Department of Computer Science

1
Department of Computer Science Engineering
University of California, San DiegoCSE-291
Ontologies in Data IntegrationSpring 2004

• Bertram Ludäscher
• LUDAESCH_at_SDSC.EDU

2
Introduction to Reasoning inFirst-order
Predicate Logicand Description Logic(s)

• Introduction to FO (aka PL)
• Syntax, Semantics
• Two decidable fragments
• propositional logic
• description logic(s)
• Reasoning w/ Tableaux
• Description Logic Reasoning
• Discussion of Topics / Assignments

3
Syntax vs Semantics

• Syntax
• a.k.a. formation rules grammar
• prescribes what a well-formed formula is
  (syntactically)
• Semantics
• the meaning of well-formed formulas
• defined via a mapping called interpretation

4
Propositional Logic Syntax
propositional logic ltlogicgt (or "propositional
calculus") A system of symbolic logic using
symbols to stand for whole propositions and
logical connectives. Propositional logic only
considers whether a proposition is true or false.
In contrast to predicate logic, it does not
consider the internal structure of propositions.
http//wombat.doc.ic.ac.uk/foldoc/foldoc.cgi?propo
sitionallogic

• Logical symbols
• conjunction ?, disjunction ?, negation ?,
• implication ?, equivalence ?, parentheses ? ?
• Non-logical symbols
• propositional variables p, q, r, ...
• signature set of propositional variables ? p,
  q, r, ...
• Formation rules for well-formed formulas (wff)
• an atomic formula (propositional variable) is a
  formula
• if F, G are formulas, so are
• F?G, F ? G, ? F, F?G , F?G, ? F ?

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Propositional Logic Semantics

• Propositions can be assigned a truth-value
• either true or false (classical 2-valued logic
  tertium non datur)
• other propositional logics exist 3-valued,
  4-valued, temporal, (modal logics), , fuzzy
  logic
• An interpretation I over a signature ? is a
  mapping
• I ? ? true, false , associating a truth
  value to every propositional variable
• Truth tables describe how to extend I from atomic
  to composite formulas (Boolean Algebra)
• F?G, F ? G, ? F, F?G , F?G

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Boolean Algebra, Truth Tables
http//wombat.doc.ic.ac.uk/foldoc/foldoc.cgi?two-v
aluedlogic
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Different Logical Bases

• Often
• ?, ? , ?
• Alternatively
• ?, ?
• ? , ?
• NAND
• NOR
• XOR
• What about ite(A,B,C) if A then B else C ?

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Reasoning in Propositional Logic

• A formula F is
• valid if it is true for all interpretations I
• satisfiable if it is true for some interpretation
  I
• unsatisfiable if it is true for no interpretation
  I
• Try these
• p ? q
• p ? ?p
• p ? ?p
• p ? p
• p ? ?p
• ? p ? p

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Reasoning in Propositional Logic

• Def. models relationship
• If a formula F evaluates to true for an
  interpretation I then I is called a model of F
  written I F
• I is a model of F1,, Fk, written I F1,,
  Fk,if I is a model of each Fj
• Automated deduction setting
• Show that A1,,, An (axioms) imply T (theorem),
  that is, every model of the axioms is also a
  model of the theorem
• That is if I A1,,, An then I T
• Short A1,,, An T
• Often Show that A1 ? ? An ? ?T is
  unsatisfiable
• We need a procedure / reasoning algorithm
• Predicate Calculus (in fact calculi resolution,
  tableaux, )

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Example

• p, p ? q q
• Truth table
• Resolution
• Tableaux

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Example Reasoning with Binary Decision
Trees(see also Binary Decision Diagrams, or
BDDs)
? B
A ? B
A
A
A
A
if-false
if-true
0
1
0
1
B
0
B
false
true
0
1
0
1
? A
? B
A ? B
A
A
A
if-false
if-true
1
0
0
1
B
B
1
false
true
0
1
0
1
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Syntax of First-Order Logic (FO)

• Logical symbols
• ?, ?, ?, ?, ?, ? ?, ? (for all), ?
  (exists), ...
• Non-logical symbols A FO signature ? consists of
• constant symbols a,b,c, ...
• function symbols f, g, ...
• predicate (relation) symbols p,q,r, ....
• function and predicate symbols have an associated
  arity
• we can write, e.g., p/3, f/2 to denote the
  ternary predicate p and the function f with two
  arguments
• First-order variables x, y, ...
• Formation rules for terms
• constants and variables are terms
• if t1,,tk are terms and f is a k-ary function
  symbols then f(t1,...,tk) is a term

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Syntax of First-Order Logic (FO)

• Formation rules for formulas
• if t1,, tk are terms and p/k is a predicate
  symbol (of arity k) then p(t1, , tk) is an
  atomic formula (short atom)
• all variable occurrences in p(t1, , tk) are free
• if F,G are formulas and x is a variable, then
  the following are formulas
• F?G, F ? G, ? F, F?G , F?G, ? F ?,
• ?x F (for all x F(x,...) is true)
• ?x F (there exists x such that F(x,...) is
  true)
• the occurrences of a variable x within the scope
  of a quantifier are called bound occurrences.

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Examples

• ?x man(x) ? person(x).
• man(bill).
• child(marriage(bill,hillary),chelsea).
• Variable x
• Constants (0-ary function symbols) bill/0,
  hillary/0, chelsea/0
• Function symbols marriage/2
• Predicate symbols man/1, person/1, child/2

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Semantics of Predicate Logic

• Let D be a non-empty domain (a.k.a. universe of
  discourse). A structure is a pair I (D,I), with
  an interpretation I that maps ...
• each constant symbols c to an element I(c)? D
• each predicate symbol p/k to a k-ary relation
  I(p) ? Dk,
• each function symbol f/k to a k-ary function
  I(f) Dk?D
• Let I be a structure, ? Vars ? D a variable
  assignment. A valuation valI,? maps Term? to D
  and Fml? to true, false
• valI,? (x) ? (x) for x ? Vars
• valI,? (f(t1,...,tk)) I(f)( valI,? (t1),...,
  valI,? (tk) ) for f(t1,...,tk) ?
  Term?
• valI,? (p(t1,...,tk)) I(p)( valI,? (t1),...,
  valI,? (tk) ) for p(t1,...,tk) ? At?
• valI,? (F ? G) valI,? (F) and valI,? (G) are
  true for F,G ?Fml?
• valI,? (F ? G) valI,? (F) or valI,? (G) is
  true for F,G ?Fml?
• valI,? (? F) true (false) if valI,? (F) is
  false (true) for F?Fml?
• valI,? (? x F) valI,?x/t (F) is true for some
  t ? D for F?Fml?
• valI,? (? x F) valI,?x/t (F) is true for all
  t ? D for F?Fml?

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Example

• Formula F ?x man(x) ? person(x).
• Domain D b, h, c, d, e
• Lets pick an interpretation I
• I(bill) b, I(hillary) h, I(chelsea) c
• I(person) b, h, c
• I(man) b
• Under this I, the formula F evaluates to true.
• If we choose I like I but I(man) b,d, then
  F evaluates to false
• Thus, I is a model of F, while I is not
• I F I / F

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FO Semantics (contd)

• F entails G (G is a logical consequence of F) if
  every model of F is also a model of G F
  G
• F is consistent or satisfiable if it has at least
  one model
• F is valid or a tautology if every interpretation
  of F is a model
• Proof Theory
• Let F,G, ... be FO sentences (no free variables).
  
• Then the following are equivalent
• F_1, ..., F_k G
• F_1 ? ... ? F_k ? G is valid
• F_1 ? ... ? F_k ? ? G is unsatisfiable
  (inconsistent)

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Proof Theory

• A calculus is formal proof system to establish
• F1,, Fk T
• via formal (syntactic) derivations
• F1,, Fk ... T, where the denotes
  allowed proof steps
• Examples
• Hilbert Calculus, Gentzen Calculus, Tableaux
  Calculus, Natural Deduction, Resolution, ...
• First-order logic is semi-decidable
• the set of valid sentences is recursively
  enumerable, but not recursive (decidable)
• Some inference engines
• http//www.semanticweb.org/inference.html

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Querying vs. Reasoning

• Querying
• given a DB instance I ( logic interpretation),
  evaluate a query expression (e.g. SQL, FO
  formula, Prolog program, ...)
• boolean query check if I ? (i.e.,
  if I is a model of ?)
• (ternary) query (X, Y, Z) I ?
  (X,Y,Z)
• gt check happyFathers in a given database
• Reasoning
• check if I ? implies I ? for all
  databases I,
• i.e., if ? gt ?
• undecidable for FO, F-logic, etc.
• Descriptions Logics are decidable fragments
• concept subsumption, concept hierarchy,
  classification
• semantic tableaux, resolution, specialized
  algorithms

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Reasoning Example

• (1) p(0)
• (2) ?x p(x) ? p(s(x))
• (3) p(s(s(0))).
• We want to show that (1) ... (2) implies (3)
• Approach assume negation of (3) and show that it
  leads to a contradiction with (1), (2)
• Question Why is this sound?

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Types of Formulas
(?) rule for F A ? B (and other disjunctions)
(?) rule for F A ? B (and other conjunctions)
(?) rule for F ?x A(X,...) substitute a
?-variable X with an arbitrary term t
(?) rules for F ?x A(X,...) substitute a
?-variable X with a new constant c
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(Semantic) Tableaux Rules

• (?) rule for F A ? B
• (?) rule for F A ? B
• (?) rule for F ?x A(X,...)
• substitute a ?-variable X with an arbitrary term
  t
• (?) rules for F ?x A(X,...)
• substitute a ?-variable X with a new constant c

• A branch is closed if it contains complementary
  formulas
• A tableaux is closed if every branch is closed

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FO Tableaux Calculus

• Theorem (Soundness, Completeness of Tableaux
  calculus)
• Let A1,..., Ak and F be first-order logic
  sentences.
• (Recall a sentence is a closed formula, i.e.,
  has no free variables)
• Then the following are equivalent
• A1, ..., Ak F
• A1 ? ... ? Ak ? F is unsatisfiable (inconsistent)
• There is a closed tableaux for A1, ..., Ak , ?
  F

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Reasoning with DLs(Shawn Bowers)