Knowledge-based engineering
Encyclopedia
Knowledge-based engineering (KBE) is a discipline with roots in computer-aided design
Computer-aided design
Computer-aided design , also known as computer-aided design and drafting , is the use of computer technology for the process of design and design-documentation. Computer Aided Drafting describes the process of drafting with a computer...

 (CAD) and knowledge-based systems
Knowledge-based systems
Knowledge based systems are artificial intelligent tools working in a narrow domain to provide intelligent decisions with justification. Knowledge is acquired and represented using various knowledge representation techniques rules, frames and scripts...

 but has several definitions and roles depending upon the context. An early role was support tool for a design engineer
Design engineer
Design Engineer is a general term that covers multiple engineering disciplines including electrical, mechanical, industrial design and civil engineering, architectural engineers in the U.S...

 generally within the context of product design. Success of early KBE prototypes was remarkable (see History); eventually this led to KBE being considered as the basis for generative design with many expectations for hands-off performance where there would be limited human involvement in the design process.

Overview

KBE can be defined as engineering on the basis of electronic knowledge models. Such knowledge models are the result of knowledge modeling
Knowledge modeling
Knowledge modeling is a process of creating a computer interpretable model of knowledge or standard specifications about a kind of process and/or about a kind of facility or product...

 that uses knowledge representation
Knowledge representation
Knowledge representation is an area of artificial intelligence research aimed at representing knowledge in symbols to facilitate inferencing from those knowledge elements, creating new elements of knowledge...

 techniques to create the computer interpretable models. The knowledge models can be imported in and/or stored in specific engineering applications that enable engineers to specify requirements or create designs on the basis of the knowledge in such models. There are various methods available for the development of knowledge models, most of them are system dependent. An example of a system-independent language for the development machine-readable ontology databases, including support for basic engineering knowledge, is called Gellish English
Gellish English
Gellish English is a variant of Gellish and is a formal language, which means that it is structured and formalised subset of natural English that is computer interpretable. Its definition includes an English dictionary of concepts that is arranged in a taxonomy and that is extended into an ontology...

. An example of a CAD-specific system that can store knowledge and use it for design is the CATIA
CATIA
CATIA is a multi-platform CAD/CAM/CAE commercial software suite developed by the French company Dassault Systemes...

 program through its KnowledgeWare module. An example of a CAD-independent, language-based KBE system with full compiler and support for runtime application deployment is General-purpose Declarative Language (GDL) from Genworks.

KBE can have a wide scope that covers the full range of activities related to Product Lifecycle Management
Product lifecycle management
In industry, product lifecycle management is the process of managing the entire lifecycle of a product from its conception, through design and manufacture, to service and disposal...

 and Multidisciplinary design optimization
Multidisciplinary design optimization
Multi-disciplinary design optimization is a field of engineering that uses optimization methods to solve design problems incorporating a number of disciplines. As defined by Prof. Carlo Poloni, MDO is "the art of finding the best compromise"...

. KBE's scope would include design, analysis (computer-aided engineering
Computer-aided engineering
Computer-aided engineering is the broad usage of computer software to aid in engineering tasks. It includes computer-aided design , computer-aided analysis , computer-integrated manufacturing , computer-aided manufacturing , material requirements planning , and computer-aided planning .- Overview...

 – CAE), manufacturing
Manufacturing
Manufacturing is the use of machines, tools and labor to produce goods for use or sale. The term may refer to a range of human activity, from handicraft to high tech, but is most commonly applied to industrial production, in which raw materials are transformed into finished goods on a large scale...

, and support. In this inclusive role, KBE has to cover a large multi-disciplinary role related to many computer aided technologies (CAx
CAx
Computer-aided technologies is a broad term that means the use of computer technology to aid in the design, analysis, and manufacture of products....

).

KBE also has more general overtones. One of its roles is to bridge knowledge management
Knowledge management
Knowledge management comprises a range of strategies and practices used in an organization to identify, create, represent, distribute, and enable adoption of insights and experiences...

 and design automation. Knowledge
Knowledge
Knowledge is a familiarity with someone or something unknown, which can include information, facts, descriptions, or skills acquired through experience or education. It can refer to the theoretical or practical understanding of a subject...

 processing is a recent advance in computing
Computing
Computing is usually defined as the activity of using and improving computer hardware and software. It is the computer-specific part of information technology...

. It has played a successful role in engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

 and is now undergoing modifications (to be explained). An example of KBE’s role is generative mechanical design. There are others. KBE can be thought of as an advanced form of computer applications (in some forms with an extreme end-user computing
End-user computing
In computing, End User Computing refer to systems in which non-programmers can create working applications. EUC is a group of approaches to computing that aim at better integrating end users into the computing environment...

 flavor) that support PLM
Product lifecycle management
In industry, product lifecycle management is the process of managing the entire lifecycle of a product from its conception, through design and manufacture, to service and disposal...

 and CAx
CAx
Computer-aided technologies is a broad term that means the use of computer technology to aid in the design, analysis, and manufacture of products....

.

There are similar techniques, such as electronic design automation
Electronic design automation
Electronic design automation is a category of software tools for designing electronic systems such as printed circuit boards and integrated circuits...

. AAAI provides a long list of engineering applications. some of which are within the KBE umbrella. At some point, the concept of KBE might split into several sub-categories as MCAD and ECAD are just two of many possible types of design automation.

History

KBE essentially was a complementary development to CAx
CAx
Computer-aided technologies is a broad term that means the use of computer technology to aid in the design, analysis, and manufacture of products....

  and can be dated from the 1980s (See Also, ICAD
ICAD
ICAD was a Knowledge-Based Engineering system that was based upon the Lisp programming language...

). CAx
CAx
Computer-aided technologies is a broad term that means the use of computer technology to aid in the design, analysis, and manufacture of products....

 has been developing along with the computer after making large strides in the 1970s.

KBE technologies suffered a downslide during AI Winter
AI winter
In the history of artificial intelligence, an AI winter is a period of reduced funding and interest in artificial intelligence research. The process of hype, disappointment and funding cuts are common in many emerging technologies , but the problem has been particularly acute for AI...

. While KBE had sufficient success stories that sustained it long enough into the 1990s, very high expectations
User expectations
User expectations refers to the consistency that users expect from products. Interaction design is very concerned with this topic. For example, our user expectations for traffic behavior is one of the more consistent ones because it is governed by traffic laws that are enforced...

 and the inability to meet them with KBE resulted in it being considered obsolete in a way similar to LISP
Lisp
A lisp is a speech impediment, historically also known as sigmatism. Stereotypically, people with a lisp are unable to pronounce sibilants , and replace them with interdentals , though there are actually several kinds of lisp...

 AI
Ai
AI, A.I., Ai, or ai may refer to:- Computers :* Artificial intelligence, a branch of computer science* Ad impression, in online advertising* .ai, the ISO Internet 2-letter country code for Anguilla...

 designs.

KBE, as implemented with ICAD
ICAD
ICAD was a Knowledge-Based Engineering system that was based upon the Lisp programming language...

 can be thought of as an advanced form of computer applications (in some forms with an extreme end-user computing
End-user computing
In computing, End User Computing refer to systems in which non-programmers can create working applications. EUC is a group of approaches to computing that aim at better integrating end users into the computing environment...

 flavor) that support PLM
Product lifecycle management
In industry, product lifecycle management is the process of managing the entire lifecycle of a product from its conception, through design and manufacture, to service and disposal...

 and CAx
CAx
Computer-aided technologies is a broad term that means the use of computer technology to aid in the design, analysis, and manufacture of products....

.

KBE and product lifecycle management

The scope of PLM involves all the steps that exist within any industry that produces goods. KBE at this level will deal with product issues of a more generic nature than it will with CAx
CAx
Computer-aided technologies is a broad term that means the use of computer technology to aid in the design, analysis, and manufacture of products....

. Some might call this level 'assembly' in orientation. However, it's much more than that as PLM covers both the technical and the business side of a product.

KBE then needs to support the decision processes involved with configuration, trades, control, management, and a number of other areas, such as optimization
Multidisciplinary design optimization
Multi-disciplinary design optimization is a field of engineering that uses optimization methods to solve design problems incorporating a number of disciplines. As defined by Prof. Carlo Poloni, MDO is "the art of finding the best compromise"...

.

Recently the Object Management Group
Object Management Group
Object Management Group is a consortium, originally aimed at setting standards for distributed object-oriented systems, and is now focused on modeling and model-based standards.- Overview :...

 released a RFP document and requested feedback.

KBE and CAX

CAx crosses many disciplinary bounds and provides a sound basis for PLM
Product lifecycle management
In industry, product lifecycle management is the process of managing the entire lifecycle of a product from its conception, through design and manufacture, to service and disposal...

. In a sense, CAx is a form of applied science that uses most of the disciplines of engineering and their associated fields. Materials science
Materials science
Materials science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. This scientific field investigates the relationship between the structure of materials at atomic or molecular scales and their macroscopic properties. It incorporates...

 comes to mind.

KBE's support of CAx may have some similarities with its support of PLM
Product lifecycle management
In industry, product lifecycle management is the process of managing the entire lifecycle of a product from its conception, through design and manufacture, to service and disposal...

 but, in a sense, the differences are going to be larger.

The KBE flavor at the CAx level may assume a strong behavioral flavor. Given the underlying object oriented focus, there is a natural use of entities possessing complicated attributes and fulfilling non-trivial roles. One vendor's approach provides a means via workbenches to embed attributes and methods within sub-parts (object) or within a joining of sub-parts into a part.

As an aggregate, the individual actions, that are event driven, can be fairly involved. This fact identifies one major problem, namely control of what is essentially a non-deterministic
Nondeterministic algorithm
In computer science, a nondeterministic algorithm is an algorithm that can exhibit different behaviors on different runs, as opposed to a deterministic algorithm. There are several ways an algorithm may behave differently from run to run. A concurrent algorithm can perform differently on different...

 mixture. This characteristic of the decision problem
Decision problem
In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem...

 will get more attention as the KBE systems subsume more levels and encompasses a broader scope of PLM
Product lifecycle management
In industry, product lifecycle management is the process of managing the entire lifecycle of a product from its conception, through design and manufacture, to service and disposal...

.

KBE and knowledge management

KBE is related to knowledge management
Knowledge management
Knowledge management comprises a range of strategies and practices used in an organization to identify, create, represent, distribute, and enable adoption of insights and experiences...

 which has many levels itself. Some approaches to knowledge
Knowledge
Knowledge is a familiarity with someone or something unknown, which can include information, facts, descriptions, or skills acquired through experience or education. It can refer to the theoretical or practical understanding of a subject...

 are reductionistic, as well they ought to be given the pragmatic focus of knowledge modeling
Knowledge modeling
Knowledge modeling is a process of creating a computer interpretable model of knowledge or standard specifications about a kind of process and/or about a kind of facility or product...

. However, due to KBE dealing with aggregates that can be quite complicated both in structure and in behavior, some holistic notions (note link to Complex systems
Complex systems
Complex systems present problems in mathematical modelling.The equations from which complex system models are developed generally derive from statistical physics, information theory and non-linear dynamics, and represent organized but unpredictable behaviors of systems of nature that are considered...

) might be apropos.

Also, given all the layers of KBE and given the fact that one part of an associated space is heavily mathematical (namely, manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

 in nature), KBE is extremely interesting from the knowledge
Knowledge
Knowledge is a familiarity with someone or something unknown, which can include information, facts, descriptions, or skills acquired through experience or education. It can refer to the theoretical or practical understanding of a subject...

 viewpoint (or one would hope).

All one has to do is note that the KBE process's goal is to produce results in the 'real world'
Reality
In philosophy, reality is the state of things as they actually exist, rather than as they may appear or might be imagined. In a wider definition, reality includes everything that is and has been, whether or not it is observable or comprehensible...

 via artifacts and to do so using techniques that are highly computational
Computational
Computational may refer to:* Computer* Computational algebra* Computational Aeroacoustics* Computational and Information Systems Laboratory* Computational and Systems Neuroscience* Computational archaeology* Computational auditory scene analysis...

. That, in essence, is the epitome of applied science/engineering, and it could never be non-interesting.

KBE methodology

The development of KBE applications concerns the requirements to identify, capture, structure, formalize and finally implement knowledge. Many different so-called KBE platforms support only the implementation step which is not always the main bottleneck in the KBE development process. In order to limit the risk associated with the development and maintenance of KBE application there is a need to rely on an appropriate methodology for managing the knowledge and maintaining it up to date.
As example of such KBE methodology the EU project MOKA "Methodology and tools Oriented to Knowledge based Applications" propose solutions which focus on the structuration and formalization steps as well as links to the implementation.

An alternative to MOKA is to use a general methodology for developing knowledge bases for expert systems and for intranet pages. Such a methodology is described in "Knowledge Acquisition in Practice: A Step-by-step Guide" by Nick Milton (click here for more details).

Languages for KBE

Some questions can be asked in regard to KBE implementation: can we represent knowledge in a vendor-neutral format? can the knowledge in our designs be retained for decades, long after a vendor system (such as CATIA) has disappeared?

These questions are addressed in the 2007 NASA-ESA Workshop on Product Data Exchange presentation A Language for Engineering Design by Walter Wilson of Lockheed Martin.

Mr. Wilson advocates using a type of programming language to define design data—operations, parameters, formulas, etc. -- instead of a proprietary file format (such as Dassault's CATIA). One's data would no longer be tied to a specific CAD system. Unlike STEP, which inevitably lags commercial CAD systems in the features it supports, programmability would allow the definition of new design features.

A logic programming language is proposed as the basis for the engineering design language because of its simplicity and extensibility. The geometric engine for the language features would be open source to give engineers control over approximation algorithms and to better guarantee long-term accessibility of the data.

Meanwhile, the commercially available General-purpose Declarative Language (GDL) from Genworks International addresses the issue of application longevity by providing a high-level declarative language kernel which is a superset of a standard dialect of the Lisp programming language (currently ANSI Common Lisp
Common Lisp
Common Lisp, commonly abbreviated CL, is a dialect of the Lisp programming language, published in ANSI standard document ANSI INCITS 226-1994 , . From the ANSI Common Lisp standard the Common Lisp HyperSpec has been derived for use with web browsers...

, or CL).

The GDL kernel follows a concise, pragmatic language specification representing something akin to a de-facto neutral format for representing KBE-style knowledge. It consists of the same Smalltalk-inspired declarative object-oriented (and object-centric) message-passing format which been a common thread among classical KBE systems for more than two decades. While CL is a multi-paradigm language (supporting procedural, object-oriented, and functional programming
Functional programming
In computer science, functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. It emphasizes the application of functions, in contrast to the imperative programming style, which emphasizes changes in state...

), the core features of KBE tend to share several aspects with Functional programming
Functional programming
In computer science, functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. It emphasizes the application of functions, in contrast to the imperative programming style, which emphasizes changes in state...

 languages "under the hood" (e.g. lazy evaluation, immutable data). However there is a consensus that pure Functional programming is too esoteric for typical engineers to practice, so one of the purposes of a declarative KBE language such as GDL is to provide an "engineer-friendly" object-centric front-end on what is essentially a Functional language
Functional programming
In computer science, functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. It emphasizes the application of functions, in contrast to the imperative programming style, which emphasizes changes in state...

 programming environment.

Because GDL applications are written as a strict superset of a standard Lisp dialect, only the high-level declarative surface syntax is GDL-specific. The bulk of application code is purely compliant with the underlying language standard. And because of Lisp's inherent (and unique) support for code transformation macros, even this surface syntax is subject to straightforward automated conversion among other variations of the de-facto standard. It is reasonable to expect that implementations following this approach will eventually converge on a true vendor-neutral Standard KBE language specification.

The Pacelab Suite addresses the problem that functional programming languages are still not widely accepted by potential KBE users. Engineers are usually trained in procedural and object-oriented programming languages; in quite a few software environments (Excel, MATLAB and FORTRAN) used by engineers the procedural programming style clearly dominates. Therefore the Pacelab Suite rebases the inherent technological advantages offered by a development and runtime environment like Common Lisp
Common Lisp
Common Lisp, commonly abbreviated CL, is a dialect of the Lisp programming language, published in ANSI standard document ANSI INCITS 226-1994 , . From the ANSI Common Lisp standard the Common Lisp HyperSpec has been derived for use with web browsers...

, on a new technological paradigm (.NET Framework
.NET Framework
The .NET Framework is a software framework that runs primarily on Microsoft Windows. It includes a large library and supports several programming languages which allows language interoperability...

) equally supporting procedural, object-oriented and functional languages.

KBE in Academia


For CAD

  • Adaptive Modeling Language from TechnoSoft Inc.
  • DriveWorks
    DriveWorks
    DriveWorks is a design automation program that runs on Microsoft Windows and is developed by DriveWorks Ltd.DriveWorks can automate the design or sales function internally or over the internet...

     A SolidWorks Certified Gold Partner http://www.solidworks.com
  • GDL from Genworks International
  • Kadviser from NIMTOTH previously edited by Kade-Tech
  • KBEWorks by VisionKBE
  • Knowledge Fusion from Siemens PLM Software
    Siemens PLM Software
    Siemens PLM Software is a computer software company specializing in 3D & 2D Product Lifecycle Management software. The company is a business unit of Siemens Industry Automation division, and is headquartered in Plano, Texas....

  • Knowledgeware from Dassault Systemes
    Dassault Systemes
    Dassault Systèmes S.A. is a leading company specializing in 3D and PLM software.Dassault Systèmes develops and markets PLM application software and services that support industrial processes and provide a 3D vision of the entire lifecycle of products from conception to maintenance to recycling...

  • Magix by Navitech
  • Pro/ENGINEER
    Pro/ENGINEER
    Creo Elements/Pro, a product formerly known as Pro/ENGINEER is a parametric, integrated 3D CAD/CAM/CAE solution created by Parametric Technology Corporation . It was the first to market with parametric, feature-based, associative solid modeling software...

     Expert Framework from Parametric Technology Corporation
    Parametric Technology Corporation
    Parametric Technology Corporation is a U.S.-based company that develops, markets and supports software for product development. Its main products are for CAD/CAM, engineering calculations, and product lifecycle management. Its customers include companies in manufacturing, publishing, services,...

  • SmartAssembly for Pro/ENGINEER from Sigmaxim Inc
  • TactonWorks Interactive design automation inside SolidWorks http://www.tacton.com
  • YVE - Your Variant Engineer from tecneos software-engineering
  • ICAD
    ICAD
    ICAD was a Knowledge-Based Engineering system that was based upon the Lisp programming language...

     from Dassault Systemes
    Dassault Systemes
    Dassault Systèmes S.A. is a leading company specializing in 3D and PLM software.Dassault Systèmes develops and markets PLM application software and services that support industrial processes and provide a 3D vision of the entire lifecycle of products from conception to maintenance to recycling...

     (no longer available)
  • KBMax Configurator by Citius Corporation

For General-purpose development of Web-deployed applications


For analysis, design and engineering processes


KBE futures, KBE theory

KBE, as a particular example of KS
Knowledge-based systems
Knowledge based systems are artificial intelligent tools working in a narrow domain to provide intelligent decisions with justification. Knowledge is acquired and represented using various knowledge representation techniques rules, frames and scripts...

, is a multi-disciplinary framework that has more than practical considerations. Not only will KBE require successful handling of issues of the computational (Ontology, Artificial Intelligence
Artificial intelligence
Artificial intelligence is the intelligence of machines and the branch of computer science that aims to create it. AI textbooks define the field as "the study and design of intelligent agents" where an intelligent agent is a system that perceives its environment and takes actions that maximize its...

, Entscheidungsproblem
Entscheidungsproblem
In mathematics, the is a challenge posed by David Hilbert in 1928. The asks for an algorithm that will take as input a description of a formal language and a mathematical statement in the language and produce as output either "True" or "False" according to whether the statement is true or false...

, Interactive computation
Interactive computation
In computer science, interactive computation is a mathematical model for computation that involves communication with the external world during the computation...

, Category Theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, ...) and logic (non-monotonic
Non-monotonic logic
A non-monotonic logic is a formal logic whose consequence relation is not monotonic. Most studied formal logics have a monotonic consequence relation, meaning that adding a formula to a theory never produces a reduction of its set of consequences. Intuitively, monotonicity indicates that learning a...

 issues related to the qualification
Qualification problem
In philosophy and AI , the qualification problem is concerned with the impossibility of listing all the preconditions required for a real-world action to have its intended effect. It might be posed as how to deal with the things that prevent me from achieving my intended result...

, frame
Frame problem
In artificial intelligence, the frame problem was initially formulated as the problem of expressing a dynamical domain in logic without explicitly specifying which conditions are not affected by an action. John McCarthy and Patrick J. Hayes defined this problem in their 1969 article, Some...

, and ramification
Ramification problem
In philosophy and artificial intelligence , the ramification problem is concerned with indirect consequences of an action. It might be posed as how to represent what happens implicitly due to an action or to control secondary and tertiary effects within the same period...

 problems)), it will touch upon all sciences that deal with matter, its manipulations, and the related decisions. In a sense, PLM
Product lifecycle management
In industry, product lifecycle management is the process of managing the entire lifecycle of a product from its conception, through design and manufacture, to service and disposal...

 allows us to have the world as a large laboratory for experimental co-evolution of our knowledge
Knowledge
Knowledge is a familiarity with someone or something unknown, which can include information, facts, descriptions, or skills acquired through experience or education. It can refer to the theoretical or practical understanding of a subject...

 and our artificial co-horts. As noted in the ACM Communications, "Computers will grow to become scientists in their own right, with intuitions and computational variants of fascination and curiosity." What better framework is there to explore the "increasingly complicated mappings between the human world and the computational"?

In terms of methodology
Methodology
Methodology is generally a guideline for solving a problem, with specificcomponents such as phases, tasks, methods, techniques and tools . It can be defined also as follows:...

 and their associated means, KBE offers support via several paradigms. These range from the home-grown all the way to strategically defined and integrated tools that cover both breadth and depth. A continuing theme will be resolving the contextual definitions for KBE into a coherent discipline (or at least attempting this) and keeping a handle on managing the necessary quantitative
Quantitative property
A quantitative property is one that exists in a range of magnitudes, and can therefore be measured with a number. Measurements of any particular quantitative property are expressed as a specific quantity, referred to as a unit, multiplied by a number. Examples of physical quantities are distance,...

 comparisons. One issue of importance considers what limits there may be to the computational; this study requires a multi-disciplinary focus and an understanding of the quasi-empirical. Given the knowledge focus of KBE, another issue involves what limits there might be to a computational basis for knowledge and whether these are overcome with the more advanced types of human-machine interface.

See also

  • Knowledge-based systems
    Knowledge-based systems
    Knowledge based systems are artificial intelligent tools working in a narrow domain to provide intelligent decisions with justification. Knowledge is acquired and represented using various knowledge representation techniques rules, frames and scripts...

  • Functional programming
    Functional programming
    In computer science, functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. It emphasizes the application of functions, in contrast to the imperative programming style, which emphasizes changes in state...

  • Multidisciplinary design optimization
    Multidisciplinary design optimization
    Multi-disciplinary design optimization is a field of engineering that uses optimization methods to solve design problems incorporating a number of disciplines. As defined by Prof. Carlo Poloni, MDO is "the art of finding the best compromise"...

  • Model
  • Model (abstract)
  • Design
    Design
    Design as a noun informally refers to a plan or convention for the construction of an object or a system while “to design” refers to making this plan...

  • Entscheidungsproblem
    Entscheidungsproblem
    In mathematics, the is a challenge posed by David Hilbert in 1928. The asks for an algorithm that will take as input a description of a formal language and a mathematical statement in the language and produce as output either "True" or "False" according to whether the statement is true or false...

  • Decision problem
    Decision problem
    In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem...

     - KBE is mainly a collection of decision points. Several aspects of KBE approach what might be called a 'decidable' bounds though it's hard to get this fact discussed (hence the node here to foster the discussion).
  • Differential geometry

External links


The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.
 
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