Automated reasoning and theorem proving

Automated reasoning and theorem proving Introduction: logic in AI Automated reasoning: Resolution Unification Normalization

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

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 ??

Logic is the ‘assembly language’ of knowledge and is closely related to natural language. In logic almost all kinds of knowledge can be represented formally and unambiguously. • 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. Automating deduction ?? Underlying premises:

1. Marcus was a man. • Was Marcus loyal to Caesar? • Did Marcus hate Caesar? Example: • The following knowledge is given : 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?

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)

( ) ~(loyal_to(x,Caesar)  hates(x,Caesar)) XOR Conversion tothe First Order Logic (2): • General representation (representation of rules): 3. All Pompeians were Romans. x Pompeian(x)  Roman(x) 5. All Romans were either loyal to Caesar or hated him. x Roman(x)  loyal_to(x,Caesar)  hates(x,Caesar) 6. Everyone is loyal to someone. x yloyal_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)

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

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

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)

The only applicable rule is: x Roman(x)  loyal_to(x,Caesar)  hates(x,Caesar) Modus ponens??? 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) “ • How do we handle x andy ?

xy person(x)  ruler(y)  try_assassinate(x,y)  ~loyal_to(x,y) In general: more complex + substitution: x/Marcus y/Caesar Problems:2) Problem solving (2): • How to compute substitutions in the general case? • 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?

The formal model semantics of Logic The meaning of “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. Semantics / Logical entailment: • Additional: inconsistency: • Given a set of formulas S: S is inconsistent if S has no models. Example:S = { p(a), ~p(a)}

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

A C V Marcus F =  y Caesar x person ruler P man try_assassinate Pompeian Roman loyal_to hates N 3 4 Marcus example: I(man) = I(person)= I(Roman) =“natural number” I(try_assassinate) =“ > ” I(Pompeian) =“even number” I(loyal_to) =“divides” I(ruler) =“prime number” I(hates) =“doesn’t divide”

YES ! 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.

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) “Logic is all form, no content” 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.

Automated reasoning and theorem proving