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

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

Why Logic is needed in AI? Solutions to many problems might require logical analysis In order to automate logical reasoning, we need a formal language So, we need a language that can represent a problem and a method to search the problem 3

Logic and AI: The Basics Representation => predicate calculus Search => resolution principle 4

Things We Will Do With Logic A two step process 1. See if there is a solution 2. Find the solution 5

“Marcia is wherever John is” and “John is at school” Example Problems John and Marcia given “Marcia is wherever John is” and “John is at school” we will find where Marcia is! 6

Resolution Proofs - Example Given: ( x) {AT(John, x) AT(Marcia, x)} AT(John, school) Prove: ( x) AT(Marcia, x) i.e., “where is Marcia?” 7

Resolution Proof - Finally! ~AT(John, x) AT(Marcia, x) ~AT(Marcia,x) ~AT(John, x) AT(John, school) The negation is false; Marcia must be somewhere. But, where! nil 8

Modified Proof Tree ~AT(John, x) AT(Marcia, x) ~AT(Marcia,x)  AT(Marcia,x) ~AT(John, x)  AT(Marcia,x) AT(John, school) AT(Marcia, school) 9

Why Logical is good Representation? The principles of correct reasoning i.e. sound and complete inference rules and semantics of predicate calculus emphasizes truth preserving operations on well formed expression.

Problem Solving Requires sufficient knowledge Makes correct inference from this knowledge Must do inference sufficiently

Building Knowledge Select the significant objects and relation in the domain Map these into a formal language

Four Categories of Knowledge Representation Logical representation schemes Procedural representation schemes Network representation schemes Structured representation schemes

Four Categories of Knowledge Representation

H/W Assignment for Assignment group 2 G2: What does “sound and complete inference rules” mean? NAiST-LAB

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

Underlying premises: 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 ??

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?

Question Is there any ambiguity of given statement? Is there any problem about predicate selection?

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)

Conversion to the 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. ( )  ~(loyal_to(x,Caesar)  hates(x,Caesar)) XOR x Roman(x)  loyal_to(x,Caesar)  hates(x,Caesar) 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)

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

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

Way of Representing Class Membership 1. man(Marcus) 2. Pompeian(Marcus) 3. x: Pompeian(x) Roman(x) 4. ruler(Caesar) 5. x: Roman(x) loyalto(x,Caesar) hate(x,Caesar)

Way of Representing Class Membership 1. instance(Marcus,man) 2. instance(Marcus,Pompeian) 3. x: instance(x, Pompeian) instance(x, Roman) 4. instance(Caesar,ruler) 5. x: instance(x, Roman) loyalto(x,Caesar) hate(x,Caesar) CS 6833/CS 680 NAiST-LAB NAiST-LAB

Way of Representing Class Membership 1. instance(Marcus,man) 2. instance(Marcus,Pompeian) 3. isa(Pompeian,Roman) 4. instance(Caesar,ruler) 5. x: instance(x, Roman) loyalto(x,Caesar) hate(x,Caesar) 6. x: y: z: instance(x, y) isa(y,z) instance(x, z) CS 6833/CS 680 NAiST-LAB NAiST-LAB

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)

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 ?

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 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?

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

Propositional logic

Basic concepts: In propositional logic:    ~   connectors ( ) The alphabet:    ~   connectors weather_is_rainy joe_has_an_umbrella Atomic propositions ( ) punctuation joe_has_an_umbrella  weather_is_rainy ~ weather_is_rainy Well-formed formulas: Notation: p, q, r : atomic propositions.

Semantics (meaning) 1. Intuitive (natural) semantics: In general (for all knowledge representation formalisms): 2 approaches to define semantics: 1. Intuitive (natural) semantics: Describe the meaning by means of a natural language Exs. (propositional logic):  : “implies” ~ : “not true that”  : “or” p  q : “p if and only if q” ~ p  r : “not p and r” every symbol and every well-formed formula gets meaning through the associated natural language

Semantics (2) 2. Transformational semantics: Logically entailed p Describe the meaning by converting to an associated “mathematical” object In propositional logic : the set of all propositional symbols that are logically entailed by the given formulas: Logically entailed p p  ~ q p q  r q r p q p r q r p q r

But how to define “logical entailment” ? NOT as: Everything that we can derive from the formulas SINCE: At this moment we do not know yet: “what is a complete set of the derivation rules” This is exactly what Automated Reasoning aims to find out! BUT by: Interpretations Models

Interpretation: = a function that assigns a truth value to each “atomic” formula Also provides (indirectly) truth values for each well-formed formula p q ~p p  q p  q p  q p  q T F T T F F T F T T F F T F Truth table

Model Given a set of formulas S: a model is an interpretation that makes all formulas in S true p  ~ q p q  r S Example: p q r T F Model: p  ~q q  r T p

Logical entailment: 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. p  ~ q p q  r S Example: F: r  p p q r T F (the only) Model: r  p T

Predicate logic

Well-formed formulas: The alphabet:  constants Yvonne Yvette variables x y z  ~   connectors  function symbols g f h  quantifiers  predicate symbols q p ( ) punctuation , Terms: Yvonne y f(x, g(Yvette)) Well-formed formulas: p(Yvonne) p(x)  ~q(x) z p(z) x y p(x)  q(y)  p(f(Yvonne, Yvette)

Formally: An alphabet consists of variables, constants, function symbols, predicate symbols (all user-defined) and of connectors, punctuations en quantifiers. Terms are either: variables, constants, function symbols provided with as many terms as arguments, as the function symbol expects. Well-formed formulas are constructed from predicate symbols, provided with terms as arguments, and from connectors, quantifiers and punctuation – according to the rules of the connectors.

Example: { {0}, {x,y}, {s}, {odd,even}, Con, Pun, Quan} Alphabet: { {0}, {x,y}, {s}, {odd,even}, Con, Pun, Quan} Terms: { 0, s(0), s(s(0)), s(s(s(0))), … x, s(x), s(s(x)), s(s(s(x))), … y, s(y), s(s(y)), s(s(s(y))), … } Well-formed formulas: odd(0), even(s(0)), … odd(x), odd(s(y)), … odd(x)  even(s(s(x))), … x ( odd(x)  even(s(x)) ), … odd(y)  x (even( s(x))), ...

Interpretation: A D = N Example: = a set D (the domain), and a function that maps constants to D, and a function that maps function symbols to functions: D -> D, and a function that maps predicate symbols to predicates: D -> Booleans. A s x y odd Example: Booleans true false “even” D = N 1 2 3

Assigning truth values: 1. To ground atomic formulas: form: p(f(a), g(a,b)) Example: I( odd( s ( s( 0 ) ) ) ) ) = ? = I (odd) ( I(s) ( I(s) ( I(0) ) ) ) = “even” ( succ ( succ ( 0 ) ) ) = “even” ( succ ( 1 ) ) = “even” ( 2 ) = true

Assigning truth values (2): 2. For closed well-formed formulas: (= no non-quantified variables) x F(x) is true if: for all d D: I( F(d) ) = true  x F(x) is true if: there exists d D such that: I( F(d) ) = true further: use the truth tables. I(x odd( s ( x ) )  odd(x) ) = ? = true if for all d in N: I (odd( s (d) )  odd(d) ) = true Example: = “even” ( succ(d) )  “even” (d) Assume: d = 0, then: = false  true Truth tables: false !

Semantics / Logical entailment: Exactly as in propositional logic ! 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)}

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

Marcus example: A x y V Marcus Caesar C F =  man person ruler Roman Pompeian hates loyal_to try_assassinate P N 4 3 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”

Model ?? YES ! 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.

“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.

Relation with respect to other courses:

Logic as a foundation for AI A much deeper and more formal study of Logic for Knowledge Representation and Reasoning !

Programming Languages and Programming Methodologies Logic-based programming languages (Prolog/CLP)

Selected Topics in Logic Programming Formal studies of semantics and formal methods (analysis, termination) of logic-based programming languages (also beyond Prolog)

Fundamentals of Artificial Intelligence First Order predicate Logic Formal semantics and analysis Logic-based programming languages (Prolog/CLP) Techniques for automated reasoning Problem-representation in logic Mostly in the exercises

Methodology for knowledge representation? Very complicated. Not many simple guidelines. Basis: Choose an alphabet that allows us to represent all objects and all relations from the problem domain: What are the basic objects, functions and relations in your problem domain ? Ontology: represent only the RELEVANT information Choose constants, function and predicate symbols to represent them. Translate every sentence in a natural language in one or more corresponding logical formulas.

Summary

Syntax of Predicate Calculus Universe - The domain of discourse, D, corresponding to the set of objects represented by logical variables. Terms - Variables are terms and f(T) is a term, where f is a function and T is a sequence of n terms. Atomic formula - P(T) is an atomic formula, where P is a predicate and T is a sequence of n terms. Literals - Atomic formulas and negated atomic formulas.

Syntax of Predicate Calculus Well-formed formulas (wffs) - Literals are wffs and wffs connected or quantified by logic symbols are also wffs. Sentence - A wff in which all variables are within the scope of corresponding quantifiers. Clause - A wff consisting of a literal or a disjunction of literals (literals connected by ORs).

Syntax of Predicate Calculus A system in first-order predicate calculus consists of a set of sentences S which represent a physical system. The system builder has complete freedom in selecting predicates and building expressions to represent the problem s/he is trying to solve.

Proposition โธมัส อัลวา เอดิสัน อัจฉริยะนั้น 99% เกิดจากความพากเพียร 1% ขึ้นอยู่ กับมันสมอง ความสำเร็จ 10% เกิดจากแรงบันดาลใจ 90% มา จากความทุ่มเทด้วยหยาดเหงื่อ