arXiv subjects

Shin-ya Katsumata

Publications and source records attributed to Shin-ya Katsumata.

At least 19 recordsLinked to original sources

arXiv2026

Programming Backpropagation with Reverse Handlers for Arrows

We introduce a new programming language and its categorical semantics in order to design and implement neural networks within the framework of algebraic effects and handlers for arrows. Our language enables us to construct neural networks symbolically, in the same manner as algebraic effects, and to assign implementations -- such as backpropagation computations -- to them via handlers. The advantage of this language design is that network descriptions become abstract and high-level, while implementations can be flexibly assigned to networks. We establish a rigorous foundation for our language by developing a type system, an operational semantics, a categorical semantics, and soundness and adequacy theorems. The technical core is the introduction of reverse handlers, a novel handler mechanism for arrows for implementing backpropagation, together with new algebras of strong promonads on reverse differential restriction categories (RDRCs), whose string diagrams provide a formal graphical syntax and semantics for neural networks.

cs.PL↗

arXiv2024

Composing Codensity Bisimulations

Proving compositionality of behavioral equivalence on state-based systems with respect to algebraic operations is a classical and widely studied problem. We study a categorical formulation of this problem, where operations on state-based systems modeled as coalgebras can be elegantly captured through distributive laws between functors. To prove compositionality, it then suffices to show that this distributive law lifts from sets to relations, giving an explanation of how behavioral equivalence on smaller systems can be combined to obtain behavioral equivalence on the composed system. In this paper, we refine this approach by focusing on so-called codensity lifting of functors, which gives a very generic presentation of various notions of (bi)similarity as well as quantitative notions such as behavioral metrics on probabilistic systems. The key idea is to use codensity liftings both at the level of algebras and coalgebras, using a new generalization of the codensity lifting. The problem of lifting distributive laws then reduces to the abstract problem of constructing distributive laws between codensity liftings, for which we propose a simplified sufficient condition. Our sufficient condition instantiates to concrete proof methods for compositionality of algebraic operations on various types of state-based systems. We instantiate our results to prove compositionality of qualitative and quantitative properties of deterministic automata. We also explore the limits of our approach by including an example of probabilistic systems, where it is unclear whether the sufficient condition holds, and instead we use our setting to give a direct proof of compositionality. ...

cs.LO↗

arXiv2023

A Categorical Framework for Program Semantics and Semantic Abstraction

Categorical semantics of type theories are often characterized as structure-preserving functors. This is because in category theory both the syntax and the domain of interpretation are uniformly treated as structured categories, so that we can express interpretations as structure-preserving functors between them. This mathematical characterization of semantics makes it convenient to manipulate and to reason about relationships between interpretations. Motivated by this success of functorial semantics, we address the question of finding a functorial analogue in abstract interpretation, a general framework for comparing semantics, so that we can bring similar benefits of functorial semantics to semantic abstractions used in abstract interpretation. Major differences concern the notion of interpretation that is being considered. Indeed, conventional semantics are value-based whereas abstract interpretation typically deals with more complex properties. In this paper, we propose a functorial approach to abstract interpretation and study associated fundamental concepts therein. In our approach, interpretations are expressed as oplax functors in the category of posets, and abstraction relations between interpretations are expressed as lax natural transformations representing concretizations. We present examples of these formal concepts from monadic semantics of programming languages and discuss soundness.

cs.PL↗

arXiv2023

Control-Data Separation and Logical Condition Propagation for Efficient Inference on Probabilistic Programs

We present a novel sampling framework for probabilistic programs. The framework combines two recent ideas -- \emph{control-data separation} and \emph{logical condition propagation} -- in a nontrivial manner so that the two ideas boost the benefits of each other. We implemented our algorithm on top of Anglican. The experimental results demonstrate our algorithm's efficiency, especially for programs with while loops and rare observations.

cs.LG↗

arXiv2022

A Semantic Account of Metric Preservation

Program sensitivity measures how robust a program is to small changes in its input, and is a fundamental notion in domains ranging from differential privacy to cyber-physical systems. A natural way to formalize program sensitivity is in terms of metrics on the input and output spaces, requiring that an $r$-sensitive function map inputs that are at distance $d$ to outputs that are at distance at most $r \cdot d$. Program sensitivity is thus an analogue of Lipschitz continuity for programs. Reed and Pierce introduced Fuzz, a functional language with a linear type system that can express program sensitivity. They show soundness operationally, in the form of a metric preservation property. Inspired by their work, we study program sensitivity and metric preservation from a denotational point of view. In particular, we introduce metric CPOs, a novel semantic structure for reasoning about computation on metric spaces, by endowing CPOs with a compatible notion of distance. This structure is useful for reasoning about metric properties of programs, and specifically about program sensitivity. We demonstrate metric CPOs by giving a model for the deterministic fragment of Fuzz.

cs.PL↗

arXiv2022

The Lattice-Theoretic Essence of Property Directed Reachability Analysis

We present LT-PDR, a lattice-theoretic generalization of Bradley's property directed reachability analysis (PDR) algorithm. LT-PDR identifies the essence of PDR to be an ingenious combination of verification and refutation attempts based on the Knaster-Tarski and Kleene theorems. We introduce four concrete instances of LT-PDR, derive their implementation from a generic Haskell implementation of LT-PDR, and experimentally evaluate them. We also present a categorical structural theory that derives these instances.

cs.LO↗

arXiv2022

Divergences on Monads for Relational Program Logics

Several relational program logics have been introduced for integrating reasoning about relational properties of programs and measurement of quantitative difference between computational effects. Towards a general framework for such logics, in this paper, we formalize quantitative difference between computational effects as divergence on monad, then develop a relational program logic acRL that supports generic computational effects and divergences on them. To give a categorical semantics of acRL supporting divergences, we give a method to obtain graded strong relational liftings from divergences on monads. We derive two instantiations of acRL for the verification of 1) various differential privacy of higher-order functional probabilistic programs and 2) difference of distribution of costs between higher-order functional programs with probabilistic choice and cost counting operations.

cs.PL↗

arXiv2021

Fibrational Initial Algebra-Final Coalgebra Coincidence over Initial Algebras: Turning Verification Witnesses Upside Down

The coincidence between initial algebras (IAs) and final coalgebras (FCs) is a phenomenon that underpins various important results in theoretical computer science. In this paper, we identify a general fibrational condition for the IA-FC coincidence, namely in the fiber over an initial algebra in the base category. Identifying (co)algebras in a fiber as (co)inductive predicates, our fibrational IA-FC coincidence allows one to use coinductive witnesses (such as invariants) for verifying inductive properties (such as liveness). Our general fibrational theory features the technical condition of stability of chain colimits; we extend the framework to the presence of a monadic effect, too, restricting to fibrations of complete lattice-valued predicates. Practical benefits of our categorical theory are exemplified by new "upside-down" witness notions for three verification problems: probabilistic liveness, and acceptance and model-checking with respect to bottom-up tree automata.

cs.LO↗

arXiv2021

Robustifying Controller Specifications of Cyber-Physical Systems Against Perceptual Uncertainty

Formal reasoning on the safety of controller systems interacting with plants is complex because developers need to specify behavior while taking into account perceptual uncertainty. To address this, we propose an automated workflow that takes an Event-B model of an uncertainty-unaware controller and a specification of uncertainty as input. First, our workflow automatically injects the uncertainty into the original model to obtain an uncertainty-aware but potentially unsafe controller. Then, it automatically robustifies the controller so that it satisfies safety even under the uncertainty. The case study shows how our workflow helps developers to explore multiple levels of perceptual uncertainty. We conclude that our workflow makes design and analysis of uncertainty-aware controller systems easier and more systematic.

cs.SE↗

arXiv2021

Architecture-Guided Test Resource Allocation Via Logic

We introduce a new logic named Quantitative Confidence Logic (QCL) that quantifies the level of confidence one has in the conclusion of a proof. By translating a fault tree representing a system's architecture to a proof, we show how to use QCL to give a solution to the test resource allocation problem that takes the given architecture into account. We implemented a tool called Astrahl and compared our results to other testing resource allocation strategies.

cs.SE↗

arXiv2021

Higher-order probabilistic adversarial computations: Categorical semantics and program logics

Adversarial computations are a widely studied class of computations where resource-bounded probabilistic adversaries have access to oracles, i.e., probabilistic procedures with private state. These computations arise routinely in several domains, including security, privacy and machine learning. In this paper, we develop program logics for reasoning about adversarial computations in a higher-order setting. Our logics are built on top of a simply typed $λ$-calculus extended with a graded monad for probabilities and state. The grading is used to model and restrict the memory footprint and the cost (in terms of oracle calls) of computations. Under this view, an adversary is a higher-order expression that expects as arguments the code of its oracles. We develop unary program logics for reasoning about error probabilities and expected values, and a relational logic for reasoning about coupling-based properties. All logics feature rules for adversarial computations, and yield guarantees that are valid for all adversaries that satisfy a fixed resource policy. We prove the soundness of the logics in the category of quasi-Borel spaces, using a general notion of graded predicate liftings, and we use logical relations over graded predicate liftings to establish the soundness of proof rules for adversaries. We illustrate the working of our logics with simple but illustrative examples.

cs.LO↗

arXiv2021

Expressivity of Quantitative Modal Logics: Categorical Foundations via Codensity and Approximation

A modal logic that is strong enough to fully characterize the behavior of a system is called expressive. Recently, with the growing diversity of systems to be reasoned about (probabilistic, cyber-physical, etc.), the focus shifted to quantitative settings which resulted in a number of expressivity results for quantitative logics and behavioral metrics. Each of these quantitative expressivity results uses a tailor-made argument; distilling the essence of these arguments is non-trivial, yet important to support the design of expressive modal logics for new quantitative settings. In this paper, we present the first categorical framework for deriving quantitative expressivity results, based on the new notion of approximating family. A key ingredient is the codensity lifting -- a uniform observation-centric construction of various bisimilarity-like notions such as bisimulation metrics. We show that several recent quantitative expressivity results (e.g. by König et al. and by Fijalkow et al.) are accommodated in our framework; a new expressivity result is derived, too, for what we call bisimulation uniformity.

cs.LO↗

arXiv2021

Graded Hoare Logic and its Categorical Semantics

Deductive verification techniques based on program logics (i.e., the family of Floyd-Hoare logics) are a powerful approach for program reasoning. Recently, there has been a trend of increasing the expressive power of such logics by augmenting their rules with additional information to reason about program side-effects. For example, general program logics have been augmented with cost analyses, logics for probabilistic computations have been augmented with estimate measures, and logics for differential privacy with indistinguishability bounds. In this work, we unify these various approaches via the paradigm of grading, adapted from the world of functional calculi and semantics. We propose Graded Hoare Logic (GHL), a parameterisable framework for augmenting program logics with a preordered monoidal analysis. We develop a semantic framework for modelling GHL such that grading, logical assertions (pre- and post-conditions) and the underlying effectful semantics of an imperative language can be integrated together. Central to our framework is the notion of a graded category which we extend here, introducing graded Freyd categories which provide a semantics that can interpret many examples of augmented program logics from the literature. We leverage coherent fibrations to model the base assertion language, and thus the overall setting is also fibrational.

cs.LO↗

arXiv2020

Relational Differential Dynamic Logic

In the field of quality assurance of hybrid systems (that combine continuous physical dynamics and discrete digital control), Platzer's differential dynamic logic (dL) is widely recognized as a deductive verification method with solid mathematical foundations and sophisticated tool support. Motivated by benchmarks provided by our industry partner, we study a relational extension of dL, aiming to formally prove statements such as "an earlier deployment of the emergency brake decreases the collision speed." A main technical challenge here is to relate two states of two dynamics at different time points. Our main contribution is a theory of suitable simulations (a relational extension of differential invariants that are central proof methods in dL), and a derived technique of time stretching. The latter features particularly high applicability, since the user does not have to synthesize a simulation out of the air. We derive new inference rules for dL from these notions, and demonstrate their use over a couple of automotive case studies.

cs.LO↗

arXiv2020

A Coalgebraic View on Reachability

Coalgebras for an endofunctor provide a category-theoretic framework for modeling a wide range of state-based systems of various types. We provide an iterative construction of the reachable part of a given pointed coalgebra that is inspired by and resembles the standard breadth-first search procedure to compute the reachable part of a graph. We also study coalgebras in Kleisli categories: for a functor extending a functor on the base category, we show that the reachable part of a given pointed coalgebra can be computed in that base category.

math.CT↗

arXiv2019

Interaction laws of monads and comonads

We introduce and study functor-functor and monad-comonad interaction laws as mathematical objects to describe interaction of effectful computations with behaviors of effect-performing machines. Monad-comonad interaction laws are monoid objects of the monoidal category of functor-functor interaction laws. We show that, for suitable generalizations of the concepts of dual and Sweedler dual, the greatest functor resp. monad interacting with a given functor or comonad is its dual while the greatest comonad interacting with a given monad is its Sweedler dual. We relate monad-comonad interaction laws to stateful runners. We show that functor-functor interaction laws are Chu spaces over the category of endofunctors taken with the Day convolution monoidal structure. Hasegawa's glueing endows the category of these Chu spaces with a monoidal structure whose monoid objects are monad-comonad interaction laws.

cs.LO↗

arXiv2019

Codensity Games for Bisimilarity

Bisimilarity as an equivalence notion of systems has been central to process theory. Due to the recent rise of interest in quantitative systems (probabilistic, weighted, hybrid, etc.), bisimilarity has been extended in various ways: notably, bisimulation metric between probabilistic systems. An important feature of bisimilarity is its game-theoretic characterization, where Spoiler and Duplicator play against each other; extension of bisimilarity games to quantitative settings has been actively pursued too. In this paper, we present a general framework that uniformly describes game characterizations of bisimilarity-like notions. Our framework is formalized categorically using fibrations and coalgebras. In particular, our characterization of bisimilarity in terms of fibrational predicate transformers allows us to derive codensity bisimilarity games: a general categorical game characterization of bisimilarity. Our framework covers known bisimilarity-like notions (such as bisimulation metric) as well as new ones (including what we call bisimulation topology).

cs.LO↗

arXiv2019

Probabilistic Relational Reasoning via Metrics

The Fuzz programming language [Reed and Pierce, 2010] uses an elegant linear type system combined with a monad-like type to express and reason about probabilistic sensitivity properties, most notably $ε$-differential privacy. We show how to extend Fuzz to capture more general relational properties of probabilistic programs, with approximate, or $(ε, δ)$-differential privacy serving as a leading example. Our technical contributions are threefold. First, we introduce the categorical notion of comonadic lifting of a monad to model composition properties of probabilistic divergences. Then, we show how to express relational properties in terms of sensitivity properties via an adjunction we call the path construction. Finally, we instantiate our semantics to model the terminating fragment of Fuzz extended with types carrying information about other divergences between distributions.

cs.PL↗