Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userModules over monads and operational semantics
103
0
0.0
(
0
)
Added by
Tom Hirschowitz
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
This paper is a contribution to the search for efficient and high-level mathematical tools to specify and reason about (abstract) programming languages or calculi. Generalising the reduction monads of Ahrens et al., we introduce transition monads, thus covering new applications such as lambda-bar-mu-calculus, pi-calculus, Positive GSOS specifications, differential lambda-calculus, and the big-step, simply-typed, call-by-value lambda-calculus. Moreover, we design a suitable notion of signature for transition monads.
rate research
Read More
We give a conceptual treatment of the notion of joints, marginals, and independence in the setting of categorical probability. This is achieved by endowing the usual probability monads (like the Giry monad) with a monoidal and an opmonoidal structure, mutually compatible (i.e. a bimonoidal structure). If the underlying monoidal category is cartesian monoidal, a bimonoidal structure is given uniquely by a commutative strength. However, if the underlying monoidal category is not cartesian monoidal, a strength is not enough to guarantee all the desired properties of joints and marginals. A bimonoidal structure is then the correct requirement for the more general case. We explain the theory and the operational interpretation, with the help of the graphical calculus for monoidal categories. We give a definition of stochastic independence based on the bimonoidal structure, compatible with the intuition and with other approaches in the literature for cartesian monoidal categories. We then show as an example that the Kantorovich monad on the category of complete metric spaces is a bimonoidal monad for a non-cartesian monoidal structure.
It has been an open question as to whether the Modular Structural Operational Semantics framework can express the dynamic semantics of call/cc. This paper shows that it can, and furthermore, demonstrates that it can express the more general delimited control operators control and shift.
We reflect on programming with complicated effects, recalling an undeservingly forgotten alternative to monadic programming and checking to see how well it can actually work in modern functional languages. We adopt and argue the position of factoring an effectful program into a first-order effectful DSL with a rich, higher-order macro system. Not all programs can be thus factored. Although the approach is not general-purpose, it does admit interesting programs. The effectful DSL is likewise rather problem-specific and lacks general-purpose monadic composition, or even functions. On the upside, it expresses the problem elegantly, is simple to implement and reason about, and lends itself to non-standard interpretations such as code generation (compilation) and abstract interpretation. A specialized DSL is liable to be frequently extended; the experience with the tagless-final style of DSL embedding shown that the DSL evolution can be made painless, with the maximum code reuse. We illustrate the argument on a simple but representative example of a rather complicated effect -- non-determinism, including committed choice. Unexpectedly, it turns out we can write interesting non-deterministic programs in an ML-like language just as naturally and elegantly as in the functional-logic language Curry -- and not only run them but also statically analyze, optimize and compile. The richness of the Meta Language does, in reality, compensate for the simplicity of the effectful DSL. The key idea goes back to the origins of ML as the Meta Language for the Edinburgh LCF theorem prover. Instead of using ML to build theorems, we now build (DSL) programs.
We consider three monads on Top, the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the Hoare hyperspace monad H, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second is the monad V of continuous valuations, also known as the extended probabilistic powerdomain. We construct both monads in a unified way in terms of double dualization. This reveals a close analogy between them, and allows us to prove that the operation of taking the support of a continuous valuation is a morphism of monads from V to H. In particular, this implies that every H-algebra (topological complete semilattice) is also a V-algebra. Third, we show that V can be restricted to a submonad of tau-smooth probability measures on Top. By composing these two morphisms of monads, we obtain that taking the support of a tau-smooth probability measure is also a morphism of monads.
We make a formal analogy between random sampling and fresh name generation. We show that quasi-Borel spaces, a model for probabilistic programming, can soundly interpret Starks $ u$-calculus, a calculus for name generation. Moreover, we prove that this semantics is fully abstract up to first-order types. This is surprising for an off-the-shelf model, and requires a novel analysis of probability distributions on function spaces. Our tools are diverse and include descriptive set theory and normal forms for the $ u$-calculus.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
