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Monash University Faculty of Information Technology 2nd Semester 2021
FIT2014 Assignment 2 Regular Languages, Context-Free Languages, Lexical
analysis, Parsing, Turing machines and Quaternions DUE: 11:55pm, Friday
8 October 2021 In these exercises, you will implement a lexical
analyser using lex (Problems 2, 4); implement parsers using lex and
yacc (Problems 1, 3–6); program a Turing machine (Problem 7); learn
about quaternions, by applying our methods to calculations with them
(Problems 3–7); practise your skills relating to context-free
languages (Problem 8). Solutions to Problem 7 must be implemented in the
simulator tuataraMonash (version 2.1 of Tuatara), which is available on
Moodle (week 8). How to manage this assignment You should start
working on this assignment now and spread the work over the time until
it is due. Do as much as possible before week 10. There will not be time
during your prac class to do the assignment from scratch; there will
only be time to get some help and clarification. Don’t be deterred by
the length of this document! Much of it is an extended tutorial to get
you started with lex and yacc (pp. 2–6) and documentation for functions,
written in C, that are provided for you to use (pp. 6–8); some sample
outputs also take up a fair bit of space. Although lex and yacc are new
to you, the questions about them only require you to modify some
existing input files for them rather than write your own input files
from scratch. Instructions Instructions are as for Assignment 1, except
that some of the filenames have changed. The file to download is now
asgn2.tar.gz, and unpacking it will create the directory asgn2 within
your FIT2014 directory. You will find three files already in the
directory: plus-times-power.l, plus-times-power.y, and quat.h. You will
not modify these files directly; you will make copies of the first two
and modify the copies, while quat.h must remain unaltered in the
directory where you do this work. You need to construct new lex files,
using plus-times-power.l as a starting point, for Problems 1, 4 & 5,
and you’ll need to construct a new yacc file from plus-times-power.y
for Problem 5. Your submission must include: a lex file prob1.l which
should be obtained by modifying a copy of plus-times-power.l a text
file prob2.txt which should contain a single line with a regular
expression in lex syntax a PDF file prob3.pdf which should contain
your solution to Problem 3 a lex file prob4.l which should also be
obtained by modifying a copy of plus-times-power.l a lex file prob5.l
which should be obtained by modifying a copy of prob4.l 1 a yacc file
prob5.y which should be obtained by modifying a copy of
plus-times-power.y a text file prob6.txt which should contain two
lines, being your solution to Problem 6 a Tuatara Turing machine file
prob7.tm a PDF file prob8.pdf which should contain your solution to
Problem 8. Each of the problem directories under the asgn2 directory
contains empty files with the required filenames. These must each be
replaced by the files you write, as described above. Before submission,
check that each of these empty files is, indeed, replaced by your own
file. To submit your work: 1. edit Makefile as described in Lab 0, 2.
enter the command ‘make’ from within the asgn2 directory, 3. submit the
resulting .tar.gz file to Moodle. As last time, make sure that you have
tested the submission mechanism and that you understand the effect of
make on your directory tree. INTRODUCTION: Lex, Yacc and the
PLUS-TIMES-POWER language In this part of the Assignment, you will use
the lexical analyser generator lex or its variant flex, initially by
itself, and then with the parser generator yacc. Some useful references
on Lex and Yacc: T. Niemann, Lex & Yacc Tutorial,
http://epaperpress.com/lexandyacc/ Doug Brown, John Levine, and Tony
Mason, lex and yacc (2nd edn.), O’Reilly, 2012. the lex and yacc
manpages We will illustrate the use of these programs with a language
PLUS-TIMES-POWER based on simple arithmetic expressions involving
nonnegative integers, using just addition, multiplication and powers.
Then you will use lex and yacc on a language QUAT of expressions based
on quaternions, which we describe later. PLUS-TIMES-POWER The language
PLUS-TIMES-POWER consists of expressions involving addition,
multiplication and powers of nonnegative integers, without any
parentheses (except for those required by the function Power). Example
expressions include: 5 + 8, 8 + 5, 3 + 5 2, 13 + 8 4 +
Power(2,Power(3, 2)), Power(1, 3) + Power(5, 3) + Power(3, 3),
Power(999, 0), 0 + 99 0 + 1, 2014, 10 14 + 74 + 10 13 73, 2 3
5 7 11 13 17 19. In these expressions, integers are written
in unsigned decimal, with no leading zeros or decimal point (so 2014,
86, 10, 7, and 0 are ok, but +2014, 2014, 86.0, A, 007, and 00 are
not). For lexical analysis, we treat every nonnegative integer as a
lexeme for the token NUMBER. Lex An input file to lex is, by convention,
given a name ending in .l. Such a file has three parts: definitions,
rules, 2 C code. These are separated by double-percent, %%. Comments
begin with /* and end with */. Any comments are ignored when lex is run
on the file. You will find an input file, plus-times-power.l, among the
files for this Assignment. Study its structure now, identifying the
three sections and noticing that various pieces of code have been
commented out. Those pieces of code are not needed yet, but some will be
needed later. We focus mainly on the Rules section, in the middle of
the file. It consists of a series of statements of the form pattern {
action } where the pattern is a regular expression and the action
consists of instructions, written in C, specifying what to do with text
that matches the pattern.1 In our file, each pattern represents a set of
possible lexemes which we wish to identify. These are: a decimal
representation of a nonnegative integer, represented as described above;
– This is taken to be an instance of the token NUMBER (i.e., a lexeme
for that token). the specific string Power, which is taken to be an
instance of the token POWER. certain specific characters: +, *, (, ),
and comma; white space, being any sequence of spaces and tabs; the
newline character. Note that all matching in lex is case-sensitive. Our
action is, in most cases, to print a message saying what token and
lexeme have been found. For white space, we take no action at all. A
character that cannot be matched by any pattern yields an error message.
If you run lex on the file plus-times-power.l, then lex generates the C
program lex.yy.c.2 This is the source code for the lexical analyser.
You compile it using a C compiler such as cc. For this assignment we use
flex, a more modern variant of lex. We generate the lexical analyser as
follows. $ flex plus-times-power.l $ cc lex.yy.c By default, cc puts
the executable program in a file usually called a.out3 but sometimes
called a.exe. This can be executed in the usual way, by just entering
./a.out at the command line. If you prefer to give the executable
program another name, such as plus-times-power-lex, then you can tell
this to the compiler using the -o option: cc lex.yy.c -o
plus-times-power-lex. When you run the program, it will initially wait
for you to input a line of text to analyse. Do so, pressing Return at
the end of the line. Then the lexical analyser will print, to standard
output, messages showing how it has analysed your input. The printing of
these messages is done by the printf statements from the file
plus-times-power.l. Note how it skips over white space, and only reports
on the lexemes and tokens. $ ./a.out 13+8 * 4 + Power(2,Power (3,2 ))
1This may seem reminiscent of awk, but note that: the pattern is not
delimited by slashes, /. . . /, as in awk; the action code is in C,
whereas in awk the actions are specified in awk’s own language, which
has similarities with C but is not the same; and the action pertains
only to the text that matches the pattern, whereas in awk the action
pertains to the entire line in which the matching text is found. 2The C
program will have this same name, lex.yy.c, regardless of the name you
gave to the lex input file. 3a.out is short for assembler output. 3
Token: NUMBER; Lexeme: 13 Token and Lexeme: + Token: NUMBER; Lexeme: 8
Token and Lexeme: * Token: NUMBER; Lexeme: 4 Token and Lexeme: + Token:
POWER; Lexeme: Power Token and Lexeme: ( Token: NUMBER; Lexeme: 2 Token
and Lexeme: , Token: POWER; Lexeme: Power Token and Lexeme: ( Token:
NUMBER; Lexeme: 3 Token and Lexeme: , Token: NUMBER; Lexeme: 2 Token and
Lexeme: ) Token and Lexeme: ) Token and Lexeme: Try running this program with some input expressions of your own. You can keep entering new expressions on new lines, and enter Control-D to stop when you are finished. Yacc We now turn to parsing, using yacc. Consider the following grammar for PLUS-TIMES-POWER. S → E S → ε E → I E → POWER(E,E) E → E E E → E + E I → NUMBER In this grammar, the non-terminals are S, E and I. Treat NUMBER and POWER as just single tokens, and hence single terminal symbols in this grammar. We now generate a parser for this grammar, which will also evaluate the expressions, with +, interpreted as the usual integer arithmetic operations and Power(. . . ,. . . ) interpreted as raising its first argument to the power of its second argument. To generate this parser, you need two files, prob1.l (for lex) and plus-times-power.y (for yacc): Copy plus-times-power.l to a new file prob1.l, and then modify prob1.l as follows: – in the Definitions section, uncomment the statement #include “y.tab.h”; – in the Rules section, in each action: uncomment the statements of the form · yylval.str = …; · yylval.num = …; · return TOKENNAME; · return *yytext; 4 · yyerror … Comment out the printf statements. These may still be handy if debugging is needed, so don’t delete them altogether, but the lexical analyser’s main role now is to report the tokens and lexemes to the parser, not to the user. – in the C code section, comment out the function main(), which in this case occupies four lines at the end of the file. plus-times-power.y, the input file for yacc, is provided for you. You don’t need to modify this yet. An input file for yacc is, by convention, given a name ending in .y, and has three parts, very loosely analogous to the three parts of a lex file but very different in their details and functionality: Declarations, Rules, Programs. These are separated by double-percent, %%. Comments begin with /* and end with */. Peruse the provided file plus-times-power.y, identify its main components, and pay particular attention to the following, since you will need to modify some of them later. in the Declarations section: – lines like int printQuaternion(Quaternion); Quaternion newQuaternion(double, double, double, double); … Quaternion rotation(double, Quaternion); which are declarations of functions (but they are defined later, in the Programs section);4 – declarations of the tokens to be used: %token NUMBER %token WHOLENUMBER %token POWER – some specifications that certain operations are left-associative (which helps determine the order in which operations are applied and can help resolve conflicts and ambiguities): %left ’+’ %left ’*’ – declarations of the nonterminal symbols to be used (which don’t need to start with an upper-case letter): %type start %type line %type expr %type int – nomination of which nonterminal is the Start symbol: %start start 4These functions for computing with quaternions are not needed by plus-times-power.y, but you will need them later, when you make a modified version of plus-times-power.y to parse expressions involving quaternions. 5 in the Rules section, a list of grammar rules in BNF, except that the colon “:” is used instead of →, and there must be a semicolon at the end of each rule. Rules with a common left-hand- side may be written in the usual compact form, by listing their right-hand-sides separated by vertical bars, and one semicolon at the very end. The terminals may be token names, in which case they must be declared in the Declarations section and also used in the lex file, or single characters enclosed in forward-quote symbols. Each rule has an action, enclosed in braces {. . . }. A rule for a Start symbol may print output, but most other rules will have an action of the form $$ = . . . . The special variable $$ represents the value to be returned for that rule, and in effect specifies how that rule is to be interpreted for evaluating the expression. The variables $1, $2, . . . refer to the values of the first, second, . . . symbols in the right-hand side of the rule. in the Programs section, various functions, written in C, that your parsers will be able to use. You do not need to modify these functions, and indeed should not try to do so unless you are an experienced C programmer and know exactly what you are doing! Most of these functions are not used yet; some will only be used later, in Problem 4. After constructing the new lex file prob1.l as above, the parser can be generated by: $ yacc -d plus-times-power.y $ flex prob1.l $ cc lex.yy.c y.tab.c -lm The executable program, which is now a parser for PLUS-TIMES-POWER, is again named a.out by default, and will replace any other program of that name that happened to be sitting in the same directory. $ ./a.out 13+8 * 4 + Power(2,Power (3,2 )) 557 13+8*4+Power(2,Power(3,2)) 557 Power(1,3)+Power(5,3)+Power(3,3) 153 1+2+3+4+5+6+7+8+9+10 55 10*9*8*7*6*5*4*3*2*1 3628800 Power(999,0) 1 Control-D Run it with some input expressions of your own. You can keep entering new expressions on new lines, as above, and enter Control-D to stop when you are finished. Problem 1. [2 marks] Construct prob1.l, as described above, so that it can be used with plus-times-power.y to build a parser for PLUS-TIMES-POWER. We now describe quaternions and a language of quaternion expressions. 6 Quaternions: introduction The quaternions are a system of four-dimensional numbers that are used in computer graphics to describe rotations in three-dimensional space, beginning with Tomb Raider in 1996 [1]. They were discovered by William Rowan Hamilton in Dublin in 1843. Every quaternion has the form w + xi+ yj + zk, where w, x, y, z ∈ R and i, j, k are special quantities, called quaternion units, satisfying i2 = 1, j2 = 1, k2 = 1, ijk = 1. Other properties that follow from these equations include: ij = k, ji = k, jk = i, kj = i, ki = j, ik = j. These properties can be used to compute any sum or product of quaternions, since the usual as- sociative and distributive laws still apply. Note, though, that multiplication of quaternions is not commutative: the order of multiplication affects the outcome, in general. You may have already met the complex numbers. These have the form x+yi, where x, y ∈ R and i2 = 1, and may be used to describe rotations in two-dimensional space. It is an intriguing math- ematical fact that, in order to extend complex numbers to describe rotations in three-dimensional space, a four -dimensional system is needed. The set of quaternions is denoted by H, just as the sets of real and complex numbers are denoted by R and C respectively. In this assignment, you will construct a quaternionic calculator for parsing and evaluating arithmetic expressions involving quaternions. Quaternions: representation We assume that, when representing the quaternion w + xi + yj + zk, the numbers w, x, y, z are represented either as integers (without a decimal point) or as numbers with a decimal point and at least one digit both before and after the decimal point. Trailing zeros are always allowed, but leading zeros are only allowed for the integer 0 or if 0 is the only digit before the decimal point. So the number 3/4 may be represented as any of 0.75, 0.750, 0.7500, etc, and all these possibilities must be accepted as valid representations of the same real number. But it must not be represented as .75 or .750, etc. The numbers 42, 42.0, 42.00 are all valid, but 042, 0042, 042.0, etc are invalid. Quaternion operations We allow all the four arithmetic operations on quaternions, denoted by the usual symbols, +, -, *, / — and grouping by parentheses, (. . . ). We also allow: exponentiation, denoted by ^, provided the exponent is a nonnegative integer. So, for example, (1-2i+3j-4k)^3 represents (1 2i+ 3j 4k)3, the cube of 1 2i+ 3j 4k. the special operation Rotation(. . . ,. . . ), written as a function, which creates a quaternion that represents a 3D rotation. The Rotation function takes two arguments, the first being a number, and the second being any quaternion expression. The functions to do these operations have all been written for you and provided in the file plus-times-power.y. You only need to modify a copy of that file, using the guidance below, to build your calculator. 7 Quaternion expressions These operations can be combined in the same way in which they are combined when used for “normal” numbers (real, complex, etc.) to make expressions. Any valid expression can be given as the second argument of Rotation(. . . ,. . . ), to give another valid expression, and expressions using Rotation(. . . ,. . . ) can be combined using arithmetic operations. We formalise the concept of a quaternion expression with the following inductive definition: 1. Each of i, j, k is a quaternion expression. 2. If r is any nonnegative integer, then each of r, ri, rj, rk is a quaternion expression. 3. If r is any nonnegative real number, then each of r, ri, rj, rk is a quaternion expression. 4. If p and q are quaternion expressions, then so are: (q), q, p+q, p q, p q, p/q. 5. If q is a quaternion expression and n is a nonnegative integer, then q n is a quaternion expres- sion. 6. If r is a nonnegative real number and q is a quaternion expression, then the following is a quaternion expression: Rotation(r, q) Notes: Negative numbers can be represented by negating positive numbers. We allow juxtaposition of any number r with any of i, j, k to form the simple expressions ri, rj and rk. This enables the succinct quaternion notation ±w ± xi± yj ± zk. However, apart from that, multiplication in our quaternion expressions is always denoted by a star, . The language QUAT Let QUAT be the language of valid quaternion expressions in which all numbers are finite decimal representations. Here are some examples of valid quaternion expressions (i.e., members of QUAT): expression evaluates to (0.5 1.618j 32i j k/(3.5 + i j))/(i/j k 48) 0.658452 + 0.033020i+ 0.000000j + 0.008664k 3i j k 3 1 + 2i 0.3 j + 4.00 k 1 + 2i 0.3j + 4.0k 0 0 ( 0.5 + 0.866j) 3 0.999934 + 0.000038j Rotation(120, 2i+ 2j + 2k) i/Rotation(120, 2i+ 2j + 2k) j Some examples of invalid quaternion expressions (i.e., not members of QUAT): expression comment 3i j k The product of j and k should use instead of juxtaposition. 1 + 2i .3 j + 4. k Need at least one digit before and after decimal point. Power( 0.5 + 0.866j, 3) Power is not valid in QUAT; we only use it in PLUS-TIMES-POWER. We first treat QUAT as a language over the fourteen-symbol alphabet {i, j, k, +, -, *, /, ^, (, ), NUMBER, WHOLENUMBER, ROTATION, , }. 5 Here, NUMBER is a token representing any finite decimal representation of a nonnegative real number, WHOLENUMBER is a token representing any nonnegative integer, ROTATION is a token representing the name of the Rotation function. 5The last symbol listed in this set is a comma. 8 Problem 2. [2 marks] Write a regular expression, using the regular expression syntax used by lex, that matches any finite decimal representation (of the type specified above) of a nonnegative real number. Problem 3. [7 marks] Write a Context-Free Grammar for the language QUAT over the fourteen-symbol alphabet {i, j, k, +, -, *, /, ^, (, ), NUMBER,WHOLENUMBER, ROTATION, , }. Now we use regular expressions (in the lex file, prob4.l) and a grammar (in the yacc file, prob5.y) to construct a lexical analyser (Problem 4) and a parser (Problem 5) for QUAT. Problem 4. [6 marks] Using the file provided for PLUS-TIMES-POWER as a starting point, construct a lex file, prob4.l, and use it to build a lexical analyser for QUAT. You’ll need to change the regular expressions associated with the NUMBER, WHOLENUM- BER and some other tokens, among other things. Sample output: $ ./a.out Rotation(120.0,2.0i+2.0j+2.0k)^2 * i / Rotation(120.0,2.0i+2.0j+2.0k)^2 Token: ROTATION; Lexeme: Rotation Token and Lexeme: ( Token: NUMBER; Lexeme: 120.0 Token and Lexeme: , Token: NUMBER; Lexeme: 2.0 Token and Lexeme: i Token and Lexeme: + Token: NUMBER; Lexeme: 2.0 Token and Lexeme: j Token and Lexeme: + Token: NUMBER; Lexeme: 2.0 Token and Lexeme: k Token and Lexeme: ) Token and Lexeme: ^ Token: WHOLENUMBER; Lexeme: 2 Token and Lexeme: * Token and Lexeme: i Token and Lexeme: / Token: ROTATION; Lexeme: Rotation Token and Lexeme: ( Token: NUMBER; Lexeme: 120.0 Token and Lexeme: , Token: NUMBER; Lexeme: 2.0 Token and Lexeme: i Token and Lexeme: + Token: NUMBER; Lexeme: 2.0 9 Token and Lexeme: j Token and Lexeme: + Token: NUMBER; Lexeme: 2.0 Token and Lexeme: k Token and Lexeme: ) Token and Lexeme: ^ Token: WHOLENUMBER; Lexeme: 2 Token and Lexeme: Control-D Problem 5. [10 marks] Make a copy of prob4.l, call it prob5.l, then modify it so that it can be used with yacc. Then construct a yacc file prob5.y from plus-times-power.y. Then use these lex and yacc files to build a parser for QUAT. Note that you do not have to program any of the quaternion functions yourself. They have already been written: see the Programs section of the yacc file. The actions in your yacc file will need to call these functions, and you can do that by using the function call for pow(. . . ) in plus-times-power.y as a template. The core of your task is to write the grammar rules in the Rules section, in yacc format, with associated actions, using the examples in plus-times-power.y as a guide. You also need to do some modifications in the Declarations section; see page 5 and further details below. When entering your grammar into the Rules section of prob5.y, it is best to leave the existing rules for the nonterminal start unchanged, as this has some extra stuff designed to allow you to enter a series of separate expressions on separate lines. So, take the Start symbol from your grammar in Problem 2 and represent it by the nonterminal line instead of by start. The specific modifications you need to do in the Declarations section should be: You need a new %token declaration for the ROTATION token. It has the same structure as the line for the NUMBER token, except that “num” is replaced by “str” (since ROTATION represents a string, being a name for a function, whereas NUMBER represents a number). For symbols that represent a binary (i.e., two-argument) arithmetic operation, it is worth including them in an appropriate %left statement. Each of these statements makes the parser treat these operations as left-associative, which helps it determine the order in which to do the operations and removes some sources of possible ambiguity. When using %left, operations having the same precedence are listed on the same line with spaces between them. So for + and – you can use the following statement: %left ’-’ ’+’ A similar line can be used for multiplication and division. For operations whose %left state- ments are on different lines, the operations with higher precedence are those with higher line numbers (i.e., later in the file). Right-associative operations can be handled similarly with a %right statement. Treat exponentiation as having higher precedence than multiplication and division. For every nonterminal symbol, you need a %type line that declares its type, i.e., the type of value that is returned when an expression generated from this nonterminal is evaluated. For example, %type start 10 Here, “qtn” is the type name we are using for quaternions. The various type names can be seen in the %union statement a little earlier in the file. But you do not need to know how that works in order to do this task. You should still use start as your Start symbol. If you use another name instead, you will need to modify the %start line too. Sample output: $ ./a.out Rotation(120,2i+2j+2k) * i / Rotation(120,2i+2j+2k) 0.000000 + 0.000000 i + 1.000000 j + 0.000000 k Control-D Now let’s apply quaternion calculations to rotate a point around an axis in 3D space. Throughout, we assume the axis goes through the origin. Points in 3D space are represented as pure quaternions, which means quaternions of the form xi + yj + zk, so they have no real part w. This is just another way of representing three- dimensional space using standard co-ordinate axes. In effect, the three basic quaternion units i, j, k are unit vectors along the x, y, z-axes, respectively, and correspond to points (1, 0, 0), (0, 1, 0), (0, 0, 1) respectively.6 A rotation is specified by giving its axis as a unit vector, i.e., a pure quaternion of length 1, and its angle as a real number. If the unit-length pure quaternion q gives the direction of the axis of rotation, and θ is the angle of rotation around that axis (clockwise, as viewed from the origin looking in the direction in which q points), then the quaternion Rotation(θ, q ) that represents the rotation is given by Rotation(θ, q ) = cos(θ/2) + sin(θ/2) · q . More generally, if q is any nonzero quaternion, then Rotation(θ, q) scales q to give the unit-length quaternion q that points in the same direction, and gives Rotation(θ, q) = cos(θ/2) + sin(θ/2) · q . It is this quaternion that is returned by the function rotation (provided in plus-times-power.y) and by the Rotation operation in quaternion expressions. In order to apply the rotation to a point p, we first represent p as a pure quaternion, p = xi+ yj + zk, and then form the expression Rotation(θ, q) p/Rotation(θ, q). So, our earlier calculation that Rotation(120,2i+2j+2k) * i / Rotation(120,2i+2j+2k) evalu- ates to j expresses the fact that, if your axis is the line with direction i + j + k, then rotating the point (1, 0, 0) by 120 clockwise around this axis gives the point (0, 1, 0). In this assignment, we restrict to angles θ ≥ 0. We lose no generality by doing this, since any rotation is equivalent to rotation about the same axis by some angle θ in the range 0 ≤ θ < 360 . Problem 6. [5 marks] Convert your eight-digit student ID number into an angle and direction as follows. Let d1d2d3d4d5d6d7d8 be the digits of your student ID number. Divide this into three single-digit numbers, followed by one two-digit number, followed by three single-digit numbers: d1, d2, d3, d4d5, d6, 6BUT the multiplication we use for quaternions is not the same as the dot product or cross product of vectors, although it is closely related to both. In fact, historically, quaternions gave rise to these products of vectors. 11 d7, d8. The axis of rotation is the line whose direction is given by d1i + d2j + d3k. Then work out the sum of the first digit d1 and the two-digit number d4d5, and use it for your angle of rotation, θ . The point to be rotated is (d6, d7, d8), which can be represented by the pure quaternion d6i+ d7j + d8k. For example, if your ID number is 12345678, then your axis of rotation is i+ 2j + 3k, your angle is 45 + 1 = 46 , and your point to be rotated is (6, 7, 8). (a) Write down the quaternion expression in QUAT that represents the calculation required to rotate the point (d6, d7, d8) by θ degrees clockwise around the axis whose direction is given by d1i+ d2j + d3k. (Your expression must use the actual numbers derived from your student ID number as specified, not the algebraic quantities used above.) (b) Run your parser on your expression from (a), and report the result of evaluating it. The answers to (a) and (b) should be copied into a single line each in the file prob6.txt. Turing machines A quaternion unit power is a string over the alphabet {1, P, i, j, k} where the first character is one of i, j, k, the second character is P (for to the power of ), and the remaning characters are each 1, with the substring after P representing a nonnegative integer n in unary. We interpret a quaternion unit power as representing qn, where q ∈ {i, j, k} and n ∈ N ∪ {0}. Our next task is to write a Turing machine to evaluate powers of this type. The following table shows some examples of quaternion unit powers and, for each, the result of evaluating it, with each given in its conventional mathematical form as well as the string represen- tation to be used in Problem 7 below. When the result is negative, we use M to denote a minus sign, as there is no dash available in Turing machine alphabets in Tuatara. input output mathematical as string for TM mathematical as string for TM i0 iP 1 1 k1 kP1 k k j2 jP11 1 M1 i3 iP111 i Mi i4 iP1111 1 1 j5 jP11111 j j Problem 7. [8 marks] Build, in Tuatara, a Turing machine to evaluate any quaternion unit power. When your Turing machine stops at the end of the computation, the tape head must be at the first square of the tape, and all tape cells beyond the output string must be blank. Context-Free Languages A finite sequence (x1, x2, . . . , xn) of positive integers is unimodal if — informally — it rises, then falls (or maybe does just one of these). Formally, we require that there exists k ∈ {1, 2, . . . , n} such that: 12 for all i such that 1 ≤ i ≤ k 1, we have xi ≤ xi+1; and for all i such that k ≤ i ≤ n 1, we have xi ≥ xi+1. So we have: x1 ≤ · · · ≤ xk 1 ≤ xk ≥ xk+1 ≥ · · · ≥ xn. The term “unimodal” means that there is one mode, i.e., one maximum value. Points to note: the mode may be at the very start or the very end, so that increasing sequences and decreasing sequences are special cases of unimodal sequences; the sequence can have consecutive equal terms before, at, or after the peak; and the peak may be broad, i.e., there may be more than one equal maximum term in the sequence. Unimodal sequences arise throughout Computer Science, Mathematics and Statistics. The bino- mial coefficients ( n k ) , for fixed n, are a notable example. In this question, sequences are represented as binary numbers separated by #, just as in Q6 of Tutorial 5. Let UNIMODAL be the language of unimodal sequences of positive integers, encoded in this way. Examples of members of UNIMODAL: 1#100#110#100#1, 11#101#101#1, 101#101#101, 1#1#10#11#101#1000, 1110#101#10#1, 101010. Some strings that do not belong to UNIMODAL: 1#100#110#100#1#0, 11#1#100, 1#101#1001#10#110, 101#11#101, 0, ε. Problem 8. [10 marks] Prove or disprove: the language UNIMODAL is context-free. References [1] Nick Bobic, Rotating objects using quaternions, Game Developer, Feb. 1998. Available at: https://www.gamasutra.com/view/feature/3278/rotating_objects_using_quaternions. php print=1 [2] Daniel Chan, Quaternions are turning tomb raiders on their heads, Parabola 40 (no. 2) (2004). Available at: https://www.parabola.unsw.edu.au/files/articles/2000-2009/volume-40- 2004/issue-2/vol40_no2_2.pdf or https://web.maths.unsw.edu.au/~danielch/talent/ quat1.pdf 13