mavii

• Here are some examples of sound rules of inference • Each can be shown to be sound using a truth table RULE PREMISE CONCLUSION Modus Ponens A, A → B B And Introduction A, B A ∧ B And Elimination A ∧ B A Double Negation ¬¬A A Unit Resolution A ∨ B, ¬B A Resolution A ∨ B, ¬B ∨ C A ∨ C

Resolution Example and Exercises. Solutions to Selected Problems. Example: Consider the following axioms: All hounds howl at night. Anyone who has any cats will not have any mice. Light sleepers do not have anything which howls at night. John has either a cat or a hound. (Conclusion) If John is a light sleeper, then John does not have any mice.

a proof strategy called Resolution Refutation, with three steps. And it goes like this. 5 Lecture 7 • 5 Propositional Resolution • Resolution rule: α v β ¬β v γ α v γ ... Propositional Resolution Example Step Formula Derivation 3 Q → R 2 P → R 1 P v Q Prove R

Resolution. Resolution is a theorem proving technique that proceeds by building refutation proofs, i.e., proofs by contradictions. It was invented by a Mathematician John Alan Robinson in the year 1965. Resolution is used, if there are various statements are given, and we need to prove a conclusion of those statements.

Resolution Refutation in FOL ... Resolution method is used to test unsatisfiability of a set S of clauses in Predicate Logic. – It is an extension of resolution method for propositional logic. The resolution principle basically checks ... Show through an example that ( ∀x) α[x] V ( ∀x)

Propositional Resolution is a refutation proof system. Just one rule of inference - the Resolution Principle. Propositional Resolution is sound and complete. The search space in propositional resolution is smaller than that of direct proof systems or natural deduction systems. Hitch: To order to use resolution, we need to transform

Refutation • When resolution is used to prove inconsistency, it is called refutation. (refute=disprove) • The aboveabove binarybinary treetree , showingshowing resolutionresolution andand resulting in the empty clause, is called a refutation tteeree. • NOTE: To avoid potential mistakes, DO NOT RESOLVE

A resolution refutation of a formula F can be seen as a proof that F is unsatisfiable. This will be made formal in the next section. Resolution can be used to prove entailments by transforming them to refutations. For example, the refutation in Example 2 can be used to show that (X ∨¬Y) ∧(Y ∨Z) ∧(¬X ∨¬Y ∨Z) |= Z .

If there is a resolution refutation of C, then we say that Cis refutable. We can now turn this into an algorithm. To prove that ϕis valid: 1. Convert: ϕto (¬ϕ) 2. Convert: (¬ϕ) to CNF 3. Find a resolution refutation of (¬ϕ) 4. If ¬ϕis refutable, then ϕmust be valid. As an example how we prove things in this system, suppose we want to ...

Resolution Proof Note: In general we have to be cautious about variable names. The X in clause 5 is NOT the same as the X in clause 9! 1. (p(X), h(Y)) 2. ( p(X), q(Y)) 3. R[1,2] (h(Y), q(Y)) This is incorrect, as now h and q seem to have the same variable. In fact the Y in 1 and the Y in 2 are different. The correct operation is to first rename

Resolution refutation proof of “not swimming” A resolution refutation proof is proof by contradiction using resolution. Like every proof by contradiction, you begin by assuming the opposite of what you wish to prove, and then show that this “fact” would lead to a contradiction. For example, the following two statements are mutually ...

Resolution Refutation • Given –a knowledge base KB (collection of true sentences) –a proposition P We wish to prove that P is true • Proof by contradiction: –Assume that P is FALSE (i.e., that ¬P is TRUE). –Show that a contradiction arises •Start with KB •Add ¬P to KB •Apply resolution rule to KB, adding results to KB

Example Harry, Ron and . Draco are students of the Hogwarts school for wizards Every student is either wicked or is a good . Quidditch player, or both No . Quidditch player likes rain and all wicked students like potions Draco . dislikes whatever Harry likes and likes . whatever Harry dislikes Draco . likes rain and potions

A unit refutation is a unit derivation of the empty clause. As an example of a unit refutation, consider the following proof. In the first two inferences, unit clauses from the initial set are resolved with binary clauses to produce two new unit clauses. These are resolved with the first clause to produce two additional unit clauses.

•Resolution refutation: •Convert all sentences to CNF •Negate the desired conclusion (converted to CNF) •Apply resolution rule until either ... Propositional Resolution Example StepFormula Derivation 9 • 4,8 8 R 5,7 7 ¬ Q 3,4 6 ¬ P 2,4 5 Q v R 1,2 Negated conclusion 4 ¬ R 3 ¬ Q v R Given 2 ¬ P v R Given 1 P v Q Given 3Q → R

If there is a refutation of F1 not using C’, then that is a refutation for F as well. If refutation of F1 uses C’, then construct a resolution of F by adding A to the first occurrence of C’ (and its descendants); that will end with {A}. From here on, follow the refutation of F2. This constructs a refutation of F. 25

Resolution-Refutation Proofs. Resolution is sound and refutation is complete; if a sentence is unsatisfiable, resolution will derive a contradiction (proof is in the text). Resolution can be used to establish that a sentence is entailed by the KB, but cannot be used to generate all logical consequences of a set of Well-Formed Formulae (WFF).

Linear Refutation •We can resolve with different clauses and keep adding new clauses forever! •To prevent this, Linear Refutation always starts with a goal (as the example showed previously). •Prolog [s computation rule: Always selects the leftmost subgoal, although logically there is no order for the subgoals.

Resolution refutation is a proof technique used in propositional and predicate logic that involves deriving a contradiction from a set of premises. This method relies on the principle that if the negation of a conclusion leads to an inconsistency, the original conclusion must be true. It connects to the completeness of resolution by showing that if a contradiction can be derived, then the set ...