CS 371: Introduction to Artificial Intelligence

First-Order Logic

Outline

When Propositional Logic Isn’t Enough

First-Order Logic (FOL)

Practice with FOL

FOL Proof Method: Resolution

[Why Representing Change in Time is Difficult]

Blocks World

Simple domain: Blocks stacked on a table

Simple problem: Stack blocks in a given configuration

Home to much research in a variety of areas:

Vision (Huffman, 1971)

Constraint Propogation (Waltz, 1975)

Machine Learning (Winston, 1970)

Natural Language Understanding (Winograd, 1972)

Planning (Fahlman, 1974)

Propositional Block World

How would you represent the following:

Block A is on Block D

Block A is not on Block B

If A is on D, A is not on B

No block is on A

If A is on D and D is on B, A is above B.

When Propositional Logic Isn’t Enough

Define Above using transitivity of On

“There’s a/there’s no block on A.”

“All blocks have red letters.”

Need for properties, relationships, and ways to generalize information compactly

Practical applications: mathematical theorem proving, common-sense reasoning (CYC image database), circuit/program design/verification

Propositional Logic

Propositions - T/F atomic sentences about the world

Boolean functions (Ø,Ù,Ú,Þ,Û) - negate and logically combine sentences

Parentheses - disambiguates grammar

First-Order Logic

Atomic sentences - more complex than propositions

Boolean functions (Ø,Ù,Ú,Þ,Û) - make compound sentences

Quantifiers (",$) - tell us first-order facts which correspond to sets of sentences

Parentheses

Building Atomic Sentences: Symbols

Domain of Discourse (DOD) - relevant things in the world that we’re reasoning about

Symbols:

variables - assigned things in the domain of discourse (DOD)

constants - things and relations relations between things in the DOD

An interpretation I maps

variables to things in the DOD.

constants to things and relationships between things in the DOD.

Blocks World Example

Domain of Discourse: blocks (for simplicity, exclude table)

Constant Symbols (A, B, C, D, On, Above, …)

Variables (x, y, z, …)

An interpretation maps variables and constant symbols to things in the world. (e.g. A means this block with an “A”.)

Building Atomic Sentences: Constant Symbols

object constants - corresponding to things in the DOD

relation constants - relationships between things in the DOD

relations also called predicates

n-ary relation - a set of n-tuples from the DOD

1-ary relation - a property constant

Blocks World Example

Object Constants: A, B, C, D correspond to blocks with the same letters on them.

Relations Constants:

OnTable: set of blocks resting on table

On: set of block pairs where the first rests on the second

Above: transitive closure of On

Clear: set of blocks with no blocks above

Building Atomic Sentences: Constant Symbols (cont.)

function constants - mappings between tuples and things in the DOD

0-ary function - an object constant

n-ary function (n>0) - special case of n+1-ary relation where n arguments uniquely determine n+1st argument

Atomic Sentences: Putting Symbols Together

terms - variables, objects, functions of terms

atomic sentence - relationship between terms

Under a given interpretation, an atomic sentence is true or false

compound (or complex) sentences are built of atomic sentences, Boolean functions (negation, connectives), and...

Blocks World Example

Terms: x, A, nextBlock(A)

Atomic sentences: Above(A,D), On(x,B), Clear(y)

These sentence are true under an interpretation where constants and relations are as we described and x, y are interpreted as D, A or D, C.

These sentences are also true under an interpretation where A,B,C,D are the numbers 16, 2, 17, 4, On is “the square of”, Above is “a power of”, Clear is “greater than 15”.

Universal Quantifier

universal (") - “For all”

"x f - “For all x, f is true.”

"Every block is a red block".

"If x>y and y>z then x>z."

Existential Quantifier

existential ($) - “There exists”

$x f - “There exists x such that f is true.”

same as Ø "x Ø f

"There is a block on the table."

Challenging: "If a block is above another block, then the block is on the other block or on a block that is above it."

Nested Quantifiers

nested quantifiers - "x $y is different than $y "x

"Everybody likes someone."

"Someone likes everyone."

Relations: Person, Time, FoolPersonAtTime

"You can fool all the people some of the time, …"

"… and some of the people all the time, …"

"… but you cannot fool all the people all the time."

Compound (Complex) Sentences

variable scope - the parts of a sentence for which a variable quantifier is relevant

free variables - unquantified variables

formula - another word for sentence

well-formed formula (wff) - a sentence with no free variables

First-Order Logic Semantics

In-Class Exercise

Express the following in FOL using relations On, Above, Clear, OnTable, and objects A, B, C, D:

If block A is on block B, then A is above B, B is not clear, and A is not on the table.

If one block is on another block, it is also above that block.

A or D or C is on the table.

If A is on the table, then D and C are not.

Validity, Satisfiability, Models

A sentence is valid if it is true for all possible interpretations.

A sentence is satisfiable if there exists a “satisfying” interpretation (i.e. an interpretation for which the sentence is true).

An interpretation which satisfies a sentence is a model of that sentence.

Propositional KB and Inference

entailment - whether a sentences follows from other sentences

inference rules - deduce new sentences from old sentences

soundness - what can be inferred is entailed

completeness - what is entailed can be inferred

Knowledge Representation and Reasoning

A logic consists of

a formal language consisting of

syntax

semantics

a proof theory

terms: sentences, KB, valid, satisfiable, model, entail, inference rule, sound, complete

Inference with FOL

Good News: There exists a complete inference procedure for FOL. (Gödel 1930-31)

Robinson’s resolution algorithm

Bad News: Can’t tell if a sentence isn’t entailed Þ FOL is semidecidable.

FOL Proof Method: Resolution

conversion to normal form

generalized resolution

unification

Why Representing Change in Time is Difficult

situation calculus - situations, actions, result function

frame problem - how to say when things don’t change (e.g. cartoon frames)

qualification problem - how to define circumstances under which an action will work (e.g. banana in tailpipe)

ramification problem - how to infer all of the implicit consequences of actions (e.g. moving luggage)


	When Propositional Logic Isn’t Enough
	First-Order Logic (FOL)
	Practice with FOL
	FOL Proof Method: Resolution
	[Why Representing Change in Time is Difficult]


	Simple domain: Blocks stacked on a table
	Simple problem: Stack blocks in a given configuration
	Home to much research in a variety of areas:
		Vision (Huffman, 1971)
		Constraint Propogation (Waltz, 1975)
		Machine Learning (Winston, 1970)
		Natural Language Understanding (Winograd, 1972)
		Planning (Fahlman, 1974)


	How would you represent the following:
		Block A is on Block D
		Block A is not on Block B
		If A is on D, A is not on B
		No block is on A
		If A is on D and D is on B, A is above B.


	Define Above using transitivity of On
	“There’s a/there’s no block on A.”
	“All blocks have red letters.”
	Need for properties, relationships, and ways to generalize information compactly
	Practical applications: mathematical theorem proving, common-sense reasoning (CYC image database), circuit/program design/verification


	Propositions - T/F atomic sentences about the world
	Boolean functions (Ø,Ù,Ú,Þ,Û) - negate and logically combine sentences
	Parentheses - disambiguates grammar


	Atomic sentences - more complex than propositions
	Boolean functions (Ø,Ù,Ú,Þ,Û) - make compound sentences
	Quantifiers (",$) - tell us first-order facts which correspond to sets of sentences
	Parentheses


	Domain of Discourse (DOD) - relevant things in the world that we’re reasoning about
	Symbols:
		variables - assigned things in the domain of discourse (DOD)
		constants - things and relations relations between things in the DOD
	An interpretation I maps
		variables to things in the DOD.
		constants to things and relationships between things in the DOD.


	Domain of Discourse: blocks (for simplicity, exclude table)
	Constant Symbols (A, B, C, D, On, Above, …)
	Variables (x, y, z, …)
	An interpretation maps variables and constant symbols to things in the world. (e.g. A means this block with an “A”.)


	object constants - corresponding to things in the DOD
	relation constants - relationships between things in the DOD
		relations also called predicates
		n-ary relation - a set of n-tuples from the DOD
		1-ary relation - a property constant


	Object Constants: A, B, C, D correspond to blocks with the same letters on them.
	Relations Constants:
		OnTable: set of blocks resting on table
		On: set of block pairs where the first rests on the second
		Above: transitive closure of On
		Clear: set of blocks with no blocks above


	function constants - mappings between tuples and things in the DOD
		0-ary function - an object constant
		n-ary function (n>0) - special case of n+1-ary relation where n arguments uniquely determine n+1st argument


	terms - variables, objects, functions of terms
	atomic sentence - relationship between terms
	Under a given interpretation, an atomic sentence is true or false
	compound (or complex) sentences are built of atomic sentences, Boolean functions (negation, connectives), and...


	Terms: x, A, nextBlock(A)
	Atomic sentences: Above(A,D), On(x,B), Clear(y)
	These sentence are true under an interpretation where constants and relations are as we described and x, y are interpreted as D, A or D, C.
	These sentences are also true under an interpretation where A,B,C,D are the numbers 16, 2, 17, 4, On is “the square of”, Above is “a power of”, Clear is “greater than 15”.


	universal (") - “For all”
		"x f - “For all x, f is true.”
	"Every block is a red block".
	"If x>y and y>z then x>z."


	existential ($) - “There exists”
		$x f - “There exists x such that f is true.”
		same as Ø "x Ø f
	"There is a block on the table."
	Challenging: "If a block is above another block, then the block is on the other block or on a block that is above it."


	nested quantifiers - "x $y is different than $y "x
		"Everybody likes someone."
		"Someone likes everyone."
	Relations: Person, Time, FoolPersonAtTime
		"You can fool all the people some of the time, …"
		"… and some of the people all the time, …"
		"… but you cannot fool all the people all the time."


	variable scope - the parts of a sentence for which a variable quantifier is relevant
	free variables - unquantified variables
	formula - another word for sentence
	well-formed formula (wff) - a sentence with no free variables


	Express the following in FOL using relations On, Above, Clear, OnTable, and objects A, B, C, D:
		If block A is on block B, then A is above B, B is not clear, and A is not on the table.
		If one block is on another block, it is also above that block.
		A or D or C is on the table.
		If A is on the table, then D and C are not.


	A sentence is valid if it is true for all possible interpretations.
	A sentence is satisfiable if there exists a “satisfying” interpretation (i.e. an interpretation for which the sentence is true).
	An interpretation which satisfies a sentence is a model of that sentence.


	entailment - whether a sentences follows from other sentences
	inference rules - deduce new sentences from old sentences
	soundness - what can be inferred is entailed
	completeness - what is entailed can be inferred


A logic consists of
	a formal language consisting of
		syntax
		semantics
	a proof theory
terms: sentences, KB, valid, satisfiable, model, entail, inference rule, sound, complete


	Good News: There exists a complete inference procedure for FOL. (Gödel 1930-31)
	Robinson’s resolution algorithm
	Bad News: Can’t tell if a sentence isn’t entailed Þ FOL is semidecidable.


	situation calculus - situations, actions, result function
	frame problem - how to say when things don’t change (e.g. cartoon frames)
	qualification problem - how to define circumstances under which an action will work (e.g. banana in tailpipe)
	ramification problem - how to infer all of the implicit consequences of actions (e.g. moving luggage)