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sav08:axioms_for_equality [2009/05/05 23:16] vkuncak |
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| - | ====== Axioms for Equality ====== | ||
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| - | For language ${\cal L}$ and a relation symbol $eq \notin {\cal L}$, the theory of equality, denoted AxEq, is the following set of formulas: | ||
| - | * Reflexivity: ++| $\forall x. eq(x,x)$ ++ | ||
| - | * Symmetry: ++| $\forall x. \forall y.\ eq(x,y) \rightarrow eq(y,x)$ ++ | ||
| - | * Transitivity: ++| $\forall x. \forall y. \forall z.\ eq(x,y) \land eq(y,z) \rightarrow eq(x,z)$ ++ | ||
| - | * Congruence for function symbols: for $f \in {\cal L}$ function symbol with $ar(f)=n$, ++ | | ||
| - | \[ | ||
| - | \forall x_1,\ldots,x_n, y_1,\ldots,y_n.\ (\bigwedge_{i=1}^n eq(x_i,y_i)) \rightarrow eq(f(x_1,\ldots,x_n),f(y_1,\ldots,y_n)) | ||
| - | \] | ||
| - | ++ | ||
| - | * Congruence for relation symbols: for $R \in {\cal L}$ relation symbol with $ar(R)=n$, ++ | | ||
| - | \[ | ||
| - | \forall x_1,\ldots,x_n, y_1,\ldots,y_n.\ (\bigwedge_{i=1}^n eq(x_i,y_i)) \rightarrow (R(x_1,\ldots,x_n) \leftrightarrow R(y_1,\ldots,y_n)) | ||
| - | \] | ||
| - | ++ | ||
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| - | **Definition:** if an interpretation $I = (D,\alpha)$ the axioms $AxEq$ are true, then we call $\alpha(eq)$ (the interpretation of eq) a //congruence// relation for interpretation $I$. | ||
| - | |||
| - | **Side remark:** Functions are relations. However, the condition above for function symbols is weaker than the condition for relation symbols. If $f$ is a function, then the relation $\{(x_1,\ldots,x_n,f(x_1,\ldots,x_n)) \mid x_1,\ldots,x_n \in D \}$ does not satisfy the congruence condition because it only has one result, namely $f(x_1,\ldots,x_n)$, and not all the results that are in relation eq with $f(x_1,\ldots,x_n)$. | ||
| - | |||
| - | ===== References ===== | ||
| - | * [[Calculus of Computation Textbook]], Section 3.1 | ||