Monadic Semantics Research Articles

Giving semantics to program-counter labels via secure effects

Type systems designed for information-flow control commonly use a program-counter label to track the sensitivity of the context and rule out data leakage arising from effectful computation in a sensitive context. Currently, type-system designers reason about this label informally except in security proofs, where they use ad-hoc techniques. We develop a framework based on monadic semantics for effects to give semantics to program-counter labels. This framework leads to three results about program-counter labels. First, we develop a new proof technique for noninterference, the core security theorem for information-flow control in effectful languages. Second, we unify notions of security for different types of effects, including state, exceptions, and nontermination. Finally, we formalize the folklore that program-counter labels are a lower bound on effects. We show that, while not universally true, this folklore has a good semantic foundation.

A monadic semantics for quantum computing in an object oriented language

Nowadays, several languages and libraries have been proposed to program and reason about quantum programs in the imperative and functional paradigms. Having in mind that the object-oriented paradigm is one of the most used for developing general purpose software systems, in this work we propose to investigate quantum programming using the object-oriented paradigm. Hence, we present a monadic semantics for FJQuantum 1 , an object-oriented language based on Featherweight Java, created to reason and to develop programs handling quantum data and quantum operations. We also provide an interpreter for this semantics and a set of examples of quantum programs to run with it. • A Monadic Semantics for Quantum Computing inside an Object Oriented Language is proposed. • The use of Featherweight Java allows the formalization and implementation of several extensions. • An interpreter for the proposed language is provided to test several examples of quantum programs.

A precise monadic dynamic slicing method

Dynamic program slicing is useful in software debugging, testing and maintenance, because it can extract more precise results than those obtained by static slicing. In this paper, we propose a precise approach for dynamic program slicing, monadic dynamic slicing, which is based on modular monadic semantics. Firstly, we abstract the computation of dynamic slicing as an object of independent language, dynamic slice-monad transformer. Then we discuss and illustrate a modular monadic dynamic slice algorithm in detail. In this monadic algorithm, dynamic slices are computed on abstract syntax directly, without the need to explicitly construct intermediate structures such as dependence graphs, or to record an execution history. Finally, we address the implementation and complexity analysis of this algorithm. We conclude that the monadic approach has excellent flexibility, combinability and parallelizability properties.

Relating computational effects by ⊤⊤-lifting

We consider the problem of establishing a relationship between two interpretations of base type terms of a λc-calculus extended with algebraic operations. We show that the given relationship holds if it satisfies a set of natural conditions. We apply this result to 1) comparing two monadic semantics related by a strong monad morphism, and 2) comparing two monadic semantics of fresh name creation: Starkʼs new name creation monad and the global counter monad. We also consider the same problem, relating semantics of computational effects, in the presence of recursive functions. We apply this additional by extending the previous monad morphism comparison result to the recursive case.

Monad transformers as monoid transformers

The incremental approach to modular monadic semantics constructs complex monads by using monad transformers to add computational features to a pre-existing monad. A complication of this approach is that the operations associated to the pre-existing monad need to be lifted to the new monad. In a companion paper by Jaskelioff, the lifting problem has been addressed in the setting of system F ω . Here, we recast and extend those results in a category-theoretic setting. We abstract and generalize from monads to monoids (in a monoidal category), and from monad transformers to monoid transformers. The generalization brings more simplicity and clarity, and opens the way for lifting of operations with applicability beyond monads.

Eager and delayed contract monitoring for call-by-value and call-by-name evaluation

Contracts are a proven tool in software development. They provide specifications for operations that may be statically verified or dynamically validated by contract monitoring. We investigate the properties of contract monitoring for languages with contracts and effects using a monadic semantics. We study three combinations of evaluation orders and contract monitoring styles: call-by-value and call-by-name with eager monitoring and call-by-name with delayed monitoring. In each case, an effect system ensures that contract monitoring does not change the meaning of a program and guarantees that contract monitoring is idempotent. The monadic semantics enables us to study design choices, to formalize implementations, to pinpoint the differences between contracts in the three combinations, and to verify algebraic laws.

Constructing language processors with algebra combinators

Modular Monadic Semantics (MMS) is a well-known mechanism for structuring modular denotational semantic definitions for programming languages. The principal attraction of MMS is that families of language constructs can be independently specified and later combined in a mix-and-match fashion to create a complete language semantics. This has proved useful for constructing formal, yet executable, semantics when prototyping languages. In this work we demonstrate that MMS has an additional software engineering benefit. In addition to composing semantics for various language constructs, we can use MMS to compose various differing semantics for the same language constructs. This capability allows us to compose and reuse orthogonal language tasks such as type checking and compilation. We describe algebra combinators, the principal vehicle for achieving this reuse, along with a series of applications of the technique for common language processing tasks.

On the relations between monadic semantics

We present a simple computational metalanguage with general recursive types and multiple notions of effects, through which a variety of concrete denotational semantics can be conveniently factored, by suitably interpreting the effects as monads. We then propose a methodology for relating two such interpretations of the metalanguage, with the aim of showing that the semantics they induce agree for complete programs. As a prototypical instance of such a relation, we use the framework to show agreement between a direct and a continuation semantics of the simple, untyped functional language from Reynolds’s original paper on the subject.

An initial algebra approach to term rewriting systems with variable binders

We present an extension of first-order term rewriting systems. It involves variable binding in the term language. We develop systems called binding term rewriting systems (BTRSs) in a stepwise manner. First we present the term language, then formulate equational logic. Finally, we define rewriting systems. This development is novel because we follow the initial algebra approach in an extended notion of ?-algebras in various functor categories. These are based on Fiore-Plotkin-Turi's presheaf semantics of variable binding and Luth-Ghani's monadic semantics of term rewriting systems. We characterise the terms, equational logic and rewrite systems for BTRSs as initial algebras in suitable categories. Then, we show an important rewriting property of BTRSs: orthogonal BTRSs are confluent. Moreover, by using the initial algebra semantics, we give a complete characterisation of termination of BTRSs. Finally, we discuss our design choice of BTRSs from a semantic perspective.

On a monadic semantics for freshness

A standard monad of continuations, when constructed with domains in the world of FM-sets [M.J. Gabbay, A.M. Pitts, A new approach to abstract syntax with variable binding, Formal Aspects Comput. 13 (2002) 341–363], is shown to provide a model of dynamic allocation of fresh names that is both simple and useful. In particular, it is used to prove that the powerful facilities for manipulating fresh names and binding operations provided by the “Fresh” series of metalanguages [M.R. Shinwell, Swapping the atom: Programming with binders in Fresh O’Caml, Proc. MER λ IN, 2003; M.R. Shinwell, A.M. Pitts, Fresh O’Caml User Manual, Cambridge University Computer Laboratory, September 2003, available at 〈 http://www.freshml.org/foc/ 〉 ; M.R. Shinwell, A.M. Pitts, M.J. Gabbay, FreshML: Programming with binders made simple, in: Proc. ICFP ’03, ACM Press, 2003, pp. 263–274] respect α -equivalence of object-level languages up to meta-level contextual equivalence.

A Monadic Semantics for Core Curry

We give a deterministic, big-step operational semantics for the essential core of the Curry language, including higher-order functions, call-by-need evaluation, non-determinism, narrowing, and residuation. The semantics is structured in modular monadic style, and is presented in the form of an executable interpreter written in Haskell. It uses monadic formulations of state, non-determinism, and resumption-based concurrency.

Algebraic operations and generic effects

Given a complete and cocomplete symmetric monoidal closed category V and a symmetric monoidal V-category C with cotensors and a strong V-monad T on C, we investigate axioms under which an ObC-indexed family of operations of the form αx:(Tx)v→(Tx)w provides semantics for algebraic operations on the computational λ-calculus. We recall a definition for which we have elsewhere given adequacy results, and we show that an enrichment of it is equivalent to a range of other possible natural definitions of algebraic operation. In particular, we define the notion of generic effect and show that to give a generic effect is equivalent to giving an algebraic operation. We further show how the usual monadic semantics of the computational λ-calculus extends uniformly to incorporate generic effects. We outline examples and non-examples and we show that our definition also enriches one for call-by-name languages with effects.

A unifying approach to goal-directed evaluation

Goal-directed evaluation, as embodied in Icon and Snobol, is built on the notions of backtracking and of generating successive results, and therefore it has always been something of a challenge to specify and implement. In this article, we address this challenge using computational monads and partial evaluation.We consider a subset of Icon and we specify it with a monadic semantics and a list monad. We then consider a spectrum of monads that also fit the bill, and we relate them to each other. For example, we derive a continuation monad as a Church encoding of the list monad. The resulting semantics coincides with Gudeman's continuation semantics of Icon.We then compile Icon programs by specializing their interpreter (i.e., by using the first Futamura projection), using type-directed partial evaluation. Through various back ends, including a run-time code generator, we generate ML code, C code, and OCaml byte code. Binding-time analysis and partial evaluation of the continuation-based interpreter automatically give rise to C programs that coincide with the result of Proebsting's optimized compiler.

A Unifying Approach to Goal-Directed Evaluation

Goal-directed evaluation, as embodied in Icon and Snobol, is built on the notions of backtracking and of generating successive results, and therefore it has always been something of a challenge to specify and implement. In this article, we address this challenge using computational monads and partial evaluation.<br /> <br />We consider a subset of Icon and we specify it with a monadic semantics and a list monad. We then consider a spectrum of monads that also fit the bill, and we relate them to each other. For example, we derive a continuation monad as a Church encoding of the list monad. The resulting semantics coincides with Gudeman's continuation semantics of Icon.<br /> <br />We then compile Icon programs by specializing their interpreter (i.e., by using the first Futamura projection), using type-directed partial evaluation. Through various back ends, including a run-time code generator, we generate ML code, C code, and OCaml byte code. Binding-time analysis and partial evaluation of the continuation-based interpreter automatically give rise to C programs that coincide with the result of Proebsting's optimized compiler.

LPS: A Language Prototyping System Using Modular Monadic Semantics

This paper describes LPS, a Language Prototyping System that facilitates the modular development of interpreters from semantic building blocks. The system is based on the integration of ideas from Modular Monadic Semantics and Generic Programming.To define a new programming language, the abstract syntax is described as the fixpoint of non-recursive pattern functors. For each functor an algebra is defined whose carrier is the computational monad obtained from the application of several monad transformers to a base monad. The interpreter is automatically generated by a catamorphism or, in some special cases, a monadic catamorphism.The system has been implemented as a domain-specific language embedded in Haskell and we have also implemented an interactive framework for language testing.

Semantics, Modularity, and Rewriting Logic

A complete formal semantic description of a practical programming language (such as Java) is likely to be a lengthy document, regardless of which semantic framework is being used. Good modularity of the description is important to the person(s) developing it, to facilitate reuse, change, and extension. Unfortunately, the conventional versions<br />of the major semantic frameworks have rather poor modularity.<br /> In this paper, we first recall some approaches that improve the modularity of denotational semantics, namely action semantics, modular monadic semantics, and a hybrid framework that combines these: modular monadic action semantics. We then address the issue of modularity in operational semantics, which appears to have received comparatively little attention so far, and report on some preliminary investigations of how one might achieve the same kind of modularity in structural operational semantics as the use of monad transformers<br />can provide in denotational semantics|this is the main technical contribution of the paper. Finally, we briefly consider the representation of structural operational semantics in rewriting logic, and speculate on the possibility of using it to interpret programs in the described language. Providing powerful meta-tools for such semantics-based interpretation<br />is an interesting potential application of rewriting logic;<br />good modularity of the semantic descriptions may be crucial for the practicality of using the tools.<br />Much of the paper consists of (very) simple examples of semantic descriptions in the various frameworks, illustrating the degree of reformulation needed when extending the described language|a strong indicator of modularity. Throughout, it is assumed that the reader has some familiarity with the concepts and notation of denotational and structural operational semantics. Familiarity with the basic notions of monads and monad transformers is not a prerequisite.

Semantics, Modularity, and Rewriting Logic

A complete formal semantic description of a practical programming language (such as Java) is likely to be a lengthy document, regardless of which semantic framework is being used. Good modularity of the description is important to the person(s) developing it, to facilitate reuse, change, and extension. Unfortunately, the conventional versions of the major semantic frameworks have rather poor modularity.In this paper, we first recall some approaches that improve the modularity of denotational semantics, namely action semantics, modular monadic semantics, and a hybrid framework that combines these: modular monadic action semantics. We then address the issue of modularity in operational semantics, which appears to have received comparatively little attention so far, and report on some preliminary investigations of how one might achieve the same kind of modularity in structural operational semantics as the use of monad transformers can provide in denotational semantics—this is the main technical contribution of the paper. Finally, we briefly consider the representation of structural operational semantics in rewriting logic, and speculate on the possibility of using it to interpret programs in the described language. Providing powerful meta-tools for such semantics-based interpretation is an interesting potential application of rewriting logic; good modularity of the semantic descriptions may be crucial for the practicality of using the tools.Much of the paper consists of (very) simple examples of semantic descriptions in the various frameworks, illustrating the degree of reformulation needed when extending the described language—a strong indicator of modularity. Throughout, it is assumed that the reader has some familiarity with the concepts and notation of denotational and structural operational semantics. Familiarity with the basic notions of monads and monad transformers is not a prerequisite.