Discrete Mathematics and Logic II. Predicate Calculus

Transcription

1 Discrete Mathematics and Logic II. Predicate Calculus SFWR ENG 2FA3 Ryszard Janicki Winter 2014 Acknowledgments: Material based on A Logical Approach to Discrete Math by David Gries and Fred B. Schneider (Chapter 9). Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 1 / 21

2 Introduction Predicate logic is an extension of propositional logic It allows the use of variables of types other than IB This extension leads to a logic with enhanced expressive Propositional calculus permits reasoning about formulas constructed from boolean variables and boolean operators Predicate calculus permits reasoning about a more expressive class of formulas Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 2 / 21

3 Introduction A predicate-calculus formula is a boolean expression in which some boolean variables may have been replaced by: Predicates : applications of boolean functions whose arguments may be of types other than IB Universal and existential quantication Example 1 x < y x = z = q(x, z + x) 2 (x, y : f (x, y) = f (y, x) ) Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 3 / 21

4 Introduction The pure predicate calculus includes 1 The axioms of propositional calculus 2 The axioms for quantications (x R : P ) and (x R : P ) 3 The inference rules of the predicate calculus 1 Substitution: E E[v:=F ] 2 Transitivity: X =Y, Y =Z X =Z 3 Leibniz: X =Y E[z:=X ]=E[z:=Y ] 4 Leibniz for quantication: P=Q (x E[z:=P] : S ) = (x E[z:=Q] : S ) R = P=Q (x R : E[z:=P] ) = (x R : E[z:=Q] ) Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 4 / 21

5 Introduction In the pure predicate calculus, the function symbols (<, >, +,, etc.) are uninterpreted except for equality = the logic provides no specic rules for manipulating function symbols general rules for manipulation that are sound independently of the meanings of the function symbols so, the pure predicate calculus is sound in all domains that may be of interest Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 5 / 21

6 Introduction We get a theory by adding axioms that give meanings to some of the (uninterpreted) function symbols The theory of integers consists of the pure predicate calculus together with axioms for manipulating the operators +,,., <,, etc. For example, the axioms say that is symmetric, associative, and has the zero 0 Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 6 / 21

7 Introduction The theory of sets provides axioms for manipulating expressions containing operators like,,,, etc. We can also form a joint theory of sets and integers, allowing us to reason about expressions that contain both The core of all these theories is the pure predicate calculus (provides the basic machinery) Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 7 / 21

8 Universal quantication Conjunction (i.e., the identity true ) is symmetric and associative and has Therefore, it is an instance of * (previous set of slides) The quantication (x R : P ) is conventionally written as (x R : P ) The symbol, which is read as "for all", is called the universal quantier All the axioms of quantition hold We now introduce additional axioms and theorems for universal quantication Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 8 / 21

9 Universal quantication Axiom, Trading: (x R : P ) (x : R = P ) Trading theorems for : 1 (x R : P ) (x : R P ) 2 (x R : P ) (x : R P R ) 3 (x R : P ) (x : R P P ) Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 9 / 21

10 Universal quantication Trading theorems for (continued): 1 (x Q R : P ) (x Q : R = P ) 2 (x Q R : P ) (x Q : R P ) 3 (x Q R : P ) (x Q : R P R ) 4 (x Q R : P ) (x Q : R P P ) Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 10 / 21

11 Universal quantication Axiom, Distributivity of over : Provided occurs( x, P ), P (x R : Q ) (x R : P Q ) Additional theorems for : 1 Provided occurs( x, P ), (x R : P ) P (x : R ) Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 11 / 21

12 Universal quantication Additional theorems for (Continued): 1 Provided occurs( x, P ), (x : R ) = ( (x R : P Q ) P (x R : Q )) 2 (x R : true ) true 3 (x R : P Q ) = ( (x R : P ) (x R : Q )) Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 12 / 21

13 Universal quantication Theorems: Weakening/strengthening and monotonicity for 1 Range weakening/strengthening : (x Q R : P ) = (x Q : P ) 2 Body weakening/strengthening: (x R : P Q ) = (x R : P ) 3 Monotonicity of : (x R : Q = P ) = ( (x R : Q ) = (x R : P )) Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 13 / 21

14 Existential quantication Disjunction is symmetric and associative and has the identity false Therefore, it is an instance of The quantication (x R : P ) is typically written as (x R : P ) The symbol, which is read as "there exists", is called the existential quantier Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 14 / 21

15 Existential quantication The expression is called an existential quantication (x R : P ) is read as "there exists an x in the range R such that P holds" A value ˆx for which (R P)[x := ˆx] is valid is called a witness for x in (x R : P ) General axioms, from previous set of slides, that hold for (x R : P ) not repeated here Note that is idempotent, so that existential quantication satises Range split for idempotent Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 15 / 21

16 Existential quantication Idea behind this generalization with an example (i 1 i 2 : p i ) (i 1 i 2 : p i ) (i 1 i 4 : p i ) (i 1 i 4 : p i ) The generalisation of De Morgan can be viewed as a denition of It can be used along with the strategy of denition elimination, to prove all theorems concerning existential quantication Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 16 / 21

17 Existential quantication Axiom, Generalised De Morgan: (x R : P ) (x R : P ) Generalised De Morgan theorems: 1 (x R : P ) (x R : P ) 2 (x R : P ) (x R : P ) 3 (x R : P ) (x R : P ) Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 17 / 21

18 Existential quantication Theorems: Trading theorems for 1 (x R : P ) (x : R P ) 2 (x Q R : P ) (x Q : R P ) More theorems for 1 Distributivity of over : Provided occurs( x, P ) P (x R : Q ) (x R : P Q ) 2 Provided occurs( x, P ) (x R : P ) P (x : R ) Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 18 / 21

19 Existential quantication Theorems: More theorems for (Continued) 1 Distributivity of over : Provided occurs( x, P ) (x : R ) = ( (x R : P Q ) P (x R : Q )) 2 (x R : false ) false Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 19 / 21

20 Existential quantication Theorems: Weakening/strengthening and monotonicity for 1 Range weakening/strengthening : (x R : P ) = (x Q R : P ) 2 Body weakening/strengthening: (x R : P ) = (x R : P Q ) 3 Monotonicity of : (x R : Q = P ) = ( (x R : Q ) = (x R : P )) Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 20 / 21

21 Universal quantication 1 -Introduction: P[x := e] = (x : P ) 2 Interchange of quantications: Provided occurs( x, Q ), = (x R : (y Q : P ) ) (y Q : (x R : P ) ) Ryszard Janicki Discrete Mathematics and Logic II. Predicate Calculus 21 / 21