Strong AI vs. Weak AI Automated Reasoning

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1 Strong AI vs. Weak AI Automated Reasoning George F Luger ARTIFICIAL INTELLIGENCE 6th edition Structures and Strategies for Complex Problem Solving 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 Weak AI: Doesn t pass the Turing Test Produces machines that act intelligently 1 2 Automated Reasoning Program An Example Logic Theorist Allows computers to reason completely, or nearly completely, automatically Uses weak problem-solving methods Prerequisite for weak method problem solving: An unambiguous and exact notation for representing information Precise inference rules for drawing conclusions, and A carefully described strategies to apply these rules The design of search strategies is an art cannot guarantee to find a useful solution 3 The first AI program Deliberately engineered to mimic the problem solving skills of a human being A weak problem solver Uses uniform representation medium propositional calculus Uses sound inference rules substitution, replacement, and detachment Uses strategies or heuristic methods to guide the solution process 4

2 Inference Rules of LT Substitution allows any expression to be substituted for every occurrence of a symbol (B B) B Substitute A for B, we have ( A A) A Replacement allows a connective to be replaced by its definition or an equivalent form Since A B A B Therefore A A can be replaced by A A Detachment is modus ponens 5 The Strategy of LT First, directly apply substitution to the goal to match it against all known axioms and theorems Second, if it fails, apply all possible detachments and replacements to the goal and test all results for success. If it fails, add them to a subproblem list Third, chaining method is used to find a new subproblem (if a c is a problem and b c is found, then a b is a new subproblem) Fourth, if the above three methods fail, go to the subproblem list and select the next untried subproblem 6 The Strategy of LT Reasoning as Search The executive routine applies the four methods repeatedly until either the solution is found or the memory and time are exhausted The LP executes a goal-driven, breath-first search of the problem space Matching process enables the substitution, replacement and detachment of the expressions Logic Theorist explored a search tree: the root was the initial hypothesis, each branch was a deduction based on the rules of logic. Somewhere "up" the tree was the goal: the proposition the program intended to prove. The pathway along the branches that led to the goal was a proof a series of statements, each deduced using the rules of logic, that led from the hypothesis to the proposition to be proved. 7 8

3 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 9 QED 10 General Problem Solver Study revealed there were many ways the LT s solution differed from those of human subjects Human behavior showed strong evidence of matching and difference reduction mechanism GPS was based on the LT GPS was intended to work as a universal problem solver to solve any formalized symbolic problem General Problem Solver The problem-solving strategies of human subject is called difference reduction The general process of using transformations to reduce problem differences is called means-ends analysis The algorithm applying means-ends analysis is called the General Problem Solver (GPS) Differences between the initial statement and the goal are found A difference table containing transformation rules is used to remove the differences 11 12

Newell and Simon (1961) 1b A proof of a theorem

(1961), generated by human subject 13 14 GPS

table for the General Problem Solver, from

The GPS model of problem solving requires two

4 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 GPS Model Fig 14.2 Flow chart and difference reduction table for the General Problem Solver, from Newell and Simon (1963b). 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

5 Resolution Method Resolution Refutation A modern and more powerful method for automated reasoning Can prove theorems in both propositional and predicate calculus The primary rule of inference in PROLOG Uses a single rule of resolution, instead of trying different rules of inference and hoping one succeeds Greatly reduces the search space It proves a theorem by negating the goal statement to be proved and adding it to the set of given axioms known to be true Proving the theorem directly may take a long time or may not work at all It uses the resolution rule of inference to show that the negated goal leads to a contradiction It follows that the original goal must be consistent this proves the theorem Resolution Refutation Proof Steps Clause Form Put the premises or axioms into clause form Add the negation of what is to be proved, in clause form, to the set of axioms Resolve these clause together, producing new clauses that logically follow from them Produce a contradiction by generating the empty clause The substitutions used to produce the empty clause are those that make the original goal true A clause is a disjunction of literals e.g. A B C B C D A literal is an atomic expression or the negation of an atomic expression The database of axioms can be represented in a conjunctive normal form using only,, and e.g. (A B C ) ( B C D ) 19 20

6 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 Axioms: Fido is a dog. All dogs are animals. All animals will die. Prove: Fido will die. Using predicate calculus: An Example 1. All dogs are animals: (X) (dog(x) animal(x)) 2. Fido is a dog: dog(fido) 3. Modus ponens and {fido/x} gives: animal(fido) 4. All animals will die: (Y) (animal(y) die(y)) 5. Modus ponens and {fido/y} gives: die(fido) Predicate Form: An Example Clause Form: 1. (X) (dog(x) animal(x)) dog(x) animal(x) 2. dog(fido) dog(fido) 3. (Y) (animal(y) die(y)) animal(y) die(y) Negate the conclusion that Fido will die: 4. die(fido) die(fido) The entire database can be represented as a conjunction of disjuncts: ( dog(x) animal(x)) ( animal(y) die(y)) (dog(fido)) ( die(fido)) 23 24

7 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 Conversion to the Clause Form Prove: Some programmers hate failures No programmer hates any success No failure is a success Let: p(x) = X is a programmer f(x) = X is a failure s(x) = X is a success h(x, Y) = X hates Y Then premises and negated conclusion are: 1) ( X) [p(x) ( Y) (f(y) h(x, Y))] 2) ( X) [p(x) ( Y) (s(y) h(x, Y))] 3) ( Y) (f(y) s(y)) 27 Conversion to the Clause Form Nine steps to convert statements to the clause form: 1. Eliminate conditionals ( ) using a b a b 1) ( X) [p(x) ( Y) ( f(y) h(x, Y))] 2) ( X) [ p(x) ( Y) ( s(y) h(x, Y))] 3) ( Y) ( f(y) s(y)) 2. When possible, eliminate negations or reduce their scope using ( a) a (a b) a b ( X) a(x) ( X) a(x) (a b) a b ( X) a(x) ( X) a(x) 3) ( Y) (f(y) s(y)) 28

8 Conversion to the Clause Form Conversion to the Clause Form 3. Standardize variables (each quantifier has unique variable name). e.g. ( X) p(x) ( X) p(x) ( X) p(x) ( Y) p(y) 4. Move all to front without changing their order 1) ( Y) ( X) [p(x) ( f(y) h(x, Y))] 2) ( X) ( Y) [ p(x) s(y) h(x, Y)] 5. Eliminate existential quantifiers using Skolem functions 1) ( Y) [p(a) ( f(y) h(a, Y))] 2) ( X) ( Y) [ p(x) s(y) h(x, Y)] 3) f(b) s(b) Drop all universal quantification 1) p(a) ( f(y) h(a, Y))] 2) p(x) s(y) h(x, Y) 3) f(b) s(b) 7. Convert the expression to conjunctive normal form using: p (q r) = (p q) (p r) Our example is already in conjunctive normal form 30 Conversion to the Clause Form Conversion to the Clause Form 8. Eliminate signs by writing the expression as a set of clauses. 1) { p(a), f(y) h(a, Y) } 2) { p(x) s(y) h(x, Y) } 3) { f(b), s(b) } Or: 1a) p(a) 1b) f(y) h(a, Y) 2a) p(x) s(y) h(x, Y) 3a) f(b) 3b) s(b) 9. Rename variables in clauses so that each clause has unique variable name. Needless sharing of variable names may cause loss of generality in the solution. 1a) p(a) 1b) f(y) h(a, Y) 2a) p(x) s(z) h(x, Z) 3a) f(b) 3b) s(b) 31 32

9 Unification The process of finding substitutions for variables to make arguments match is called unification. Without unification it is impossible to resolve clauses such as: f(y) h(a, Y) f(b) For (1a) (3b): {a/x, b/y, b/z} 33 Proof by Resolution Refutation p(a) ( f(b) h(a, b) ) ( p(a) s(b) h(a, b) ) f(b) s(b) = nil Since the root is nil, the conclusion is valid. 34 The Binary Resolution Proof Procedure One resolution proof for an example from the propositional calculus. Prove a from the following axioms: Propositions b c a b d e c e f d f Clause Form a b c b c d e e f d f a We first convert the propositions to the clause form using: p q p q and (p q) p q Then the negated goal, a, is added to the clause set

10 Happy Student Problem Example Anyone passing his history exams and winning the lottery is happy. But anyone who studies or is lucky can pass all his exams. John did not study but he is lucky. Anyone who is lucky wins the lottery. Is John happy? To solve the problem, we first change the sentences to the predicate form Fig 14.5 One refutation for the happy student problem. Exciting Life Problem Example All people who are not poor and are smart are happy. Those people who read are smart. John can read and is not poor. Happy people have exciting lives. Can anyone be found with an exciting life? To apply resolution refutation method we change the sentences first to the predicate form and then to the clause form 39 40

11 Fig 14.6 Resolution proof for the exciting life problem Strategies and Simplification Techniques for Resolution Fig 14.7 another resolution refutation for the example of Fig There is a different resolution refutation proof tree for the exciting life problem (Fig. 17) For N clauses, there are up to N 2 ways of combining them at just the first level. In a large problem the exponential growth of the combinatorics will quickly get out of control Therefore search heuristics or strategies are used to control and simplify the proof process 43 44

and all clauses previously produced It guarantees to find the shortest solution It guarantees to find a refutation if one exists A good strategy for a small problem 45 46 Fig 14.

12 The Breadth-First Strategy Fig 14.8 Complete state space for the exciting life problem generated by breadth-first search (to two levels). Each clause is compared for resolution with every clause in the clause space on the first round The clauses at level n are generated by resolving all clauses at level n-1 against the original clauses and all clauses previously produced It guarantees to find the shortest solution It guarantees to find a refutation if one exists A good strategy for a small problem Fig 14.9 Using the unit preference strategy on the exciting life problem. The Unit Preference Strategy A clause of one literal is called a unit clause The unit preference strategy uses units for resolving whenever they are available Resolving with a unit clause guarantees the resolvent is smaller than the largest parent clause, which makes it closer to producing the clause of no literals 47 48

13 Answer Extraction from Resolution Refutation Unification substitutions of Fig 14.6 applied to the original query Retaining information on the unification substitution made in the resolution refutation gives information for the correct answer To record the answer, retain the original conclusion to be approved and, into that conclusion, introduce each unification made in the resolution process Fido Dog Problem Example Fig Answer extraction process on the finding fido problem. Fido the dog goes wherever John, his master, goes. John is at the library. Where is Fido? First represent the sentences in predicate calculus expressions and then reduce them to clause form. Predicates: at(john, X) at(fido, X) at(john, library) Clauses: at(john, X) at(fido, X) at(john, library) Negated goal: at(fido, Z) 51 52

14 Grandparent Problem Example Fig Skolemization as part of the answer extraction process. Everyone has a parent. The parent of a parent is a grandparent. Given the person John, prove that John has a grandparent. The following are the sentences and negated goal in predicate calculus expressions and the clause form: ( X)( Y) p(x, Y) ( X)( Y)( Z) p(x, Y) p(y, Z) gp(x, Z) gp(john, W) p(x, pa(x)) p(w, Y) p(y, Z) gp(w, Z) gp(john, V) Prolog and Automated Reasoning One problem with resolution proof procedure is when predicate calculus descriptors are transformed to clause form, important heuristic problem-solving information is left out. For example: a b c d (provides heuristic information) a b c d (no heuristic information) Prolog uses Horn clauses whose procedural interpretation provides an explicit strategy that preserves the heuristic information Horn Clause A Horn clause contains at most one positive literal a b 1 b 2 b n To emphasize the positive literal (a), the Horn clause is generally written as an implication: a b 1 b 2 b n Horn clause allows only restricted representation of the clauses, but provides very efficient search strategy for refutation 55 56

15 Three Forms of the Horn Clause 1. The original clause has no positive literals b 1 b 2 b n called a headless clause or goals 2. It has no negative literals a 1 a 2 a 3 called the facts 3. It has one positive and one or more negative literals a b 1 b 2 b n Reducing Clauses into Horn Form It takes three steps Select and move the positive literal to the very left a b 1 b 2 b n Change the clause to Horn form by the rule: a b 1 b 2 b n a ( b 1 b 2 b n ) Use de Morgan s law to change it to a b 1 b 2 b n It may not always be possible to transform a clause to the Horn form (e.g. p q) called a rule Search Strategy for Refutation Prolog implements left-to-right, depth-first search of clause for refutation Given a goal: a 1 a 2 a n and a program P, the Prolog interpreter searches for the first clause in P whose head unifies with a 1 a 1 b 1 b 2 b m After reducing a 1, the goal clause becomes: b 1 b 2 b m a 2 a n Search Strategy for Refutation The Prolog interpreter continues to reduce the leftmost goal, b 1, using the first clause in P that unifies with b 1 b 1 c 1 c 2 c p The goal then becomes: c 1 c 2 c p b 2 b n a 2 a n The search continues from left to right and from top to down until the goal reduces to the null clause 59 60

16 Depth-First Search Strategy goal a 1 a 2 a 3 a n b 1 b 2 b 3 b m c 1 c 2 c 3 c p Homework Assignment 1. Put the following predicate calculus expression in clause form: (X) (p(x) { (Y) [p(y) p(f(x, Y))] (Y) [q(x, Y) p(y)]}) 2. Prove the following using resolution refutation method. Draw an inversed binary tree to show the resolution process. Fact: d(f) [b(f) c(f)] Rules: d(x) a(x) b(y) e(y) g(w) c(w) Prove: a(z) e(z) 61 62

Logic. Knowledge Representation & Reasoning Mechanisms. Logic. Propositional Logic Predicate Logic (predicate Calculus) Automated Reasoning

Logic. Knowledge Representation & Reasoning Mechanisms. Logic. Propositional Logic Predicate Logic (predicate Calculus) Automated Reasoning Logic Knowledge Representation & Reasoning Mechanisms Logic Logic as KR Propositional Logic Predicate Logic (predicate Calculus) Automated Reasoning Logical inferences Resolution and Theorem-proving Logic