1. Logics for Data and Knowledge Representation
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

3. Summary about PL so far PROPOSITIONS
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 ν is a mapping L  {T, F}
Logical implication ()
Normal Forms: CNF - DNF 3

4. Reasoning Services Basic reasoning tasks for a PL-based system:
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:

5. Reasoning in PL Reasoning in PL is the simplest case of reasoning
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 can reduce 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-complete The 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 Like PSAT, CNFSAT is NP-complete.
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 clauses θi 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 Input: a proposition P in CNF
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); } 14

15. DPLL algorithm Input: a proposition P in CNF
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. 15

16. DPLL algorithm Input: a proposition P in CNF
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. 16

17. DPLL algorithm Input: a proposition P in CNF
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. 17

18. DPLL algorithm Input: a proposition P in CNF
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. 18

19. DPLL algorithm Input: a proposition P in CNF
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. 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 PROS CONS
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

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) If we codify it as a database: No negations
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