first order logic (fol) -...

First Order Logic (FOL)CE417: Introduction to Artificial IntelligenceSharif University of TechnologySpring 2017

Soleymani

“Artificial Intelligence: A Modern Approach”, 3rd Edition, Chapter 8

Why FOL?

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To represent knowledge of complex environments concisely

Expressive enough to represent a good deal of of our knowledge

E.g., “pits cause breezes in adjacent squares” needs many rules in

propositional logic but one rule in FOL

Natural language elements & FOL elements

Some basic elements of natural language (included also in FOL):

Nouns and noun phrases referring to objects (squares, pits, wumpus)

Some of objects are defined as functions of other objects

Verbs and verb phrases referring to relation among objects (is breezy,is adjacent to, shoot)

Examples:

Objects: people, houses, numbers, baseball games,…

Relations

Unary relation or property: red, round, prime,…

𝑛-ary: brother of, bigger than, inside, part of, owns, comes between,…

Functions: father of, best friend, one more than, beginning of,…

FOL does not include all elements of natural language.

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Symbols & interpretations

Types of symbols

Constant for objects

Predicate for relations

Function for functional relations

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Example

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𝐵𝑟𝑜𝑡ℎ𝑒𝑟(𝑅𝑖𝑐ℎ𝑎𝑟𝑑, 𝐽𝑜ℎ𝑛)

𝐹𝑎𝑡ℎ𝑒𝑟(𝐽𝑜ℎ𝑛) = 𝐻𝑒𝑛𝑟𝑦

𝑀𝑎𝑟𝑟𝑖𝑒𝑑(𝐹𝑎𝑡ℎ𝑒𝑟(𝑅𝑖𝑐ℎ𝑎𝑟𝑑), 𝑀𝑜𝑡ℎ𝑒𝑟(𝐽𝑜ℎ𝑛))

∀𝑥 𝐾𝑖𝑛𝑔 𝑥 ⇒ 𝑃𝑒𝑟𝑠𝑜𝑛 𝑥

∃𝑥 𝐶𝑟𝑜𝑤𝑛 𝑥 ⇒ 𝑂𝑛𝐻𝑒𝑎𝑑 𝑥, 𝐽𝑜ℎ𝑛

∃𝑥, 𝑦 𝐵𝑟𝑜𝑡ℎ𝑒𝑟 𝑥, 𝑅𝑖𝑐ℎ𝑎𝑟𝑑 ∧ 𝐵𝑟𝑜𝑡ℎ𝑒𝑟 𝑦, 𝑅𝑖𝑐ℎ𝑎𝑟𝑑 ∧ ¬(𝑥 = 𝑦)

¬𝐾𝑖𝑛𝑔(𝑅𝑖𝑐ℎ𝑎𝑟𝑑) ⇒ 𝐾𝑖𝑛𝑔(𝐽𝑜ℎ𝑛)

𝑥, 𝑦 𝑆𝑖𝑏𝑙𝑖𝑛𝑔(𝑥, 𝑦) [ 𝑥 = 𝑦 ∃𝑚, 𝑓 (𝑚 = 𝑓) 𝑃𝑎𝑟𝑒𝑛𝑡(𝑚, 𝑥) 𝑃𝑎𝑟𝑒𝑛𝑡(𝑓, 𝑥) 𝑃𝑎𝑟𝑒𝑛𝑡(𝑚, 𝑦) 𝑃𝑎𝑟𝑒𝑛𝑡(𝑓, 𝑦)]

Syntax of FOL: Basic elements

Logical elements

Connectives ,,,,

Quantifiers ,

Domain specific elements

Constants 𝐽𝑜ℎ𝑛, 𝑅𝑖𝑐ℎ𝑎𝑟𝑑, …

Predicates 𝐵𝑟𝑜𝑡ℎ𝑒𝑟, ...

Functions 𝐿𝑒𝑓𝑡𝐿𝑒𝑔𝑂𝑓,...

Non-logical general elements

Variables 𝑥, 𝑦, 𝑎, 𝑏, …

Equality =

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Universal quantification

𝑥 𝑃(𝑥) is true in a model 𝑚 iff 𝑃(𝑥) is true with 𝑥 beingeach possible object in the model

conjunction of instantiations of 𝑃(𝑥)

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Existential quantification

𝑥 𝑃(𝑥) is true in a model 𝑚 iff 𝑃(𝑥) is true with 𝑥 beingsome possible object in the model

disjunction of instantiations of 𝑃(𝑥)

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Properties of quantifiers

𝑥 𝑦 𝑃 is the same as 𝑦 𝑥 𝑃 𝑥 𝑦 𝑃 is the same as 𝑦 𝑥 𝑃 𝑥 𝑦 𝑃 is not the same as 𝑦 𝑥 𝑃

𝑥 𝑦 𝑀𝑜𝑡ℎ𝑒𝑟(𝑥, 𝑦) “There is a person who is mother of everyone in the world”

𝑦 𝑥 𝑀𝑜𝑡ℎ𝑒𝑟(𝑥, 𝑦) “Everyone in the world has a mother”

Quantifier duality 𝑥 𝑃 ⇔ 𝑥𝑃 𝑥 𝑃 ⇔ ¬∀𝑥 𝑃

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FOL: Kinship domain example

Objects people

Functions

Predicates

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Objects: people

Functions:𝑀𝑜𝑡ℎ𝑒𝑟, 𝐹𝑎𝑡ℎ𝑒𝑟

Predicates:

Unary:𝑀𝑎𝑙𝑒, 𝐹𝑒𝑚𝑎𝑙𝑒

Binary:𝑃𝑎𝑟𝑒𝑛𝑡, 𝑆𝑖𝑏𝑙𝑖𝑛𝑔, 𝐵𝑟𝑜𝑡ℎ𝑒𝑟, 𝑆𝑖𝑠𝑡𝑒𝑟, 𝐶ℎ𝑖𝑙𝑑, 𝐷𝑎𝑢𝑔ℎ𝑡𝑒𝑟, 𝑆𝑜𝑛, 𝑆𝑝𝑜𝑢𝑠𝑒,

𝑊𝑖𝑓𝑒, 𝐻𝑢𝑠𝑏𝑎𝑛𝑑, 𝐺𝑟𝑎𝑛𝑑𝑝𝑎𝑟𝑒𝑛𝑡, 𝐺𝑟𝑎𝑛𝑑𝑐ℎ𝑖𝑙𝑑, 𝐶𝑜𝑢𝑠𝑖𝑛, 𝐴𝑢𝑛𝑡, 𝑈𝑛𝑐𝑙𝑒

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Sample sentences:

𝑚, 𝑐 𝑀𝑜𝑡ℎ𝑒𝑟 𝑐 = 𝑚 ⇔ (𝐹𝑒𝑚𝑎𝑙𝑒(𝑚) ∧ 𝑃𝑎𝑟𝑒𝑛𝑡(𝑚, 𝑐))

𝑝, 𝑐 𝑃𝑎𝑟𝑒𝑛𝑡 𝑝, 𝑐 ⇔ 𝐶ℎ𝑖𝑙𝑑(𝑐, 𝑝)

𝑔, 𝑐 𝐺𝑟𝑎𝑛𝑑𝑝𝑎𝑟𝑒𝑛𝑡 𝑔, 𝑐 ⇔ ∃𝑝 𝑃𝑎𝑟𝑒𝑛𝑡 𝑔, 𝑝 ∧ 𝑃𝑎𝑟𝑒𝑛𝑡 𝑝, 𝑐

𝑥, 𝑦 𝑆𝑖𝑏𝑖𝑙𝑖𝑛𝑔 𝑥, 𝑦 ⇔ 𝑥 ≠ 𝑦 ∧ ∃𝑝 𝑃𝑎𝑟𝑒𝑛𝑡 𝑝, 𝑥 ∧ 𝑃𝑎𝑟𝑒𝑛𝑡 𝑝, 𝑥

Assertions & queries in FOL KBs

Assertions: sentences added to 𝐾𝐵 using 𝑇𝐸𝐿𝐿

𝑇𝑒𝑙𝑙(𝐾𝐵, 𝐾𝑖𝑛𝑔(𝐽𝑜ℎ𝑛))

𝑇𝑒𝑙𝑙(𝐾𝐵, (∀𝑥 𝐾𝑖𝑛𝑔 𝑥 ⇒ 𝑃𝑒𝑟𝑠𝑜𝑛 𝑥 ))

Queries or goals: questions asked from 𝐾𝐵 using 𝐴𝑆𝐾

𝐴𝑆𝐾 𝐾𝐵, 𝐾𝑖𝑛𝑔 𝐽𝑜ℎ𝑛

𝐴𝑆𝐾 𝐾𝐵, ∃𝑥 𝐾𝑖𝑛𝑔 𝑥

Substitution or binding list:AskVars

In KB of Horn clauses, every way of making the query true will bind thevariables to specific values

AskVars(𝐾𝐵, 𝐾𝑖𝑛𝑔(𝑥)): answer {𝑥/𝐽𝑜ℎ𝑛}

If KB contains 𝐾𝑖𝑛𝑔 𝐽𝑜ℎ𝑛 ∨ 𝐾𝑖𝑛𝑔(𝑅𝑖𝑐ℎ𝑎𝑟𝑑), there is no binding although𝐴𝑆𝐾 𝐾𝐵, ∃𝑥 𝐾𝑖𝑛𝑔 𝑥 is true

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FOL: Set domain example

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Objects: sets, elements

Functions: 𝑠1 ∩ 𝑠2, 𝑠1 ∪ 𝑠2, {𝑥|𝑠}

Predicates: Unary: Set

Binary: 𝑥 ∈ 𝑠, 𝑠1 ⊆ 𝑠2

𝑠 𝑆𝑒𝑡(𝑠) ⇔ (𝑠 = {}) ∨ (∃𝑥, 𝑠2 𝑆𝑒𝑡(𝑠2) 𝑠 = {𝑥|𝑠2})

𝑥, 𝑠 {𝑥|𝑠} = {} 𝑥, 𝑠 𝑥 ∈ 𝑠 ⇔ 𝑠 = {𝑥|𝑠}

𝑥, 𝑠 𝑥 ∈ 𝑠 ⇔ ∃𝑦, 𝑠2 (𝑠 = {𝑦|𝑠2} (𝑥 = 𝑦 ∨ 𝑥 ∈ 𝑠2))] 𝑠1, 𝑠2 𝑠1𝑠2 (𝑥 𝑥𝑠1 𝑥 ∈ 𝑠2) 𝑠1, 𝑠2 (𝑠1 = 𝑠2) (𝑠1 𝑠2 𝑠2 𝑠1) 𝑥, 𝑠1, 𝑠2 𝑥 ∈ (𝑠1𝑠2) (𝑥 ∈ 𝑠1 𝑥 ∈ 𝑠2)

𝑥, 𝑠1, 𝑠2 𝑥 ∈ (𝑠1𝑠2) (𝑥 ∈ 𝑠1 𝑥 ∈ 𝑠2)

FOL: Wumpus world example

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Environment

Objects: pairs identifying squares [𝑖, 𝑗], 𝐴𝑔𝑒𝑛𝑡, 𝑊𝑢𝑚𝑝𝑢𝑠

Relations:𝑃𝑖𝑡, 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡, 𝐵𝑟𝑒𝑒𝑧𝑦, 𝑆𝑡𝑒𝑛𝑐ℎ𝑦,

𝑃𝑒𝑟𝑐𝑒𝑝𝑡, 𝐴𝑐𝑡𝑖𝑜𝑛, 𝐴𝑡, 𝐻𝑎𝑣𝑒𝐴𝑟𝑟𝑜𝑤, …


Perceptions: perceives a smell, a breeze, and glitter at 𝑡 = 5:

𝑃𝑒𝑟𝑐𝑒𝑝𝑡 𝑆𝑡𝑒𝑛𝑐ℎ, 𝐵𝑟𝑒𝑒𝑧𝑒, 𝐺𝑙𝑖𝑡𝑡𝑒𝑟, 𝑁𝑜𝑛𝑒, 𝑁𝑜𝑛𝑒 , 5

Actions: 𝑇𝑢𝑟𝑛(𝐿𝑒𝑓𝑡), 𝑇𝑢𝑟𝑛(𝑅𝑖𝑔ℎ𝑡), 𝐹𝑜𝑟𝑤𝑎𝑟𝑑, 𝑆ℎ𝑜𝑜𝑡, 𝐺𝑟𝑎𝑏, 𝐶𝑙𝑖𝑚𝑏

Perceptions implies facts about current state

∀𝑡, 𝑠, 𝑔, 𝑚, 𝑐 𝑃𝑒𝑟𝑐𝑒𝑝𝑡( 𝑠, 𝐵𝑟𝑒𝑒𝑧𝑒, 𝑔, 𝑚, 𝑐 , 𝑡) ⇒ 𝐵𝑟𝑒𝑒𝑧𝑒(𝑡)

Simple reflex behavior

∀𝑡 𝐺𝑙𝑖𝑡𝑡𝑒𝑟(𝑡) ⇒ 𝐵𝑒𝑠𝑡𝐴𝑐𝑡𝑖𝑜𝑛(𝐺𝑟𝑎𝑏, 𝑡)

AskVars(∃𝑎 𝐵𝑒𝑠𝑡𝐴𝑐𝑡𝑖𝑜𝑛(𝑎, 5))

Binding list: e.g., {𝑎/𝐺𝑟𝑎𝑏}

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Samples of rules:

∀𝑥, 𝑦, 𝑎, 𝑏 𝐴𝑑𝑗 𝑥, 𝑦 , 𝑎, 𝑏 ⇔ 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 − 1 ∨ 𝑦 = 𝑏 − 1

∨ 𝑦 = 𝑏 ∧ 𝑥 = 𝑎 − 1 ∨ 𝑥 = 𝑎 + 1

𝐴𝑡 𝐴𝑔𝑒𝑛𝑡, 𝑠, 𝑡

∃𝑥, 𝑦 ∀𝑡 𝐴𝑡(𝑊𝑢𝑚𝑝𝑢𝑠, [𝑥, 𝑦], 𝑡)

∀𝑠, 𝑡 𝐴𝑡 𝐴𝑔𝑒𝑛𝑡, 𝑠, 𝑡 ∧ 𝐵𝑟𝑒𝑒𝑧𝑒 𝑡 ⇒ 𝐵𝑟𝑒𝑒𝑧𝑦 𝑠

∀𝑥, 𝑠1, 𝑠2, 𝑡 𝐴𝑡 𝐴𝑔𝑒𝑛𝑡, 𝑠1, 𝑡 ∧ 𝐴𝑡 𝐴𝑔𝑒𝑛𝑡, 𝑠2, 𝑡 ⇒ 𝑠1 = 𝑠2

∀𝑠, 𝑡 𝐵𝑟𝑒𝑒𝑧𝑦 𝑠 ⇒ ∃𝑟 𝐴𝑑𝑗 𝑟, 𝑠 ∧ 𝑃𝑖𝑡 𝑟

One successor-state axiom for each predicate

∀𝑡 𝐻𝑎𝑣𝑒𝐴𝑟𝑟𝑜𝑤 𝑡 + 1 ⇔ ( 𝐻𝑎𝑣𝑒𝐴𝑟𝑟𝑜𝑤 𝑡 ∧ ¬𝐴𝑐𝑡𝑖𝑜𝑛 𝑆ℎ𝑜𝑜𝑡, 𝑡 )

Knowledge Engineering (KE) in FOL

1) Identify the task

2) Assemble the relevant knowledge

3) Decide on a vocabulary of predicates, functions, and

constants (Ontology)

4) Encode general knowledge about the domain

5) Encode a description of the specific problem instance

6) Pose queries to the inference procedure and get answers

7) Debug the knowledge base

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KE: electronic circuits example

One-bit full adder

XORXOR

ORAND

AND

Carry

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KE: electronic circuits example (steps)

1) Identify the tasko Does the circuit add properly? (circuit verification)

2) Assemble the relevant knowledgeo Signals, input and output terminals, gates, wires connecting gates,

types of gates (AND,OR, XOR,NOT)

3) Decide on a vocabulary (ontology)o types of gates, terminals of gates, connections between gates, signal

on terminals of gates

o e.g., alternatives for determining the type of a gate:o 𝑇𝑦𝑝𝑒(𝑋1) = 𝑋𝑂𝑅o 𝑋𝑂𝑅(𝑋1)o 𝑇𝑦𝑝𝑒(𝑋1, 𝑋𝑂𝑅)

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4) Encode general knowledge of the domain

𝑡1, 𝑡2 𝐶𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑(𝑡1, 𝑡2) 𝑆𝑖𝑔𝑛𝑎𝑙(𝑡1) = 𝑆𝑖𝑔𝑛𝑎𝑙(𝑡2)

𝑡 𝑆𝑖𝑔𝑛𝑎𝑙 𝑡 = 1 ∨ 𝑆𝑖𝑔𝑛𝑎𝑙 𝑡 = 0

𝑡1, 𝑡2 𝐶𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑡1, 𝑡2 ⇔ 𝐶𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑡2, 𝑡1

𝑔 𝑇𝑦𝑝𝑒 𝑔 = 𝑂𝑅 ⇒ (𝑆𝑖𝑔𝑛𝑎𝑙 𝑂𝑢𝑡 1, 𝑔 = 1 ⇔ ∃𝑛 𝑆𝑖𝑔𝑛𝑎𝑙 𝐼𝑛 𝑛, 𝑔 = 1)

𝑔 𝑇𝑦𝑝𝑒 𝑔 = 𝐴𝑁𝐷 ⇒ (𝑆𝑖𝑔𝑛𝑎𝑙 𝑂𝑢𝑡 1, 𝑔 = 0 ⇔ ∃𝑛 𝑆𝑖𝑔𝑛𝑎𝑙 𝐼𝑛 𝑛, 𝑔 = 0)

𝑔 𝑇𝑦𝑝𝑒 𝑔 = 𝑋𝑂𝑅 ⇒ (𝑆𝑖𝑔𝑛𝑎𝑙 𝑂𝑢𝑡 1, 𝑔 = 0 ⇔ 𝑆𝑖𝑔𝑛𝑎𝑙 𝐼𝑛 1, 𝑔

= 𝑆𝑖𝑔𝑛𝑎𝑙 𝐼𝑛 2, 𝑔 )

𝑔 𝑇𝑦𝑝𝑒 𝑔 = 𝑁𝑂𝑇 (𝑆𝑖𝑔𝑛𝑎𝑙 𝑂𝑢𝑡 1, 𝑔 ≠ 𝑆𝑖𝑔𝑛𝑎𝑙 𝐼𝑛 1, 𝑔 )

1 ≠ 0

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[You can see a more accurate encoding in 3rd edition of AIMA]


5) Encode the specific problem instance𝑇𝑦𝑝𝑒(𝑋1) = 𝑋𝑂𝑅 𝑇𝑦𝑝𝑒(𝑋2) = 𝑋𝑂𝑅

𝑇𝑦𝑝𝑒(𝐴1) = 𝐴𝑁𝐷 𝑇𝑦𝑝𝑒(𝐴2) = 𝐴𝑁𝐷𝑇𝑦𝑝𝑒(𝑂1) = 𝑂𝑅

𝐶𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑂𝑢𝑡 1, 𝑋1 , 𝐼𝑛 1, 𝑋2 𝐶𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑(𝐼𝑛(1, 𝐶1), 𝐼𝑛(1, 𝑋1))𝐶𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑂𝑢𝑡 1, 𝑋1 , 𝐼𝑛 2, 𝐴2 𝐶𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑(𝐼𝑛(1, 𝐶1), 𝐼𝑛(1, 𝐴1))𝐶𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑂𝑢𝑡 1, 𝐴2 , 𝐼𝑛 1, 𝑂1 𝐶𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑(𝐼𝑛(2, 𝐶1), 𝐼𝑛(2, 𝑋1))𝐶𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑂𝑢𝑡 1, 𝐴1 , 𝐼𝑛 2, 𝑂1 𝐶𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑(𝐼𝑛(2, 𝐶1), 𝐼𝑛(2, 𝐴1))𝐶𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑂𝑢𝑡 1, 𝑋2 , 𝑂𝑢𝑡 1, 𝐶1 𝐶𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑(𝐼𝑛(3, 𝐶1), 𝐼𝑛(2, 𝑋2))𝐶𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑂𝑢𝑡 1, 𝑂1 , 𝑂𝑢𝑡 2, 𝐶1 𝐶𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑(𝐼𝑛(3, 𝐶1), 𝐼𝑛(1, 𝐴2))

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6) Pose queries to the inference procedure

𝑖1, 𝑖2, 𝑖3 𝑆𝑖𝑔𝑛𝑎𝑙 𝐼𝑛 1, 𝐶1 = 𝑖1 𝑆𝑖𝑔𝑛𝑎𝑙 𝐼𝑛 2, 𝐶1 = 𝑖2

𝑆𝑖𝑔𝑛𝑎𝑙 𝐼𝑛 3, 𝐶1 = 𝑖3 𝑆𝑖𝑔𝑛𝑎𝑙 𝑂𝑢𝑡 1, 𝐶1 = 0

𝑆𝑖𝑔𝑛𝑎𝑙(𝑂𝑢𝑡(2, 𝐶1)) = 1

Bindings= {𝑖1/1, 𝑖2/1, 𝑖3/0}, {𝑖1/1, 𝑖2/0, 𝑖3/1}, {𝑖1/0, 𝑖2/1, 𝑖3/1}

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6) Pose queries to the inference procedure

𝑖1, 𝑖2, 𝑖3, 𝑜1, 𝑜2 𝑆𝑖𝑔𝑛𝑎𝑙 𝐼𝑛 1, 𝐶1 = 𝑖1 𝑆𝑖𝑔𝑛𝑎𝑙 𝐼𝑛 2, 𝐶1 = 𝑖2

𝑆𝑖𝑔𝑛𝑎𝑙 𝐼𝑛 3, 𝐶1 = 𝑖3 𝑆𝑖𝑔𝑛𝑎𝑙 𝑂𝑢𝑡 1, 𝐶1 = 𝑜1

𝑆𝑖𝑔𝑛𝑎𝑙(𝑂𝑢𝑡(2, 𝐶1)) = 𝑜2

Bindings = whole input-output table

7) Debug the knowledge base

Has 1 ≠ 0 been asserted?

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Summary

First-order logic:

objects and relations are semantic primitives

syntax: constants, variables, functions, predicates, quantifiers

Knowledge engineering

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first order logic (fol) -...

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