Strong AI vs. Weak AI Automated Reasoning • Artificial intelligence can be classified into two categories: strong AI and weak AI • Strong AI: – Can pass the Turing Test – Has an intelligence that matches or exceeds that of human beings George F Luger • Weak AI: ARTIFICIAL INTELLIGENCE 6th edition – Doesn’t pass the Turing Test Structures and Strategies for Complex Problem Solving – Produces machines that act intelligently 1 2 Automated Reasoning Program An Example – Logic Theorist • Allows computers to reason completely, or nearly • The first AI program completely, automatically • Deliberately engineered to mimic the problem • Uses weak problem-solving methods solving skills of a human being • Prerequisite for weak method problem solving: • A weak problem solver – An unambiguous and exact notation for representing • Uses uniform representation medium – information propositional calculus – Precise inference rules for drawing conclusions, and • Uses sound inference rules – substitution, – A carefully described strategies to apply these rules replacement, and detachment • The design of search strategies is an art – cannot • Uses strategies or heuristic methods to guide the guarantee to find a useful solution solution process 3 4 The Strategy of LT Inference Rules of LT • First, directly apply substitution to the goal to match • Substitution allows any expression to be it against all known axioms and theorems substituted for every occurrence of a symbol • Second, if it fails, apply all possible detachments (B ∨ B) → B Substitute ¬A for B, we have and replacements to the goal and test all results for (¬A ∨ ¬A) → ¬A success. If it fails, add them to a subproblem list • Replacement allows a connective to be replaced by • Third, chaining method is used to find a new its definition or an equivalent form subproblem (if a → c is a problem and b → c is – Since ¬A ∨ B ≡ A → B found, then a → b is a new subproblem) – Therefore ¬A ∨ ¬A can be replaced by A → ¬A • Fourth, if the above three methods fail, go to the • Detachment is modus ponens subproblem list and select the next untried subproblem 5 6 The Strategy of LT Reasoning as Search • The executive routine applies the four methods • Logic Theorist explored a search tree: the root was repeatedly until either the solution is found or the the initial hypothesis, each branch was a deduction memory and time are exhausted based on the rules of logic. Somewhere "up" the • The LP executes a goal -driven, breath -first search tree was the goal: the proposition the program of the problem space intended to prove. The pathway along the • Matching process enables the substitution, branches that led to the goal was a proof – a series replacement and detachment of the expressions of statements, each deduced using the rules of logic, that led from the hypothesis to the proposition to be proved. 7 8 An Example Suppose we wish to prove: (p → ¬p) → ¬p 1. (A ∨ A) → A Matching identifies the best axiom out of five available ones 2. (¬A ∨ ¬A) → ¬A Substitute ¬A for A in order to apply 3. (A → ¬A) → ¬A Replacement of → for ∨ and ¬, followed by 4. (p → ¬p) → ¬p substitution of p for A QED 9 10 General Problem Solver General Problem Solver • The problem-solving strategies of human subject • Study revealed there were many ways the LT’s is called difference reduction solution differed from those of human subjects • The general process of using transformations to • Human behavior showed strong evidence of reduce problem differences is called means-ends matching and difference reduction mechanism analysis • GPS was based on the LT • The algorithm applying means-ends analysis is • GPS was intended to work as a universal problem called the General Problem Solver (GPS) solver to solve any formalized symbolic problem – Differences between the initial statement and the goal are found – A difference table containing transformation rules is used to remove the differences 11 12 Fig 14.1a Transformation rules for logic problems, from Newell and Simon (1961) Fig 14.1b A proof of a theorem in propositional calculus, from Newell and Simon (1961), generated by human subject 13 14 Fig 14.2 Flow chart and difference reduction table for the General Problem Solver, from Newell and Simon (1963b). GPS Model • The GPS model of problem solving requires two components – General procedure for comparing two state descriptions and reducing their differences – The table of connections, giving links between problem differences and transformations that reduce them 15 16 Resolution Method Resolution Refutation • A modern and more powerful method for • It proves a theorem by negating the goal automated reasoning statement to be proved and adding it to the set of • Can prove theorems in both propositional and given axioms known to be true predicate calculus • Proving the theorem directly may take a long time • The primary rule of inference in PROLOG or may not work at all • Uses a single rule of resolution, instead of trying • It uses the resolution rule of inference to show that different rules of inference and hoping one the negated goal leads to a contradiction succeeds • It follows that the original goal must be consistent • Greatly reduces the search space – this proves the theorem 17 18 Resolution Refutation Proof Steps Clause Form • Put the premises or axioms into clause form • A clause is a disjunction of literals • Add the negation of what is to be proved, in clause – e.g. A ∨ B ∨ C form, to the set of axioms ¬ B ∨ C ∨ D • Resolve these clause together, producing new • A literal is an atomic expression or the negation of clauses that logically follow from them an atomic expression • Produce a contradiction by generating the empty • The database of axioms can be represented in a clause conjunctive normal form using only ∨, ∧, and ¬ • The substitutions used to produce the empty – e.g. clause are those that make the original goal true (A ∨ B ∨ C ) ∧ (¬ B ∨ C ∨ D ) 19 20 Clause Form Resolution Rules of Reference • Resolution is an operation on two clauses (one containing a literal and the other containing its negation) to produce a new and simpler clause where two literals are “resolved away” – e.g. (A ∨ B ∨ C ) ∧ (¬ B ∨ C ∨ D ) ≡ A ∨ C ∨ D (p ∨ q ) ∧ (¬ q ∨ r) ≡ p ∨ r (f ∨ g ) ∧ (f ∨ ¬ g) ≡ f ∨ f ≡ f • Unification may be required before resolution 21 22 An Example An Example Axioms: Predicate Form: Clause Form: Fido is a dog. All dogs are animals. 1. ∀(X) (dog(X) → animal(X)) ¬ dog(X) ∨ animal(X) All animals will die. 2. dog(fido) dog(fido) Prove: 3. ∀(Y) (animal(Y) → die(Y)) ¬ animal(Y) ∨ die(Y) Fido will die. Negate the conclusion that Fido will die: Using predicate calculus: 4. ¬ die(fido) ¬ die(fido) 1. All dogs are animals: ∀(X) (dog(X) → animal(X)) The entire database can be represented as a conjunction 2. Fido is a dog: dog(fido) 3. Modus ponens and {fido/X} gives: animal(fido) of disjuncts: 4. All animals will die: ∀(Y) (animal(Y) → die(Y)) (¬ dog(X) ∨ animal(X)) ∧ (¬ animal(Y) ∨ die(Y)) 5. Modus ponens and {fido/Y} gives: die(fido) ∧ (dog(fido)) ∧ (¬ die(fido)) 23 24 Resolution proof for the “dead dog” problem Substitution of Variables • The sequence of substitutions used gives values of variables to make the goal true • e.g. – Find out if “something will die” is true – Negated goal: ¬ (∃ Z die(Z)) – The substitution {fido/Z} shows that fido is an instance of animal that will die 25 26 Conversion to the Clause Form Conversion to the Clause Form Prove: Nine steps to convert statements to the clause form: Some programmers hate failures 1. Eliminate conditionals (→) using a → b ≡ ¬ a ∨ b No programmer hates any success ⇒ ∃ ∧ ∀ ¬ ∨ ∴ No failure is a success 1) ( X) [p(X) ( Y) ( f(Y) h(X, Y))] ⇒ ∀ ¬ ∨ ∀ ¬ ∨ ¬ Let: 2) ( X) [ p(X) ( Y) ( s(Y) h(X, Y))] ⇒ ¬ ∀ ¬ ∨ ¬ p(X) = X is a programmer s(X) = X is a success 3) ( Y) ( f(Y) s(Y)) f(X) = X is a failure h(X, Y) = X hates Y 2. When possible, eliminate negations or reduce their scope using ¬ (¬ a) ≡ a Then premises and negated conclusion are: ¬ (a ∨ b) ≡ ¬ a ∧ ¬ b ¬ (∃X) a(X) ≡ (∀X) ¬ a(X) ∃ ∧ ∀ → 1) ( X) [p(X) ( Y) (f(Y) h(X, Y))] ¬ (a ∧ b) ≡ ¬ a ∨ ¬ b ¬ (∀X) a(X) ≡ (∃X) ¬ a(X) 2) (∀X) [p(X) → (∀Y) (s(Y) → ¬ h(X, Y))] ⇒ ∃ ∧ 3) ¬ (∀Y) (f(Y) → ¬ s(Y)) 3) ( Y) (f(Y) s(Y)) 27 28 Conversion to the Clause Form Conversion to the Clause Form 3. Standardize variables (each quantifier has unique 6. Drop all universal quantification variable name). 1) ⇒ p(a) ∧ (¬ f(Y) ∨ h(a, Y))] e.g. ( ∃X) ¬ p(X) ∨ (∀X) p(X) ⇒ (∃X) ¬ p(X) ∨ (∀Y) p(Y) 2) ⇒ ¬ p(X) ∨ ¬ s(Y) ∨ ¬ h(X, Y) ∀ 4. Move all to front without changing their order 3) ⇒ f(b) ∧ s(b) 1) ⇒ (∀Y) ( ∃X) [p(X) ∧ (¬ f(Y) ∨ h(X, Y))] 2) ⇒ (∀X) ( ∀Y) [ ¬ p(X) ∨ ¬ s(Y) ∨ ¬ h(X, Y)] 7. Convert the expression to conjunctive normal 5. Eliminate existential quantifiers using Skolem form using: p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r) functions Our example is already in conjunctive normal ⇒ ∀ ∧ ¬ ∨ 1) ( Y) [p(a) ( f(Y) h(a, Y))] form 2) ⇒ (∀X) ( ∀Y) [ ¬ p(X) ∨ ¬ s(Y) ∨ ¬ h(X, Y)] 3) ⇒ f(b) ∧ s(b) 29 30 Conversion to the Clause Form Conversion to the Clause Form 8. Eliminate ∧ signs by writing the expression as a 9. Rename variables in clauses so that each clause set of clauses. has unique variable name. Needless sharing of 1) ⇒ { p(a), ¬ f(Y) ∨ h(a, Y) } variable names may cause loss of generality in 2) ⇒ {¬ p(X) ∨ ¬ s(Y) ∨ ¬ h(X, Y) } the solution.