1. Automated reasoning and theorem proving Introduction: logic in AI Automated reasoning: ResolutionUnificationNormalization

2. Introduction: Motivating example Automated reasoning Logic:Syntax Model semantics Logical entailment

3. 3 The AI dream in the 60’s:  Logic allows to express almost everything ‘formally’.  Logic also allows to prove “theorems” based on the information given.  Can we exploit this to build automated reasoning systems ??

4. 4 Underlying premises:  Logic is the ‘assembly language’ of knowledge and is closely related to natural language.  Since computers are supposed to process the knowledge, it should be expressed formally and unambiguously.  Logical deduction allows us to derive systematically new knowledge from the existing one. In logic almost all kinds of knowledge can be represented formally and unambiguously. Automating deduction ??

5. 5 Example:  The following knowledge is given : 1. Marcus was a man. 2. Marcus was a Pompeian. 3. All Pompeians were Romans. 4. Caesar was a ruler. 5. All Romans were either loyal to Caesar or hated him. 6. Everyone is loyal to someone. 7. People only try to assassinate rulers to whom they are not loyal. 8. Marcus tried to assassinate Caesar.  Can we automatically answer the following questions?  Was Marcus loyal to Caesar?  Did Marcus hate Caesar?

6. 6 Conversion to the First Order Logic:  Representation of facts: 1. Marcus was a man. man(Marcus) 2. Marcus was a Pompeian. Pompeian(Marcus) 4. Caesar was a ruler. ruler(Caesar) 8. Marcus tried to assassinate Caesar. try_assassinate(Marcus, Caesar)

7. 7 5. All Romans were either loyal to Caesar or hated him. Conversion to the First Order Logic (2):  General representation (representation of rules): 3. All Pompeians were Romans.  x Pompeian(x)  Roman(x) 6. Everyone is loyal to someone.  x  y loyal_to(x,y) 7. People only try to assassinate rulers to whom they are not loyal.  x  y person(x)  ruler(y)  try_assassinate(x,y)  ~loyal_to(x,y) ( )  ( )  ~(loyal_to(x,Caesar)  hates(x,Caesar)) XOR  x Roman(x)  loyal_to(x,Caesar)  hates(x,Caesar)

8. 8 Prove that he did: The “theorem” ? Was Marcus loyal to Caesar? Did Marcus hate Caesar? hates(Marcus,Caesar) Try, for example, to prove that he was not : ~loyal_to(Marcus,Caesar)

9. 9 A proof using backward- reasoning problem-reduction: ~loyal_to(Marcus,Caesar)  x  y person(x)  ruler(y)  try_assassinate(x,y)  ~loyal_to(x,y) + substitution: x/Marcus y/Caesar y/Caesar person(Marcus)  ruler(Caesar)  try_assassi- nate(Marcus,Caesar)  ~loyal_to(Marcus,Caesar) + Modus ponens person(Marcus) ruler(Cesar) AND try_assassinate(Marcus, Caesar) Done! 4. Done! 8. Extra rule:  x man(x)  person(x)  person(x) man(Marcus) Done!1.

10. 10 Problems: 1) knowledge representation:  Natural language is imprecise / ambiguous  see “People only try …”  Obvious information is easily forgotten.  see man person  Some information is more difficult to represent in logic.  Vb.: “perhaps …”, “possibly…”, “probably…”, “the chance of … is 45%”,  Logic is inconvenient from a software engineering perspective.  too ‘fine-grained’ (like an assembly language)

11. 11 Problems: 2) Problem solving:  All trade-offs that we had with search methods based on states space representation:  backward/forward, tree/graph, OR-tree/AND-OR, control aspects,...  What deduction rules are needed in general?  Example: prove “ hates(Marcus,Caesar) “  x Roman(x)  loyal_to(x,Caesar)  hates(x,Caesar) The only applicable rule is: Modus ponens???  How do we handle  x and  y ?

12. 12 Problems: 2) Problem solving (2):  How to compute substitutions in the general case ?  x  y person(x)  ruler(y)  try_assassinate(x,y)  ~loyal_to(x,y) + substitution: x/Marcus y/Caesar y/Caesar In general: more complex  Which theorem do we try to prove?  Ex.: loyal_to(Marcus,Caesar) or ~loyal_to(Marcus,Caesar)  How to handle equality of objects?  Problem: combinatorial explosion of the derived equalities (reflexivity, symmetry, transitivity, …)  How to guarantee correctness/completeness?

13. The formal model semantics of Logic The meaning of “Logical entailment”

14. 14 Semantics / Logical entailment:  Given a set of formulas S: a model is an interpretation that makes all formulas in S true.  Given a set of formulas S and a formula F: F is logically entailed by S ( S |= F ), if all models of S also make F true.  Additional: inconsistency:  Given a set of formulas S: S is inconsistent if S has no models. Example: S = { p(a), ~p(a)}

15. 15 Marcus example: x yV Marcus CaesarCA F =  man person ruler Roman Pompeian hates loyal_to try_assassinate P D = world of ~40 VC. “Marcus” “Caesar” “Intended” interpretation: Is a model IF ALL FORMULAS ARE CORRECT Boolean true false “was_ruler” “was_pompeian” Boolean true false

16. 16 Marcus example: x y V Marcus Caesar CA F =  man personruler Roman Pompeian hates loyal_to try_assassinate P N4 3 I(man) = I(person)= I(Roman) = “natural number” I(Pompeian) = “even number” I(ruler) = “prime number” I(try_assassinate) = “ > ” I(loyal_to) = “divides” I(hates) = “doesn’t divide”

17. 17 Model ?? 1. Marcus was a man. 4 is a natural number 2. Marcus was Pompeian. 4 is an even number 4. Caesar was a ruler. 3 is a prime number 8. Marcus tried to assassinate Caesar. 4 > 3 3. All Pompeians were Romans. Even numbers are naturals. 5. All Romans were either loyal to Caesar or hated him. A number either divides 3 or doesn’t divide 3. 6. Everybody is loyal to somebody. Each number is a divisor of some number. 7. People try to assassinate only those rulers to whom they are not loyal. A natural number that is greater than a prime number doesn’t divide the prime number. YES !

18. 18 “Logic is all form, no content” person(x)  mortal(x) person(Socrates) mortal(Socrates) January(x)  cold(x) January(21/1/01) cold(21/1/01) P(x)  Q(x) P(A) Q(A) Only the underlying structure of a set of logical formulas is important for the conclusions! (up to the names isomorphism) But from the knowledge representation perspective also the ‘contents’ is important.