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Slides and Assignments - COM SFWR 707: Formal Specification...

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Unformatted text preview: COM SFWR 707: Formal Specification Techniques Dr. R. Khedri Outline Introduction Basic Definitions L-terms, Interpretation, L-formulas, 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 L-terms, Interpretation, L-formulas, and Satisfiability Constructions Elementary Equivalence and Isomorphism 1 Introduction 2 Basic Definitions Maps that preserve the interpretation of L 3 L-terms, Interpretation, L-formulas, and Satisfiability 4 Constructions L-Substructure (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 L-terms, Interpretation, L-formulas, and Satisfiability Constructions Elementary Equivalence and Isomorphism Structures and Theories Introduction In mathematical logic, we use first-order 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 L-terms, Interpretation, L-formulas, 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 L-terms, Interpretation, L-formulas, 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.

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Slides and Assignments - COM SFWR 707: Formal Specification...

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