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sav07_lecture_22 [2008/05/21 01:17] vkuncak |
sav07_lecture_22 [2015/04/21 17:32] (current) |
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| ==== Semantic ==== | ==== Semantic ==== | ||
| - | We define the semantic of our syntax recursively by defining each of its elements over a language $L = \lbrace a, \ldots, f_1, \ldots, g_1, \ldots \rbrace$. We first have that : \[ [[S]] \in H \] where $H$ is the Herbrand univers, or "set of all ground terms", or also called the "term model". As we already saw, it's the set of all possibly constructed trees using language $L$ : \[ H = \lbrace a, f_1(a, a), f_1(g_1(a, a), a), \ldots \rbrace \] | + | We define the semantic of our syntax recursively by defining each of its elements over a language $L = \lbrace a, \ldots, f_1, \ldots, g_1, \ldots \rbrace$. We first have that : \begin{equation*} [[S]] \in H \end{equation*} where $H$ is the Herbrand univers, or "set of all ground terms", or also called the "term model". As we already saw, it's the set of all possibly constructed trees using language $L$ : \begin{equation*} H = \lbrace a, f_1(a, a), f_1(g_1(a, a), a), \ldots \rbrace \end{equation*} |
| Given that, if the meaning of $v$ is given (i.e. we have $[[v]] = \alpha(v) \subseteq H$), then we can estimate the meaning of our language as following : | Given that, if the meaning of $v$ is given (i.e. we have $[[v]] = \alpha(v) \subseteq H$), then we can estimate the meaning of our language as following : | ||
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| ==== ==== | ==== ==== | ||
| - | An important property we would like to have in this semantic is a very intuitive one : \[ [[f^{-1}(f(S_1, S_2))]] = [[S_1]] \] | + | An important property we would like to have in this semantic is a very intuitive one : \begin{equation*} [[f^{-1}(f(S_1, S_2))]] = [[S_1]] \end{equation*} |
| This property is in fact not conserved as we can easily see using a very simple counter-example : | This property is in fact not conserved as we can easily see using a very simple counter-example : | ||
| - | \[ [[f^{-1}(f(S_1, \emptyset))]] = [[f^{-1}(\lbrace f(t_1, t_2) | t_1 \in [[S_1]] \wedge t_2 \in \emptyset \rbrace)]] = [[f^{-1}(\emptyset)]] = \emptyset \] | + | \begin{equation*} [[f^{-1}(f(S_1, \emptyset))]] = [[f^{-1}(\lbrace f(t_1, t_2) | t_1 \in [[S_1]] \wedge t_2 \in \emptyset \rbrace)]] = [[f^{-1}(\emptyset)]] = \emptyset \end{equation*} |
| The correct interpretation of this property is : | The correct interpretation of this property is : | ||
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| From this we can deduce, as we defined previously, the following contraints : | From this we can deduce, as we defined previously, the following contraints : | ||
| - | \[\begin{array}[c] | + | \begin{equation*}\begin{array}[c] |
| \left { \lambda_x \right } \subseteq [[\lambda x.x]] \\ | \left { \lambda_x \right } \subseteq [[\lambda x.x]] \\ | ||
| \left { \lambda_y \right } \subseteq [[\lambda y.y]] \\ | \left { \lambda_y \right } \subseteq [[\lambda y.y]] \\ | ||
| \lambda_x \subseteq [[\lambda x.x]] \Rightarrow \left ( [[\lambda y.y]] \subseteq [[x]] \wedge [[x]] \subseteq [[\left ( \lambda x.x \right ) \lambda y.y]] \right ) \\ | \lambda_x \subseteq [[\lambda x.x]] \Rightarrow \left ( [[\lambda y.y]] \subseteq [[x]] \wedge [[x]] \subseteq [[\left ( \lambda x.x \right ) \lambda y.y]] \right ) \\ | ||
| \lambda_y \subseteq [[\lambda x.x]] \Rightarrow \left ( [[\lambda y.y]] \subseteq [[y]] \wedge [[y]] \subseteq [[\left ( \lambda x.x \right ) \lambda y.y]] \right ) | \lambda_y \subseteq [[\lambda x.x]] \Rightarrow \left ( [[\lambda y.y]] \subseteq [[y]] \wedge [[y]] \subseteq [[\left ( \lambda x.x \right ) \lambda y.y]] \right ) | ||
| - | \end{array} \] | + | \end{array} \end{equation*} |
| Which leads to the following solutions : | Which leads to the following solutions : | ||