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The refinement calculus is a formalized approach to stepwise refinement for program construction. The required behaviour of the final executable program
Refinement_calculus
alternative technique when formal refinement is not possible. The opposite of refinement is abstraction. Refinement calculus is a formal system (inspired from
Refinement_(computing)
Topics referred to by the same term
processes Relational calculus, a calculus for the relational data model Domain relational calculus Tuple calculus Refinement calculus, a way of refining
Calculus_(disambiguation)
Formalism in computer science
calculus considered (see kinds below). From a certain point of view, typed lambda calculi can be seen as refinements of the untyped lambda calculus,
Typed_lambda_calculus
Reformulation of Floyd-Hoare logic
Repetition is a generalization of while statement in a similar way. Refinement calculus extends GCL with the notion of specification statement. Syntactically
Predicate transformer semantics
Predicate_transformer_semantics
Mathematical-logic system based on functions
In mathematical logic, the lambda calculus (also written as λ-calculus) is a formal system for expressing computation based on function abstraction and
Lambda_calculus
Formal specification language used for describing and modelling computing systems
commands. These ideas were later formalized by Carroll Morgan in his refinement calculus. Z schema boxes were added by Carroll Morgan for the structuring
Z_notation
Topics referred to by the same term
Join-calculus, a process calculus developed at INRIA for the design of distributed programming languages Join-pattern, generalization of Join-calculus Joins
Join
"Hennessy-Milner Logic with recursion as a specification language, and a refinement calculus based on it". Specification and Verification of Concurrent Systems
Hennessy–Milner_logic
Dijkstra notation with non-deterministic conditionals
observational congruence of Guarded Commands into a lattice has led to Refinement Calculus. This has been mechanized in Formal Methods like B-Method that allow
Guarded_Command_Language
Verification modeling language
(PDF) on 7 October 2015. CAVA project website Sharma, Asankhaya. "A Refinement Calculus for Promela." Engineering of Complex Computer Systems (ICECCS), 2013
Promela
Increasing sequence of numbers that span an interval
partition x. Another partition Q of the given interval [a, b] is defined as a refinement of the partition P, if Q contains all the points of P and possibly some
Partition_of_an_interval
Rules to verify computer program correctness
calculus for a simple while language. j-Algo Hoare Calculus module (j-Algo on GitHub, j-Algo on SourceForge) – A visualisation of the Hoare calculus in
Hoare_logic
Relational calculus, a calculus for the relational data model Domain relational calculus Tuple calculus, inspired the SQL language Refinement calculus, a way
List_of_formal_systems
Specification language for Java programs
designers' intentions. JML inherits ideas from Eiffel, Larch and the Refinement Calculus, with the goal of providing rigorous formal semantics while still
Java_Modeling_Language
American computer scientist
proponent of a formalized approach to program development called the refinement calculus. He has authored many papers. He is involved with developing international
Carroll Morgan (computer scientist)
Carroll_Morgan_(computer_scientist)
Finnish computer scientist
Back originated the refinement calculus, an important approach to the formal development of programs using stepwise refinement, in his 1978 PhD thesis
Ralph-Johan_Back
Graph data structure
Testing Formal methods Concepts Curry–Howard correspondence Loop invariant Refinement Side effect Soundness and completeness Specification Languages Verification
E-graph
Indian American mathematician (born 1940)
processes with an orientation towards the refinement and further development of Itô’s stochastic calculus. In the year 2007, he won the Abel Prize. Srinivasa
S._R._Srinivasa_Varadhan
Concepts in theoretical computer science
semantics Operational Big-step Small-step Models Lambda calculus Petri net Process calculus Rewriting system State machine Turing machine Analyses Static
Safety and liveness properties
Safety_and_liveness_properties
Computer science field
used for verification when two descriptions are not equivalent. During refinement, the specification is complemented with details that are unnecessary in
Model_checking
Analysis of computer programs without executing them
semantics Operational Big-step Small-step Models Lambda calculus Petri net Process calculus Rewriting system State machine Turing machine Analyses Static
Static_program_analysis
Ralph-Johan Back, originated the refinement calculus, used in the formal development of programs using stepwise refinement Roland Backhouse, work on the
List of programming language researchers
List_of_programming_language_researchers
Integral using products instead of sums
integral is any product-based counterpart of the usual sum-based integral of calculus. The product integral was developed by the mathematician Vito Volterra
Product_integral
Family of formalisms in natural language syntax
grammar shares some features with the simply typed lambda calculus. Whereas the lambda calculus has only one function type A → B {\displaystyle A\rightarrow
Categorial_grammar
Branch of mathematical logic
topics. This has led, in particular, to: Refinement of Gödel's result, particularly J. Barkley Rosser's refinement, weakening the above requirement of ω-consistency
Proof_theory
Of a function, an additional effect besides returning a value
semantics Operational Big-step Small-step Models Lambda calculus Petri net Process calculus Rewriting system State machine Turing machine Analyses Static
Side effect (computer science)
Side_effect_(computer_science)
Area of mathematics
is expected to emerge as a limit of refinement of the discretization. Furthermore, in the process of refinement, DDG places an emphasis on the so-called
Discrete differential geometry
Discrete_differential_geometry
Branch of functional analysis
In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function f of a
Holomorphic functional calculus
Holomorphic_functional_calculus
Basic integral in elementary calculus
published in 1868. It is the integral most commonly introduced in elementary calculus, although in advanced analysis it is often replaced by more general notions
Riemann_integral
Swiss mathematician (1707–1783)
mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and notation
Leonhard_Euler
Computer program for the Boolean satisfiability problem
Testing Formal methods Concepts Curry–Howard correspondence Loop invariant Refinement Side effect Soundness and completeness Specification Languages Verification
SAT_solver
Manifold upon which it is possible to perform calculus
allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within
Differentiable_manifold
Type system used in computer programming and mathematics
Hindley–Milner (HM) type system is a classical type system for the lambda calculus with parametric polymorphism. It is also known as Damas–Milner or Damas–Hindley–Milner
Hindley–Milner_type_system
Integral constructed using Darboux sums
that of the Riemann integral. Consequently, introductory textbooks on calculus and real analysis often develop Riemann integration using the Darboux integral
Darboux_integral
Irish computer scientist
Åbo Akademi University in Turku, Finland with Ralph-Johan Back on refinement calculus. He joined the University of Southampton in 1995 as a lecturer, rising
Michael Butler (computer scientist)
Michael_Butler_(computer_scientist)
Book on mathematical analysis by Camille Jordan published in 1882
Camille Jordan. The three volumes (tomes) address differential calculus, integral calculus, and differential equations. The book is remembered for introducing
Cours d'analyse de l'École polytechnique
Cours_d'analyse_de_l'École_polytechnique
Proof assistant
Ricciotti, C Sacerdoti Coen, Enrico Tassi. "A Bi-Directional Refinement Algorithm for the Calculus of (Co)Inductive Constructions" Logical Methods in Computer
Matita
Approach to static program analysis
Testing Formal methods Concepts Curry–Howard correspondence Loop invariant Refinement Side effect Soundness and completeness Specification Languages Verification
Abstract_interpretation
Danish computer scientist (1949–2012)
commands. These ideas were later formalized by Carroll Morgan in his refinement calculus. In 1985 Sørensen established the Z Users Meeting series at Rewley
Ib_Holm_Sørensen
Theorem in mathematical measure theory
μ . {\displaystyle h={\frac {d\lambda _{a}}{d\mu }}.} An alternative refinement is that of the decomposition of a regular Borel measure ν = ν a c + ν
Lebesgue's decomposition theorem
Lebesgue's_decomposition_theorem
Validates computer program operations
of the object. Typestates are capable of representing behavioral type refinements such as "method A must be invoked before method B is invoked, and method
Typestate_analysis
of relational semantic and graph-based operational semantics, a refinement calculus and tool support for model construction, model analysis and verification
RCOS_(computer_sciences)
Approximation of a function by a polynomial
In calculus, Taylor's theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree
Taylor's_theorem
Concept in computer science
Testing Formal methods Concepts Curry–Howard correspondence Loop invariant Refinement Side effect Soundness and completeness Specification Languages Verification
Separation_logic
System of resource-aware logic
substructural logic proposed by French logician Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former
Linear_logic
Concept in geometry
Modern mathematics can obtain the area using the methods of integral calculus or its more sophisticated offspring, real analysis. However, the area of
Area_of_a_circle
and infinite descending chains. A subset of this lattice is known as a refinement graph. θ-subsumption was first introduced by J. Alan Robinson in 1965
Theta-subsumption
Formal semantics and 1998 book
2006. The semantic foundation of the UTP is the first-order predicate calculus, augmented with fixed-point constructs from second-order logic. Following
Unifying Theories of Programming
Unifying_Theories_of_Programming
Process of analyzing computer program behavior
Testing Formal methods Concepts Curry–Howard correspondence Loop invariant Refinement Side effect Soundness and completeness Specification Languages Verification
Program_analysis
Partition of an interval Partition of a set Ordered partition Partition refinement Disjoint-set data structure Partition problem 3-partition problem Partition
List_of_partition_topics
Mathematics of real numbers and real functions
that develops calculus rigorously over the real numbers and Euclidean spaces. Introductory real analysis is sometimes called advanced calculus, and studies
Real_analysis
cardinal numbers) Cantor's theorem (set theory) Church–Rosser theorem (lambda calculus) Compactness theorem (mathematical logic) Conservativity theorem (mathematical
List_of_theorems
Set of software engineering methods
Testing Formal methods Concepts Curry–Howard correspondence Loop invariant Refinement Side effect Soundness and completeness Specification Languages Verification
Program_slicing
Type whose definition depends on a value
extensional. In 1934, Haskell Curry noticed that the types used in typed lambda calculus, and in its combinatory logic counterpart, followed the same pattern as
Dependent_type
Formal methods terminology
reflection in programming languages. It can be applied as a stepwise refinement at system design time. Reification is one of the most frequently used
Reification (computer science)
Reification_(computer_science)
Notion in calculus
In calculus, the differential represents the principal part of the change in a function y = f ( x ) {\displaystyle y=f(x)} with respect to changes in the
Differential_of_a_function
Theorem in electrical engineering
also applies when the source and/or load are not purely resistive. A refinement of the maximum power theorem says that any reactive components of source
Maximum power transfer theorem
Maximum_power_transfer_theorem
semantics Operational Big-step Small-step Models Lambda calculus Petri net Process calculus Rewriting system State machine Turing machine Analyses Static
Hyperproperty
Family of type systems based on substructural logic
view, there is no K-combinator which can destroy states. From the lambda calculus point of view, a variable x can appear exactly once in a term. Linear type
Substructural_type_system
Conceptual model in philosophy of science
operator is applied (the value is fixed) to multiple variables. The do calculus is the set of manipulations that are available to transform one expression
Causal_model
Set of charts that describes a manifold
such that ( U i ) i ∈ I {\displaystyle \left(U_{i}\right)_{i\in I}} is a refinement of V {\displaystyle {\mathcal {V}}} . M {\displaystyle M} U α {\displaystyle
Atlas_(topology)
Sub-area of scientific computing for solving General Relativity equations
methods have received a majority of the attention, characteristic and Regge calculus based methods have also been used. All of these methods begin with a snapshot
Numerical_relativity
Solution of some Diophantine equation
Matiyasevich's theorem has since been used to prove that many problems from calculus and differential equations are unsolvable. One can also derive the following
Diophantine_set
Computer science and logic conference
Abadi, Leslie Lamport, "The existence of refinement mappings" Eugenio Moggi, "Computational lambda-calculus and monads" Rajeev Alur, Costas Courcoubetis
Symposium on Logic in Computer Science
Symposium_on_Logic_in_Computer_Science
Task to construct a program meeting a formal specification
synthesis are usually non-algorithmic statements in an appropriate logical calculus. The primary application of program synthesis is to relieve the programmer
Program_synthesis
Programming paradigm restricted to provably terminating programs
those inputs; or excluding them by use of type system features such as refinement types. These restrictions mean that total functional programming is not
Total_functional_programming
Formal model in concurrency theory
was essentially a concurrent programming language rather than a process calculus. It had a substantially different syntax than later versions of CSP, did
Communicating sequential processes
Communicating_sequential_processes
About mathematical functions
function dates from the 17th century in connection with the development of calculus; for example, the slope d y / d x {\displaystyle dy/dx} of a graph at a
History of the function concept
History_of_the_function_concept
Fundamental problem in computer science
semantics Operational Big-step Small-step Models Lambda calculus Petri net Process calculus Rewriting system State machine Turing machine Analyses Static
Path_explosion
Formalization of the natural numbers
system. The rule of induction in Curry's system was unusual. A later refinement was given by Goodstein (1954). The rule of induction in Goodstein's system
Primitive recursive arithmetic
Primitive_recursive_arithmetic
semantics Operational Big-step Small-step Models Lambda calculus Petri net Process calculus Rewriting system State machine Turing machine Analyses Static
Dependence_analysis
for describing patterns of interaction in concurrent systems. FDR2 is a refinement checking tool for CSP, comparing two models for compatibility. DVE input
List_of_model_checking_tools
Data type for values having two types
not necessarily composite. A restricted form of intersection types are refinement types. Intersection types are useful for describing overloaded functions
Intersection_type
Country in South Asia
and comparative abundance that marked the Mewari landscape fostered refinements unattainable in other lands. Ameri, Marta (2018). "Letting the Pictures
India
goal Several policy refinement approaches have been developed. The most notable ones are based on linear temporal logic, event calculus, and utility computing
Policy-based_management
Term in stochastic calculus
Monte‑Carlo paths N_list = [50, 100, 200, 500, 1000, 2000, 5000] # mesh refinements # dataframe to hold absolute errors index = pd.Index(N_list, name="N")
Itô_isometry
Process of developing trajectory performance
idea of trajectory optimization has been around for hundreds of years (calculus of variations, brachistochrone problem), it only became practical for real-world
Trajectory_optimization
Construction in functional analysis, useful to solve differential equations
L^{2}(\mathbb {R} ,\mu ),} the decomposition of σ(T) from Borel functional calculus is a refinement of the Banach space case. The preceding comments can be extended
Decomposition of spectrum (functional analysis)
Decomposition_of_spectrum_(functional_analysis)
Cohomology with real coefficients computed using differential forms
speaking) measures precisely the extent to which the fundamental theorem of calculus fails in higher dimensions and on general manifolds. — Terence Tao, Differential
De_Rham_cohomology
Quantity standard
standard system of units facilitates this. Scientific systems of units are a refinement of the concept of weights and measures historically developed for commercial
Unit_of_measurement
GUI and CLI interfaces mesh adaptive-refinement: Yes, full adaptive mesh refinement (h-refinement); no p-refinement but several higher-order elements are
List of finite element software packages
List_of_finite_element_software_packages
Property of a mathematical space
smallest integer n for which the following holds: any open cover has an open refinement (a second open cover in which each element is a subset of an element in
Dimension
Euler presented Lagrange's approach to the Berlin Academy in 1756 as the "calculus of variations". Unlike Euler, Lagrange's approach was purely analytic rather
History of variational principles in physics
History_of_variational_principles_in_physics
Universal type in logic and computer science
propositional calculus, linking to a formula which is true in every possible interpretation. It has a similar meaning in predicate calculus. In description
Any_type
semantics Operational Big-step Small-step Models Lambda calculus Petri net Process calculus Rewriting system State machine Turing machine Analyses Static
Polyvariance
Algebraic structure of set algebra
algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of
Σ-algebra
Type of search algorithm
1961 by Martin Davis, George Logemann and Donald W. Loveland and is a refinement of the earlier Davis–Putnam algorithm, which is a resolution-based procedure
DPLL_algorithm
parameterization), geometric invariants do not. The moving frames method, which is a refinement of Élie Cartan's method of moving frames, gives several new powerful tools
Differential_invariant
Topological model
been used to express spatial predicates; for example region connection calculus was introduced in 1992 by Randell, Cohn and Cohn. The spatial predicates
DE-9IM
Formal method for the development of computer-based systems
been validated can be transformed into detailed system designs through a refinement process. The language has a formal semantics, enabling proof of the properties
Vienna_Development_Method
Form of type polymorphism
often allow the subtyping of records. Consequently, simply typed lambda calculus extended with record types is perhaps the simplest theoretical setting
Subtyping
Field of electrical engineering
analysis techniques of the 17th century. They further state that the digital refinement of these techniques can be found in the digital control systems of the
Signal_processing
Ancient algorithm for generating prime numbers
additions are needed, no multiplications or divisions are used. As a refinement, it is sufficient to mark the numbers in step 3 starting from p2, as all
Sieve_of_Eratosthenes
Mathematical function conceived as a crude model
they activate simultaneously and weakens if they activate separately. A refinement of Hebbian learning, known as spike-timing-dependent plasticity, was developed
Artificial_neuron
Matrix decomposition
D. R.; Hayden, A. F. (2004). Shen, Sylvia S.; Lewis, Paul E. (eds.). "Refinement and generalization of the extension method of covariance matrix inversion
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
British-American psychologist (1905–1998)
and achievement, and many multivariate research methods including the refinement of factor analytic methods for exploring and measuring these domains.
Raymond_Cattell
System for representing and reasoning about time
logic PAT is a powerful free model checker, LTL checker, simulator and refinement checker for CSP and its extensions (with shared variable, arrays, wide
Temporal_logic
Study of parts and the wholes they form
needed] This evolved into the "calculus of individuals" of Goodman and Leonard (1940). Goodman revised and elaborated this calculus in the three editions of
Mereology
Fundamental object of geometry
to the covering dimension because every open cover of the space has a refinement consisting of a single open set. Let X be a metric space. If S ⊂ X and
Point_(geometry)
REFINEMENT CALCULUS
REFINEMENT CALCULUS
Girl/Female
Arabic, Muslim
Purity; Refinement; Clarity
Male
Scottish
Scottish form of Irish Gaelic Cormac, CORMAG means "son of defilement."
Boy/Male
Muslim/Islamic
Refinement
Girl/Female
Muslim
Purity, Refinement, Clarity
Girl/Female
Bengali, Hindu, Indian, Kannada, Marathi, Sanskrit, Sindhi, Telugu
Non-attachment; Bliss; Retirement
Boy/Male
Arabic, Muslim, Sindhi
Refinement
Girl/Female
Arabic, Muslim
Purity; Refinement
REFINEMENT CALCULUS
REFINEMENT CALCULUS
Girl/Female
Tamil
Mudrika | மூதà¯à®°à®¿à®•à®¾
Ring
Female
English
Anglicized form of Irish BrÃd, BREEDA means "exalted one."
Female
Dutch
, pure.
Male
English
Anglicized form of Hebrew Chanan, HANAN means "compassionate, merciful." In the bible, this is the name of many characters, including a chief of the tribe of Benjamin. Compare with feminine Hanan.
Female
Irish
(pronounced kee-va) Irish name derived from Gaelic caomh, CAOIMHE means "beloved, comely."
Girl/Female
American, British, English
Island of Linden Trees; From the Linden Tree Island
Boy/Male
Indian
Lord Vishnu, Imperishable, Indestructible, Immovable
Girl/Female
Tamil
Nectar
Girl/Female
English
Beautiful and Intelligent
Girl/Female
Tamil
Silk
REFINEMENT CALCULUS
REFINEMENT CALCULUS
REFINEMENT CALCULUS
REFINEMENT CALCULUS
REFINEMENT CALCULUS
n.
That which is refined, elaborated, or polished to excess; an affected subtilty; as, refinements of logic.
n.
Concealment; seclusion; retirement.
n.
Privacy; retirement.
a.
Affording retirement from society.
n.
Seclusion; privacy; retirement.
n.
A spot; a defilement.
n.
Hence, figuratively, lacking refinement, gentleness, or polish.
n.
A soiling; defilement.
n.
Defilement.
n.
Refinement; delicacy.
n.
Refinement; subtlety; extreme attenuation.
n.
Excessive refinement.
n.
Refinement; extreme acuteness; subtlety.
n.
The act of retiring, or the state of being retired; withdrawal; seclusion; as, the retirement of an officer.
n.
Defilement; obscuration.
n.
The act of refining, or the state of being refined; as, the refinement or metals; refinement of ideas.
n.
Fig.: Refinement; elegance of manners.
n.
Retirement; retreat; secession.
n.
The act of defining; definition; description.
a.
Of or pertaining to retirement; causing retirement; suited to, or belonging to, retirement.