scholarly journals The geometry of Bayesian programming

2021 ◽  
pp. 1-49
Author(s):  
Ugo Dal Lago ◽  
Naohiko Hoshino
Keyword(s):  
Uniform Distribution ◽  
Operational Semantics ◽  
Higher Order ◽  
Interaction Models ◽  
Measurable Functions

Abstract We give two geometry of interaction models for a typed λ-calculus with recursion endowed with operators for sampling from a continuous uniform distribution and soft conditioning, namely a paradigmatic calculus for higher-order Bayesian programming. The models are based on the category of measurable spaces and partial measurable functions, and the category of measurable spaces and s-finite kernels, respectively. The former is proved adequate with respect to both a distribution-based and a sampling-based operational semantics, while the latter is proved adequate with respect to a sampling-based operational semantics.

scholarly journals Measurable cones and stable, measurable functions: a model for probabilistic higher-order programming

2018 ◽  
Vol 2 (POPL) ◽  
pp. 1-28 ◽  
Author(s):  
Thomas Ehrhard ◽  
Michele Pagani ◽  
Christine Tasson
Keyword(s):  
Higher Order ◽  
Measurable Functions

Principal signatures for higher-order program modules

1994 ◽  
Vol 4 (3) ◽  
pp. 285-335 ◽  
Author(s):  
Mads Tofte
Keyword(s):  
Programming Languages ◽  
Functional Programming ◽  
Operational Semantics ◽  
Higher Order ◽  
Principal Type ◽  
Functional Programming Languages ◽  
Standard Ml ◽  
Static Semantics ◽  
Program Modules

AbstractIn this paper we present a language for programming with higher-order modules. The language HML is based on Standard ML in that it provides structures, signatures and functors. In HML, functors can be declared inside structures and specified inside signatures; this is not possible in Standard ML. We present an operational semantics for the static semantics of HML signature expressions, with particular emphasis on the handling of sharing. As a justification for the semantics, we prove a theorem about the existence of principal signatures. This result is closely related to the existence of principal type schemes for functional programming languages with polymorphism.

Mechanizing proofs with logical relations – Kripke-style

2018 ◽  
Vol 28 (9) ◽  
pp. 1606-1638 ◽  
Author(s):  
ANDREW CAVE ◽  
BRIGITTE PIENTKA
Keyword(s):  
Case Studies ◽  
Operational Semantics ◽  
Lambda Calculus ◽  
Equational Theory ◽  
Higher Order ◽  
Abstract Syntax ◽  
Completeness Proof ◽  
Typed Lambda Calculus ◽  
Contextual Equivalence ◽  
The Right

Proofs with logical relations play a key role to establish rich properties such as normalization or contextual equivalence. They are also challenging to mechanize. In this paper, we describe two case studies using the proof environmentBeluga: First, we explain the mechanization of the weak normalization proof for the simply typed lambda-calculus; second, we outline how to mechanize the completeness proof of algorithmic equality for simply typed lambda-terms where we reason about logically equivalent terms. The development of these proofs inBelugarelies on three key ingredients: (1) we encode lambda-terms together with their typing rules, operational semantics, algorithmic and declarative equality using higher order abstract syntax (HOAS) thereby avoiding the need to manipulate and deal with binders, renaming and substitutions, (2) we take advantage ofBeluga's support for representing derivations that depend on assumptions and first-class contexts to directly state inductive properties such as logical relations and inductive proofs, (3) we exploitBeluga's rich equational theory for simultaneous substitutions; as a consequence, users do not need to establish and subsequently use substitution properties, and proofs are not cluttered with references to them. We believe these examples demonstrate thatBelugaprovides the right level of abstractions and primitives to mechanize challenging proofs using HOAS encodings. It also may serve as a valuable benchmark for other proof environments.

scholarly journals HOPLA--A Higher-Order Process Language

2002 ◽  
Vol 9 (49) ◽  
Author(s):  
Mikkel Nygaard ◽  
Glynn Winskel
Keyword(s):  
Operational Semantics ◽  
Lambda Calculus ◽  
Expressive Power ◽  
Higher Order ◽  
Domain Theory ◽  
Order Process ◽  
Encode Process ◽  
Mobile Ambients ◽  
Computation Path ◽  
Congruence Properties

A small but powerful language for higher-order nondeterministic processes is introduced. Its roots in a linear domain theory for concurrency are sketched though for the most part it lends itself to a more operational account. The language can be viewed as an extension of the lambda calculus with a ``prefixed sum'', in which types express the form of computation path of which a process is capable. Its operational semantics, bisimulation, congruence properties and expressive power are explored; in particular, it is shown how it can directly encode process languages such as CCS, CCS with process passing, and mobile ambients with public names.

scholarly journals New-HOPLA--A Higher-Order Process Language with Name Generation

2004 ◽  
Vol 11 (21) ◽  
Author(s):  
Glynn Winskel ◽  
Francesco Zappa Nardelli
Keyword(s):  
Operational Semantics ◽  
Expressive Power ◽  
Higher Order ◽  
Domain Theory ◽  
Process Algebras ◽  
Order Process ◽  
Pi Calculus ◽  
Mobile Ambients ◽  
Congruence Properties

This paper introduces new-HOPLA, a concise but powerful language for higher-order nondeterministic processes with name generation. Its origins as a metalanguage for domain theory are sketched but for the most part the paper concentrates on its operational semantics. The language is typed, the type of a process describing the shape of the computation paths it can perform. Its transition semantics, bisimulation, congruence properties and expressive power are explored. Encodings are given of well-known process algebras, including pi-calculus, Higher-Order pi-calculus and Mobile Ambients.

scholarly journals On the class gamma and related classes of functions

2013 ◽  
Vol 93 (107) ◽  
pp. 1-18 ◽  
Author(s):  
Edward Omey
Keyword(s):  
Uniform Convergence ◽  
Auxiliary Function ◽  
Higher Order ◽  
Representation Theorems ◽  
Order Relations ◽  
Measurable Functions ◽  
Local Uniform Convergence

The gamma class ??(g) consists of positive and measurable functions that satisfy f(x + yg(x))/f(x) ? exp(?y). In most cases the auxiliary function g is Beurling varying and self-neglecting, i.e., g(x)/x ? 0 and g??0(g). Taking h = log f, we find that h?E??(g, 1), where E??(g, a) is the class of positive and measurable functions that satisfy (f(x + yg(x))? f(x))/a(x) ? ?y. In this paper we discuss local uniform convergence for functions in the classes ??(g) and E??(g, a). From this, we obtain several representation theorems. We also prove some higher order relations for functions in the class ??(g) and related classes. Two applications are given.

scholarly journals Higher Order Oscillation and Uniform Distribution

2019 ◽  
Vol 14 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Shigeki Akiyama ◽  
Yunping Jiang
Keyword(s):  
Number Theory ◽  
Real Number ◽  
Uniform Distribution ◽  
Differentiable Function ◽  
Mild Condition ◽  
Higher Order ◽  
Real Numbers ◽  
Mobius Function ◽  
Real Polynomials ◽  
Almost All

AbstractIt is known that the Möbius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form (e2πiαβn g(β))n∈𝕅, for a non-decreasing twice differentiable function g with a mild condition. This follows the result we prove in this paper that for a fixed non-zero real number α and almost all real numbers β> 1 (alternatively, for a fixed real number β> 1 and almost all real numbers α) and for all real polynomials Q(x), sequences (αβng(β)+ Q(n)) n∈𝕅 are uniformly distributed modulo 1.

Typing and subtyping for mobile processes

1996 ◽  
Vol 6 (5) ◽  
pp. 409-453 ◽  
Author(s):  
Benjamin Pierce ◽  
Davide Sangiorgi
Keyword(s):  
Process Algebra ◽  
Operational Semantics ◽  
Type System ◽  
Higher Order ◽  
Order Process ◽  
Process Calculi ◽  
Mobile Processes

The π-calculus is a process algebra that supports mobility by focusing on the communication of channels. Milner's presentation of the π-calculus includes a type system assigning arities to channels and enforcing a corresponding discipline in their use. We extend Milner's language of types by distinguishing between the ability to read from a channel, the ability to write to a channel, and the ability both to read and to write. This refinement gives rise to a natural subtype relation similar to those studied in typed λ-calculi. The greater precision of our type discipline yields stronger versions of standard theorems on the π-calculus. These can be used, for example, to obtain the validity of β-reduction for the more efficient of Milner's encodings of the call-by-value λ-calculus, which fails in the ordinary π-calculus. We define the syntax, typing, subtyping, and operational semantics of our calculus, prove that the typing rules are sound, apply the system to Milner's λ-calculus encodings, and sketch extensions to higher-order process calculi and polymorphic typing.

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