Homework 06 - Due April 9

Problem 1

(Recall Definition of Resolution for FOL.)

Let $F_0$ denote formula

$\begin{equation*} \forall x. (A_1(x) \rightarrow B_1(x)) \land (A_2(x) \rightarrow B_2(x)) \leftrightarrow (A_1(x) \land B_1(x)) \lor (A_2(x) \land B_2(x)) \end{equation*}$

For each of the following formulas, if the formula is valid, use resolution to prove it; if it is invalid, construct at least one Herbrand model for its negation.

a): Formula $F_0$

b): Formula

$\begin{equation*} \begin{array}{l} (\forall y. \lnot (A_1(y) \land A_2(y))) \rightarrow F_0 \end{array} \end{equation*}$

c): Formula

$\begin{equation*} \begin{array}{l} (\forall y. A_1(y) \leftrightarrow \lnot A_2(y)) \rightarrow F_0 \end{array} \end{equation*}$

d): Formula

$\begin{equation*} \begin{array}{l} (\forall y. A_1(y) \leftrightarrow \lnot A_2(y)) \land (\forall z. B_1(z) \leftrightarrow \lnot B_2(z)) \rightarrow F_0 \end{array} \end{equation*}$

e): Formula:

$\begin{equation*} \begin{array}{l} (\forall x. \lnot R(x,x)) \land (\forall x. R(x,f(x)) \rightarrow (\exists x,y,z.\ R(x,y) \land R(y,z) \land \lnot R(x,z)) \end{array} \end{equation*}$

Problem 2

(Recall Sets and Relations.)

We say that a binary relation is a partial order iff it is reflexive, antisymmetric, and transitive. Let $D$ be a non-empty set and $r_0 \subseteq D \times D$ a binary relation on $D$ . Let $r = r_0^* = \bigcup_{i\ge 0} r_0^n$ be the reflexive transitive closure of $r_0$ .

a) Give an example $r_0$ for which $r$ is not necessarily a partial order.

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 $x,x',y,y' \in D$ ,

$\begin{equation*} (x,x') \in s \land (y,y') \in s \rightarrow ((x,y) \in r \leftrightarrow (x',y') \in r) \end{equation*}$

c) For each $x \in D$ let $[x] = \{ y \mid (x,y) \in s \}$ . Let $[D] = \{ [x] \mid x \in D\}$ . Define a new relation, $[r] \subseteq [D] \times [D]$ , by

$\begin{equation*} [r] = \{ ([x],[y]) \mid (x,y) \in r \} \end{equation*}$

Show that $[r]$ is a partial order on $[D]$ .

Optional: Explain this constructions using terminology of graphs and strongly connected components.

Problem 3

(Recall Substitutions for First-Order Logic, Unification.)

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 $subst(\sigma_2) = subst(\sigma_1) \circ subst(\tau)$ where $\circ$ is the standard relation composition.

a) Compute $r = r_0^*$ . What is its relationship to $r_0$ ?

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 $V$ .

Optional: 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 $a \in [U]$ such that for all $b \in [U]$ , we have $(a,b) \in [r]$ (that is, $a$ is the least element of $[U]$ with respect to $[r]$ defined as in Problem 2).