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First-Order Logic
CSE 573
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© Daniel S. Weld 2
Logic-Based KR
Propositional logicSyntax (CNF, Horn clauses, …)Semantics (Truth Tables)Inference (FC, Resolution, DPLL, WalkSAT)Restricted Subsets
First-order logic Syntax (quantifiers, skolem functions, …Semantics (Interpretations)Inference (FC, Resolution, Compilation)Restricted Subsets (e.g. Frame Systems)
Representing events, action & change
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Propositional. Logic vs. First Order
Ontology
Syntax
Semantics
Inference Algorithm
Complexity
Objects, Properties, Relations
Atomic sentencesConnectives
Variables & quantificationSentences have structure: termsfather-of(mother-of(X)))
UnificationForward, Backward chaining Prolog, theorem proving
DPLL, GSATFast in practice
Semi-decidableNP-Complete
Facts (P, Q)
Interpretations (Much more complicated)Truth Tables
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FOL Definitions• Constants: a,b, dog33.
Name a specific object. • Variables: X, Y.
Refer to an object without naming it.• Functions: dad-of
Mapping from objects to objects.• Terms: dad-of(dog33)
Refer to objects• Atomic Sentences: in(dad-of(dog33), food6)
Can be true or false Correspond to propositional symbols P, Q
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More Definitions• Logical connectives: and, or, not, =>• Quantifiers:
Forall There exists
• Examples Dumbo is grey
Elephants are grey
There is a grey elephant
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Quantifier / Connective Interaction
1. x E(x) G(x)
2. x E(x) G(x)
3. x E(x) G(x)
4. x E(x) G(x)
E(x) == “x is an elephant”G(x) == “x has the color grey”
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Nested Quantifiers: Order matters!
• Examples Every dog has a tail
x y P(x,y) y x P(x,y)
d t has(d,t)
Someone is loved by everyone
t d has(d,t)?
x y loves(y, x)
Every dog shares a tail!
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Semantics• Syntax: a description of the legal
arrangements of symbols (Def “sentences”)
• Semantics: what the arrangement of symbols means in the world
Sentences
ModelsModels
Sentences
Representation
World
Semantics
Semantics
Inference
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Propositional Logic: SEMANTICS
• “Interpretation” (or “possible world”)• Specifically, TRUTH TABLES
Assignment to each variable either T or F Assignment of T or F to each connective
PT
T
F
F
Q
P Q
T
F F
F
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Models• Depiction of one possible “real-world” model
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Interpretations=Mappingssyntactic tokens model
elementsDepiction of one possible interpretation, assuming Constants: Functions: Relations:
Richard John Leg(p,l) On(x,y) King(p)
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Interpretations=Mappingssyntactic tokens model
elementsAnother interpretation, same assumptions Constants: Functions: Relations:
Richard John Leg(p,l) On(x,y) King(p)
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Satisfiability, Validity, & Entailment
•S is valid if it is true in all interpretations
•S is satisfiable if it is true in some interp
•S is unsatisfiable if it is false all interps
•S1 entails S2 if forall interps where S1 is true, S2 is also true
|=
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Skolemization• Existential quantifiers aren’t necessary!
Existential variables can be replaced by• Skolem functions (or constants)• Args to function are all surrounding vars
d t has(d, t)
x y loves(y, x)
d has(d, f(d) )
y loves(y, f() )
y loves(y, f97 )
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FOL Reasoning
• FO Forward & Backward Chaining • FO Resolution• Many other types of theorem proving• Restricted representations
Description logics Horn Clauses
• Compilation to SAT
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Forward Chaining • Given
?x lifeform(?x) => mortal(?x)?x mammal(?x) => lifeform(?x)?x dog(?x) => mammal(?x) dog(fido)
• Prove mortal(fido)
?x dog(?x) => mammal(?x)dog(fido)mammal(fido) ?
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Unification• Emphasize variables with ?• Useful for FO inference (modus ponens, …)
Also for compilation of FOPC -> propositional
• Unify(, ) returns “mgu” Unify(city(?a), city(kent)) returns ?a/kent
• Substitute(expr, mapping) returns new expr Substitute(connected(?a, ?b), {?a/kent})returns connected(kent, ?b)
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Unification Examples
• Unify(road(?a, kent), road(seattle, ?b))
• Unify(road(?a, ?a), road(seattle, kent))
• Unify(f(g(?x, dog), ?y)), f(g(cat, ?y), dog)
• Unify(f(g(?x)), f(?x))
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Compilation to Prop. Logic I
• Typed Logiccity a,b connected(a,b)
• Universe Cities: seattle, tacoma, enumclaw
• Equivalent propositional formula:
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Compilation to Prop. Logic II
•Universe•Cities: seattle, tacoma, enumclaw•Firms: IBM, Microsoft, Boeing
•First-Order formulacity c firm f hasHQ(c, f)
•Equivalent propositional formula
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Hey!
• You said FO Inference is semi-decidable• But you compiled it to SAT
Which is NP Complete • So now we can always do the inference?!?
Tho it might take exponential time…
• Something seems wrong here….????
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Restricted Forms of FO Logic
• Known, Finite Universes Compile to SAT
• Frame Systems Ban certain types of expressions
• Horn Clauses Aka Prolog
• Function-Free Horn Clauses Aka Datalog