Homework 10, Due May 7th

Problem 1

Let ${\cal F}$ be the set of all first-order formulas (viewed as syntax trees) and let $r$ be the implication relation on formulas:

   r = \{ (F_1,F_2) \mid \models F_1 \rightarrow F_2 \}

Check whether $r$ is reflexive, antisymmetric, and transitive relation.

Problem 2

Let $(A,\sqsubseteq)$ be a partial order such that every set $S \subseteq A$ has the greatest lower bound. Prove that then every set $S \subseteq A$ has the least upper bound.

Problem 3

Let $A = [0,1] = \{ x \in \mathbb{R} \mid 0 \le x \le 1 \}$ be the interval of real numbers. Recall that, by definition of real numbers and complete lattice, $(A,\le)$ is a complete lattice with least lattice element $0$ and greatest lattice element $1$. Here $\sqcup$ is the least upper bound operator on sets of real numbers, also called supremum and denoted sup in real analysis.

Let function $f : A \to A$ be given by

    f(x) = \left\{\begin{array}{l}
\frac{1}{2} + \frac{1}{4}x, \mbox{ if } x \in [0,\frac{2}{3}) \\
\ \\
\frac{3}{5} + \frac{1}{5}x, \mbox{ if } x \in [\frac{2}{3},1]

(It may help you to try to draw $f$.)

Part a)

Prove that $f$ is monotonic and injective (so it is strictly monotonic).

Part b)

Compute the set of fixpoints of $f$.

Part c)

Define $iter(x) = \sqcup \{ f^n(x) \mid n \in \{0,1,2,\ldots \}\}$. (This is in fact equal to $\lim_{n\to\infty} f^n(x)$ when $f$ is a monotonic bounded function.)

Compute $iter(0)$ (prove that the computed value is correct by definition of $iter$, that is, that the value is indeed $\sqcup$ of the set of values). Is $iter(0)$ a fixpoint of $f$? Is $iter(iter(0))$ a fixpoint of $f$? Is $f$ an $\omega$-continuous function?

Optional part d)

Define a monotonic function $f : A \to A$ such that, for every natural number $k$, the value $iter^k(0)$ is not a fixpoint of $f$. (It may be difficult to draw $f$.)