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sav08:homework06 [2008/04/02 16:47] vkuncak |
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- | ====== Homework 06 - DRAFT ====== | + | ====== Homework 06 - Due April 9 ====== |
===== Problem 1 ===== | ===== Problem 1 ===== | ||
+ | |||
+ | (Recall [[Definition of Resolution for FOL]].) | ||
Let $F_0$ denote formula | Let $F_0$ denote formula | ||
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**a)** Give an example $r_0$ for which $r$ is not necessarily a partial order. | **a)** Give an example $r_0$ for which $r$ is not necessarily a partial order. | ||
- | **b)** Define $s = r \cup r^{-1}$. Show that $s$ is a congruence with respect to $r$, that is: $s$ is reflexive, symmetric, and transitive and for all $x,x',y,y' \in D$, | + | **b)** Define $s = r \cap r^{-1}$. Show that $s$ is a congruence with respect to $r$, that is: $s$ is reflexive, symmetric, and transitive and for all $x,x',y,y' \in D$, |
\[ | \[ | ||
(x,x') \in s \land (y,y') \in s \rightarrow ((x,y) \in r \leftrightarrow (x',y') \in r) | (x,x') \in s \land (y,y') \in s \rightarrow ((x,y) \in r \leftrightarrow (x',y') \in r) | ||
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\] | \] | ||
Show that $[r]$ is a partial order on $[D]$. | Show that $[r]$ is a partial order on $[D]$. | ||
+ | |||
+ | Optional: Explain this constructions using terminology of graphs and strongly connected components. | ||
===== Problem 3 ===== | ===== Problem 3 ===== | ||
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Let $V$ be an infinite set of variables. Let ${\cal L}$ be some first-order language. We will consider terms that contain variables from $V$ and function symbols from ${\cal L}$. | Let $V$ be an infinite set of variables. Let ${\cal L}$ be some first-order language. We will consider terms that contain variables from $V$ and function symbols from ${\cal L}$. | ||
- | Following Problem 2 above, let $(\sigma_1,\sigma_2) \in r_0$ iff there exists substitution $\tau$ such that $\sigma_2 = \sigma_1 \circ \tau$ where $\circ$ is the standard relation composition. | + | Following Problem 2 above, let $(\sigma_1,\sigma_2) \in r_0$ iff there exists substitution $\tau$ such that $subst(\sigma_2) = subst(\sigma_1) \circ subst(\tau)$ where $\circ$ is the standard relation composition. |
**a)** Compute $r = r_0^*$. What is its relationship to $r_0$? | **a)** Compute $r = r_0^*$. What is its relationship to $r_0$? | ||
- | **b)** Compute $s = r \cup r^{-1}$. Show that relation $s$ holds iff $\sigma_2 = \sigma_1 \circ b$ where $b$ is a relation which is bijection on the set $V$. | + | **b)** Compute $s = r \cap r^{-1}$. Show that relation $s$ holds iff $subst(\sigma_2) = subst(\sigma_1) \circ subst(b)$ where $b$ is a relation which is bijection on the set $V$. |
- | + | ||
- | **c)** Let $E$ be a fixed set of syntactic equations. Let $U$ be the set of unifiers for $E$ and $[U] = \{ [\sigma] \mid \sigma \in U \}$. Show that if $U$ is non-empty, then there exists a least element of $[U]$, that is, there exists $a \in [U]$ such that for all $b \in [U]$, we have $(a,b) \in [r]$. | + | |
+ | **c)** Let $E$ be a fixed set of syntactic equations. Let $U$ be the set of unifiers for $E$ and $[U] = \{ [\sigma] \mid \sigma \in U \}$. Show that if $U$ is non-empty, $a \in [U]$ such that for all $b \in [U]$, we have $(a,b) \in [r]$ (that is, $a$ is the least element of $[U]$). |