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sav08:substitutions_for_first-order_logic [2008/03/19 22:31] damien |
sav08:substitutions_for_first-order_logic [2015/04/21 17:30] |
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- | ====== Substitutions for First-Order Logic ====== | ||
- | (Recall [[Substitution Theorems for Propositional Logic]].) | ||
- | |||
- | ===== Motivating Example ===== | ||
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- | It is important to be precise about substitutions in first-order logic. For example, we would like to derive from formula $\forall x.F(x)$ formula $F(t)$, denoted $subst(\{x \mapsto t\})(F)$ that results from substituting a term $t$ instead of $x$. For example, from $\forall x. x < x + 5$ we would like to derive $y - 1 < (y - 1) + 5$. Consider, however formula | ||
- | \[ | ||
- | \forall x. \exists y. x < y | ||
- | \] | ||
- | Consider an interpretation in integers. This formula is true in this domain. Now substitute instead of x the term y+1. We obtain | ||
- | \[ | ||
- | \exists y. y + 1 < y | ||
- | \] | ||
- | This formula is false. We say that the variable $y$ in term $y+1$ was captured during substitution. When doing substitution in first-order logic we must avoid variable capture. One way to do this is to rename bound variables. Suppose we want to instantiate the formula $\forall x. \exists y. x<y$ with $y+1$. Then we first rename variables in the formula, obtaining | ||
- | \[ | ||
- | \forall x_1. \exists y_1. x_1 < y_1 | ||
- | \] | ||
- | and then after substitution $\{x_1 \mapsto y+1\}$ we obtain $\exists y_1. y + 1 < y_1$, which is a correct consequence of $\forall x. \exists y. x < y$. | ||
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- | |||
- | ===== Naive and Safe Substitutions ===== | ||
- | |||
- | Let $U$ be a set of variables (that is, $U \subseteq V$). A //variable substitution// is a function $\sigma : U \to T$ where $T$ is the set of [[First-Order Logic Syntax|terms first-order logic]]. | ||
- | |||
- | We define naive substitution recursively, first for terms: | ||
- | \[\begin{array}{rcl} | ||
- | subst(\sigma)( x ) &=& \sigma( x ),\ \sigma \mbox{ d{}efined at } x \\ | ||
- | subst(\sigma)( x ) &=& x,\ \sigma \mbox{ not d{}efined at } x \\ | ||
- | subst(\sigma)(f(t_1,\ldots,t_n)) &=& f(subst(\sigma)(t_1),\ldots,subst(\sigma)(t_n)) | ||
- | \end{array}\] | ||
- | |||
- | and then for formulas: | ||
- | \[\begin{array}{rcl} | ||
- | nsubst(\sigma)(R(t_1,\ldots,t_n)) &=& R(nsubst(\sigma)(t_1),\ldots,nsubst(\sigma)(t_n)) \\ | ||
- | nsubst(\sigma)(t_1 = t_2) &=& (nsubst(\sigma)(t_1) = nsubst(\sigma)(t_n)) \\ | ||
- | nsubst(\sigma)(\lnot F) &=& \neg nsubst(\sigma)(F) \\ | ||
- | nsubst(\sigma)(F_1 \land F_2) &=& nsubst(\sigma)(F_1) \wedge nsubst(\sigma)(F_2) \\ | ||
- | nsubst(\sigma)(F_1 \lor F_2) &=& nsubst(\sigma)(F_1) \vee nsubst(\sigma)(F_2) \\ | ||
- | nsubst(\sigma)(\forall x.F) &=& \forall x.\ nsubst(\sigma)(F)\\ | ||
- | nsubst(\sigma)(\exists x.F) &=& \exists x.\ nsubst(\sigma)(F) | ||
- | \end{array} | ||
- | \] | ||
- | |||
- | **Lemma:** Let $\sigma = \{x_1 \mapsto t_1,\ldots,x_n \mapsto t_n\}$ be a variable substitution and $t$ a term. Then for every interpretation $I$, | ||
- | \[ | ||
- | e_T(nsubst(\sigma)(t))(I) = e_T(t)(I[x_1 \mapsto e_T(t_1)(I),\ldots, x_n \mapsto e_T(t_n)(I)]) | ||
- | \] | ||
- | |||
- | To avoid variable capture, we introduce in addition to $subst$ a safe substitution, $sfsubst$. | ||
- | \[ | ||
- | \begin{array}{rcl} | ||
- | sfsubst(\sigma)(R(t_1,\ldots,t_n)) &=& R(nsubst(\sigma)(t_1),\ldots,nsubst(\sigma)(t_n)) \\ | ||
- | sfsubst(\sigma)(t_1 = t_2) &=& (nsubst(\sigma)(t_1) = nsubst(\sigma)(t_n)) \\ | ||
- | sfsubst(\sigma)(\lnot F) &=& \neg sfsubst(\sigma)(F) \\ | ||
- | sfsubst(\sigma)(F_1 \land F_2) &=& sfsubst(\sigma)(F_1) \wedge sfsubst(\sigma)(F_2) \\ | ||
- | sfsubst(\sigma)(F_1 \lor F_2) &=& sfsubst(\sigma)(F_1) \vee sfsubst(\sigma)(F_2) \\ | ||
- | sfsubst(\sigma)(\forall x.F) &=& \left\{ \begin{array}{ll} | ||
- | \forall x. sfsubst(\sigma)(F) & \text{if} ~ x \notin \text{domain}(\sigma) \wedge x \notin \bigcup_{v \in \text{domain}(\sigma)} FV(v) \\ | ||
- | sfsubst(\sigma)(\forall x^\prime. sfsubst(\{x \mapsto x^\prime \})(F)) & \text{else}} | ||
- | \end{array} \right.\\ | ||
- | sfsubst(\sigma)(\exists x.F) &=& \left\{ \begin{array}{ll} | ||
- | \exists x. sfsubst(\sigma)(F) & \text{if} ~ x \notin \text{domain}(\sigma) \wedge x \notin \bigcup_{v \in \text{domain}(\sigma)} FV(v) \\ | ||
- | sfsubst(\sigma)(\exists x^\prime. sfsubst(\{x \mapsto x^\prime \})(F)) & \text{else}} | ||
- | \end{array} \right. | ||
- | \end{array} | ||
- | \] | ||
- | |||
- | **Lemma:** Let $\sigma = \{x_1 \mapsto t_1,\ldots,x_n \mapsto t_n\}$ be a variable substitution and $t$ a term. Then for every interpretation $I$, | ||
- | \[ | ||
- | e_F(sfsubst(\sigma)(F))(I) = e_F(F)(I[x_1 \mapsto e_T(t_1)(I),\ldots, x_n \mapsto e_T(t_n)(I)]) | ||
- | \] | ||
- | |||
- | **Lemma:** $(\forall x.F) \models sfsubst(\{x \mapsto t\}(F)$. |