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sav08:interpretation_quotient_under_congruence [2008/04/16 09:22] maysam |
sav08:interpretation_quotient_under_congruence [2015/04/21 17:30] |
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- | ====== Quotient of an Interpretation under a Congruence ====== | ||
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- | ===== Example: quotient on pairs of natural numbers ===== | ||
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- | Let ${\cal N} = \{0,1,2,\ldots, \}$. Consider a structure with domain $N^2$, with functions | ||
- | \[ | ||
- | p((x_1,y_1),(x_2,y_2)) = (x_1 + x_2, y_1 + y_2) | ||
- | \] | ||
- | \[ | ||
- | m((x_1,y_1),(x_2,y_2)) = (x_1 + y_2, y_1 + x_2) | ||
- | \] | ||
- | Relation $r$ defined by | ||
- | \[ | ||
- | r = \{((x_1,y_1),(x_2,y_2)) \mid x_1 + y_2 = x_2 + y_1 \} | ||
- | \] | ||
- | is a congruence with respect to operations $p$ and $m$. | ||
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- | Congruence is an equivalence relation. What are equivalence classes for elements: | ||
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- | $[(1,1)] = $ ++| $\{ (x,y) \mid x=y \}$++ | ||
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- | $[(10,1)] = $ ++| $\{ (x,y) \mid x=y+9 \}$++ | ||
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- | $[(1,10)] = $ ++| $\{ (x,y) \mid x+9=y \}$++ | ||
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- | Whenever we have a congruence in an interpretation, we can shrink the structure to a smaller one by merging elements that are in congruence. | ||
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- | In the resulting structure $([N^2], I_Q)$ we define operations $p$ and $m$ such that the following holds: | ||
- | \[ | ||
- | \begin{array}{l} | ||
- | I_Q(p)( [(x_1,y_1)] , [(x_2,y_2)] ) = [(x_1 + x_2, y_1 + y_2)] \\ | ||
- | I_Q(m)( [(x_1,y_1)] , [(x_2,y_2)] ) = [(x_1 + y_2, y_1 + x_2)] | ||
- | \end{array} | ||
- | \] | ||
- | This construction is an algebraic approach to construct from natural numbers one well-known structure. Which one? ++| $({\cal Z}, + , -)$ where ${\cal Z}$ is the set of integers. ++ | ||
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- | Note: this construction can be applied whenever we have an associative and commutative operation $*$ satisfying the cancelation law $x * z = y * z \rightarrow x=y$. It allows us to construct a structure where operation $*$ has an inverse. What do we obtain if we apply this construction to multiplication of strictly positive integers? | ||
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- | ===== Definition of Quotient of an Interpretation ===== | ||
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- | (Recall notation in [[First-Order Logic Semantics]].) | ||
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- | Let $I = (D,\alpha)$ be an interpretation of language ${\cal L}$ with $eq \in {\cal L}$ for which [[Axioms for Equality]] hold, that is, $\alpha(eq)$ is a congruence relation for $I$. | ||
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- | For each element $x \in D$, define | ||
- | \[ | ||
- | [x] = \{ y \mid (x,y) \in \alpha(eq) \} | ||
- | \] | ||
- | Let | ||
- | \[ | ||
- | [D] = \{ [x] \mid x \in D \} | ||
- | \] | ||
- | The constructed model will be $I_Q = ([D],\alpha_Q)$ where | ||
- | \[ | ||
- | \alpha_Q(R) = \{ ([x_1],\ldots,[x_n]) \mid (x_1,\ldots,x_n) \in \alpha(R) \} | ||
- | \] | ||
- | In particular, when $R$ is $eq$ we have | ||
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- | $\alpha_Q(eq) = $ ++| $\{ ([x_1],[x_2]) \mid (x_1,x_2) \in \alpha(eq) \} = \{ (a,a) \mid a \in D \}$ | ||
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- | that is, the interpretation of eq in $([D],I_Q)$ is diagonal relation - equality. | ||
- | ++ | ||
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- | Functions are special case of relations: | ||
- | \[ | ||
- | \alpha_Q(f) = \{ ([x_1],\ldots,[x_n],[x_{n+1}]) \mid (x_1,\ldots,x_n,x_{n+1}) \in \alpha(f) \} | ||
- | \] | ||
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- | Interpretation of variables is analogous to interpretation of constants: | ||
- | \[ | ||
- | \alpha_Q(x) = [\alpha(x)] | ||
- | \] | ||
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- | **Lemma 0:** For all $x_1,\ldots,x_n \in D$, | ||
- | \[ | ||
- | ([x_1],\ldots,[x_n]) \in \alpha_Q(R) \mbox{ iff } (x_1,\ldots,x_n) \in \alpha(R) | ||
- | \] | ||
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- | **Lemma 1:** For each function symbol $f$ with $ar(f)=n$, the relation $\alpha_Q(f)$ is a total function $[D]^n \to [D]$ and for all $x_1,\ldots,x_n \in D$, | ||
- | \[ | ||
- | \alpha_Q(f)([x_1],\ldots,[x_n]) = [\alpha(f)(x_1,\ldots,x_n)] | ||
- | \] | ||
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- | **Lemma 2:** For each term $t$ we have $e_T(t)(I_Q) = [e_T(t)(I)]$. | ||
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- | **Theorem:** For each formula $F$ that contains no '=' symbol, we have $e_F(F)(I) = e_F(F)(I_Q) = e_F(F_0)(I_Q)$ where $F_0$ is result of replacing 'eq' with '=' in $F$. | ||