Logics for Data and Knowledge Representation

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Logics for Data and KnowledgeRepresentation

• Propositional Logic

Originally by Alessandro Agostini and Fausto
Giunchiglia Modified by Fausto Giunchiglia, Rui
Zhang and Vincenzo Maltese
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Outline

• Syntax
• Semantics
• Entailment and logical implication
• Reasoning Services

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Logical Modeling
Language L
Theory T
Data Knowledge
Interpretation
Modeling
Entailment
World
Mental Model
?
I
Realization
Domain D
Model M
Meaning
SEMANTIC GAP
NOTE the key point is that in logical modeling
we have formal semantics
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Language (Syntax)
SYNTAX SEMANTICS ENTAILMENT AND LOGICAL
IMPLICATION REASONING SERVICES

• The first step in setting up a formal language is
  to list the symbols of the alphabet

• Auxiliary symbols parentheses ( )
• Defined symbols
• ? (falsehood symbol, false, bottom) ? df P?P
• T (truth symbol, true, top) T df ?

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Formation Rules (FR) well formed formulas
SYNTAX SEMANTICS ENTAILMENT AND LOGICAL
IMPLICATION REASONING SERVICES

• Well formed formulas (wff) in PL can be described
  by the following BNF grammar (codifying the
  rules)
• ltAtomic Formulagt A B ... P Q ...
  ? ?
• ltwffgt ltAtomic Formulagt ltwffgt ltwffgt?
  ltwffgt ltwffgt ? ltwffgt
• Atomic formulas are also called atomic
  propositions
• Wff are propositional formulas (or just
  propositions)
• A formula is correct if and only if it is a wff
• S0 FR define a propositional language

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Propositional Theory
SYNTAX SEMANTICS ENTAILMENT AND LOGICAL
IMPLICATION REASONING SERVICES

• Propositional (or sentential) theory
• A set of propositions
• It is a (propositional) knowledge base (true
  facts)
• It corresponds to a TBox (terminology) only,
  where no meaning is specified yet it is a
  syntactic notion

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Semantics formal model
SYNTAX SEMANTICS ENTAILMENT AND LOGICAL
IMPLICATION REASONING SERVICES

• Intensional interpretation
• We must make sure to assign the formal meanings
  out of our intended interpretation to the
  (symbols of the) language, so that formulas
  (propositions) really express what we intended.
• The mental model What we have in mind?
• In our mind (mental model) we have a set of
  properties that we associate to propositions. We
  need to make explicit (as much as possible) what
  we mean.
• The formal model
• This is done by defining a formal model M.
  Technically we have to define a pair (M,?) for
  our propositional language
• Truth-values
• In PL a sentence A is true (false) iff A denotes
  a formal object which satisfies (does not
  satisfy) the properties of the object in the real
  world.

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Truth-values
SYNTAX SEMANTICS ENTAILMENT AND LOGICAL
IMPLICATION REASONING SERVICES

• Definition a truth valuation on a propositional
  language L is a mapping ? assigning to each
  formula A of L a truth value ?(A), namely in the
  domain D T, F
• ?(A) T or F according to the modeler, with A
  atomic
• ?(A) T iff ?(A) F
• ?(A?B) T iff ?(A) T and ?(B) T
• ?(A?B) T iff ?(A) T or ?(B) T
• ?(?) F (since ?df P?P)
• ?(?) T (since ?df ?)

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Truth Relation (Satisfaction Relation)
SYNTAX SEMANTICS ENTAILMENT AND LOGICAL
IMPLICATION REASONING SERVICES

• Let ? be a truth valuation on language L, we
  define the truth-relation (or satisfaction-relatio
  n) ? and write
• ? ? A
• (read ? satisfies A) iff ?(A) True
• Given a set of propositions G, we define
• ? ? G
• iff if ? ? ? for all formulas ? ? G

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Model, Satisfiability, truth and validity
SYNTAX SEMANTICS ENTAILMENT AND LOGICAL
IMPLICATION REASONING SERVICES

• Let ? be a truth valuation on language L.
• ? is a model of a proposition P (set of
  propositions G) iff ? satisfies P (G).
• P (G) is satisfiable if there is some (at least
  one) truth valuation ? such that ? ? P (? ? G).
• Let ? be a truth valuation on language L.
• P is true under ? if ? ? P
• P is valid if ? ? P for all ? (notation ? P).
• P is called a tautology

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Entailment and implication
SYNTAX SEMANTICS ENTAILMENT AND LOGICAL
IMPLICATION REASONING SERVICES

• Propositional entailment ? ? ?
• where ? ?1, ..., ?n is a finite set of
  propositions
• ? ? ?i for all ?i in ? implies ? ? ?
• Entailment can be seen as the logical implication
• (?1 ? ?2 ? ... ? ?n) ? ?
• to be read ?1 ? ?2 ? ... ? ?n logically implies
  ?
• ? is a new symbol that we add to the language

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Implication and equivalence
SYNTAX SEMANTICS ENTAILMENT AND LOGICAL
IMPLICATION REASONING SERVICES

• We extend our alphabet of symbols with the
  following defined logical constants?
  (implication)? (double implication or
  equivalence)
• ltAtomic Formulagt A B ... P Q ...
  ? ?
• ltwffgt ltAtomic Formulagt ltwffgt ltwffgt?
  ltwffgt ltwffgt? ltwffgt
• ltwffgt ? ltwffgt ltwffgt ? ltwffgt
  (new rules)
• Let propositions ?, ?, and finite set ?1,...,?n
  of propositions be given. We define
• ? ? ? ? iff ? ? ?
• ? (?1?...??n) ? ? iff ?1,...,?n ? ?
• ? ? ? ? iff ? ? ? and ? ? ?

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Reasoning Services
SYNTAX SEMANTICS ENTAILMENT AND LOGICAL
IMPLICATION REASONING SERVICES

• Model Checking (EVAL)
• Is a proposition P true under a truth-valuation
  ?? Check ? ? P

Satisfiability (SAT) Is there a truth-valuation
? where P is true? find ? such that ? ?
P Unsatisfiability (UnSAT) the impossibility to
find a truth-valuation ?
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Reasoning Services
SYNTAX SEMANTICS ENTAILMENT AND LOGICAL
IMPLICATION REASONING SERVICES
Validity (VAL) Is P true according to all
possible truth-valuation ?? Check if ? ? P for
all ?
Entailment (ENT) All ? ? G true in ? (in all ?)
implies ? true in ? (in all ?). check G ? ? in ?
(in all ? ) by checking that given that ? ? ?
for all ? ? G implies ? ? ?
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Reasoning Services properties
SYNTAX SEMANTICS ENTAILMENT AND LOGICAL
IMPLICATION REASONING SERVICES

• EVAL is the easiest task. We just test one
  assignment.
• SAT is NP complete. We need to test in the worst
  case all the assignments. We stop when we find
  one which is true.
• UnSAT is CO-NP. We need to test in the worst case
  all the assignments. We stop when we find one
  which is true.
• VAL is CO-NP. We need to test all the assignments
  and verify that they are all true. We stop when
  we find one which is false.
• ENT is CO-NP. It can be computed using VAL (see
  next slide)

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Using DPLL for reasoning tasks
SYNTAX SEMANTICS ENTAILMENT AND LOGICAL
IMPLICATION REASONING SERVICES

• DPLL solves the CNFSAT-problem by searching a
  truth-assignment that satisfies all clauses ?i in
  the input proposition P ?1 ? ? ?n
• Model checking Does ? satisfy P? (? ? P?)
• Check if ?(P) true
• Satisfiability Is there any ? such that ? ? P?
• Check that DPLL(P) succeeds and returns a ?
• Unsatisfiability Is it true that there are no ?
  satisfying P?
• Check that DPLL(P) fails
• Validity Is P a tautology? (true for all ?)
• Check that DPLL(?P) fails

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