Shin-ya Katsumata

Parametric effect monads and semantics of effect systems

We study fundamental properties of a generalisation of monad called parametric effect monad, and apply it to the interpretation of general effect systems whose effects have sequential composition operators. We show that parametric effect monads admit analogues of the structures and concepts that exist for monads, such as Kleisli triples, the state monad and the continuation monad, Plotkin and Powers algebraic operations, and the categorical ┬┬-lifting. We also show a systematic method to generate both effects and a parametric effect monad from a monad morphism. Finally, we introduce two effect systems with explicit and implicit subeffecting, and discuss their denotational semantics and the soundness of effect systems.

Proof-Directed De-compilation of Low-Level Code

We present a proof theoretical method for de-compiling low-level code to the typed lambda calculus. We first define a proof system for a low-level code language based on the idea of Curry-Howard isomorphism. This allows us to regard an executable code as a proof in intuitionistic propositional logic. As being a proof of intuitionistic logic, it can be translated to an equivalent proof of natural deduction proof system. This property yields an algorithm to translate a given code into a lambda term. Moreover, the resulting lambda term is not a trivial encoding of a sequence of primitive instructions, but reflects the behavior of the given program. This process therefore serves as proof-directed decompilation of a low-level code language to a high-level language. We carry out this development for a subset of Java Virtual Machine instructions including most of its features such as jumps, object creation and method invocation. The proof-directed de-compilation algorithm has been implemented, which demonstrates the feasibility of our approach.

A semantic formulation of ⊤⊤-lifting and logical predicates for computational metalanguage

A semantic formulation of Lindley and Starks ⊤⊤-lifting is given. We first illustrate our semantic formulation of the ⊤⊤-lifting in Set with several examples, and apply it to the logical predicates for Moggis computational metalanguage. We then abstract the semantic ⊤⊤-lifting as the lifting of strong monads across bifibrations with lifted symmetric monoidal closed structures.

Comparing type-based and proof-directed decompilation

In the past couple of years interest in decompilation has widened from its initial concentration on reconstruction of control flow into well-founded-in-theory methods to reconstruct type information. A. Mycroft (1999) described Type-Based Decompilation and S. Katsumata and A. Ohori (2001) described Proof-Directed Decompilation. The article summarises the two approaches and identifies their commonality, strengths and weaknesses; it concludes by suggesting how they may be integrated.

Attribute Grammars and Categorical Semantics

We give a new formulation of attribute grammars (AG for short) called monoidalAGsin traced symmetric monoidal categories. Monoidal AGs subsume existing domain-theoretic, graph-theoretic and relational formulations of AGs. Using a 2-categorical aspect of monoidal AGs, we also show that every monoidal AG is equivalent to a synthesised one when the underlying category is closed, and that there is a sound and complete translation from local dependency graphs to relational AGs.

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 · 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.

Preorders on monads and coalgebraic simulations

We study the construction of preorders on Set-monads by the semantic ⊤⊤-lifting. We show the universal property of this construction, and characterise the class of preorders on a monad as a limit of a Cardop-chain. We apply these theoretical results to identifying preorders on some concrete monads, including the powerset monad, maybe monad, and their composite monad. We also relate the construction of preorders and coalgebraic formulation of simulations.

Combining effects and coeffects via grading

Effects and coeffects are two general, complementary aspects of program behaviour. They roughly correspond to computations which change the execution context (effects) versus computations which make demands on the context (coeffects). Effectful features include partiality, non-determinism, input-output, state, and exceptions. Coeffectful features include resource demands, variable access, notions of linearity, and data input requirements. The effectful or coeffectful behaviour of a program can be captured and described via type-based analyses, with fine grained information provided by monoidal effect annotations and semiring coeffects. Various recent work has proposed models for such typed calculi in terms of graded (strong) monads for effects and graded (monoidal) comonads for coeffects. Effects and coeffects have been studied separately so far, but in practice many computations are both effectful and coeffectful, e.g., possibly throwing exceptions but with resource requirements. To remedy this, we introduce a new general calculus with a combined effect-coeffect system. This can describe both the changes and requirements that a program has on its context, as well as interactions between these effectful and coeffectful features of computation. The effect-coeffect system has a denotational model in terms of effect-graded monads and coeffect-graded comonads where interaction is expressed via the novel concept of graded distributive laws. This graded semantics unifies the syntactic type theory with the denotational model. We show that our calculus can be instantiated to describe in a natural way various different kinds of interaction between a program and its evaluation context.

Towards a Formal Theory of Graded Monads

We initiate a formal theory of graded monads whose purpose is to adapt and to extend the formal theory of monads developed by Street in the early 1970’s. We establish in particular that every graded monad can be factored in two different ways as a strict action transported along a left adjoint functor. We also explain in what sense the first construction generalizes the Eilenberg-Moore construction while the second construction generalizes the Kleisli construction. Finally, we illustrate the Eilenberg-Moore construction on the graded state monad induced by any object V in a symmetric monoidal closed category \(\mathscr {C}\).

Extensional Models of Untyped Lambda-mu Calculus

This paper proposes new mathematical models of the untyped Lambda-mu calculus. One is called the stream model, which is an extension of the lambda model, in which each term is interpreted as a function from streams to individual data. The other is called the stream combinatory algebra, which is an extension of the combinatory algebra, and it is proved that the extensional equality of the Lambda-mu calculus is equivalent to equality in stream combinatory algebras. In order to define the stream combinatory algebra, we introduce a combinatory calculus SCL, which is an abstraction-free system corresponding to the Lambda-mu calculus. Moreover, it is shown that stream models are algebraically characterized as a particular class of stream combinatory algebras.