Announcements. Today s Menu

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Cleopatra Wheeler
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1 Announcements Reading Assignment: > Nilsson chapters Announcements: > LISP and Extra Credit Project Assigned Today s Handouts in WWW: > Homework > Outline for Class 26 > > Software and Notes 1 Today s Menu Informal Introduction to the Predicate Calculus The Propositional Calculus > Expressing Constraints in Feature Values > The Language > Rules of Inference > Proofs > Semantics Interpretations Truth Tables Satisfiability and Models Validity Equivalence Entailment 2

2 > Plusses (+) Concise rich notation. Universally understood A lot is known about its limitations: decidability, admissibility We can place bounds on expected performance > Minuses (-) May direct attention away from the problem specification phase Many problems do not map well to mathematical analysis > Concepts Notation: e.g., x y P(x) Q(y) Proofs by Refutation Rules of Inference modus ponens modus tolens resolution Operators Alternatives for Proofs: constraint propagation and justification 3 > A mathematical language that allows us to specify Rules of Inference as ways (algorithms) to reason. Example 1 It is a bird it it has feathers, or it flies and lays eggs Rule 1a: If animal has feathers then animal is bird Rule 1b: If animal flies and animal lays eggs then animal is bird > In Robotics we need to capture the idea that something has properties. Definition: A predicate is a function that returns (whose range is) {T,F} for a specified domain. Example 2 In the domain of birds these are true predicates Feathers(Robin) Bird(Robin) 4

3 > To say that Feathers(Julie) is true means that Julie is constrained to what it stands for. Other constraints are: Flies(Julie) & Lays-Eggs(Julie) Expressions joined by & or by are called conjunctions Expressions joined by or by are called disjunctions > Logical Connectives are: ~, & ( ), ( ), and ( ) Example 3 ~Feathers(Sue) > Sue is an object for which Feathers(Sue) is not satisfiable (i.e., cannot be made true) 5 Example 4 Feathers(Squicky) Bird(Squicky) note: F G F G F G or better yet Feathers(X) Bird(X) X is an object in the domain of interest constrained to satisfy the following definition: If Feathers(X) then Bird(X) else true > Example 4 is true if Feathers(X) is true and if Bird(X) is true > However the definition allows Feathers(X) and Bird(X) to be both false > and the definition allows Feathers(X) is false and Bird(X) to be true > If Feathers(X) is true and Bird(X) is false then Example 4 is false A K-Map of E 1 E 2 demonstrates that it is equivalent to E 1 E 2 E 2 E E 1 E 2 E 1 E

4 > Precedence Rules: Use parentheses ( ) to clarify ~ has the highest precedence & ( ), ( ) equal precedence and lower than ~ ( ) lowest precedence > Logical Connectors are commutative: E 1 E 2 E 2 E 1 E 1 E 2 E 2 E 1 > Logical Connectors are distributive: E 1 (E 2 E 3 ) (E 1 E 2 ) (E 1 E 3 ) E 1 (E 2 E 3 ) (E 1 E 2 ) (E 1 E 3 ) > Logical Connectors are associative: E 1 (E 2 E 3 ) (E 1 E 2 ) E 3 E 1 (E 2 E 3 ) (E 1 E 2 ) E 3 > Identity Property: ~ ~ E 1 = E 1 > De Morgan s Law: ~(E 1 E 2 ) ~E 1 ~E 2 ~(E 1 E 2 ) ~E 1 ~E 2 7 > Quantifiers (, ) are used to determine when things are true Example 5 ( x)[feathers(x) Bird(x)] Quantifier variable For all objects x in the domain of interest (x D) we get a true expression when we substitute any object x inside the square brackets. Notice a sort of domain independence, i. e., Anything that has feathers is a bird (in the domain of interest) > The expression surrounded by square brackets associated with a quantifier is said to lie in the scope of the quantifier. (In other words Feathers(x) Bird(x) lies in the scope of ( x)[ ] ) > is called the universal quantifier, it is true for all objects > is called the existential quantifier, it is true for some objects (at least 1 object). It is pronounced there exists. 8

5 > ( x)bird(x) there exists an x (at least one) for which Bird(x) is true > Vocabulary of the PC Terms Domain Objects (A,B, Robin, etc.) Variables (x, y, z, w,...) they range over domain objects Functions of objects that return objects, e. g., f(x) Arguments to functions, e.g., (x), (x,y), (x,y,z), etc. Arguments to predicates, e.g., (x), (h(x)), (x,f(z)), etc. Atomic Formulas (Atf) the atoms T or F Predicates with Arguments, e.g., P(x), Q(w,y) <term>=<term> or <term> <term> Literals Atf or ~Atf Well-Formed-Formula (wff) is any literal or <wff> {& ( ), ( ), ( )} <wff> ( v i )[<wff>] ( v i )[<wff>] ~ <wff> 9 > Sentences are closed wffs a wff that has no free variables Example 6 6a ( x)[feathers(x) Bird(x)] 6b Feathers(Oriole) Bird(Oriole) x in 6a is said to be bound, that is, it appears within the scope of its corresponding quantifier. Further, variables that are not bound are free. Both 6a and 6b are sentences. Example 7 ( x)[feathers(x) ~ Feathers(y)] This example is not a sentence. Can you explain why? > If variables are allowed to represent only objects the logic is 1 st order. It is called the First Order Predicate Calculus. > Second Order Predicate Calculus: variables allowed to represent predicates and object functions. > ZOPC/Boolean Algebra/Propositional Calculus - No variables 10

6 > Clause: A wff consisting of a disjunction of literals, e.g., ( x)[f(x) B(x)] ( P)[P(x) Q(x)] ( f)[~p(x) H(x, f(x))] > Interpretations: Tie logic symbols to words. The goal is to say something about the world, e.g., ON(A,B). We do this by associating functions, predicates and objects with tangible things. Thus, objects in some domain D correspond to object symbols in logic. Example 9 Logic Side Real World Symbolic Object A Block A P(A,B) Symbolic Object B Block B _ ON(B,A) implies the relation that block B is on block A P(x,y) is read as x P y 11 B Functions B=whats_on(A) A B=is_on(A) is_whats_on(b,a) Def: Interpretation: A triple written as I[D, I v, I c ] which is a full accounting of the correspondence between the logic world and the real world. D is the domain of interest, I v the assignments to variables and I c is the assignment to constants. Real World Logic objects object symbols relations predicates object functions symbolic object functions > BIG CONCEPT: Proofs tie axioms to consequences Suppose we are given a set of wffs which are assumed to be true, this set is called axioms (or given/assumed) set. 12

Example 9 9a Feathers(Julie) Γ: 9b ( x)[feathers(x) Bird(x)] When we say these are axioms we are restricting the interpretations to the objects, symbols, predicates and functions for which the

7 Example 9 9a Feathers(Julie) Γ: 9b ( x)[feathers(x) Bird(x)] When we say these are axioms we are restricting the interpretations to the objects, symbols, predicates and functions for which the implied imaginable world holds. These interpretations for which the axiom set holds are said to models of the wffs. > Suppose we are asked to show that all interpretations that make the axioms true make some other given wff true also. If we succeed, we say we have proven that this extra wff is a theorem with respect to the axioms. We prove that a wff is a theorem when we show that the theorem MUST be true give that the axioms are true. We prove that a wff is a theorem when every model for the axioms is also a model for the wff. If so, we say the wff logically follows from the axioms or alternatively, the axioms logically imply the wff. 13 > Given the set of axioms in Ex. 9 can we prove Bird (Julie) is a theorem w.r.t. the axioms? Yes! HOW? > We need a procedure or algorithm to do it. A proof is a procedure consisting of manipulations based on equivalences which are called Sound Rules of Inference which produce a new wff from old ones guaranteeing that models which make the old wffs true also make the new (derived) wffs true. > A straightforward proof procedure applies sound rules of inference to the axioms recursively until the desired wff is produced. Do you like this? Why or why not? 14

> Proving a theorem is not the same as showing that a wff is valid. Why? A valid wff is true independent of interpretation > Proving a theorem is not the same as showing that a wff is satisfiable.

8 > Proving a theorem is not the same as showing that a wff is valid. Why? A valid wff is true independent of interpretation > Proving a theorem is not the same as showing that a wff is satisfiable. Why? A satisfiable wff is true for some interpretation > Modus Ponens : the most straightforward sound rule of inference when applied recursively or successively. Def: Given the set Γ : {E 1 E 2 ; E 1 } then E 2 is a theorem w.r.t. Γ thus, we can add E 2 to the axioms if it was obtained by mp Example 10 Using mp prove Bird(Julie) Γ: 10a Feathers(Julie) 10b ( x)[feathers(x) Bird(x)] Since Γ is true for all x D we can let x=julie and obtain 10b Feathers(Julie) Bird(Julie) and thus Bird(Julie) m.p. 15 > We observe that this proof was merely a syntactic (mechanical) procedure. We obtained Bird(Julie) because it follows from the axioms logically. The possibility exists that the given axioms or derived theorem may clash with our sense of the world. > Modus Tolens : A sound rule of inference when applied recursively or successively produces a derived wff as follows: Def: Given the set Γ: {E 1 E 2 ; ~E 2 } then ~E 1 is a theorem w.r.t. Γ we can add ~E 1 to the axioms if it was obtained by mt Example 11 11a ~Bird(Larry) 11b ( x)[feathers(x) Bird(x)] Since Γ is true for all x D we can let x=larry and obtain 11b Feathers(Larry) Bird(Larry) and therefore ~Feathers(Larry) m.t. 16

9 > Resolution : A sound rule of inference when applied recursively or successively produces a derived wff as follows: Def: Given the set Γ: {~E 1 E 2 ; E 1 E 3 } then E 2 E 3 is a theorem w.r.t. Γ we can add E 2 E 3 to the axioms if it was obtained by Resolution Resolution subsumes both mp and mt > Γ: {~E 1 E 2 ; E 1 E 3 } then E 2 E 3 let E 3 =null and Resolution=mp > Γ: {~E 1 E 2 ; E 1 E 3 } then E 2 E 3 let E 1 =~E 2 and E 3 =~E 1 and Resolution=mt [check: {E 2 ; ~E 2 ~E 1 } yields ~E 1 Q.E.D.] 17 The End! 18

Přednáška 12. Důkazové kalkuly Kalkul Hilbertova typu. 11/29/2006 Hilbertův kalkul 1

Přednáška 12. Důkazové kalkuly Kalkul Hilbertova typu. 11/29/2006 Hilbertův kalkul 1 Přednáška 12 Důkazové kalkuly Kalkul Hilbertova typu 11/29/2006 Hilbertův kalkul 1 Formal systems, Proof calculi A proof calculus (of a theory) is given by: A. a language B. a set of axioms C. a set of