Chapter9 · Chapter9 2. Existential instantiation (EI) For any sentence α, variable v, and...

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Inference in first-order logic Chapter 9 Chapter 9 1

Transcript of Chapter9 · Chapter9 2. Existential instantiation (EI) For any sentence α, variable v, and...

Page 1: Chapter9 · Chapter9 2. Existential instantiation (EI) For any sentence α, variable v, and constant symbol k that does not appear elsewhere intheknowledgebase:

Inference in first-order logic

Chapter 9

Chapter 9 1

Page 2: Chapter9 · Chapter9 2. Existential instantiation (EI) For any sentence α, variable v, and constant symbol k that does not appear elsewhere intheknowledgebase:

Universal instantiation (UI)

Every instantiation of a universally quantified sentence is entailed by it:

∀ v α

Subst({v/g}, α)

for any variable v and ground term g

E.g., ∀ x King(x) ∧Greedy(x) ⇒ Evil(x) yields

King(John) ∧Greedy(John) ⇒ Evil(John)King(Richard) ∧Greedy(Richard) ⇒ Evil(Richard)King(Father(John)) ∧Greedy(Father(John)) ⇒ Evil(Father(John))...

Chapter 9 2

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Existential instantiation (EI)

For any sentence α, variable v, and constant symbol kthat does not appear elsewhere in the knowledge base:

∃ v α

Subst({v/k}, α)

E.g., ∃ x Crown(x) ∧OnHead(x, John) yields

Crown(C1) ∧OnHead(C1, John)

provided C1 is a new constant symbol, called a Skolem constant

Another example: from ∃ x d(xy)/dy= xy we obtain

d(ey)/dy= ey

provided e is a new constant symbol

Chapter 9 3

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Existential instantiation contd.

UI can be applied several times to add new sentences;the new KB is logically equivalent to the old

EI can be applied once to replace the existential sentence;the new KB is not equivalent to the old,but is satisfiable iff the old KB was satisfiable

Chapter 9 4

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Inference...

Bob loves someone.

If someone loves someone, then that person loves them back.

Prove:

Someone loves bob.

Chapter 9 5

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Unification

Suppose the KB contains just the following:

∀x King(x) ∧Greedy(x) ⇒ Evil(x)King(John)∀ y Greedy(y)Brother(Richard, John)

We can get the inference immediately if we can find a substitution θsuch that King(x) and Greedy(x) match King(John) and Greedy(y)

θ = {x/John, y/John} works

Unify(α, β) = θ if αθ= βθ

Chapter 9 6

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p q θKnows(John, x) Knows(John, Jane)Knows(John, x) Knows(y, OJ)Knows(John, x) Knows(y,Mother(y))Knows(John, x) Knows(x,OJ)

Chapter 9 7

Page 8: Chapter9 · Chapter9 2. Existential instantiation (EI) For any sentence α, variable v, and constant symbol k that does not appear elsewhere intheknowledgebase:

Unification

We can get the inference immediately if we can find a substitution θsuch that King(x) and Greedy(x) match King(John) and Greedy(y)

θ = {x/John, y/John} works

Unify(α, β) = θ if αθ= βθ

p q θKnows(John, x) Knows(John, Jane) {x/Jane}Knows(John, x) Knows(y, OJ)Knows(John, x) Knows(y,Mother(y))Knows(John, x) Knows(x,OJ)

Chapter 9 8

Page 9: Chapter9 · Chapter9 2. Existential instantiation (EI) For any sentence α, variable v, and constant symbol k that does not appear elsewhere intheknowledgebase:

Unification

We can get the inference immediately if we can find a substitution θsuch that King(x) and Greedy(x) match King(John) and Greedy(y)

θ = {x/John, y/John} works

Unify(α, β) = θ if αθ= βθ

p q θKnows(John, x) Knows(John, Jane) {x/Jane}Knows(John, x) Knows(y, OJ) {x/OJ, y/John}Knows(John, x) Knows(y,Mother(y))Knows(John, x) Knows(x,OJ)

Chapter 9 9

Page 10: Chapter9 · Chapter9 2. Existential instantiation (EI) For any sentence α, variable v, and constant symbol k that does not appear elsewhere intheknowledgebase:

Unification

We can get the inference immediately if we can find a substitution θsuch that King(x) and Greedy(x) match King(John) and Greedy(y)

θ = {x/John, y/John} works

Unify(α, β) = θ if αθ= βθ

p q θKnows(John, x) Knows(John, Jane) {x/Jane}Knows(John, x) Knows(y, OJ) {x/OJ, y/John}Knows(John, x) Knows(y,Mother(y)) {y/John, x/Mother(John)}Knows(John, x) Knows(x,OJ)

Chapter 9 10

Page 11: Chapter9 · Chapter9 2. Existential instantiation (EI) For any sentence α, variable v, and constant symbol k that does not appear elsewhere intheknowledgebase:

Unification

We can get the inference immediately if we can find a substitution θsuch that King(x) and Greedy(x) match King(John) and Greedy(y)

θ = {x/John, y/John} works

Unify(α, β) = θ if αθ= βθ

p q θKnows(John, x) Knows(John, Jane) {x/Jane}Knows(John, x) Knows(y, OJ) {x/OJ, y/John}Knows(John, x) Knows(y,Mother(y)) {y/John, x/Mother(John)}Knows(John, x) Knows(x,OJ) fail

Standardizing apart eliminates overlap of variables, e.g., Knows(z17, OJ)

Chapter 9 11

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Generalized Modus Ponens (GMP)

p1′, p2

′, . . . , pn′, (p1 ∧ p2 ∧ . . . ∧ pn⇒ q)

qθwhere pi

′θ= piθ for all i

p1′ is King(John) p1 is King(x)

p2′ is Greedy(y) p2 is Greedy(x)

θ is {x/John, y/John} q is Evil(x)qθ is Evil(John)

GMP used with KB of definite clauses (exactly one positive literal)All variables assumed universally quantified

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Soundness of GMP

Need to show that

p1′, . . . , pn

′, (p1 ∧ . . . ∧ pn⇒ q) |= qθ

provided that pi′θ= piθ for all i

Lemma: For any definite clause p, we have p |= pθ by UI

1. (p1 ∧ . . . ∧ pn⇒ q) |= (p1 ∧ . . . ∧ pn⇒ q)θ= (p1θ ∧ . . . ∧ pnθ ⇒ qθ)

2. p1′, . . . , pn

′ |= p1′ ∧ . . . ∧ pn

′ |= p1′θ ∧ . . . ∧ pn

′θ

3. From 1 and 2, qθ follows by ordinary Modus Ponens

Chapter 9 13

Page 14: Chapter9 · Chapter9 2. Existential instantiation (EI) For any sentence α, variable v, and constant symbol k that does not appear elsewhere intheknowledgebase:

Example knowledge base

The law says that it is a crime for an American to sell weapons to hostilenations. The country Nono, an enemy of America, has some missiles, andall of its missiles were sold to it by Colonel West, who is American.

Prove that Col. West is a criminal

Chapter 9 14

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Example knowledge base contd.

. . . it is a crime for an American to sell weapons to hostile nations:

Chapter 9 15

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Example knowledge base contd.

. . . it is a crime for an American to sell weapons to hostile nations:American(x)∧Weapon(y)∧Sells(x, y, z)∧Hostile(z)⇒ Criminal(x)

Nono . . . has some missiles

Chapter 9 16

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Example knowledge base contd.

. . . it is a crime for an American to sell weapons to hostile nations:American(x)∧Weapon(y)∧Sells(x, y, z)∧Hostile(z)⇒ Criminal(x)

Nono . . . has some missiles, i.e., ∃ x Owns(Nono, x) ∧Missile(x):Owns(Nono,M1) and Missile(M1)

. . . all of its missiles were sold to it by Colonel West

Chapter 9 17

Page 18: Chapter9 · Chapter9 2. Existential instantiation (EI) For any sentence α, variable v, and constant symbol k that does not appear elsewhere intheknowledgebase:

Example knowledge base contd.

. . . it is a crime for an American to sell weapons to hostile nations:American(x)∧Weapon(y)∧Sells(x, y, z)∧Hostile(z)⇒ Criminal(x)

Nono . . . has some missiles, i.e., ∃ x Owns(Nono, x) ∧Missile(x):Owns(Nono,M1) and Missile(M1)

. . . all of its missiles were sold to it by Colonel West∀x Missile(x) ∧Owns(Nono, x) ⇒ Sells(West, x,Nono)

Missiles are weapons:

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Example knowledge base contd.

. . . it is a crime for an American to sell weapons to hostile nations:American(x)∧Weapon(y)∧Sells(x, y, z)∧Hostile(z)⇒ Criminal(x)

Nono . . . has some missiles, i.e., ∃ x Owns(Nono, x) ∧Missile(x):Owns(Nono,M1) and Missile(M1)

. . . all of its missiles were sold to it by Colonel West∀x Missile(x) ∧Owns(Nono, x) ⇒ Sells(West, x,Nono)

Missiles are weapons:Missile(x)⇒ Weapon(x)

An enemy of America counts as “hostile”:

Chapter 9 19

Page 20: Chapter9 · Chapter9 2. Existential instantiation (EI) For any sentence α, variable v, and constant symbol k that does not appear elsewhere intheknowledgebase:

Example knowledge base contd.

. . . it is a crime for an American to sell weapons to hostile nations:American(x)∧Weapon(y)∧Sells(x, y, z)∧Hostile(z)⇒ Criminal(x)

Nono . . . has some missiles, i.e., ∃ x Owns(Nono, x) ∧Missile(x):Owns(Nono,M1) and Missile(M1)

. . . all of its missiles were sold to it by Colonel West∀x Missile(x) ∧Owns(Nono, x) ⇒ Sells(West, x,Nono)

Missiles are weapons:Missile(x)⇒ Weapon(x)

An enemy of America counts as “hostile”:Enemy(x,America) ⇒ Hostile(x)

West, who is American . . .American(West)

The country Nono, an enemy of America . . .Enemy(Nono,America)

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Forward chaining algorithm

function FOL-FC-Ask(KB,α) returns a substitution or false

repeat until new is empty

new←{}

for each sentence r in KB do

( p1 ∧ . . . ∧ pn ⇒ q)←Standardize-Apart(r)

for each θ such that (p1 ∧ . . . ∧ pn)θ = (p ′1 ∧ . . . ∧ p ′n)θ

for some p ′1, . . . , p′n in KB

q ′←Subst(θ, q)

if q ′ is not a renaming of a sentence already in KB or new then do

add q ′ to new

φ←Unify(q ′,α)

if φ is not fail then return φ

add new to KB

return false

Chapter 9 21

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Forward chaining proof

Enemy(Nono,America)Owns(Nono,M1)Missile(M1)American(West)

Chapter 9 22

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Forward chaining proof

Hostile(Nono)

Enemy(Nono,America)Owns(Nono,M1)Missile(M1)American(West)

Weapon(M1) Sells(West,M1,Nono)

Chapter 9 23

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Forward chaining proof

Hostile(Nono)

Enemy(Nono,America)Owns(Nono,M1)Missile(M1)American(West)

Weapon(M1)

Criminal(West)

Sells(West,M1,Nono)

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Properties of forward chaining

Sound and complete for first-order definite clauses(proof similar to propositional proof)

Datalog = first-order definite clauses + no functions (e.g., crime KB)FC terminates for Datalog in poly iterations: at most p · nk literals

May not terminate in general if α is not entailed

This is unavoidable: entailment with definite clauses is semidecidable

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Efficiency of forward chaining

Simple observation: no need to match a rule on iteration kif a premise wasn’t added on iteration k − 1

⇒ match each rule whose premise contains a newly added literal

Matching itself can be expensive

Database indexing allows O(1) retrieval of known factse.g., query Missile(x) retrieves Missile(M1)

Matching conjunctive premises against known facts is NP-hard

Forward chaining is widely used in deductive databases

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Hard matching example

Victoria

WA

NT

SA

Q

NSW

V

T

Diff(wa, nt) ∧ Diff(wa, sa) ∧

Diff(nt, q)Diff(nt, sa) ∧

Diff(q, nsw) ∧ Diff(q, sa) ∧

Diff(nsw, v) ∧ Diff(nsw, sa) ∧

Diff(v, sa) ⇒ Colorable()

Diff(Red,Blue) Diff(Red,Green)

Diff(Green,Red) Diff(Green,Blue)

Diff(Blue,Red) Diff(Blue,Green)

Colorable() is inferred iff the CSP has a solutionCSPs include 3SAT as a special case, hence matching is NP-hard

Chapter 9 27

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Backward chaining algorithm

function FOL-BC-Ask(KB, goals,θ) returns a set of substitutions

inputs: KB, a knowledge base

goals, a list of conjuncts forming a query (θ already applied)

θ, the current substitution, initially the empty substitution { }

local variables: answers, a set of substitutions, initially empty

if goals is empty then return {θ}

q ′←Subst(θ,First(goals))

for each sentence r in KB

where Standardize-Apart(r) = ( p1 ∧ . . . ∧ pn ⇒ q)

and θ′←Unify(q, q ′) succeeds

new goals← [ p1, . . . , pn|Rest(goals)]

answers←FOL-BC-Ask(KB,new goals,Compose(θ′,θ)) ∪ answers

return answers

Chapter 9 28

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Backward chaining example

Criminal(West)

Chapter 9 29

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Backward chaining example

Criminal(West)

Weapon(y)American(x) Sells(x,y,z) Hostile(z)

{x/West}

Chapter 9 30

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Backward chaining example

Criminal(West)

Weapon(y) Sells(x,y,z) Hostile(z)

{x/West}

{ }

American(West)

Chapter 9 31

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Backward chaining example

Hostile(Nono)

Criminal(West)

Missile(y)

Weapon(y) Sells(West,M1,z)American(West)

{ }

Sells(x,y,z) Hostile(z)

{x/West}

Chapter 9 32

Page 33: Chapter9 · Chapter9 2. Existential instantiation (EI) For any sentence α, variable v, and constant symbol k that does not appear elsewhere intheknowledgebase:

Backward chaining example

Hostile(Nono)

Criminal(West)

Missile(y)

Weapon(y) Sells(West,M1,z)American(West)

{ }

Sells(x,y,z) Hostile(z)

y/M1{ }

{x/West, y/M1}

Chapter 9 33

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Backward chaining example

Owns(Nono,M1)Missile(M1)

Criminal(West)

Missile(y)

Weapon(y) Sells(West,M1,z)American(West)

y/M1{ }

{ } z/Nono{ }

Hostile(z)

{x/West, y/M1, z/Nono}

Chapter 9 34

Page 35: Chapter9 · Chapter9 2. Existential instantiation (EI) For any sentence α, variable v, and constant symbol k that does not appear elsewhere intheknowledgebase:

Backward chaining example

Hostile(Nono)

Enemy(Nono,America)Owns(Nono,M1)Missile(M1)

Criminal(West)

Missile(y)

Weapon(y) Sells(West,M1,z)American(West)

y/M1{ } { }{ }{ }

{ } z/Nono{ }

{x/West, y/M1, z/Nono}

Chapter 9 35

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Properties of backward chaining

Depth-first recursive proof search: space is linear in size of proof

Incomplete due to infinite loops⇒ fix by checking current goal against every goal on stack

Inefficient due to repeated subgoals (both success and failure)⇒ fix using caching of previous results (extra space!)

Widely used (without improvements!) for logic programming

Chapter 9 36

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Logic programming

Sound bite: computation as inference on logical KBs

Logic programming Ordinary programming1. Identify problem Identify problem2. Assemble information Assemble information3. Tea break Figure out solution4. Encode information in KB Program solution5. Encode problem instance as facts Encode problem instance as data6. Ask queries Apply program to data7. Find false facts Debug procedural errors

Should be easier to debug Capital(NewY ork, US) than x := x + 2 !

Chapter 9 37

Page 38: Chapter9 · Chapter9 2. Existential instantiation (EI) For any sentence α, variable v, and constant symbol k that does not appear elsewhere intheknowledgebase:

Prolog systems

Basis: backward chaining with Horn clauses + bells & whistlesWidely used in Europe, Japan (basis of 5th Generation project)Compilation techniques ⇒ approaching a billion LIPS

Program = set of clauses = head :- literal1, . . . literaln.

criminal(X) :- american(X), weapon(Y), sells(X,Y,Z), hostile(Z).

Efficient unification by open codingEfficient retrieval of matching clauses by direct linkingDepth-first, left-to-right backward chainingBuilt-in predicates for arithmetic etc., e.g., X is Y*Z+3

Closed-world assumption (“negation as failure”)e.g., given alive(X) :- not dead(X).

alive(joe) succeeds if dead(joe) fails

Chapter 9 38

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Prolog examples

Depth-first search from a start state X:

dfs(X) :- goal(X).

dfs(X) :- successor(X,S),dfs(S).

No need to loop over S: successor succeeds for each

Appending two lists to produce a third:

append([],Y,Y).

append([X|L],Y,[X|Z]) :- append(L,Y,Z).

query: append(A,B,[1,2]) ?

answers: A=[] B=[1,2]

A=[1] B=[2]

A=[1,2] B=[]

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Page 40: Chapter9 · Chapter9 2. Existential instantiation (EI) For any sentence α, variable v, and constant symbol k that does not appear elsewhere intheknowledgebase:

Resolution: brief summary

Full first-order version:

ℓ1 ∨ · · · ∨ ℓk, m1 ∨ · · · ∨mn

(ℓ1 ∨ · · · ∨ ℓi−1 ∨ ℓi+1 ∨ · · · ∨ ℓk ∨m1 ∨ · · · ∨mj−1 ∨mj+1 ∨ · · · ∨mn)θ

where Unify(ℓi,¬mj) = θ.

For example,

¬Rich(x) ∨ Unhappy(x)Rich(Ken)

Unhappy(Ken)

with θ = {x/Ken}

Apply resolution steps to CNF (KB ∧ ¬α); complete for FOL

Chapter 9 40

Page 41: Chapter9 · Chapter9 2. Existential instantiation (EI) For any sentence α, variable v, and constant symbol k that does not appear elsewhere intheknowledgebase:

Conversion to CNF

Everyone who loves all animals is loved by someone:∀x [∀ y Animal(y) ⇒ Loves(x, y)] ⇒ [∃ y Loves(y, x)]

1. Eliminate biconditionals and implications

∀x [¬∀ y ¬Animal(y) ∨ Loves(x, y)] ∨ [∃ y Loves(y, x)]

2. Move ¬ inwards: ¬∀x, p ≡ ∃ x ¬p, ¬∃x, p ≡ ∀x ¬p:

∀x [∃ y ¬(¬Animal(y) ∨ Loves(x, y))] ∨ [∃ y Loves(y, x)]∀x [∃ y ¬¬Animal(y) ∧ ¬Loves(x, y)] ∨ [∃ y Loves(y, x)]∀x [∃ y Animal(y) ∧ ¬Loves(x, y)] ∨ [∃ y Loves(y, x)]

Chapter 9 41

Page 42: Chapter9 · Chapter9 2. Existential instantiation (EI) For any sentence α, variable v, and constant symbol k that does not appear elsewhere intheknowledgebase:

Conversion to CNF contd.

3. Standardize variables: each quantifier should use a different one

∀x [∃ y Animal(y) ∧ ¬Loves(x, y)] ∨ [∃ z Loves(z, x)]

4. Skolemize: a more general form of existential instantiation.Each existential variable is replaced by a Skolem functionof the enclosing universally quantified variables:

∀x [Animal(F (x)) ∧ ¬Loves(x, F (x))] ∨ Loves(G(x), x)

5. Drop universal quantifiers:

[Animal(F (x)) ∧ ¬Loves(x, F (x))] ∨ Loves(G(x), x)

6. Distribute ∧ over ∨:

[Animal(F (x)) ∨ Loves(G(x), x)] ∧ [¬Loves(x, F (x)) ∨ Loves(G(x), x)]

Chapter 9 42

Page 43: Chapter9 · Chapter9 2. Existential instantiation (EI) For any sentence α, variable v, and constant symbol k that does not appear elsewhere intheknowledgebase:

Resolution proof: definite clauses

American(West)

Missile(M1)

Missile(M1)

Owns(Nono,M1)

Enemy(Nono,America) Enemy(Nono,America)

Criminal(x)Hostile(z)LSells(x,y,z)LWeapon(y)LAmerican(x)L > > > >

Weapon(x)Missile(x)L >

Sells(West,x,Nono)Missile(x)L Owns(Nono,x)L> >

Hostile(x)Enemy(x,America)L >

Sells(West,y,z)LWeapon(y)LAmerican(West)L > > Hostile(z)L>

Sells(West,y,z)LWeapon(y)L > Hostile(z)L>

Sells(West,y,z)L> Hostile(z)L>L Missile(y)

Hostile(z)L>L Sells(West,M1,z)

> > L Hostile(Nono)L Owns(Nono,M1)L Missile(M1)

> L Hostile(Nono)L Owns(Nono,M1)

L Hostile(Nono)

Criminal(West)LChapter 9 43


Appendix A2: Sentence analogy task - Home - Springer10.1007/s11145... · Web viewAppendix A1: Judgment of pseudo-word inflection Singular Plural Correct Incorrect Examples: a. Αυτό

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static-content.springer.com10.1007...  · Web viewwere downloaded after purchasing the right to use them from . Appendix A2: Sentence analogy task. Examples: Correct

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The z Transform - UTKweb.eecs.utk.edu/~mjr/ECE316/PresentationSlides/Chapter9.pdf · n−− ()() () (−)(−) (− n))!

The z Transform - UTKweb.eecs.utk.edu/~mjr/ECE316/PresentationSlides/Chapter9.pdf · n−− ()() () (−)(−) (− n))!

Exegetical Notes for Galatians 3:1-5 - Tony A. Bartolucci Exegesis/Galatians 3.1-5.… · Sentence Structure B. THE THESIS IS PROVEN BY THE GALATIANS’ EXPERIENCE (3:1–5)* vocaddress

Exegetical Notes for Galatians 3:1-5 - Tony A. Bartolucci Exegesis/Galatians 3.1-5.… · Sentence Structure B. THE THESIS IS PROVEN BY THE GALATIANS’ EXPERIENCE (3:1–5)* vocaddress

Mechanics of Materials - University of Pittsburghqiw4/Academic/MEMS1082/Chapter9-3.pdf · 2011. 4. 4. · Chapter 9 . Resistive Sensors 9-3. Department of Mechanical Engineering.

Mechanics of Materials - University of Pittsburghqiw4/Academic/MEMS1082/Chapter9-3.pdf · 2011. 4. 4. · Chapter 9 . Resistive Sensors 9-3. Department of Mechanical Engineering.

Main results. -Presburger sentence alternating quanti erspak/papers/HNP-alphapa.pdf · Theorem 1.2. Let 2Q be a quadratic irrational number, and let r 1. An -PA sentence Swith ralternating

Main results. -Presburger sentence alternating quanti erspak/papers/HNP-alphapa.pdf · Theorem 1.2. Let 2Q be a quadratic irrational number, and let r 1. An -PA sentence Swith ralternating

She has a good knowledge of.. · A link/ linking verb does not add much to the meaning of a sentence. Link verb èπ ¢√éπuç Å®Ωnç O’ü¿ Åçûª v°æ¶μ«´ç Öçúøü¿’.

She has a good knowledge of.. · A link/ linking verb does not add much to the meaning of a sentence. Link verb èπ ¢√éπuç Å®Ωnç O’ü¿ Åçûª v°æ¶μ«´ç Öçúøü¿’.

ARTICULATORY PHONETICS AND PROSODY Phonology framework Browman & Goldstein 1992 and elsewhere • The gesture is the basic phonological unit. • A linguistically relevant, goal oriented

ARTICULATORY PHONETICS AND PROSODY Phonology framework Browman & Goldstein 1992 and elsewhere • The gesture is the basic phonological unit. • A linguistically relevant, goal oriented

The z Transform - UTKweb.eecs.utk.edu/~hli31/ECE316_2015_files/Chapter9.pdf · Existence of the z Transform! The z transform of x[n]=αnun−n [0], α∈ is X(z)=αnun−n [0]z−n

The z Transform - UTKweb.eecs.utk.edu/~hli31/ECE316_2015_files/Chapter9.pdf · Existence of the z Transform! The z transform of x[n]=αnun−n [0], α∈ is X(z)=αnun−n [0]z−n

Acoustofluidic particle trapping, manipulation, and ... · modeled as a loss factor (η = 0.05). Additional details are pro-vided elsewhere.46 FEA of two-dimensional (2D) single-phase

Acoustofluidic particle trapping, manipulation, and ... · modeled as a loss factor (η = 0.05). Additional details are pro-vided elsewhere.46 FEA of two-dimensional (2D) single-phase

J= , the , and charm - Hitoshi Murayamahitoshi.berkeley.edu/129a/cahn-goldhaber/chapter9.pdf · the experiment much more sensitive than an earlier one at Brookhaven led by Leon Lederman.

J= , the , and charm - Hitoshi Murayamahitoshi.berkeley.edu/129a/cahn-goldhaber/chapter9.pdf · the experiment much more sensitive than an earlier one at Brookhaven led by Leon Lederman.

Constructing many atomic models inlaskow/Pubs/RESUBMIT.pdf · order theories (equivalently models of a complete sentence of L! 1;!). Theorem: Let Tbe any complete first-order theory

Constructing many atomic models inlaskow/Pubs/RESUBMIT.pdf · order theories (equivalently models of a complete sentence of L! 1;!). Theorem: Let Tbe any complete first-order theory

Chapter9 - ce.sharif.educe.sharif.edu/courses/96-97/1/ce417-2/resources/root/Slides/chapter… · Chapter9 7. Problems with propositionalization Propositionalization seems to generate

Chapter9 - ce.sharif.educe.sharif.edu/courses/96-97/1/ce417-2/resources/root/Slides/chapter… · Chapter9 7. Problems with propositionalization Propositionalization seems to generate

PROGRAMME - Σύγχρονη Γαλλική Φιλοσοφία Τhe ase of the Sentence ... Jazz Blues Night . ... Plenary Lecture F • Theory of Thinking: Methods of Inquiry ...

PROGRAMME - Σύγχρονη Γαλλική Φιλοσοφία Τhe ase of the Sentence ... Jazz Blues Night . ... Plenary Lecture F • Theory of Thinking: Methods of Inquiry ...

ΚΑΝΟΝΙΚΗ ΚΑΤΑΝΟΜΗjpan/statistiki-skepsi-II/chapter9.pdf · 179 Παρατήρηση: Σε προηγούμενη παρατήρηση σχετική με την ανισότητα

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Inference in First-Order Logic · • Every FOL KB can be propositionalized so as to preserve entailment – A ground sentence is entailed by new KB iff entailed by original KB •

Inference in First-Order Logic · • Every FOL KB can be propositionalized so as to preserve entailment – A ground sentence is entailed by new KB iff entailed by original KB •

The Genesis Manifesto: Story Understanding and Human ... · the Genesis story understanding system and by explaining how Genesis goes about understanding short, 20-to 100-sentence

The Genesis Manifesto: Story Understanding and Human ... · the Genesis story understanding system and by explaining how Genesis goes about understanding short, 20-to 100-sentence