First Order Logic
● Propositional logic represented states of the world– Raining, Sunny, a cookie being sweet
● FOL adds objects, relations, and functions– Objects - kim, my car, the duration of time between
1pm and 2pm today
– Relations – Kim and I are Friends, 1pm-2pm is before 3pm
– Functions – Kim’s father → some dog● Many to 1 mapping from Objects to Objects
FOL Syntax
● Constant Symbols – represent Objects– kim01, truck32, timeSpan7
● N-ary predicate - represent relations/states– 0-ary – states (propositional logic)
– 1-ary – properties (Red(marker1)
– 2-ary – binary relationship (Ate(kim0, apple56))
– n-ary - Buy(kim0,truck45,odie2)
● n-ary functions – some object based on its arguments– Owner(truck45) = kim0
– SteeringWheel(truck32) returns steeringWheel98
Syntax
● Terms – refer to objects– Constants – kim0,
house45
– Variables - ∀ x happy(x)
– Functions - motherOf(kim0)
● Atoms – Single Predicates– Raining
– Dog(kim0)
– kim0 = zorri41
● Universal Quantifier– ∀ x P(x)
– For all x it is true P(x)
● Existential Quantifier– ∃ x Q(x)
– There exists an assignment of x to an object such that Q(x)
Syntax
Semantics
● Domain – The (non empty) set of objects we are trying to model
● Interpretation of a domain, d, )– links (�constants, predicates, and functions to the possible world the model represents– c C | (C): C→ D∈ �– f Functions | (f) D x D...x D → D∈ � ∈– p Predicates | (p) D x D...x D→ {true, false}∈ � ∈
Semantics
Quantifier
● ∀ x P(x):– For every assignment of x to an object in our
domain it is true P(x)
– P(a) P(b) … P(z)∧ ∧
● ∃ x P(x):– For some assignment of x to an object in our
domain it is true P(x)
– P(a) P(b) … P(z)∨ ∨
FOL De Morgan
Knowledge Representation
● Types and properties– dog(kim0)
– red(ball56)
● Relationships– friendOf(kim0,odie2)
● Functions– birthMotherOf(kim0)
● Quantification– Everything that has a property also has some other property
● ∀ x. dog(x) → animal(x)
– Something has a property● ∃ x. dog(x) red(x)∧
– Be careful of mixing existential with implications● ∃ x. dog(x) → red(x) ≡ x. ¬dog(x) red(x)∃ ∨
Knowledge Representation
● All Cookies are Sweet● All Fudge cookies are Yummy● All Dark Cookies are either Burnt or Fudge● Sweet Cookies are not Burnt● Bread is either sweet or savory● Onion Bread is savory● There is no such thing as a bread that is both sweet and savory.● This Bread is Burnt.● The Baker of this Bread is bad at their job.
Knowledge Representation
● Kim is both a Golden Retriever and a Labrador Retriever● Kim’s birth mother is either a Golden or Lab● Kim and Odie are friends● Some dogs have a ball● All dogs have a tail● Kim is a black dog with a brown tail● All Golden Retrievers are red except for Kim.● All dogs love some person● Every person who owns a dog loves it● Kim has two toys
Resolution in FOL
Similar to propositional logic but we have to account for Terms and Quantifiers– Constants – kim0, house45
– Variables - ∀ x happy(x)
– Functions - motherOf(kim0)
● Skolemization● Unification
FOL CNF
● Start just like Propositional CNF– Eliminate bi-conditionals, implication, move negation
inward, distribute over ∨ ∧
● Treat universally quantified variables as free variables – Give variables unique names and remove the quantifier, :∀– [ x. p(x) ] [ x. f(x) , ¬p(x)]∀ ∧ ∀– [ x∀ 1. p(x1) ] [ x∧ ∀ 2. f(x2), ¬p(x2) ]
– [ p(x1) ] [ f(x∧ 2), ¬p(x2) ]
● Handle with Skolemize∃
Unification
● θ is a set of variable substitutions <variable>/<term>– e.g. θ = {v/w, x/Kim01, y/motherOf(z)}
– Combined with a sentence they create a new sentence
– f(x,y) Substituted with {y/Kim} = f(x,Kim)
● Two sentences unify if we can find a substitution of variables that make both look syntactically identical to each other– Unify( f(x,y), f(Kim,Kim) )?
– Unify( f(x,y), f(Kelly,z) )?
– Unify( f(Kim,y), f(Kelly,x) )?
Skolemization
● Remove Existential Instantiation∃– Introduce a single new literal – Skolem constant
– ∃ y. q(y) {y/sk1}
– We name that y sk1
– q(sk1)
Nested ∃
● Introduce a function that points to a new literal ● Arguments contextualize the literal
– ∀ x y. r1(x, y)∃
● r1(x, skF1(x))– ∀ x,z y q(y)∃
● q(skF2(x,z))
Unification
● Unify( f(w,x), f(y,z)) – {w/y,z/x} or {w/x, y/x, z/x} or {w/y, x/Kim, z/Kim} or {w/Kim,
x/Kelley, y/Kim,z/Kelley}?
● Among a set of valid unifiers for two sentences, a Most General Unifier produces a sentence that can unify with every other sentence produced by the other possible substitutions– f(y,x): {w/y,z/x}
– f(y,Kim): {w/y, x/Kim, z/Kim}
– f(Kim,Kelley): {w/Kim, x/Kelley, y/Kim,z/Kelley}
“An angry man bought a truck”
∃ x, y Man(x) Angry(x) Truck(y) Buy(x,y)∧ ∧ ∧
CNF
FOL Resolution
KB:Buy(x1,y1) → Acquire(x1,y1)
Woman(x2) → Person(x2)
Man(x3) → Person(x3)
Truck(x4) → Vehicle(x4)
Man(SK1)
Angry(SK1)
Truck(SK2)
Buy(SK1,SK2)
● Did a person buy a vehicle?
● Prove:
KB ¬( x y ∧ ∃Person(x) ∧Vehicle(y) Buy(x,y)) ∧
is unsatisfiable
FOL Resolution
KB:Buy(x,y) → Acquire(x,y)
Man(x) → Person(x)
Man(Mike)
Truck(SK1)
Buy(Mike,SK1)
Person(Kelly)
Truck(SK2)
Buy(Kelly,SK2)
● What person bought a truck?
● Prove:
KB ¬( x y Person(x) ∧ ∃ ∧Truck(y) Buy(x,y)) ∧is unsatisfiable
● Add ANS(x) to query clause
● Resolution stops when ANS(x) is the only literal left
FOL Resolution
KB:Buy(x1,y1) → Acquire(x1,y1)
Woman(x2) → Person(x2)
Man(x3) → Person(x3)
Truck(x4) → Vehicle(x4)
Man(SK1)
Angry(SK1)
Truck(SK2)
Buy(SK1,SK2)
● Did every person buy a truck?
● Prove:
KB ¬( x y ∧ ∀ ∃Person(x) → Truck(y)
Buy(x,y)) ∧is unsatisfiable
See R&N p 350 for a discussion on the completeness of FOL resolution
FOL Resolution
KB:Ancestor(mom(Kim),Kim)
∀ x Ancestor(x, Kim) →Ancestor(mom(x),Kim)
See R&N p 350 for a discussion on the completeness of FOL resolution
Representing Situations
● Often in natural language understanding we are faced with designing predicates to represent situations– “Kim bought a car”
– “Kim bought a car from Odie for $100”
● Verbs are predicates, roles related to those verbs are arguments– Buy(<buyer>,<item>)
– bought(kim0, car3)
● Naively, we creating a predicate dealing with with every conceivable combination of arguments– bought(kim0, car3, odie2, $100)
Representing Situations
● In linguistics we can characterize situations compositionally – e.g. VerbNet or Framenet
● Reify situations into objects with role relations instead of arguments
● Represent situations more concisely and deal with composition more elegantly
● Instead we can represent every event as an object and treat each role as a relationship
buy(e1) ∧buyer(e1,kim0) seller(e1,odie2) ∧ ∧ money(e1, $100)
Special Assumptions
● Unique name– Literals map to objects 1:1 (not many:1)– e.g. we cannot say: kim0 = zori41
● Closed-world– Any atomic sentences not known (i.e. an axiom or proven to be entailed) to be true
are false– Proof by failure – a sentence is false if we fail to prove it
– KB = {Dog(kim), Cat(kelly)} ¬Cat(kim), ¬Dog(kelly), kim != kelley⊢
● Domain closure– The only objects in the Domain are those named by constant symbols (much easier
to talk about than what we’ve been doing)
Database Semantics: closed-world, unique-names, domain closure
Horn Clauses
● Horn Clause - disjunction of literals; at most 1 literal is positive
● Definite Clause – exactly 1 is positive● Fact – a definite clause with 0 negative literals● Goal Clause – none are positive
Horn Clauses
● P ¬Q∨
P ← Q
P :- Q● Head ← Body● Head - positive literal● Body – clause of negative literals
Horn Clauses
● If it’s raining then the grass is wet – W ← R (Definite Clause)
● It is raining– R (Fact)
● Prove that the grass is wet– Prove KB W⊨– Prove KB ¬W is unsatisfiable∧– ¬W (Goal Clause)
Logic Programming
● Prolog – a declarative programming language based on logic– Describe a problem in horn logic
– Control flow is handled by resolution
● Uses an efficient resolution refutation algorithm– SLD resolution
Blocks World
● Assume we are using Database Semantics – Closed-world, unique-names, domain closure
● n blocks, 1 table● A block can be on top of another block or on the
table● At most 1 block can be on top of another● A block can only be picked up iff there isn’t a
another block on top of it
Blocks World with Actions
● n blocks ● A block can be on top of another block or on a
table
● A block can only be picked up iff there isn’t a another block on top of it
● If a block can be picked up and moved to the table or on top of another block
Actions and Time
● Fluent – predicates who’s truth value depends on time– on(b1,b2,t1) : b1 is on b2 at time t1
● In a simple case, time is treated as discrete steps that occur one after the other.– on(greenB,blueB, 1) on(∧ greenB,blueB, 2)
● Actions:– Action(t) → Precondition(t) PostCondition(t+1)∧– ∀b1,b2,t. move(b1,b2, table, t) →
( ¬ On(b1,table,t+1) On(∧ b1,b2, t) ) ∧ ( On(b1,table,t+1) ¬On(∧ b1,b2, t+1) )
– Don’t forget to say what doesn’t change! ● ∀b1,b2,t. move(b1,b2, table, t) →
On(b1,table,t+1) ¬On(∧ b1,b2, t+1) ∧ x,y. x !=∀ b1 →( On(x,y,t) ↔ On(x,y,t+1) )● Dog(kim,t+1) ?
Planning with Logic
● PDDL – Planning Domain Definition Language● Database Semantics – Closed-world, unique-names,Domain
closure● Each state is a set of logical statements (factored representation)
– No need for temporal arguments – time is signified by our current state in the graph
● Actions change states ● Preconditions tell us what must be true to perform an action● Effects tell us what changed
– Everything else remains the same
– Circumvents the frame problem
