Changed layout for more manageable volumes

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-#LyX 2.4 created this file. For more info see https://www.lyx.org/
-\lyxformat 620
-\begin_document
-\begin_header
-\save_transient_properties true
-\origin unavailable
-\textclass book
-\begin_preamble
-\input defs
-\end_preamble
-\use_default_options true
-\maintain_unincluded_children no
-\language english
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-\index Index
-\shortcut idx
-\color #008000
-\end_index
-\secnumdepth 3
-\tocdepth 3
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-\end_header
-
-\begin_body
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-exerc1[10]
-\end_layout
-
-\end_inset
-
-The text showed how to interchange the values of variables 
-\begin_inset Formula $m$
-\end_inset
-
- and 
-\begin_inset Formula $n$
-\end_inset
-
-,
- using the replacement notation,
- by setting 
-\begin_inset Formula $t\gets m$
-\end_inset
-
-,
- 
-\begin_inset Formula $m\gets n$
-\end_inset
-
-,
- 
-\begin_inset Formula $n\gets t$
-\end_inset
-
-.
- Show how the values of the 
-\emph on
-four
-\emph default
- variables 
-\begin_inset Formula $(a,b,c,d)$
-\end_inset
-
- can be rearranged to 
-\begin_inset Formula $(b,c,d,a)$
-\end_inset
-
- by a sequence of replacements.
- In other words,
- the new value of 
-\begin_inset Formula $a$
-\end_inset
-
- is to be the original value of 
-\begin_inset Formula $b$
-\end_inset
-
-,
- etc.
- Try to use the minimum number of replacements.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-answer 
-\end_layout
-
-\end_inset
-
-
-\begin_inset Formula $t\gets a,a\gets b,b\gets c,c\gets d,d\gets t$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-rexerc5[12]
-\end_layout
-
-\end_inset
-
-Show that the 
-\begin_inset Quotes eld
-\end_inset
-
-Procedure for Reading This Set of Books
-\begin_inset Quotes erd
-\end_inset
-
- that appears in the preface actually fails to be a genuine algorithm on at least three of our five counts!
- Also mention some differences in format between it and Algorithm E.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-answer 
-\end_layout
-
-\end_inset
-
-It fails on finiteness (it doesn't have an ending condition),
- definiteness (the instructions are not unambiguous),
- and effectiveness (solving the Fermat's theorem on the exercises is not 
-\begin_inset Quotes eld
-\end_inset
-
-sufficiently basic that it can be done exactly and in a finite length of time
-\begin_inset Quotes erd
-\end_inset
-
-),
- and it may have no output (although,
- this time,
- this is the output).
- It also doesn't have a header,
- an ending mark,
- or a letter prefix on the steps.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-rexerc7[HM21]
-\end_layout
-
-\end_inset
-
-Let 
-\begin_inset Formula $U_{m}$
-\end_inset
-
- be the average number of times that step E1 is executed in Algorithm E,
- if 
-\begin_inset Formula $m$
-\end_inset
-
- is known and 
-\begin_inset Formula $n$
-\end_inset
-
- is allowed to range over all positive integers.
- Show that 
-\begin_inset Formula $U_{m}$
-\end_inset
-
- is well defined.
- Is 
-\begin_inset Formula $U_{m}$
-\end_inset
-
- in any way related to 
-\begin_inset Formula $T_{m}$
-\end_inset
-
-?
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-answer 
-\end_layout
-
-\end_inset
-
-On average,
- it will be the case that 
-\begin_inset Formula $n>m$
-\end_inset
-
-,
- so after one step,
- the two numbers will exchange positions and,
- after two steps,
- we'll have the same value for 
-\begin_inset Formula $m$
-\end_inset
-
- but 
-\begin_inset Formula $n$
-\end_inset
-
- will be the remainder of the original values of 
-\begin_inset Formula $n$
-\end_inset
-
- and 
-\begin_inset Formula $m$
-\end_inset
-
-,
- which is uniformly distributed on the integers between 
-\begin_inset Formula $0$
-\end_inset
-
- and 
-\begin_inset Formula $m-1$
-\end_inset
-
-.
- This means 
-\begin_inset Formula $U_{m}$
-\end_inset
-
- is two plus the average number of times that E1 is executed if 
-\begin_inset Formula $m$
-\end_inset
-
- is known and 
-\begin_inset Formula $0\leq n<m$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Further,
- since after one step,
- the two numbers interchange,
- it is clear that 
-\begin_inset Formula $U_{m}=1+T_{m}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-rexerc9[M30]
-\end_layout
-
-\end_inset
-
-Suppose that 
-\begin_inset Formula $C_{1}=(Q_{1},I_{1},\Omega_{1},f_{1})$
-\end_inset
-
- and 
-\begin_inset Formula $C_{2}=(Q_{2},I_{2},\Omega_{2},f_{2})$
-\end_inset
-
- are computational methods.
- For example,
- 
-\begin_inset Formula $C_{1}$
-\end_inset
-
- might stand for Algorithm E as in Eqs.
- (2),
- except that 
-\begin_inset Formula $m$
-\end_inset
-
- and 
-\begin_inset Formula $n$
-\end_inset
-
- are restricted in magnitude,
- and 
-\begin_inset Formula $C_{2}$
-\end_inset
-
- might stand for a computer program implementation of Algorithm E.
- (This 
-\begin_inset Formula $Q_{2}$
-\end_inset
-
- might be the set of all states of the machine,
- i.e.,
- all possible configurations of its memory and registers;
- 
-\begin_inset Formula $f_{2}$
-\end_inset
-
- might be the definition of single machine actions;
- and 
-\begin_inset Formula $I_{2}$
-\end_inset
-
- might be the set of initial states,
- each including the program that determines the greatest common divisor as well as the particular values of 
-\begin_inset Formula $m$
-\end_inset
-
- and 
-\begin_inset Formula $n$
-\end_inset
-
-.)
-\end_layout
-
-\begin_layout Standard
-Formulate a set-theoretic definition for the concept 
-\begin_inset Quotes eld
-\end_inset
-
-
-\begin_inset Formula $C_{2}$
-\end_inset
-
- is a representation of 
-\begin_inset Formula $C_{1}$
-\end_inset
-
-
-\begin_inset Quotes erd
-\end_inset
-
- or 
-\begin_inset Quotes eld
-\end_inset
-
-
-\begin_inset Formula $C_{2}$
-\end_inset
-
- simulates 
-\begin_inset Formula $C_{1}$
-\end_inset
-
-
-\begin_inset Quotes erd
-\end_inset
-
-.
- This is to mean intuitively that any computation sequence of 
-\begin_inset Formula $C_{1}$
-\end_inset
-
- is mimicked by 
-\begin_inset Formula $C_{2}$
-\end_inset
-
-,
- except that 
-\begin_inset Formula $C_{2}$
-\end_inset
-
- might take more steps in which to do the computation and it might retain more information in its states.
- (We thereby obtain a rigorous interpretation of the statement,
- 
-\begin_inset Quotes eld
-\end_inset
-
-Program 
-\begin_inset Formula $X$
-\end_inset
-
- is an implementation of Algorithm 
-\begin_inset Formula $Y$
-\end_inset
-
-.
-\begin_inset Quotes erd
-\end_inset
-
-)
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-answer 
-\end_layout
-
-\end_inset
-
-We could state that 
-\begin_inset Formula $C_{2}$
-\end_inset
-
- is a representation of 
-\begin_inset Formula $C_{1}$
-\end_inset
-
- if there are functions 
-\begin_inset Formula $h:Q_{2}\to Q_{1}$
-\end_inset
-
-,
- 
-\begin_inset Formula $g:I_{1}\to I_{2}$
-\end_inset
-
-,
- and 
-\begin_inset Formula $j:Q_{2}\to\mathbb{N}^{*}$
-\end_inset
-
- such that 
-\begin_inset Formula $h\circ g=\text{Id}_{I_{1}}$
-\end_inset
-
-,
- 
-\begin_inset Formula $\Omega_{2}=h^{-1}(\Omega_{1})$
-\end_inset
-
-,
- and for all 
-\begin_inset Formula $q\in Q_{2}$
-\end_inset
-
-,
- 
-\begin_inset Formula $f_{1}(h(q))=h(f_{2}^{j(q)}(q))$
-\end_inset
-
-.
-\end_layout
-
-\end_body
-\end_document