Logics for Data and Knowledge Representation

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Logics for Data and KnowledgeRepresentation

• Propositional Logic Reasoning

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

• Review of PL Syntax and semantics
• Reasoning in PL
• Typical tasks
• Calculus
• Problems with reasoning
• Calculus using tableaux
• The DPLL Procedure for PSAT
• Main steps
• The algorithm
• Examples
• Observations about the DPLL
• Conclusions on PL
• Pros and cons
• Examples

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Summary about PL so far
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• PROPOSITIONS
• Propositional logic (PL) is the simplest logic
  which deals with propositions (no individuals, no
  quantifiers)
• Propositions are something true or false
• SYNTAX
• We need to provide a language, including the
  alphabet of symbols and formation rules to
  articulate complex formulas (sentences)
• A propositional theory is formed by a set of PL
  formulas
• SEMANTICS
• Providing semantics means providing a pair (M,?),
  namely a formal model satisfying the theory
• A truth valuation ? is a mapping L ? T, F
• Logical implication (?)
• Normal Forms CNF - DNF

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Reasoning Services
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• Basic reasoning tasks for a PL-based system
• Model checking (EVAL)
• Satisfiability (SAT) reduced to model checking
  (we choose an assignment first)
• Validity (VAL) reduced to model checking (try for
  all possible assignments)
• Unsatisfiability (unSAT) reduced to model
  checking (try for all possible assignments)
• Entailment (ENT) reduced to previous problems
• NOTE SAT/UNSAT/VAL on generic formulas can be
  reduced to SAT/UNSAT/VAL on CNF formulas
• See for instance http//www.satisfiability.org

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Reasoning in PL
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• Reasoning in PL is the simplest case of reasoning
  
• We use truth tables
• In model checking we just verify a given
  assignment ?
• In SAT we try with all possible assignments but
  we stop when we find the first one which makes
  the formula true.

Given ?(A) T, ?(B) F is the formula A ? B
true? YES!
Is A ? B satisfiable? With ?(A)T, ?(B)F we
have ?(A ? B) F With ?(A)F, ?(B)T we have ?(A
? B) F With ?(A)T, ?(B)T we have ?(A ? B) T
STOP!
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Calculus
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• Semantic tableau is a decision procedure to
  determine the satisfiability of finite sets of
  formulas.
• There are rules for handling each of the logical
  connectives thus generating a truth tree.
• A branch in the tree is closed if a contradiction
  is present along the path (i.e. of an atomic
  formula, e.g. B and ?B)
• If all branches close, the proof is complete and
  the set of formulas are unsatisfiable, otherwise
  are satisfiable.
• With refutation tableaux the objective is to show
  that the negation of a formula is unsatisfiable.

G (A ??B), B
B
A ? ?B
A
?B
closed
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Rules of the semantic tableaux
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• Conjunctions lie on the same branch
• Disjunctions generate new branches
• The initial set of formulas are considered in
  conjunction and are put in the same branch of the
  tree
• (?) A ? B (?) A ? B
• --------- ---------
• A A B
• B
• (?) ? ? A A
• --------- ---------
• A ? ? A

??(A ??B) ? B
B
??(A ? ?B)
A ? ?B
A
?B
closed
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What is the problem in reasoning in PL? (I)
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• Given a proposition P with n atomic formulas, we
  have 2n possible assignments ? !
• SAT is NP-complete
• In the worst case reasoning time is exponential
  in n (in the case we test all possible
  assignments), but potentially less if we find a
  way to look for good assignments (if there is
  at least an assignment such that ? ? P. We stop
  when we find it.)
• NOTE the worst case is when the formula is
  unsatisfiable
• Testing validity (VAL) of a formula P is even
  harder, since we necessarily need to try ALL the
  assignments. We stop when we find a ? such that ?
  ? P

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What is the problem in reasoning in PL? (II)
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• Reducing unSAT to SAT
• Unsatisfiability is the opposite of SAT. We stop
  when we find an assignment ? such that ? ? P.
• SAT, VAL, unSAT are search problems
• How complex is the task?
• Notice that what makes reasoning exponential are
  disjunctions (?) because we need to test all
  possible options
• Trivially, in case of conjunctions (?) all the
  variables must be true
• IMPORTANT Often (when we do not use individuals
  or quantifiers) we can reduce reasoning in
  complex logics to reasoning in PL

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PSAT-Problem (Boolean SAT)
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• Definition PSAT find ? such that ? ? P
  (Satisfiability problem)
• Is PSAT decidable? YES, BUT EXPENSIVE!
• Theorem Cook,1971 PSAT is NP-completeThe
  theorem established a limitative result of PL
  (and Logic). A problem is NP-complete when it is
  very difficult to be computed!
• DPLL (Davis-Putnam-Logemann-Loveland, 1962)
• It is the most widely used algorithm for PSAT
• It works on CNF formulas
• It can take from constant to exponential time

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CNFSAT-Problem
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• Definition CNFSAT find ? s.t. ? ? P, with P in
  CNF
• Is CNFSAT decidable? YES, BUT STILL EXPENSIVE!
• Like PSAT, CNFSAT is NP-complete.
• Converting a formula in CNF
• It is always possible to convert a generic
  formula in CNF, but in exponential time
  (polynomial in most of the cases).
• It causes an exponential blow up in the length of
  the formula.

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The DPLL Procedure
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• DPLL employs a backtracking search to explore the
  space of propositional variables truth-valuations
  of a proposition P in CNF, looking for a
  satisfying truth-valuation of P
• DPLL solves the CNFSAT-Problem by searching a
  truth-assignment that satisfies all clauses ?i in
  the input proposition P ?1 ? ? ?n
• The basic intuition behind DPLL is that we can
  save time if we first test for some assignments
  before others

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DPLL Procedure Main Steps
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• 1. It identifies all literal in the input
  proposition P
• 2. It assigns a truth-value to each variable to
  satisfy them
• 3. It simplifies P by removing all clauses in P
  which become true under the truth-assignments at
  step 2 and all literals in P that become false
  from the remaining clauses (this may generate
  empty clauses)
• 4. It recursively checks if the simplified
  proposition obtained in step 3 is satisfiable if
  this is the case then P is satisfiable, otherwise
  the same recursive checking is done assuming the
  opposite truth value ().

B ? C ? (B ? A ? C) ? ( B ? D)
B ? C ? (B ? A ? C) ? ( B ? D) ?(B) T
?(C) F
D
D YES, it is satisfiable for ?(D) T. NOTE
?(A) can be T/F
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DPLL algorithm
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• Input a proposition P in CNF
• Output true if "P satisfiable" or false if "P
  unsatisfiable"
• boolean function DPLL(P)
• if consistent(P) then return true
• if hasEmptyClause(P) then return false
• foreach unit clause C in P do
• P unit-propagate(C, P)
• foreach pure-literal L in P do
• P pure-literal-assign(L, P)
• L choose-literal(P)
• return DPLL(P ? L) OR DPLL(P ? ?L)

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DPLL algorithm
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• Input a proposition P in CNF
• Output true if "P satisfiable" or false if "P
  unsatisfiable"
• boolean function DPLL(P)
• if consistent(P) then return true
• if hasEmptyClause(P) then return false
• foreach unit clause C in P do
• P unit-propagate(C, P)
• foreach pure-literal L in P do
• P pure-literal-assign(L, P)
• L choose-literal(P)
• return DPLL(P ? L) OR DPLL(P ? ?L)

It tests the formula P for consistency, namely it
does not contain contradictions (e.g. A ? ?A)
and all clauses are unit clauses.
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DPLL algorithm
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• Input a proposition P in CNF
• Output true if "P satisfiable" or false if "P
  unsatisfiable"
• boolean function DPLL(P)
• if consistent(P) then return true
• if hasEmptyClause(P) then return false
• foreach unit clause C in P do
• P unit-propagate(C, P)
• foreach pure-literal L in P do
• P pure-literal-assign(L, P)
• L choose-literal(P)
• return DPLL(P ? L) OR DPLL(P ? ?L)

An empty clause does not contain literals. It
can be due to previous iterations of the
algorithm where some simplifications has been
done. If any of them exists then P is
unsatisfiable.
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DPLL algorithm
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• Input a proposition P in CNF
• Output true if "P satisfiable" or false if "P
  unsatisfiable"
• boolean function DPLL(P)
• if consistent(P) then return true
• if hasEmptyClause(P) then return false
• foreach unit clause C in P do
• P unit-propagate(C, P)
• foreach pure-literal L in P do
• P pure-literal-assign(L, P)
• L choose-literal(P)
• return DPLL(P ? L) OR DPLL(P ? ?L)

(a) It assigns the right truth value to each
literal (true for positives and false for
negatives). (b) It simplifies P by removing all
clauses in P which become true under the
truth-assignment and all literals in P that
become false from the remaining clauses.
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DPLL algorithm
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• Input a proposition P in CNF
• Output true if "P satisfiable" or false if "P
  unsatisfiable"
• boolean function DPLL(P)
• if consistent(P) then return true
• if hasEmptyClause(P) then return false
• foreach unit clause C in P do
• P unit-propagate(C, P)
• foreach pure-literal L in P do
• P pure-literal-assign(L, P)
• L choose-literal(P)
• return DPLL(P ? L) OR DPLL(P ? ?L)

For all literals which appear pure in the formula
(i.e. with only one polarity) assign the
corresponding value - true if positive
literal - false if negative Not all DPLL
versions perform this step.
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DPLL algorithm
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• Input a proposition P in CNF
• Output true if "P satisfiable" or false if "P
  unsatisfiable"
• boolean function DPLL(P)
• if consistent(P) then return true
• if hasEmptyClause(P) then return false
• foreach unit clause C in P do
• P unit-propagate(C, P)
• foreach pure-literal L in P do
• P pure-literal-assign(L, P)
• L choose-literal(P)
• return DPLL(P ? L) OR DPLL(P ? ?L)

The splitting rule Select a variable whose
value is not assigned yet. Recursively call
DPLL for the cases in which the literal is true
or false.
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DPLL Procedure Example 1
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• P A ? (A ? A) ? B
• There are still variables and clauses to analyze,
  go ahead
• P does not contain empty clauses, go ahead
• It assigns the right truth-value to A and B ?(A)
  T, ?(B) T
• It simplifies P by removing all clauses in P
  which become true under ?(A) T and ?(B) T
• This causes the removal of all the clauses in P
• It simplifies P by removing all literals in the
  clauses of P that become false from the remaining
  clauses nothing to remove
• It assigns values to pure literals. nothing to
  assign
• All variables are assigned it returns true

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DPLL Procedure Example 2
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• P C ? (A ? A) ? B
• There are still variables and clauses to analyze,
  go ahead
• P does not contain empty clauses, go ahead
• It assigns the right truth-value to C and B ?(C)
  T, ?(B) T
• It simplifies P by removing all clauses in P
  which become true under ?(C) T and ?(B) T.
• P is then simplified to (A ? A)
• It simplifies P by removing all literals in the
  clauses of P that become false from the remaining
  clauses nothing to remove
• It assigns values to pure literals nothing to
  assign
• It selects A and applies the splitting rule by
  calling DPLL on
• A ? (A ? A) AND A ? (A ? A)
• which are both true (the first call is enough).
  It returns true

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DPLL Procedure Example 3
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• P A ? B ? (A ? B)
• There are still variables and clauses to analyze,
  go ahead
• P does not contain empty clauses, go ahead
• It assigns the right truth-value to A and B
• ?(A) T, ?(B) F
• It simplifies P by removing all clauses in P
  which become true under ?(A) T and ?(B) F.
• P is simplified to (A ? B)
• It simplifies P by removing all literals in the
  clauses of P that become false from the remaining
  clauses the last clause becomes empty
• It assigns values to pure literals nothing to
  assign
• All variables are assigned but there is an empty
  clause it returns false

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The branching literal
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• The branching literal is the literal considered
  in the backtracking step (the one chosen for the
  splitting rule)
• The DPLL algorithm (and corresponding efficiency)
  highly depends on the choice of the branching
  literal
• DPLL as a family of algorithms
• One for each possible way of choosing the
  branching literal
• The running time can be constant or exponential
  depending on the choice of the branching literals
• Researches mainly focus on smart choices for the
  branching literal

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Final observations on the DPLL
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• There are several versions of the DPLL. We
  presented one.
• Finding solutions to propositional logic formulas
  is an NP-complete problem
• A DPLL SAT solver
• works on formulas in CNF
• employs a systematic backtracking search
  procedure to explore the (exponentially-sized)
  space of variable assignments looking for
  satisfying assignments
• Modern SAT solvers (developed in the last ten
  years) come in two flavors "conflict-driven" and
  "look-ahead approaches

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Using DPLL for reasoning tasks
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• 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
• NOTE typical DPLL implementations take two
  parameters the proposition P and a model ?.
  Therefore, in case of model checking the real
  call would be DPLL(P, ?) and check that it
  succeeds

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Pros and Cons of PL
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• PROS
• PL is declarative the syntax captures facts
• PL allows disjunctive (partial) and negated
  knowledge (unlike most databases)
• PSAT is fundamental in important applications

• CONS
• PL has limited expressive power (yet useful in
  lots of applications)
• No enumerations
• No qualifiers (exists, for all)
• No instances

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Example (KB)
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• Consider the following propositions
• AreaManager ? Manager TopManager ? Manager
• Manager ? Employee TopManager(John)
• Since we cannot reason on instances, we cant
  deduce the following
• Manager(John), Employee(John)

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Example (KB)
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• Consider the following
• AreaManager(x) ? Manager(x) Manager(x) ?
  Employee(x)
• TopManager(x) ? Manager(x) TopManager(John)
• They are not propositions because of the
  variables.

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Example (DB)
REVIEW REASONING IN PL THE DPLL PROCEDURE
OBSERVATIONS CONCLUSIONS

• If we codify it as a database
• No negations
• No disjunctions
• the reasoning is polynomial.

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