Artificial Intelligence Notes Lecture : Propositional Logic and Inference

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1 Page 1 of 7 Introduction Artificial Intelligence Notes Lecture : Propositional Logic and Inference Logic is a natural bridge between man and machine. This is because: Logic is well-defined, which makes it possible to be manipulated using a computer. Logic is symbolic, which is closer to the way humans reason than the numeric types of computation normally associated with computers. The automation of logical deduction has been the most successful forms of artificial intelligence, resulting in a large number of useful expert systems, as well as a programming language (Prolog) designed to automate the process. Propositional Logic This simplest form of logic is propositional logic. A statement in propostional logic may contain the following: Propositions These are statements which are either true or false. Examples include: "It is sunny." "It is winter." "The patient has a fever" "The patient has the " Logical Connectives These include the "logical operators" associated with most programming languages, such as: Λ V "and" "or" "not" > "implies" also called " if.. then..." <-> "if and only if" Some examples of logical statements include: summer V winter ("It is summer or it is winter") cold Λ summer ("It is cold and it is not summer") winter > cold ("If it is winter, then it is cold") (note however if its cold it doesn't imply its winter) A B A > Β 1 T T T 2 T F F A B B V A F T T F F F

2 Page 2 of 7 3 F T T 4 F F T T T T T F T One important thing to note is that implies can be rewritten in terms of not and or; specifically, A > B may be rewritten as B V A. The output column of the two truth tables show that they are logically equivalent. The relationship A > B is particularly important for certain proofs we will see later. The validility of the statement A > B can be hard to get your head around. Consider that someone tells you " If I am hungry then I eat". One day you find that he is hungry and he's not eating (see #2 in the table)! Obviously the (general) assertion that he made to you about eating when he's hungry isn't true. Note however if you had met him eating when he wasn't hungry (see line #3 in the table) this would not make him a lier because his general statement does not preclude him from eating when he's not hungry. So the assertion A > B is a lie if you find: A is true, and B is false Therefore, if either B is true or A is false then the assertion A > B cannot be proven to be false. But please note that some crazy "If...Then" statements are possible. If (3<5) then (5>3)... This is valid per line #1 above If Paris is in Ireland then Sydney is a lovely city... This is valid per line #3 above This is possible simply because the logic is pure and the statements do not have to make any sence in their association with the real-world. Deductive Methods for Propositional Logic Modus Ponens: And Elimination: Double Negative: Or Introdution: And Introduction: (A > B) Λ A B A1_Λ_Α2_Λ...Λ_Α Λ_Α2_Λ...Λ_Αn Ai A A A1 A1 V A2 V... An A1, Α2,..., Αn A1 Λ Α2 Λ...Λ Αn "if the 'if' part of an implication is true, then so is the 'then' part" (see line #1 of table above) "if a lot of things are true, than each one is indivually true" "if something is true, than any statement "or-ed" with it is also true" "if a bunch of things are true individually, then their conjunction is also true" Unit A V B, B "if A or B is true, and we know that B is not true, then A

3 Page 3 of 7 Resolution: A must be true" Resolution: A V B, B V C A V C If B is true, then C must be true for the B V C part of the statement to be true. If B is false, then A must be true for the A V B part of the statement to be true. For example, consider the following set of facts and rules for a simple diagnosis domain (the knowledge base for that domain): Facts: innocuated, fever Rules: fever > flu V innoculated > Goal: prove flu fever, fever > flu V flu V innoculated, innoculated > flu V, flu Given that we know for certain (the fact) that someone has a fever and we know (the rule) fever > flu V is a valid rule then we can say for certain that its either flu or measels. How Holmes? Because my dear fellow we used the logical deductive method modus ponens. modus ponens modus ponens unit resolution Cannonical Statement Forms In order to automate the process of deduction, it helps greatly if all of are statements are in some "cannonical" or "uniform" form. The idea is to be able to use the same deductive method all of the time, rather than having to choose between several (as in the above example). One such form is based on resolution. This requires that all statements be represented as disjunctions (that is, in the form A V B V...). This can be done by, as noted above, converting statements of the form A > B to B V A. For example, the above knowledge base may be rewritten as: Facts: innocuated, fever Rules: fever > flu V becomes fever V flu V innoculated > becomes innoculated V The above proof may then be rewritten as:

4 Page 4 of 7 fever, fever V flu V flu V innoculated, innoculated V flu V, flu resolution resolution resolution In Practice While this is very simple from a mathematical point of view, it is not a particularly intuitive representation. Users are more comfortable with rules of the form A > B, which is why most expert systems use that form. However, most place some restrictions on the form of these rules, such as: The statement on the right of the > can only be a single proposition. The statement on the right of the > must not be a negation. For example, one way to rewrite the above rules is: Facts: innocuated, fever Rules: fever > flu fever Λ innoculated > Search and Logic Search is an important part of the process of logical deduction, with respect to choosing which rules to apply next. In particular, there are two different ways to approach deduction -- either working forward from facts to conclusions, or backwards from potential conclusions to facts. Forward chaining is the most intuitive way to think about deduction; start with a set of rules and facts and combine them, creating new knowledge. That is, we match the "if" part of each rule against the current set of facts. If there is a match, then the "then" part of the rule is added to the total knowledge: Facts Rules New Knowledge fever, innoculated (fever > flu) fever, flu There are some major weaknesses with forward chaining, however: It is not "goal directed". That is, we usually have some purpose in performing decuction, rather than just to blindly add knowledge. For example, we may want to know whether some specific proposition ("does the patient have ") is true, or which of a small number of propositions is true. If the knowledge base is large enough, a forward chaining algorithm can perfoem a lot of deduction about things that the user considers irrelevant. It forces us to gather all facts at the start of the process in order to perform deductions. This can be be a problem if some of the facts are expensive to gather (such as medical tests).

5 Page 5 of 7 Backward Chaining is often preferred by most expert systems because it allows us to work backwards from goal propositions (such as whether or not the patient has ), gathering facts only when needed to prove/disprove those propositions. The basic ideas are as follows: Suppose that we are currently trying to prove the proposition A: If we have a rule of the form B > A, then B becomes a current proposition to be proven. Or if we have a rule of the form B Λ C > A, then both B and C become a current propositions to be proven. Note that if we fail to prove one, then there is no point in trying to prove the other (which will save time). Or if we have a rule of the form B V C > A, then either B or C become current propositions to be proven. Note that we only try to prove the second if we fail to prove the first one (which will save time). Or if there are no such rules with A on the left side of the >, then A is considered to be a "fact", which we then try to find the value of (usually by asking the user). For example, consider the following knowledge base for loan determination: 1. low_risk V good_assets > loan 2. has_job Λ good_credit Λ high_income > low_risk 3. owns_home V high_balance > good assets Suppose the goal of the expert system is to determine whether loan is true (that is, whether or not to give a loan to this particular applicant). This means that loan is the initial goal to be proven. The search would look like the following: 1. The expert system would search the knowledge base for any rule with loan on the right of the >. It would find rule 1, and make low_risk the current goal (assuming it tries the leftmost proposition of an or first). 2. The expert system would search the knowledge base for any rule with low_risk on the right of the >. It would find rule 2, and make has_job the current goal (assuming it tries the leftmost proposition of an or first). 3. The expert system would search the knowledge base for any rule with low_risk on the right of the >. It would find none, so it would ask the user whether or not the applicant has a job. We will assume that it is told that this is true. 4. The expert system would then make good_credit the current goal, as that is the next proposition in the conjunction. 5. The expert system would search the knowledge base for any rule with good_credit on the right of the >. It would find none, so it would ask the user whether or not the applicant has good credit. We will assume that it is told that this is false. 6. This means that the proposition low_risk is also false. The user would not be asked about high_income. 7. The expert system would then try to prove good_assets, since that is another way to prove loan. 8. The expert system would search the knowledge base for any rule with good_assets on the right of the >. It would find rule 3, and make owns_home the current goal. 9. The expert system would search the knowledge base for any rule with owns_home on the right of the >. It would find none, so it would ask the user whether or not the applicant owns a home. We will assume that it is told that this is true. 10. This means that the proposition good_assets is also true. The user would not be asked about high_balance. 11. This means that loan is also true. Since our goal has been proven, we do no further work.

6 Page 6 of 7 Note that the above process is a depth-first search, in which we explore one branch completely before starting another. This is the algorithm generally used by expert systems. loan / / \ false low_risk good_assets true / \ / \ / \ / \ has_job good_credit high_income owns_home high_balance true false truetrue If there are several possible goals which may or may nor be true, most expert systems simply try to prove them one at a time. For example, if the patient may have either or flu, the expert system would first try to prove, and if that failed, it would try to prove flu (without, of course, reasking any of the qusetions it used to try to prove ). This strategy is sometimes refererred to as generate and test. Finally, most of the complex reasoning done by humans often combines forward and backward chaining: Starting with an initial problem/symptoms/set of facts, do forward chaining to find a set of hypotheses. For example, if a patient comes in complaining of fever, we might forward chain to or flu as potential hypotheses. We would then do backward chaining from those hypotheses in order to gather additional information to prove or disprove them. For example, we might ask the user whether or not innoculated is true in order to prove/disprove the hypotheses of. Expert Systems and Knowledge Acquisition As mentioned above, propositional logic is the basis of expert systems. An expert system is meant to duplicate the knowledge of a human expert in some particular domain. The major components of an expert system are the knowledge base and the inference engine: Knowledge base A knowledge base is a set of rules in some logic (such as propositional logic) which apply to some particular domain. For example, the rules given above are a very simple knowledge base for medicine. Inference engine Software that automates the process of deduction. This usually involves backward chaining from some given goal or set of goals, asking the user questions where necessary, and reporting whether the goal is true or false. One of the nice things about this model is that the same inference engine can be used for any knowledge base. Many expert system shells are complete programs (including inference engines, editors, and user interfaces) that allow a programmer to create a complete expert system for a particular domain by just adding the knowledge base for that domain. The major difficulty with creating an expert system is acquiring the domain knowledge. As mentioned above, most knowlege bases are meant to duplicate the knowledge of a human expert. This means that the domain knowledge must be acquired from that expert, usually by performing interviews of some sort. This has some major problems: The expert probably has little understanding of programming and formal logics. They will

7 Page 7 of 7 almost certainly have difficulty expressing their knowledge in the kind of logical and complet form required for an expert system. The programmer probably has little understanding of the domain, but will still have to try to translate the expert's knowledge into formal rules. There will almost certainly be miscommunication, resulting in errors in the way the knowledge is represented. Since many of the errors will involve the expert knowledge itself, the expert will have to be available during the long process of debugging. This will probably make it very expensive and time consuming.