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# Slides and Assignments - COM SFWR 707 Formal Specication...

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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

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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

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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 R , + , · , exp , <, 0 , 1 What are the components of this structure? We would use a language where we have symbols for + , · , exp , <, 0 , 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 . τ = F , R , C , n F , n R The numbers n f and n R tell us that f is a function of n f variables and R is an n R -ary relation Any or all of the sets F , R and C may be empty

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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 Example ( Language
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• Spring '10
• RidhaKhedri
• Equivalence relation, Formal specification techniques, Constructions Elementary Equivalence and Isomorphism

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Slides and Assignments - COM SFWR 707 Formal Specication...

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