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Unformatted text preview: COM SFWR 707: Formal Specification Techniques Dr. R. Khedri Outline Introduction Basic Definitions Lterms, Interpretation, Lformulas, and Satisfiability Constructions Elementary Equivalence and Isomorphism COM SFWR 707: Formal Specification Techniques Dr. Ridha Khedri Department of Computing and Software, McMaster University Canada L8S 4L7, Hamilton, Ontario COM SFWR 707: Formal Specification Techniques Dr. R. Khedri Outline Introduction Basic Definitions Lterms, Interpretation, Lformulas, and Satisfiability Constructions Elementary Equivalence and Isomorphism 1 Introduction 2 Basic Definitions Maps that preserve the interpretation of L 3 Lterms, Interpretation, Lformulas, and Satisfiability 4 Constructions LSubstructure (revisited) L Quotient Structure Direct Product Structure 5 Elementary Equivalence and Isomorphism 6 Theories 7 Definable Sets and Interpretability 8 The Compactness Theorem 9 Complete Theories COM SFWR 707: Formal Specification Techniques Dr. R. Khedri Outline Introduction Basic Definitions Lterms, Interpretation, Lformulas, and Satisfiability Constructions Elementary Equivalence and Isomorphism Structures and Theories Introduction In mathematical logic, we use firstorder languages to describe mathematical structures Intuitively, a structure is a set that we wish to study equipped with a collection of distinguished functions , relations , and elements After that, we choose a language where we can talk about them (Funct., rel., and elements) and nothing more COM SFWR 707: Formal Specification Techniques Dr. R. Khedri Outline Introduction Basic Definitions Lterms, Interpretation, Lformulas, and Satisfiability Constructions Elementary Equivalence and Isomorphism Structures and Theories Introduction Example When we study the ordered field of real numbers with the exponential function We study the structure h R , + , , exp , <, , 1 i What are the components of this structure? We would use a language where we have symbols for + , , exp , <, , 1 We can write statements such as: ( x , y  x , y R : exp( x ) exp( y ) = exp( x + y ) ) That we interpret as the assertion: e x e y = e ( x + y ) for all x and y in real numbers. COM SFWR 707: Formal Specification Techniques Dr. R. Khedri Outline Introduction Basic Definitions Maps Lterms, Interpretation, Lformulas, and Satisfiability Constructions Elementary Equivalence and Isomorphism Structures and Theories Basic Definitions Definition ( Language ) A language L is given by specifying the following data: 1 a set of function symbols F and positive integers n f for each f F 2 a set of relation symbols R and positive integers n R for each R R 3 a set of constant symbols C ....
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This note was uploaded on 03/31/2010 for the course CAS 707 taught by Professor Ridhakhedri during the Spring '10 term at McMaster University.
 Spring '10
 RidhaKhedri

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