Ontology Engineering: Chapter 14

Chapter 14: Ontology Engineering

• Studies the principles and methodologies for developing and maintaining ontologies.
• Foundational ontologies offer a generic framework for representing knowledge.
• Domain ontologies capture specific knowledge for a particular field or application.
• Development methodologies provide guidance for creating ontologies.
• Evaluation ensures the quality and consistency of ontologies.
• Evolution adapts ontologies to changing knowledge and requirements.

Foundational Ontologies

• Known as upper ontologies or top-level ontologies.
• Give a generic framework for representing knowledge across domains.
• Define fundamental concepts, relations, and axioms for specialized ontologies.
• Examples include BFO, DOLCE, and SUMO.

Basic Formal Ontology (BFO)

• Is a small, upper-level ontology developed by Barry Smith and collaborators.
• It promotes interoperability in science, emphasizes realism, and provides a clear foundation for representing real-world entities.

Descriptive Ontology for Linguistic and Cognitive Engineering (DOLCE)

• Is an upper ontology focusing on cognitive and linguistic aspects of knowledge representation.
• Distinguishes between endurants (objects persisting through time) and perdurants (events unfolding over time).

Suggested Upper Merged Ontology (SUMO)

• Is a broad upper ontology aiming to provide a comprehensive framework for knowledge representation.
• Includes concepts from time, space, and quantity domains.

Domain Ontologies

• Capture knowledge specific to a field or application.
• Define concepts, properties, and relationships relevant to the domain.
• Provide a structured representation of domain-specific knowledge.
• Serve applications in biomedical, geographic, and e-commerce areas.

Biomedical Ontologies

• Cover biology, medicine, and healthcare.
• Examples include the Gene Ontology (GO) which describes gene function, and the National Cancer Institute Thesaurus (NCIt) which has standardized cancer terminology.

Geographic Ontologies

• Represent geographic concepts (locations, regions, spatial relationships).
• Are used in geographic information systems (GIS) and location-based services.

E-commerce Ontologies:

• Capture knowledge about products, services, and customer preferences.
• Used in product recommendation systems and semantic search engines.

Ontology Development Methodologies

• Offer structured approaches to guide ontology creation.
• Provide steps, guidelines, and best practices for building high-quality ontologies.
• Common examples include Methontology, NeOn Methodology, and Ontology 101.

Methontology

• Is a comprehensive methodology covering the entire ontology development lifecycle.
• Emphasizes defining clear requirements and involving domain experts.

NeOn Methodology

• Is a scenario-based methodology supporting collaborative ontology development.
• Offers guidelines for reusing existing ontologies, modularizing ontologies and aligning ontologies with knowledge resources.

Ontology 101

• Is a lightweight methodology focusing on core ontology development steps.
• Defining classes, properties, and instances are included.
• Suited for small to medium-sized projects with limited resources.

Ontology Evaluation

• Assesses the quality and consistency of ontologies.
• Involves evaluating accuracy, completeness, and usability.
• Techniques include competency questions, consistency checking, and user evaluation.

Competency Questions:

• Are used to evaluate the ability of an ontology to answer specific questions about the domain.

Consistency Checking

• Involves verifying that the ontology does not contain any logical contradictions.

User Evaluation

• Involves gathering feedback from users to assess the usability and understandability of the ontology.

Ontology Evolution

• Addresses adapting ontologies to changing knowledge and requirements.
• As the domain evolves, the ontology needs updating with new concepts, relationships etc.
• Techniques include versioning, modularization, and alignment.

Versioning

• Involves creating new versions of the ontology to capture changes over time.

Modularization

• Is breaking the ontology into smaller, self-contained modules for independent updates.

Alignment

• Involves mapping concepts and relationships between ontologies to integrate it with other knowledge.

Semantic Web Rule Language (SWRL)

• It combines OWL with RuleML, allowing rules to infer new knowledge in ontologies.

Introduction to SWRL

• A rule-based language for the Semantic Web.
• It combines the expressive power of OWL (Web Ontology Language) with that of RuleML (Rule Markup Language).
• Lets users write rules to infer new knowledge based on existing knowledge in an ontology.
• SWRL rules are typically used to define relationships between concepts, classify individuals, and reason about data.

Syntax and Semantics of SWRL

• Rules are written as implication, antecedent (body), and consequent (head).
• Antecedent and consequent are both conjunctions of atoms; atoms are class assertions, property assertions, or built-in predicates.
• Example: hasParent(?x, ?y) ^ hasBrother(?y, ?z) -> hasUncle(?x, ?z) ; if x has parent y, and y has brother z, then x has uncle z

SWRL Reasoning

• The process of inferring new knowledge from existing knowledge using SWRL rules.
• Reasoners use forward chaining, backward chaining, and hybrid approaches.

Forward Chaining

• Starts with known facts and applies the rules to infer new facts.
• Continues until no new facts can be inferred.

Backward Chaining

• Starts with a query and finds the rules that can prove the query.
• Works backwards from the consequent to the antecedent.

Hybrid Approaches

• Combines forward and backward chaining to improve performance.

Advantages of SWRL

• Expressive power: SWRL allows users to express complex relationships and constraints that cannot be expressed in OWL alone.
• Flexibility: SWRL rules can be easily modified and extended to accommodate new knowledge and requirements.
• Integration with OWL: SWRL is tightly integrated with OWL, allowing users to combine the advantages of both languages.

Disadvantages of SWRL

• Complexity: SWRL can be complex to learn and use, especially for users who are not familiar with rule-based programming.
• Performance: SWRL reasoning can be computationally expensive, especially for large ontologies and complex rule sets.
• Limited tool support: SWRL tool support is still relatively limited, compared to OWL tool support.

Ontology Editors

• Software applications designed to create, modify, and manage ontologies.

Protégé

• Is a free, open-source ontology editor and knowledge-base framework developed by Stanford University.
• Offers a graphical user interface for creating, editing, and managing ontologies in formats like OWL, RDF, and XML.

Features of Protégé

• User-friendly interface.
• Supports multiple ontology formats (OWL, RDF, XML).
• Integrates with reasoners for consistency checking and inference.
• Has plugin architecture allowing users to add custom plugins.
• Collaboration features include versioning and change tracking.

Usage of Protégé

• Creating classes, properties, and individuals in an ontology.
• Modifying classes, properties, or individuals in existing ontologies.
• Reasoning with ontologies: integrates with reasoners to perform consistency checking and inference.
• Querying ontologies: queries ontologies using SPARQL, a query language for RDF data.
• Visualizing ontologies: provides visualization tools for exploring and understanding ontologies.

Advantages of Protégé

• Free and open-source.
• User-friendly interface.
• Supports multiple ontology formats (OWL, RDF, XML).
• Reasoning support: Integrates with various reasoners to perform consistency checking and inference.
• Plugin architecture: Protégé's plugin architecture allows users to extend its functionality with custom plugins.

Disadvantages of Protégé

• Steep learning curve.
• Performance issues with large ontologies.
• Limited support for some OWL features (complex property restrictions).

Summary

• Ontology engineering creates and maintains ontologies.
• Foundational ontologies give high-level frameworks, and domain ontologies capture specific knowledge.
• Methodologies guide creation, evaluation ensures quality, and evolution adapts to change.
• Protégé is an editor for creating, editing, and managing ontologies.

Algorithmic Game Theory

• Examines strategic interactions among rational agents using mathematical models
• Used in economics, political science, biology, computer science

Example: Prisoner’s Dilemma

• Two suspects are arrested for a crime, held separately
• If one confesses, they are released while the other gets a 10-year sentence.
• If both confess, they get a 5-year sentence. If neither confesses, they each get a 1-year sentence.
• Payoff indicated as (-years for suspect A, -years for suspect B)

Strategic Choice

• Each suspect's dominant strategy is to confess, even though both would be better off if they both stayed silent.

Algorithmic Game Theory Focuses on:

• Brings algorithmic thinking into game theory.
• Examines computational aspects of finding Nash equilibria.
• Examines computational aspects of mechanism design, coalition formation and social choice

Computing a Nash Equilibrium

• A Nash equilibrium is a set of strategies for each player, where no player benefits from changing their strategy unilaterally.
• Computationally difficult in games with many players, some having none, others having many

Mechanism Design

• Is used to design the rules of a game to achieve a desired outcome
• A standard auction allocates a good to the player who values it most

Braess's Paradox

• Adding a road can increase travel time for all drivers, even with a reduced network load

Example: Travel time, Original Network

• 4000 drivers going from A to B
• A -> C: $T_1(x) = x/100$
• A -> D: $T_2(x) = 45$
• C -> B: $T_3(x) = 45$
• D -> B: $T_4(x) = x/100$
• Equilibrium: 2000 drivers choose A->C->B, 2000 choose A->D->B, and each takes 65 minutes

Example: Travel time, New Network

• Add C -> D: $T_5(x) = 0$
• Now all drivers pick A -> C -> D -> B, each taking 80 minutes

Notes on Functional Analysis

• Focuses on normed vector spaces, linear operators, and the Riemann Zeta Function.

Normed Vector Spaces

• A norm on vector space $X$ is a function $| \cdot | : X \to \mathbb{R}$ where:
  • $|x| \geq 0$ for all $x \in X$, and $|x| = 0$ if and only if $x = 0$.
  • $|\alpha x| = |\alpha| |x|$ for all $x \in X$ and $\alpha \in \mathbb{F}$.
  • $|x + y| \leq |x| + |y|$ for all $x, y \in X$
• $(X, |\cdot|)$ is called a normed vector space.
• Examples include $\mathbb{R}^n$, $\mathbb{C}^n$, $\ell^p$ spaces, $\ell^\infty$, $C[a, b]$, $L^p(a, b)$

Supremum Norm or Uniform Norm

• $C[a, b]$ (continuous functions on $[a, b]$) with $|f|\infty = \sup{x \in [a, b]} |f(x)|$

Remarks

• A norm induces a metric (distance) on $X$ as $d(x, y) = |x - y|$. Includes concepts like open sets, convergence etc.

Banach Space

• A complete normed vector space regarding the metric induced by the norm.
• e.g., $\mathbb{R}^n, \mathbb{C}^n, \ell^p$ spaces, $C[a, b]$, $L^p(a, b)$

Linear Isometry

• If $(X, |\cdot|)$ is a normed space, there is a Banach space $\tilde{X}$ and a linear isometry $i: X \to \tilde{X}$ such that $i(X)$ is dense in $\tilde{X}$.

Definition; Linear Operators

• Let $X$ and $Y$ be vector spaces.
• A linear operator is a map $T: X \to Y$ such that $T(\alpha x + \beta y) = \alpha T(x) + \beta T(y)$.

Bounded Linear Operator

• Let $X$ and $Y$ be normed spaces and $T: X \to Y$.
• Exists $M > 0$ with $|T(x)| \leq M |x|$ for all $x \in X$

Operator Norm

• The operator norm of a bounded linear operator $T$ is defined as $|T| = \sup_{|x| \leq 1} |T(x)|$.

Theorem ; Linear Operator

Let $T: X \to Y$ be a linear operator with $X$ and $Y$ normed spaces Then:

• $T$ is continuous if and only if $T$ is bounded.
• Every linear operator $T: X \to Y$ is bounded if $X$ is finite-dimensional

Banach Space

$B(X, Y)$ is a Banach space with the operator norm if $Y$ is a Banach space, $B(X, Y)$ is of all bounded linear operators from $X$ to $Y$.

• $B(X, Y)$ is a vector space with operations defined pointwise.

Definition: Dual Space

• The dual space of $X$ is denoted by $X'$, and the space $B(X, \mathbb{F})$ of all bounded linear functionals on $X$. $X'$ is a Banach space because $\mathbb{F}$ is complete.

Lecture 28: The Riemann Zeta Function

Euler's Crazy Idea

$$\sum_{n=1}^{\infty} \frac{1}{n^{s}}=\prod_{p \text { prime }} \frac{1}{1-p^{-s}}$$

• A Product over primes equal to an infinite sum which holds true for $\operatorname{Re}(s)>1$.

Analytic Continuation

• The analytic continuation of $\zeta(s)$ to $\mathbb{C} \backslash{1}$ given by $\zeta(s)=\frac{1}{1-2^{1-s}} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{s}}$

Functional Equation

$$\zeta(s)=2^{s} \pi^{s-1} \sin \left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s)$$

Zeros & Riemann Hypothesis

• "Trivial zeros" exist at $s=-2,-4,-6, \ldots$.
• Other zeros ("nontrivial zeros") lie in the "critical strip" where ${s \in \mathbb{C}: 0 \leq \operatorname{Re}(s) \leq 1}$
• Riemann Hypothesis suggests that nontrivial zeros of $\zeta(s)$ lie on $\operatorname{Re}(s)=\frac{1}{2}$

Chapter 9: Momentum and Collisions

• Examines linear momentum, impulse, collisions, center of mass, and rocket propulsion.

9. 1 Linear Momentum

Definition of Linear Momentum $\vec{p}$ of an object

• Mass $m$ moving with velocity $\vec{v}$ as $\vec{p} = m\vec{v}$
• Vector quantity with same direction as that of $\vec{v}$. Magnitude is $p = mv$.
• SI units= kg·m/s

Newton's Second Law (Alternative Statement)

• $\vec{F}_{net} = \frac{\Delta \vec{p}}{\Delta t}$; The time rate of change is the net force.

Isolated System

• An isolated system is one for which the net external force acting on the system is zero.

Momentum of Isolated System

• Constant in time if no net external force acts on a system (conserved).
• $\vec{p}{total} = \vec{p}1 + \vec{p}2 = \text{constant}$ and $\vec{p}{1i} + \vec{p}{2i} = \vec{p}{1f} + \vec{p}_{2f}$

Conservation of Momentum

• Total momentum remains constant when no net external force acts on it.
• $\vec{p}{total, i} = \vec{p}{total, f}$

9. 2 Impulse and Momentum

• $\vec{F} = \frac{d\vec{p}}{dt} \Rightarrow d\vec{p} = \vec{F}dt$ (Newton's Second Law)

Impulse

• Impulse $\vec{I}$ of the force $\vec{F}$ acting on object: $\vec{I} \equiv \int_{t_i}^{t_f} \vec{F} dt$
• Integrates to change in momentum over time interval: $\Delta \vec{p} = \vec{p}_f - \vec{p}i = \int{t_i}^{t_f} \vec{F} dt \equiv \vec{I}$

Impulse-Momentum Theorem

• Change in momentum equals the impulse: $\Delta \vec{p} = \vec{I}$
• Impulse equals area under force-time curve.

9. 3 Collisions in One Dimension

• Collision represents when 2 particles interact via forces
• The forces are stronger than any external forces present.
• Use impulse approximation
  • Assume force is strong, but acts briefly with $\vec{p}_i$ and $\vec{p}_f$ are the momenta immediately before and after the collision.

Types of Collisions

• Elastic collision: Total kinetic energy is conserved.
• Inelastic collision: Total kinetic energy is NOT conserved.
• Perfectly inelastic collision: Objects stick together after colliding.

Elastic Collisions conservation

• Both momentum and kinetic energy are kept the same throughout
• $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$
• $\frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2$
• $v_{1i} - v_{2i} = -(v_{1f} - v_{2f})$
• $v_{1f} = (\frac{m_1 - m_2}{m_1 + m_2})v_{1i} + (\frac{2m_2}{m_1 + m_2})v_{2i}$
• $v_{2f} = (\frac{2m_1}{m_1 + m_2})v_{1i} + (\frac{m_2 - m_1}{m_1 + m_2})v_{2i}$

Inelastic Collisions

• Only momentum is conserved: $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$
• Kinetic energy is not conserved.

Perfectly Inelastic Collisions

• Objects stick together after colliding
• $m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2) v_f$
• $v_f = \frac{m_1 v_{1i} + m_2 v_{2i}}{m_1 + m_2}$

9. 4 The Center of Mass

• Moves as if all of the mass concentrates there when external forces are applied to it.

Center of Mass (CM) of a System of Particles

$\qquad x_{CM} = \frac{\sum_i m_i x_i}{\sum_i m_i}= \frac{m_1 x_1 + m_2 x_2 + m_3 x_3 +...}{m_1 + m_2 + m_3 +...}$ $\qquad y_{CM} = \frac{\sum_i m_i y_i}{\sum_i m_i}= \frac{m_1 y_1 + m_2 y_2 + m_3 y_3 +...}{m_1 + m_2 + m_3 +...}$ $\qquad \vec{r}_{CM} = \frac{\sum_i m_i \vec{r}_i}{\sum_i m_i}$

Center of Mass of an Extended Object

$x_{CM} = \frac{1}{M} \int x dm$ $y_{CM} = \frac{1}{M} \int y dm$, where $M$ is the total mass.

If object is homogeneous, then: $\vec{r}_{CM} = \frac{1}{V} \int \vec{r} dV$, where $V$ is the total volume of the object.

Velocity of CM system is $\vec{v}_{CM} =\frac{\sum_i m_i \vec{v}_i}{M}$

Acceleration of the Center of Mass

$\qquad \vec{a}{CM} = \frac{d\vec{v}{CM}}{dt} = \frac{1}{M} \sum_i m_i \frac{d\vec{v}_i}{dt} = \frac{\sum_i m_i \vec{a}_i}{M}$

$\qquad M\vec{a}_{CM} = \sum_i m_i \vec{a}_i = \sum_i \vec{F}i = \vec{F}{net}$

$\qquad \vec{F}{net} = M\vec{a}{CM} = M \frac{d\vec{v}{CM}}{dt} = \frac{d\vec{p}{total}}{dt}$

Total Linear Momentum of System

• Equals total mass multiplied by the velocity: $M\vec{v}_{CM} =\sum_i m_i \vec{v}i = \vec{p}{total}$

9. 5 Systems of Many Particles

• Net external force on the CM system; the rate of change equal to the total linear momentum.

Conservation of Momentum for a System of Particles

• The CM linear momentum is conserved: $\vec{p}{total} = M\vec{v}{CM} = constant$

Isolated System

• No external forces acting on the system

Deformable Systems

• The CM deforms as if all of the mass had been concentrated there.

9. 6 Collisions in Two Dimensions

• Momentum is a vector quantity, conserved in the x and y directions.

• Be extended to 3D collisions. $\qquad m_1 v_{1ix} + m_2 v_{2x} = m_1 v_{1fx} + m_2 v_{2fx}$ $\qquad m_1 v_{1iy} + m_2 v_{2y} = m_1 v_{1fy} + m_2 v_{2fy}$

• Choose the coordinate system so that the velocity of one of the particles is along one of the axes.

• If the collision is elastic, then kinetic energy is also conserved.

Kinetic Energy KE

$\qquad \frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2$

9. 7 Rocket Propulsion

• Rocket's depend on momentum conservation, $M \Delta \vec{v} = - \vec{v}_e \Delta M$
• Integrated over time, $\Delta v = v_e \ln (\frac{M_i}{M_f})$

Thrust

• Thrust is force from ejecting gas. $\text{Thrust} = |M \frac{dv}{dt}| = |v_e \frac{dM}{dt}|$