KAND Lecture 2

Knowledge Graphs

A representation of information and knowledge that goes beyond tables and includes explicit relationships and attributes.

Formal Systems

A set of rules and symbols used to represent and manipulate knowledge in a structured and logical way.

Simple Knowledge Graph Logic

A formal system used to represent knowledge graphs before using RDF format.

Entailment

The logical relationship between statements where one statement can be inferred or implied by another.

Assignment 1

A task or project given to students to apply the concepts and principles learned in the course.

Tables

A format commonly used to store and present data in rows and columns.

Attributes

Characteristics or properties of a thing or entity in a knowledge graph.

Relations

Connections or associations between different entities or things in a knowledge graph.

Metadata

Information about the data, such as labels or descriptions, that provides additional context and meaning.

Inference

The process of deriving new information or conclusions based on existing knowledge and logical reasoning.

Knowledge graph

A representation of information using circles and arrows to connect related concepts.

Semantics

Adding additional information to the relations in a knowledge graph, such as domains and ranges, to provide more context and meaning.

Machine readable language

A formal and unambiguous language that can be understood and interpreted by machines.

Syntax

The rules and structure that define the valid expressions and sentences in a formal language.

Semantics

The meaning and interpretation of sentences in a formal language, typically defined by mapping the language to another universe or domain.

Calculus

The rules and methods for deriving new information or statements from existing ones in a formal system.

Logic

A structured way of describing and reasoning about concepts and relationships in a formal system.

Arithmetic

A branch of mathematics that deals with numbers and basic operations like addition, subtraction, multiplication, and division.

Syntax

The rules that determine which statements are well formed in a language.

Semantics

The process of deriving meaning from statements in a language.

Well formed sentence

A sentence that follows the syntax rules of a language, consisting of two terms with a comparator between them.

Term

The components of a well formed sentence, which can be a natural number, an ordered variable, or a complex term.

Complex term

A term that consists of an operator (such as plus, minus, times, or divided by) applied to two terms.

Comparator

A symbol used to compare two terms, such as equals, greater than, or smaller than.

Variable

A symbol that represents an unknown value in a mathematical expression.

Natural number

A positive whole number (e.g., 1, 2, 3, etc.).

Assignment

A function that assigns a value to each variable in a set of variables.

Truth

The property of a sentence being true with respect to a specific assignment of values to variables.

Interpretation

The act of assigning meaning to a sentence or formula.

Assignment

A specific mapping of variables to values in an interpretation.

Model

An interpretation or assignment that makes a formula true.

Formula

A statement or sentence in logic.

True

A property of a formula that holds in a given interpretation or assignment.

World

A possible scenario or state of affairs in which a sentence or formula can be true or false.

Entailment

The relationship between two formulas where one formula logically follows from another.

Variable

A symbol that represents an unspecified value in a formula.

Syntax

The rules and structure of a formal language.

Semantics

The meaning and interpretation of symbols and formulas in a formal language.

Assignment

A mapping of concepts to subsets in a logic.

Model

An assignment that makes a sentence true in a logic.

Entailment

If for all models of f, g is true in a logic.

Syntax

The rules and structure of a language or logic.

Semantics

The meaning and interpretation of sentences in a language or logic.

Axiom

A statement or formula that is considered true in a logic.

Knowledge base

A set of axioms or multiple axioms in a logic.

Concept

An abstraction or generalization from experience in a logic.

Subclass

A relationship between concepts where one concept is a subset of another concept.

LCH

Logic of Concept Hierarchies, a logic for organizing concepts in subclass hierarchies.

Model

In the context of knowledge bases, a model refers to an assignment that satisfies all the axioms in the knowledge base.

Assignment

An assignment is a mapping of elements in the universe to their respective interpretations in a knowledge base.

Axiom

An axiom is a statement or rule that is considered to be true in a knowledge base.

Interpretation

An interpretation is the mapping of elements in the universe to their corresponding meanings or values in a knowledge base.

Subset

A subset refers to a set that contains only elements that are also present in another set.

Knowledge Base

A knowledge base is a collection of axioms and facts that represent a specific domain of knowledge.

True

In the context of knowledge bases, an axiom is considered true if the interpretation of the left side of the axiom is a subset of the interpretation of the right side.

Counterexample

A counterexample is an example that disproves a statement or rule by providing a case where it does not hold true.

Universe

The universe refers to the set of all possible elements or objects in a given context or domain.

Empty set

An empty set is a set that contains no elements.

Model

A representation or interpretation of a knowledge base or system.

Subclass

A class that is a subset of another class.

Interpretation

The assignment of meaning or value to symbols or statements.

Subset

A set that contains only elements that are also in another set.

Axiom

A statement or proposition that is assumed to be true without proof.

Assignment

The act of assigning values or meanings to variables or symbols.

Model checking

The process of verifying whether a given model satisfies a set of specified properties or axioms.

Syntax

The rules and structure that govern the formation of valid statements or expressions in a language or formal system.

Semantics

The meaning or interpretation of statements or expressions in a language or formal system.

Propositional logic

A formal system that deals with propositions or statements that can be either true or false.

Propositional logic

A logic system that focuses on the logical form of arguments, regardless of their actual content.

Syntax

The set of rules that define the structure and formation of valid sentences in a formal language.

Propositional variables

Variables that represent declarative statements in propositional logic.

Connectives

Symbols used to connect propositional variables and form compound statements, such as "and," "or," "not," "implies," and "if and only if."

Inductive definition

A definition that recursively defines valid sentences in a formal language by building formulas using propositional variables and connectives.

Legal sentences

Sentences that adhere to the syntax rules and can be generated using the inductive definition.

Serializations

Different ways of writing down valid sentences in a formal language, such as using symbols, parentheses, or prefix/infix notation.

Prefix notation

A way of writing compound statements where the connective appears before the propositional variables, e.g., "not p" or "and p q."

Infix notation

A way of writing compound statements where the connective appears between the propositional variables, e.g., "p and q" or "p implies q."

Semantics

The meaning or interpretation of a sentence or statement.

Truth tables

Tables used to determine the truth value of a composite formula in propositional logic.

Composite formula

A formula in propositional logic that is made up of simpler formulas.

Not

A logical operator that negates the truth value of a formula.

Conjunction

A logical operator that returns true if both formulas it connects are true.

Assignment

A possible assignment of true or false to individual propositions in a truth table.

Evaluation

The process of calculating the truth values of formulas in propositional logic.

Valuation

A specific assignment of true or false to the variables in a formula.

Finite

Having a limited or countable number of elements or possibilities.

Proposition logic

A branch of logic that deals with the relationships between propositions and their truth values.

Semantically equivalent

Two formulas are semantically equivalent if they have identical notions in the truth table.

Tautology

A tautology is a single formula that is always true regardless of the truth values of its variables.

Contradiction

A contradiction is a formula that can never be true regardless of the truth values of its variables.

Semantic entailment

A formula psi is semantically entailed by the premises phi1 to phin if every assignment function that makes the premises true also makes the conclusion true.

Valid reasoning

The process of deriving a conclusion from premises that are all true.

Knowledge base

A collection of sentences or axioms that represent information or knowledge.

Entailment

The relationship between a set of premises and a conclusion, where the conclusion logically follows from the premises.

Counterexample

An example that disproves a statement or argument by providing a case where the premises are true but the conclusion is false.

Propositional logic

A branch of logic that deals with propositions or statements and their logical relationships.

Syntax

The rules and structure of a formal language, specifying how sentences or statements are formed.

Semantics

The meaning or interpretation of sentences or statements in a formal language.

Truth table

A table that shows the truth values of a logical expression for all possible combinations of truth values of its variables.

Calculus

A method or system of calculation or reasoning.