Logics for Data and Knowledge Representation

1. Logics for Data and KnowledgeRepresentation Propositional Logic: Reasoning Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

2. 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

3. 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 modelsatisfying the theory • Atruth valuation νis a mapping L  {T, F} • Logical implication () • Normal Forms: CNF - DNF 3

4. Reasoning Services REVIEW :: REASONING IN PL :: THE DPLL PROCEDURE :: OBSERVATIONS :: CONCLUSIONS • Basicreasoning 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

5. 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 ν • In SAT we try with all possible assignments but we stop when we find the first one which makes the formula true. Given ν(A) = T, ν(B) = F is the formula A  B true? YES! Is A  B satisfiable? With ν(A)=T, ν(B)=F we have ν(A  B) = F With ν(A)=F, ν(B)=T we have ν(A  B) = F With ν(A)=T, ν(B)=T we have ν(A  B) = T STOP! 5

6. 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. Γ = {(A B), B} B A  B A B closed 6

7. 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 7

8. 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 ν ! • 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 ν ⊨ 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 ν such that ν ⊭ P 8

9. 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 ν such that ν ⊨ 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 canreduce reasoning in complex logics to reasoning in PL 9

10. PSAT-Problem (Boolean SAT) REVIEW :: REASONING IN PL :: THE DPLL PROCEDURE :: OBSERVATIONS :: CONCLUSIONS • Definition: PSAT = find ν such that ν⊨ 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

11. CNFSAT-Problem REVIEW :: REASONING IN PL :: THE DPLL PROCEDURE :: OBSERVATIONS :: CONCLUSIONS • Definition: CNFSAT = find ν s.t. ν⊨ 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.

12. 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 clausesθi in the input proposition P = θ1 … θn • The basic intuition behind DPLL is that we can save time if we first test for some assignments before others

13. 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) ν(B) = T; ν(C) = F D D YES, it is satisfiable for ν(D) = T. NOTE: ν(A) can be T/F

14. 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 returntrue; 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); } 14

15. 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 returntrue; 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. 15

16. 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 returntrue; 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. 16

17. 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 returntrue; 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. 17

18. 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 returntrue; 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. 18

19. 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 returntrue; 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. 19

20. 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: ν(A) = T, ν(B) = T • It simplifies P by removing all clauses in P which become true under ν(A) = T and ν(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 20

21. 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: ν(C) = T, ν(B) = T • It simplifies P by removing all clauses in P which become true under ν(C) = T and ν(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 21

22. 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 ν(A) = T, ν(B) = F • It simplifies P by removing all clauses in P which become true under ν(A) = T and ν(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 22

23. 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

24. 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 24

25. Using DPLL for reasoning tasks REVIEW :: REASONING IN PL :: THE DPLL PROCEDURE :: OBSERVATIONS :: CONCLUSIONS • Model checking Does ν satisfy P? (ν⊨ P?) Check if ν(P) = true • Satisfiability Is there any ν such that ν⊨ P? Check that DPLL(P) succeeds and returns a ν • Unsatisfiability Is it true that there are no ν satisfying P? Check that DPLL(P) fails • Validity Is P a tautology? (true for all ν) Check that DPLL(P) fails NOTE: typical DPLL implementations take two parameters: the proposition P and a model ν. Therefore, in case of model checking the real call would be DPLL(P, ν) and check that it succeeds 25

26. Pros and Cons of PL REVIEW :: REASONING IN PL :: THE DPLL PROCEDURE :: OBSERVATIONS :: CONCLUSIONS CONS • PL has limited expressive power (yet useful in lots of applications) • No enumerations • No qualifiers (exists, for all) • No instances PROS • PL is declarative: the syntax captures facts • PL allows disjunctive (partial) and negated knowledge (unlike most databases) • PSAT is fundamental in important applications

27. 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 can’t deduce the following: Manager(John), Employee(John)

28. 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. 28

29. 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. 29