RDM Chapter 2 : Introduction to Logic

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RDM Chapter 2 Introduction to Logic

• prepared for COMP422/522-2008, Bernhard
  Pfahringer

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2.1 A relational DB example

• citation database (like CiteSeer)
• authorOf(lloyd,logic_for_learning) lt-
• reference(logic_for_learning,foundations_of_lp)
  lt-
• Predicates authorOf/2 pname/arity
• Constants start with lowercase letter
• Facts p(t1,..,tn) lt- p/n name, ti terms,
  facts are also called tupels

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Example 2.2

• query
• lt- authorOf(lloyd,logic_for_learning)
• yes/no question
• lt- authorOf(lloyd,Pub)
• answer substitutions, which make the query
  true Pub/logic_for_learning, but also
  Pub/foundations_of_lp

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Logical queries vs. SQL

• lt- authorOf(lloyd,Pub)
• SELECT Pub FROM authorOf WHERE Author lloyd
• lt- authorOf(A,logic_for_learning), authorOf(A,
  foundations_of_lp)
• SELECT A FROM authorOf t1, authorOf t2 WHERE
  t1.Author t2.Author AND t1.Pub
  logic_for_learning AND t2.Pub
  foundations_of_lp

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More examples

• who cites who
• lt- authorOf(A,P),reference(P,Q),authorOf(B,Q)
• turn into new predicate/clause
• cites(A,B) lt- authorOf(A,P),reference(P,Q),authorO
  f(B,Q)
• similar to a view (intensional relation) in SQL
• CREATE VIEW cites AS SELECT a1.Author, a2.Author
  FROM authorOf a1, autherOf a2, reference r WHERE
  a1.Pub r.Pub1 AND a2.Pub r.Pub2 (try
  Exercise 2.3)

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2.2 Syntax of Clausal Logic

• term can be a
• constant, like John
• variable, like Pub
• compound term, like card(queen,hearts) can be
  nested succ(succ(0)), , short cut for lists
  1,2,3 is actually cons(1,cons(2,cons(3,nil)))
• Atom p(t1,..,tn), p/n is a predicate name and ti
  are terms
• Ground atoms variable-free, are true or false
  pair(card(queen,hearts),card(queen,spades))

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A clause

• h1 .. hn lt- b1, .. , bm where all hi and bj
  are atoms
• means disjuntion / or
• , means conjunction / and
• lt- means implication
• variables are universally quantified / for all
• female(X) male(X) lt- human(X)

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Set notation

• h1, ,hn, b1, , m or
• h1 OR .. OR hn OR b1 OR .. OR bm
• facts n 1, m 0
• definite clause n 1
• Horn clause n 1 or n 0
• denial (query) n 0

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more notation

• a set of clauses also called
• a (clausal) theory
• a database (all ground facts ?)
• a knowledge base
• Substitution theta ? V1/t1, .. assigns term
  t1 to variable V1
• card(X,Y)?
• with ? X/queen,Y/hearts
• denotes card(queen,hearts)

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2.3 Semantics - Model Theory

• Interpretation uses a domain (set of objects
  that exist), maps
• constants gt objects
• functions f/n gt objects
• predicates p/n gt n-tupels of objects
• e.g. john gt John Doe Jr.
• fatherOf(john) gt John Doe Sr.
• parent(fatherOf(john),john) lt- gt true (provided
  John Doe Sr. is really the father of John Doe
  Jr.)

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Herbrand interpretation

• Herbrand domain set of all ground terms
  constructed from all constants and functions
  occuring in any clause
• Citation DB only constants lloyd,
  foundations_of_lp,
• natural number theory
• nat(0) lt-
• nat(succ(X)) lt- nat(X)
• Herbrand domain 0,succ(0), succ(succ(0)), ,
  i.e. is infinite
• Herbrand interpretation maps every ground term
  onto itself

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Definition 2.7. Herbrand interpretation

• Herbrand interpretation of a set of clauses is a
  set of ground atoms over the constants,
  functions, and predicate symbols occuring in
  these clauses
• e.g. for natural numbers
• I3
• I4 nat(0),nat(succ(0))
• I5 nat(0),nat(succ(0)), nat(succ(succ(0))),
  

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Def 2.10. Herbrand model

• A Herbrand interpretation I is a Herbrand model
  for a set of clauses C if and only if for all
  clauses h1hn lt- bi,,bm in C and for all
  ground substitutions ? b1?, , bm? in I -gt
  h1?,,hn? ? I ?
• I.e. if the preconditions of any rule are
  satisfied, then at least one of its consequences
  must also be part of the interpretation
• cf. Examples 2.11 and 2.12

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Def 2.13. Logical Entailment based on model theory

• A set of clauses C logically entails another
  clause c (C c), if and only if all models of C
  are also models of c.
• Theorem 2.14. A set of clauses C has a model if
  and only if C has a Herbrand model
• Theorem 2.15. C c if and only if C /\ c is
  unsatisfiable, I.e. C /\ c
• cf. Examples 2.16, 2.17

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Range-restriced clauses

• A clause is range-restricted if all variables in
  the head also appear in its body. This also
  implies that all facts are ground (because they
  have no body, so cannot have any variables)
• cites(A,B) lt- authorOf(A,P),reference(P,Q),authorO
  f(B,Q)
• cites(A,B,X) lt- authorOf(A,P),reference(P,Q),autho
  rOf(B,Q)

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Algorithm 2.1. Generating models

• M
• while M is not a model of C do
• if there is a denial lt- b1,,bm that is violated
  by M then backtrack
• select a clause where all body atoms are
  satisfied (are in M) but none of the consequences
  is (are not in M), and non-deterministically add
  one of the consequences to M (choice-point)

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Properties

• can fail if no model exists
• can loop infinitely for infinite models
• cf Examples 2.18 - 2.20, exercise 2.21
• there may be multiple models, and some may be
  subsets of others
• a model is minimal if no subset is a model

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Least Herbrand model

• for definite clauses the minimal model is unique,
  called least Herbrand model, denoted as M(C)
• Property 2.24 If C is a set of definite clauses
  and f some ground fact, then C f if and only
  if f is in M(C)
• can improve Algorithm 2.1

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Algorithm 2.2 Forward chaining

• no backtracking
• M1 f f lt- is a fact in C
• I 1
• while Mi ! Mi-1 do
• Mi1 Mi
• for all h lt- b1,,bn in C do
• for all ? such that b1?,,bn? ? Mi do
• add h? to Mi1
• i i1
• cf. Example 2.25, exercise 2.26

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Inference - Proof theory

• Resolution, Robinson (1965) was a break-through
  in logic,
• here specifically Horn clauses and
  SLD-resolution
• propositional (simpler, cf. Ex.2.27)
• l lt- b1,,bn and
• h lt- c1,ci-1,l,ci,,cm
• THEN hlt- c1,ci-1,b1,,bm,ci,,cm

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Resolution proofs

• Resolution is sound, I.e. c1 /\ c2 -res c then
  also c1 /\ c2 c
• (cf. Example 2.28, Fig 2.2., 2.3.)
• Resolution is NOT complete (see 2.29) but it is
  refutation-complete
• for Horn clauses C hlt-b1,,bn
• negate c into T lt-h, b1lt-,,bnlt-
• derive from C /\ T, I.e. C /\ T -

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First-order terms Unification

• father(X,Y) lt- parent(X,Y), male(X)
• lt- father(luc,maarten)
• s X/luc,Y/marteen
• lt- parent(luc,maarten),male(luc)
• cf. Examples 2.32
• unifiers do not always exist
• most general unifier (def 2.33, example 2.34,
  algorithm 2.3)

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First-order resolution

• l lt- b1,,bn and
• h lt- c1,ci-1,l,ci,,cm
• AND ? mgu(l,l)
• THEN h? lt- c1?,ci-1?, b1?, , bm?, ci?,
  ,cm?
• can still do refutation proofs, but need to
  introduce skolem-constants

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Decidability

• propositional logic decidable
• first-order semi-decidable, I.e. may not
  terminate if C / c
• but if we only allow constants and variables (no
  function symbols), also called relational
  definite clause logic or Datalog, then this is
  decidable!