First-Order Logic Introduction Syntax and Semantics Using First-Order Logic Summary

Introduction Propositional logic: o Less expressive than predicate logic. o Is declarative (as opposed to procedural) o It lacks power to describe things concisely. o Assumes facts are true or false

Compared to First-Order Logic… Builds around objects and relations. Relations may hold or not. Even more complex languages: Temporal logic. Facts hold at particular times High-order logic. Relations and functions are objects themselves.

Languages LanguageElementsBelief PropositionsFactsTrue/false/unknown First-order logicFacts,objects, relations True/false/unknown Temporal logicFacts,objects, relations, time True/false/unknown ProbabilityFactsDegree of belief [0,1] Fuzzy logicFacts with degree of truth Interval value

First-Order Logic Introduction Syntax and Semantics Using First-Order Logic Summary

Syntax and Semantics What is a model? Propositional logic: sets of truth values for symbols First-order logic: a.Domain of the model (objects it contains) b.Relations (tuples of objects related)

Example Stars, Galaxies, Quasars Surrounded by planets Made of stars A kind of AGN in galaxies

Symbols and Interpretations Symbols: 1. Constants: earth, moon, sun, milky-way 2. Predicate Symbols: Neighbor-planets(x,y) 3. Function Symbols: 3rdPlanet(sun)

Syntax Sentence  AtomicSentence | ( Sentence Connective Sentence ) | ( Quantifier Variable,… Sentence ) | ~Sentence AtomicSentence  Predicate(Term) | Term = Term Term  Function(Term) | Constant | Variable

Syntax Connective   | ^ | V |  Quantifier  V | Constant  A | X1 | Sun Variable  a | x | s | … Predicate  NeighborPlanets | TypeGalaxy | ColorStar Function  1stPlanet, 2ndPlanet

Semantics Interpretation: Symbol “sun” refers to star sun Symbol “earth” refers to planet earth Star 3x45f refers to “specific star”

Terms Logical expression that refers to an object. Examples: Constant symbols: sun, earth, mars, venus. Function symbols: 3rdplanet(sun)

Sentences Statements or facts. Examples: Neighbor-Planets(earth,mars) ^ Neighbor-Galaxies(milky-way,andromeda)

Universal Quantifiers How do we express properties of entire collections of objects? Universal quantification V All stars are burning hydrogen: V x Star(x)  burning-hydrogen(x) True in all extended interpretations.

Existential Quantifiers x Star(x) ^ burning-hydrogen(x) Normally: Universal quantifier connects with  Existential quantifiers connect to ^

More on Quantifiers We can use multiple quantifiers: Vx Vy Star(x) ^ Galaxy(y) ^ InGalaxy(x,y)  Contains(y,x) Vx y Star(x) ^ Star(y) ^ Neighbor(x,y) All stars have a neighbor star

More on Quantifiers De Morgan’s rules apply: Vx ~P = ~ x P ~Vx P = x ~P Vx P = ~ x ~P x P = ~Vx ~P

Equality x,y NeighborStars(x,y) ^ NeighborStars(y,x) ^ ~(x = y)

First-Order Logic Introduction Syntax and Semantics Using First-Order Logic Summary

Using First-Order Logic Rational Agent: TELL(KB,Star(sun)) ASK(KB,Star(sun)) ASK(KB, x Star(x))

First-Order Logic Introduction Syntax and Semantics Using First-Order Logic Summary

Propositional logic talks about facts; predicate logic talks about relations (more expressive) A model is made of objects,relations, and functions. An interpretation maps symbols to the model Complex sentences use connectives and quantifiers Steps to develop a knowledge base.

Robert Kowalski Born 1941 in Connecticut. Professor at Imperial College London since He is well known for his many achievements in logic programming. Co-developed “event calculus’ and “legal reasoning”

Exercises Under propositional logic, assume you have 4 propositions named A, B, C, and D (all Boolean). Observe that V is disjunction or OR, & is conjunction or AND, and ~ is negation or NOT. Answer the following: What is the total number of models defined in this language? Is the statement A V B valid? why? Is the statement A V ~A valid? why? How many models are true under the statement B & D ?