Lecture Slides for MAT-60556 PART II: Propositional logic, deduction and other topics Henri Hansen September 11, 2014 1
Questions to ponder & review • What do completeness and soundness mean in logic? • Is it correct to say that a valid propositional formula is uninformative? • Is it correct to say that an unsatisfiable formula is trivial? 2
Deductive proofs? • Let U = { A 1 , . . . , A n } . We have already shown that U | = A if and only if A 1 ∧ · · · ∧ A n → A is valid • Tableaux method could be used to show this valid- ity, but what if the set of axioms is infinite? • A decision procedure may not give any information about the relationship between axioms and theo- rems • We may be interested in the intermediate steps (lemmas) 3
Deductive systems • A deductive system is a set of formulas called ax- ioms and a set of rules of inference • A proof in a deductive system is a sequence of for- mulas, A 1 , . . . , A n such that for each i , A i is either an axiom or inferred from a subset of A 1 , . . . A i − 1 . • Given a proof S = A 1 , . . . , A n , A n is a theorem and S is a proof of A n . • If a formula A has a proof, we say A is provable in the given system, denoted ⊢ A 4
Genzen System G • An axiom of G is a set of literals U that contains a complementary pair • Rules of inference are similar to the α and β -rules of tableux, but they are ”‘reversed”’ (i.e., α in G is a β in tableaux, and vice versa) 1. If { α 1 , α 2 } ⊆ U 1 , then ( U 1 \{ α 1 , α 2 } ) ∪{ α } can be inferred 2. If { β 1 } ⊆ U 1 and { β 2 } ⊆ U 2 , then we can be infer ( U 1 \ { β 1 } ∪ U 2 \ { β 2 } ) ∪ { β } 5
Genzen System (cont.) • The proofs in G can be thought of as ”upside down tableaux” • I.e., a closed tableaux that shows φ is UNSAT cor- responds to a proof of ¬ φ in G 1. axioms correspond to closed leaves (with literals negated) 2. Every node has a formula that is a negation of the corresponding formula in the tableaux 3. The ”‘theorem”’ is the negation of the root of the tableau 6
Hilbert System H • The axioms of H (with the letters denoting arbitrary formulas) are 1. ⊢ ( A → ( B → A )) 2. ⊢ ( A → ( B → C )) → (( A → B ) → ( A → C )) 3. ⊢ ( ¬ B → ¬ A ) → ( A → B ) • The rule of inference of H is modus ponens 1. ( ⊢ A, ⊢ A → B ) ⇒ ⊢ B 7
Hilbert System (cont.) • Proofs in H tend to be complicated, so we may wish to have additional rules, called derived rules • An example of a derived rule is the deduction rule , which says that ( U ∪ { A } ⊢ B ) ⇒ U ⊢ ( A → B ) • This is proven by induction over the length of the proof for U ∪ { A } ⊢ B , to show that the proof can always be transformed into a proof of U ⊢ ( A → B ) • Other important derived rules in H can be found in the book, such as the contrapositive rule, transitiv- ity rule, etc. 8
Hilbert System (cont. II) • H is sound and complete, i.e., ⊢ A if and only if | = A • Soundness (only if) is proven by structural induc- tion, i.e., by showing that modus ponens preserves validity • Completeness is proven by transforming a closed tableau for ¬ A first into a proof of A in G and then showing that every proof in G can be transformed into a proof in H 9
Proof of A → A in H 1. ⊢ ( A → (( A → A ) → A )) → (( A → ( A → A ) → ( A → A )) (axiom 2) 2. ⊢ A → (( A → A ) → A ) (axiom 1) 3. ⊢ ( A → ( A → A )) → ( A → A ) (MP 1, 2) 4. ⊢ A → ( A → A ) (axiom 1) 5. ⊢ A → A (MP 3, 4) 10
Proving derived rules for H • Let us prove the deduction rule, i.e.: ( U ∪ { A } ⊢ B ) ⇒ U ⊢ ( A → B ) – This is done on the induction on the length of ( U ∪ { A } ⊢ B ) (which we call ”‘the proof”’) – For n = 1, B is proved in one step, so B must be an element of U ∪ { A } , an axiom, or a previously proved theorem – If B = A then ⊢ A → A by the previous theorem – Otherwise U ⊢ B is an axiom or theorem, U ⊢ B → ( A → B ) by axiom 1, and U ⊢ A → B by modus ponens 11
• if n > 1, the last step of the proof is either one-step inference of B or the use of modus ponens. In the first case, it is equivalent to n = 1 • If MP is used, there must be a formula C such that U ∪ { A } ⊢ C and U ∪ { A } ⊢ C → B have been proven before B • By inductive hypothesis, U ⊢ A → C and U ⊢ A → ( C → B ). • We can then construct a proof of U ⊢ A → B : 1. U ⊢ A → C (inductive hypothesis)
2. U ⊢ A → ( C → B ) (inductive hypothesis) 3. U ⊢ ( A → ( C → B )) → (( A → C ) → ( A → B )) (axiom 2) 4. U ⊢ ( A → C ) → ( A → B ) (MP 2, 3) 5. U ⊢ A → B (MP 1, 4)
Consistency • A set of formulas U is inconsistent if there exists a formula A such that U ⊢ A and U ⊢ ¬ A • U is inconsistent if and only if U ⊢ A for every for- mula A • U is consistent if and only if there exists some for- mula A such that U �⊢ A • U ⊢ A if and only if U ∪ {¬ A } is inconsistent 12
Strong completenes and Compactness • A set of formulas S = { A 1 , . . . } is unsatisfiable if and only if some tableau for S closes • Let U be a countable set of formulas and A an arbitrary formula. If U | = A , then U ⊢ A • If S is a countable set of formulas such that every finite subset of S is satisfiable, then S is satisfiable • These (infinitary) properties of propositional logic will become important with first order logic 13
Conjuctive normal form • A formula is in conjunctive normal form iff it is a conjunction of a disjunction of literals. • For example ( p ∨ ¬ q ∨ r ) ∧ ( ¬ p ∨ q ∨ r ) is CNF • Every formula in propositional logic can be given in CNF. This can be proven by giving rules that transform other operators to ∧ and ∨ , and then showing that these can be arranged into CNF • Conjunctive normal form is perhaps the most com- mon normal form; others exist 14
Clausal form 1. A clause is a set of literals, considered as a disjunc- tion. A unit clause is a single literal, and the empty clause is the empty set of literals, denoted � . If the clause contains a literal and its negation it is trivial 2. A formula in clausal form is given as a set of clauses, and the formula is considered to be the conjunction of its clauses, and the empty set of clauses is de- noted ∅ • Every formula in propositional logic can be trans- formed into clausal form (like CNF) 15
Clausal form notation • We denote a clause as a concatenation of literals so that an overbar denotes negation. For example p ¯ qr is a clause for ( p ∨ ¬ q ∨ r ) • The clausal form for a formula is given as a set of such clauses. For instance { p ¯ qr, ¯ pqr } • if l is a literal, we denote its complement with l c , i.e., if l = p then l c = ¯ p and vice versa • we generalize the concept of interpretation to clauses in the natural way 16
Resolution • Resolution is a refutation procedure used to check if a clausal formula is unsatisfiable • The resolution rule: Let C 1 and C 2 be clauses such that l ∈ C 1 and l c ∈ C 2 . We say that C 1 and C 2 are clashing clauses , and they clash on l, l c . • The resolvent of clauses C 1 and C 2 that clash on l amd l c is the clause ( C 1 \ { l } ) ∪ ( C 2 \ { l c } ) • If two clauses clash on more than one literal, their resolvent is a trivial clause! 17
Resolution (cont.) • The resolvent is satisfiable iff the parent clauses are satisfiable • Resolution algorithm for a set of clauses S : 1. Find clashing clauses { C 1 , C 2 } ⊆ S and compute their resolvent C 2. Discard C if trivial, otherwise add it to S (DO not remove C 1 and C 2 !!) 3. Terminate if C = � or no more clashing clauses, otherwise goto 1 18
Example of resolution • Consider the formula ( p ∨ q ) ∧ ( p ∨ r ) ∧ ( q ∨ r ) ∧ ( ¬ p ∨ q ) ∧ ( ¬ r ∨ ¬ p ) ∧ ( r ∨ ¬ p ) • In clausal form, this is { pq, pr, qr, ¯ pq, ¯ r ¯ p, ¯ pr } • Resolution: 1. pq and ¯ pq clash on p , the resolvent is q : { pq, pr, qr, ¯ pr, q } pq, ¯ r ¯ p, ¯ 19
2. q and ¯ p ¯ q resolve into p , { pq, pr, qr, ¯ pq, ¯ r ¯ p, ¯ pr, q, p } 3. p and ¯ pr resolve into r , { pq, pr, qr, ¯ pq, ¯ r ¯ p, ¯ pr, q, p, r } 4. r and ¯ r ¯ p resolve into ¯ p , { pq, pr, qr, ¯ pq, ¯ r ¯ p, ¯ pr, q, p, r, ¯ p } 5. p and ¯ p resolve into � , so the formula is unsat.
Resolution (cont. II) • We can think of the resolution algorithm as pro- ducing an “upside down” tree: The root is the last resolvent • Given two clashing clauses resolved, they are the children and the resolvent is the parent • The derivation of � from a set of clauses proves that the set is not satisfiable • Soundness and completeness are a bit hard to prove 20
Binary Decision Diagrams • We make the distinction here: a formula is a tree, but the semantics (i.e., the meaning) of the formula is a boolean function • Binary decision diagram, or BDD, is data structure for representing the semantics of a formula • A formula is represented by a directed graph and an algorithm is used for reducing the graph • A formula is valid iff the BDD represents trivial T and unsatisfiable iff the BDD represents trivial F 21
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