfy composability as the ones constructed from the Cartesian lifting of a monad. We moreover show that Cartesian liftings of monads along lax slice categories bijectively correspond to Eilenberg–Moore monotone algebras. We then instantiate our techniques by deriving wppts for commonplace effects such as the maybe monad, the nonempty powerset monad, the counter monad, or the distribution monad. We also show how to combine them to derive the wppts appearing in the literature of verification of probabilistic programs.

Abstract We present a novel decision tree-based synthesis algorithm of ranking functions for verifying program termination. Our algorithm is integrated into the workflow of CounterExample Guided Inductive Synthesis (CEGIS). CEGIS is an iterative learning model where, at each iteration, (1) a synthesizer synthesizes a candidate solution from the current examples, and (2) a validator accepts the candidate solution if it is correct, or rejects it providing counterexamples as part of the next examples. Our main novelty is in the design of a synthesizer: building on top of a usual decision tree learning algorithm, our algorithm detectscyclesin a set of example transitions and uses them for refining decision trees. We have implemented the proposed method and obtained promising experimental results on existing benchmark sets of (non-)termination verification problems that require synthesis of piecewise-defined lexicographic affine ranking functions.

Abstract Dependent refinement types are types equipped with predicates that specify preconditions and postconditions of underlying functional languages. We propose a general semantic construction of dependent refinement type systems from underlying type systems and predicate logic, that is, a construction of liftings of closed comprehension categories from given (underlying) closed comprehension categories and posetal fibrations for predicate logic. We give sufficient conditions to lift structures such as dependent products, dependent sums, computational effects, and recursion from the underlying type systems to dependent refinement type systems. We demonstrate the usage of our construction by giving semantics to a dependent refinement type system and proving soundness.

Abstract We provide graded extensions of algebraic theories and Lawvere theories that correspond to graded monads. We prove that graded algebraic theories, graded Lawvere theories, and finitary graded monads are equivalent via equivalence of categories, which extends the equivalence for monads. We also give sums and tensor products of graded algebraic theories to combine computational effects as an example of importing techniques based on algebraic theories to graded monads.

Programs with randomization constructs is an active research topic, especially after the recent introduction of martingale-based analysis methods for their termination and runtimes. Unlike most of the existing works that focus on proving almost-sure termination or estimating the expected runtime, in this work we study the tail probabilities of runtimes—such as “the execution takes more than 100 steps with probability at most 1%.” To this goal, we devise a theory of supermartingales that overapproximate higher moments of runtime. These higher moments, combined with a suitable concentration inequality, yield useful upper bounds of tail probabilities. Moreover, our vector-valued formulation enables automated template-based synthesis of those supermartingales. Our experiments suggest the method’s practical use.