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Truth tables, msb/lsb, alignment, sign-extend.

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book/binary/chapter.tex
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@ -11,6 +11,17 @@ RISC-V assembly language uses binary to represent all values, be they
boolean or numeric. It is the context within which they are used that
determines whether they are boolean or numeric.
\enote{Add some diagrams here showing bits, bytes and the MSB,
LSB,\ldots\ perhaps relocated from the RV32I chapter?}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Boolean Functions}
Boolean functions apply on a per-bit basis.
When applied to multi-bit values, each bit position is operated upon
independently of the other bits.
RISC-V assembly language uses zero to represent {\em false} and one
to represent {\em true}. In general, however, it is useful to relax
this and define zero {\bf and only zero} to be {\em false} and anything
@ -19,27 +30,11 @@ that is not {\em false} is therefore {\em true}.%
many other languages as well as the common assembly language idioms
discussed in this text.}
\enote{Add some diagrams here showing bits, bytes and the MSB,
LSB,\ldots\ perhaps relocated from the RV32I chapter?}%
The reason for this relaxation is because, while a single binary digit
(\gls{bit}) can represent the two values zero and one, the vast majority
of the time data is processed by the CPU in groups of bits. These
groups have names like \gls{byte}, \gls{halfword} and \gls{fullword}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Boolean Functions}
\enote{Probably should add basic truth table diagrams.}%
Boolean functions apply on a per-bit basis.
%in that they do not impact neighboring bits.
%by generating things like a carry or a borrow.
When applied to multi-bit values, each bit position is operated upon
independently of the other bits.
groups have names like \gls{byte} (8 bits), \gls{halfword} (16 bits)
and \gls{fullword} (32 bits).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -54,9 +49,35 @@ If the input is 1 then the output is 0. If the input is 0 then the
output is 1. In other words, the output value is {\em not} that of the
input value.
This text will use the operator used in the C language when discussing
Expressing the {\em not} function in the form a a truth table:
\begin{center}
\begin{tabular}{c|c}
A & $\overline{\mbox{A}}$\\
\hline
0 & 1 \\
1 & 0 \\
\end{tabular}
\end{center}
A truth table is drawn by indicating all of the possible input values on
the left of the vertical bar with each row displaying the output values
that correspond to the input for that row. The column headings are used
to define the illustrated operation expressed using a mathematical
notation. The {\em not} operation is indicated by the presense of
an {\em overline}.
In computer programming languages, things like an overline can not be
efficiently expressed using a standard keyboard. Therefore it is common
to use a notation such as that used by the C language when discussing
the {\em NOT} operator in symbolic form. Specifically the tilde: `\verb@~@'.
It is also uncommon to for programming languages to express boolean operations
on single-bit input(s). A more generalized operation is used that applies
to a set of bits all at once. For example, performing a {\em not} operation
of eight bits at once can be illustrated as:
\begin{verbatim}
~ 1 1 1 1 0 1 0 1 <== A
-----------------
@ -73,12 +94,30 @@ The boolean {\em and} function has two or more inputs and the output is a
single bit. The output is 1 if and only if all of the input values are 1.
Otherwise it is 0.
This function works like it does in spoken language. For example
if A is 1 {\em AND} B is 1 then the output is 1 (true).
Otherwise the output is 0 (false).
In mathamatical notion, the {\em and} operator is expressed the same way
as is {\em multiplication}. That is by a raised dot between, or by
juxtaposition of, two variable names. It is also worth noting that,
in base-2, the {\em and} operation actually {\em is} multiplication!
\begin{center}
\begin{tabular}{cc|c}
A & B & AB \\
\hline
0 & 0 & 0 \\
0 & 1 & 0 \\
1 & 0 & 0 \\
1 & 1 & 1 \\
\end{tabular}
\end{center}
This text will use the operator used in the C language when discussing
the {\em AND} operator in symbolic form. Specifically the ampersand: `\verb@&@'.
This function works like it does in spoken language. For example
if A is 1 {\em AND} B is 1 then the output is 1 (true).
Otherwise the output is 0 (false). For example:
An eight-bit example:
\begin{verbatim}
1 1 1 1 0 1 0 1 <== A
@ -96,12 +135,28 @@ In a line of code the above might read like this: \verb@output = A & B@
The boolean {\em or} function has two or more inputs and the output is a
single bit. The output is 1 if at least one of the input values are 1.
This function works like it does in spoken language. For example
if A is 1 {\em OR} B is 1 then the output is 1 (true).
Otherwise the output is 0 (false).
In mathamatical notion, the {\em or} operator is expressed using the plus
($+$).
\begin{center}
\begin{tabular}{cc|c}
A & B & A$+$B \\
\hline
0 & 0 & 0 \\
0 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 1 \\
\end{tabular}
\end{center}
This text will use the operator used in the C language when discussing
the {\em OR} operator in symbolic form. Specifically the pipe: `\verb@|@'.
This function works like it does in spoken language. For example
if A is 1 {\em OR} B is 1 then the output is 1 (true).
Otherwise the output is 0 (false). For example:
An eight-bit example:
\begin{verbatim}
1 1 1 1 0 1 0 1 <== A
@ -120,14 +175,29 @@ The boolean {\em exclusive or} function has two or more inputs and the
output is a single bit. The output is 1 if only an odd number of inputs
are 1. Otherwise the output will be 0.
This text will use the operator used in the C language when discussing
the {\em XOR} operator in symbolic form. Specifically the carrot: `\verb@^@'.
Note that when {\em XOR} is used with two inputs, the output
is set to 1 (true) when the inputs have different values and 0
(false) when the inputs both have the same value.
For example:
In mathamatical notion, the {\em xor} operator is expressed using the plus
in a circle ($\oplus$).
\begin{center}
\begin{tabular}{cc|c}
A & B & A$\oplus{}$B \\
\hline
0 & 0 & 0 \\
0 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 0 \\
\end{tabular}
\end{center}
This text will use the operator used in the C language when discussing
the {\em XOR} operator in symbolic form. Specifically the carrot: `\verb@^@'.
An eight-bit example:
\begin{verbatim}
1 1 1 1 0 1 0 1 <== A
@ -200,9 +270,9 @@ $10^2$ & $10^1$ & $10^0$ & $2^7$ & $2^6$ & $2^5$ & $2^4$ & $2^3$ & $2^2$ & $2^1$
One way to look at this table is on a per-row basis where each place
value is represented by the base raised to the power of the place value
position (shown in the column headings.) This is useful when
converting arbitrary values between bases. For example to interpret
the decimal value on the fourth row:
position (shown in the column headings.)
%This is useful when converting arbitrary numeric values between bases.
For example to interpret the decimal value on the fourth row:
\begin{equation}
0 \times 10^2 + 0 \times 10^1 + 3 \times 10^0 = 3_{10}
@ -220,6 +290,15 @@ Interpreting the hexadecimal value on the fourth row by converting it to decimal
0 \times 16^1 + 3 \times 16^0 = 3_{10}
\end{equation}
\index{Most significant bit}\index{MSB|see {Most significant bit}}%
\index{Least significant bit}\index{LSB|see {Least significant bit}}%
We refer to the place values with the largest exponenet (the one furthest to the
left for any given base) as the {\em most significant} digit and the place value
with the lowest exponenet as the {\em least significant} digit. For binary
numbers these are the \acrfull{msb} and \acrfull{lsb} respectively.%
\footnote{Changing the value of the MSB will have a more {\em significant}
impact on the numeric value than changing the value of the LSB.}
Another way to look at this table is on a per-column basis. When
tasked with drawing such a table by hand, it might be useful
@ -242,8 +321,8 @@ for readability.
The relationship between binary and hex values is also worth taking
note. Because $2^4 = 16$, there is a clean and simple grouping
of 4 \gls{bit}s to 1 \gls{hit}. There is no such relationship
between binary and decimal.
of 4 \gls{bit}s to 1 \gls{hit} (aka \gls{nybble}).
There is no such relationship between binary and decimal.
Writing and reading numbers in binary that are longer than 8 bits
is cumbersome and prone to error. The simple conversion between
@ -283,8 +362,9 @@ To convert from binary to decimal, put the decimal value of the place values
{\ldots8 4 2 1} over the binary digits like this:
\begin{verbatim}
128 64 32 16 8 4 2 1
0 0 0 1 1 0 1 1
Base-2 place values: 128 64 32 16 8 4 2 1
Binary: 0 0 0 1 1 0 1 1
Decimal: 16 +8 +2 +1 = 27
\end{verbatim}
Now sum the place-values that are expressed in decimal for each
@ -303,9 +383,9 @@ extend the binary to the left with zeros to make it so.
Grouping the bits into sets of four and summing:
\begin{verbatim}
Place: 8 4 2 1 8 4 2 1 8 4 2 1 8 4 2 1
Binary: 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 0
Decimal: 4+2 =6 8+4+ 1=13 8+ 2 =10 8+4+2 =14
Base-2 place values: 8 4 2 1 8 4 2 1 8 4 2 1 8 4 2 1
Binary: 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 0
Decimal: 4+2 =6 8+4+ 1=13 8+ 2 =10 8+4+2 =14
\end{verbatim}
After the summing, convert each decimal value to hex. The decimal
@ -339,10 +419,9 @@ method discussed in \autoref{section:bindec}.
For example:
\begin{verbatim}
Hex: 4 C
Binary: 0 1 0 0 1 1 0 0
Decimal: 128 64 32 16 8 4 2 1
Sum: 64+ 8+4 = 76
Hex: 7 C
Decimal Sum: 4+2+1=7 8+4 =12
Binary: 0 1 1 1 1 1 0 0
\end{verbatim}
@ -356,8 +435,9 @@ of the place values that would yield a non-negative result.
For example, to convert $1234_{10}$ to binary:
\begin{verbatim}
Place values: 2048-1024-512-256-128-64-32-16-8-4-2-1
Base-2 place values: 2048-1024-512-256-128-64-32-16-8-4-2-1
0 2048 (too big)
1 1234 - 1024 = 210
@ -429,20 +509,46 @@ negate the place value of the \acrshort{msb}. For example, the
number one is represented the same as discussed before:
\begin{verbatim}
-128 64 32 16 8 4 2 1
0 0 0 0 0 0 0 1
Base-2 place values: -128 64 32 16 8 4 2 1
Binary: 0 0 0 0 0 0 0 1
\end{verbatim}
The \acrshort{msb} of any negative number in this format will always
be 1. For example the value $-1_{10}$ is:
\begin{verbatim}
-128 64 32 16 8 4 2 1
1 1 1 1 1 1 1 1
Base-2 place values: -128 64 32 16 8 4 2 1
Binary: 1 1 1 1 1 1 1 1
\end{verbatim}
\ldots because: $-128+64+32+16+8+4+2+1=-1$.
Calculating $4+5 = 9$
\begin{verbatim}
1 <== carries
000100 <== 4
+000101 <== 5
------
001001 <== 9
\end{verbatim}
Calculating $-4+ -5 = -9$
\begin{verbatim}
1 11 <== carries
111100 <== -4
+111011 <== -5
---------
1 110111 <== -9 (with a truncation)
-32 16 8 4 2 1
1 1 0 1 1 1
-32 + 16 + 4 + 2 + 1 = -9
\end{verbatim}
This format has the virtue of allowing the same addition logic
discussed above to be used to calculate $-1+1=0$.
@ -452,11 +558,11 @@ discussed above to be used to calculate $-1+1=0$.
1 1 1 1 1 1 1 1 <== addend (-1)
+ 0 0 0 0 0 0 0 1 <== addend (1)
----------------------
1 0 0 0 0 0 0 0 0 <== sum (0 with an overflow)
1 0 0 0 0 0 0 0 0 <== sum (0 with a truncation)
\end{verbatim}
In order for this to work, the \gls{overflow} carry out of the
sum of the MSBs is ignored.
In order for this to work, the carry out of the sum of the MSBs is
ignored.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Converting between Positive and Negative}
@ -623,10 +729,78 @@ the results of ADD and ADDW on the operands.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Sign and Zero Extension}
\enote{Refactor the sx() and zx() discussion in the RV32I chapter
and locate the details here.}%
Seems like a good place to discuss extension.
\index{sign extension}
\label{SignExtension}
Due to the nature of the two's complement encoding scheme, the following
numbers all represent the same value:
\begin{verbatim}
1111 <== -1
11111111 <== -1
11111111111111111111 <== -1
1111111111111111111111111111 <== -1
\end{verbatim}
As do these:
\begin{verbatim}
01100 <== 12
0000001100 <== 12
00000000000000000000000000000001100 <== 12
\end{verbatim}
The phenomenon illustrated here is called {\em sign extension}. That is
any signed number can have any quantity of additional MSBs added to it,
provided that they repeat the value of the sign bit.
\autoref{Figure:SignExtendNegative} illustrates extending the negative sign
bit of {\em val} to the left by replicating it.
When {\em val} is negative, its \acrshort{msb} (bit 19 in this example) will
be set to 1. Extending this value to the left will set all the new bits
to the left of it to 1 as well.
\begin{figure}[ht]
\centering
\DrawBitBoxSignExtendedPicture{32}{10100000000000000010}
\captionof{figure}{Sign-extending a negative integer from 20 bits to 32 bits.}
\label{Figure:SignExtendNegative}
\end{figure}
\autoref{Figure:SignExtendPositive} illustrates extending the positive sign
bit of {\em val} to the left by replicating it.
When {\em val} is positive, its \acrshort{msb} will be set to 0. Extending this
value to the left will set all the new bits to the left of it to 0 as well.
\begin{figure}[ht]
\centering
\DrawBitBoxSignExtendedPicture{32}{01000000000000000010}
\captionof{figure}{Sign-extending a positive integer from 20 bits to 32 bits.}
\label{Figure:SignExtendPositive}
\end{figure}
\label{ZeroExtension}
In a similar vein, any {\em unsigned} number also may have any quantity of
additional MSBs added to it provided that htey are all zero. For example,
the following all represent the same value:
\begin{verbatim}
1111 <== 15
01111 <== 15
00000000000000000000000001111 <== 15
\end{verbatim}
The observation here is that any {\em unsigned} number may be
{\em zero extended} to any size.
\autoref{Figure:ZeroExtend} illustrates zero-extending a 20-bit {\em val} to the
left to form a 32-bit fullword.
\begin{figure}[ht]
\centering
\DrawBitBoxZeroExtendedPicture{32}{10000000000000000010}
\captionof{figure}{Zero-extending an unsigned integer from 20 bits to 32 bits.}
\label{Figure:ZeroExtend}
\end{figure}
%Sign- and zero-extending binary numbers are common operations used to
%fit a byte or halfword into a fullword.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -694,28 +868,59 @@ value and TRUE if it is a conditional.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Alignment}
Draw a diagram showing the overlapping data types when they are all aligned.
\enote{Include the obligatory diagram showing the overlapping data types
when they are all aligned.}%
With respect to memory and storage, {\em \gls{alignment}} refers to the
{\em location} of a data element when the address that it is stored is
a precise multiple of a power-of-2.
The primary alignments of concern are typically 2 (a halfword),
4 (a fullword), 8 (a double word) and 16 (a quad-word) bytes.
For example, any data element that is aligned to 2-byte boundary
must have an (hex) address that ends in any of: 0, 2, 4, 6, 8, A,
C or E.
Any 4-byte aligned element must be located at an address ending
in 0, 4, 8 or C. An 8-byte aligned element at an address ending
with 0 or 8, and 16-byte aligned elements must be located at
addresses ending in zero.
Such alignments are important when exchanging data between the CPU
and memory because the hardware implementations are optimized to
transfer aligned data. Therefore, aligning data used by any program
will reap the benefit of running faster.
An element of data is considered to be {\em aligned to its natural size}
when its address is an exact multiple of the number of bytes used to
represent the data. Note that the ISA we are concerned with {\em only}
operates on elements that have sizes that are powers of two.
For example, a 32-bit integer consumes one full word. If the four bytes
are stored in main memory at an address than is a multiple of 4 then
the integer is considered to naturally aligned.
The same would apply to 16-bit, 64-bit, 128-bit and other such values
as they fit into 2, 8 and 16 byte elements respectively.
Some CPUs can deliver four (or more) bytes at the same time while others
might only be capable of delivering one or two bytes at a time. Such
differences in hardware typically impact the cost and performance of a
system.%
\footnote{The design and implementation
choices that determine how any given system operates are part of what is
called a system's {\em organization} and is beyond the scope of this text.
See~\cite{codriscv:2017} for more information on computer organization.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Instruction Alignment}
\enote{Rewrite this section for data rather than instructions and then
note here that instructions must be naturally aligned. For RV32 that
is on a 4-byte boundary}%
The RISC-V ISA requires that all instructions be aligned to their
natural boundaries.
Every possible instruction that an RV32I CPU can execute contains
exactly 32 bits. Therefore each one must be stored in four bytes
of the main memory.
To simplify the hardware, each instruction must be placed into four
adjacent bytes whose numeric address sequence begins with a multiple
four. For example, an instruction might be located in bytes
4, 5, 6 and 7 (but not in 5, 6, 7 and 8 nor in 9, 3, 1, and 0\ldots).
This sort of addressing requirement is common and is referred to as
\gls{alignment}. An aligned instruction begins at a memory address
that is a multiple of four. An {\em unaligned} instruction would
be one beginning at any other address and is {\em illegal}.
exactly 32 bits. Therefore they are always stored on a full word
boundary. Any {\em unaligned} instruction would is {\em illegal}.
An attempt to fetch an instruction from an unaligned address
will result in an error referred to as an alignment {\em \gls{exception}}.
@ -724,15 +929,3 @@ current instruction and start executing a different set of instructions
that are prepared to handle the problem. Often an exception is
handled by completely stopping the program in a way that is commonly
referred to as a system or application {\em crash}.
Given a properly aligned instruction address, the CPU can request
that the main memory locate and deliver the values of the four bytes
in the address sequence to the CPU using what is called a memory
read operation. Some systems can deliver four (or more) bytes at the
same time while others might only be capable of delivering one or
two bytes at a time. These differences in hardware typically impact the
cost and performance of a system.\footnote{The design and implementation
choices that determine how any given system operates are part of what is
called a system's {\em organization} and is beyond the scope of this text.
See~\cite{codriscv:2017} for more information on computer organization.}

44
book/rv32/chapter.tex
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@ -27,37 +27,8 @@ This is used to convert a signed integer value expressed using some number of
bits to a larger number of bits by adding more bits to the left. In doing so,
the sign will be preserved. In this case {\em val} represents the least
\acrshort{msb}s of the value.
For more on binary numbers see \autoref{chapter:NumberSystems}.
\autoref{Figure:SignExtendNegative} illustrates extending the negative sign
bit of {\em val} to the left by replicating it.
When {\em val} is negative, its \acrshort{msb} (bit 19 in this example) will
be set to 1. Extending this value to the left will set all the new bits
to the left of it to 1 as well.
\begin{figure}[ht]
\centering
\DrawBitBoxSignExtendedPicture{32}{10100000000000000010}
\captionof{figure}{Sign-extending a negative integer from 20 bits to 32 bits.}
\label{Figure:SignExtendNegative}
\end{figure}
\autoref{Figure:SignExtendPositive} illustrates extending the positive sign
bit of {\em val} to the left by replicating it.
When {\em val} is positive, its \acrshort{msb} will be set to 0. Extending this
value to the left will set all the new bits to the left of it to 0 as well.
\begin{figure}[ht]
\centering
\DrawBitBoxSignExtendedPicture{32}{01000000000000000010}
\captionof{figure}{Sign-extending a positive integer from 20 bits to 32 bits.}
\label{Figure:SignExtendPositive}
\end{figure}
For more on sign-extension see \autoref{SignExtension}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{zx(val)}
@ -69,19 +40,8 @@ This is used to convert an unsigned integer value expressed using some number of
bits to a larger number of bits by adding more bits to the left. In doing so,
the new bits added will all be set to zero. As is the case with \verb@sx(val)@,
{\em val} represents the \acrshort{lsb}s of the final value.
\autoref{Figure:ZeroExtend} illustrates zero-extending a 20-bit {\em val} to the
left to form a 32-bit fullword.
For more on binary numbers see \autoref{chapter:NumberSystems}.
\begin{figure}[ht]
\centering
\DrawBitBoxZeroExtendedPicture{32}{10000000000000000010}
\captionof{figure}{Zero-extending an unsigned integer from 20 bits to 32 bits.}
\label{Figure:ZeroExtend}
\end{figure}
For more on zero-extension see \autoref{ZeroExtension}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%