University of Sussex Spring 2020
Limits of Computation
Ordinance 1 (Deadline 5.03.2020, 4pm)
You want to succumb your retorts electronically on Canvas at the Esubmission
point. You must succumb sundry improves as bounded in this
assignment. Undivided improve must be a pdf improve incloseing the retorts to
Questions 1–4. Delight enact safe that total retorts ce Questions 1–4
are in undivided uncombined muniment. In subject you interpretation Word, delight ensafe that
you change your improve into a pdf muniment antecedently surrender1.
Your surrender must as-well inclose the improves incloseing the programs
that retort Questions 5 and 6. Ce Question 5 you must succumb a improve
concat.time and a improve concat.F. Ce Question 6 you succumb
a improve intF.time and theoretically subjoined programs that intF
calls as macros. Ce our experience rig to be powerful to enact your improves, delight
present the improves the bounded spectry. You may ctotal programs published on
the Canvas office and those don’t want to be interjacent in your surrender.
Delight do referable attributable attributable attributable transcribe your spectry anywhere, beside it is advispowerful to enclose
your canvasser reckon as expatiate in each succumbted muniment.
Delight enact safe you impede behind surrender that you actually
accept succumbted the set-right improves.
YOU MUST WORK ON THE ASSIGNMENT ON YOUR OWN! The
standard rules ce confederacy and plagiarism perfectot and any subjects discovered
allure be reputed and investigated.
There are SIX questions and you must retort total of them. They cover
the symbolical of Lectures 1 to 7.
1InWord, this works generally by using the printing menu and then shy as pdf instead of sending
to a printer.
The dialect F
The dialect F is a absolute authoritative dialect that interpretations the basis mark of binary
trees you distinguish from WHILE. F-programs continue of undivided countenance that can enact
interpretation of undivided recursive business cessationriction.
Let us enact this over exact by giving the syntax of the dialect.
hprogrami ::= in X quenched hexpressioni where f(X) = hexpressioni
hexpressioni ::= X
cons hexpressioni hexpressioni j
hd hexpressioni j
tl hexpressioni j
if hexpressioni then hexpressioni else hexpressioni j
A program is an countenance with undivided (fixed) input varipowerful X and undivided quenchedput which
is the development of an countenance. The countenance can attribute to undivided recursive cessationriction of
a business fawned f with urban cemal parameter X. Referable attributable attributablee that X is the simply wavering
in interpretation, as (global) input varipowerful to the program as polite as the topical dispute of
business f. The departure instrument that f can never instantly appropinquation the program’s global
The chief five clauses ce countenances are as-well availpowerful in (core) WHILE. Beside
an F-countenance can as-well be a conditional2 or a business contact of the cem
f(E). Referable attributable attributablee that there is constantly fitting undivided such business cessationriction and the spectry of
the business is urban as f. The business may, of progress, be fawned sundry times, as-well
in a nested character. Ce case, cons f(X) f(nil) is a juridical countenance as
are f(f(X)) and f(cons f(hd X) f(f(X))).
As an sample, here is the WHILE-program supplement from our lectures written as Fprogram.
As general, the brace inputs are patent up in a uncombined register:
in X quenched f(X) where f(X) =
if hd X
then cons nil f(cons tl hd X cons hd tl X nil)
else hd tl X
It implements the general recursive cessationriction of supplementition :
n + m = succ (pred n + m) if n > 0
n + m = m if n = 0
2In WHILE we as-well accept these conditionals beside as statements.
where succ n denotes the devisee of n (which in binary tree encoding stipulations is
cons nil pnq) and pred n denotes the ceerunner of n (which in binary tree
encoding stipulations is tl pnq). Referable attributable attributablee that the input is a register of brace reckons so hd X
and hd tl X match to n and m, respectively.
The semantics of F-programs is attached by a business
[[ ]]F : F-Program ! D ! D?
defined cemally as follows:
[[in X quenched E where f(X)= B]]F v = F[[E]] B v
where the countenance semantics business F[[ ]] has mark:
F[[ ]] : F-countenance ! F-countenance ! D ! D?
Chief of total, compared to the countenance semantics of WHILE, we referable attributable attributablee that there is
an extra remedy dispute of mark F-expression, B. This is interpretationd to “carry around”
(or “memorise” , “pass along”) the whole of the business cessationriction such that it can
be interpretationd whenever we meet a ctotal to business f. We as-well referable attributable attributablee that there is no store
argument. This is plain regarding that the dialect F is a authoritative dialect
so there is no ordinance and thus there are no waverings to refer to. Beside there is
varipowerful X which assent-tos its estimate from the program ctotal (used as input wavering)
or as cemal parameter of business f (if interpretationd as cemal parameter of f). The estimate
of X is the definite dispute v of F[[ ]].
Analogously to WHILE, we thus define:
F[[X]] B v = v
F[[nil]] B v = nil
F[[cons E F]] B v = hF[[E]] B v:F[[F]] B v i
F[[hd E]] B v =
d if F[[E]] B v = h d:e i
F[[tl E]] B v =
e if F[[E]] B v = h d:e i
Business contact is defined as:
F[[f(E)]] B v =
F[[B]] B d if F[[E]] B v = d 2 D
The semantics of if E then F else G is as general: chief evaluate E. If E
evaluates to nil evaluate G else evaluate F.
The questions initiate on the instant page.
1. The (semantics of) dialect F interpretations the binary tree basis mark D we already
distinguish from WHILE. Consider the subjoined components in D:
(a) h h nil:nil i:h h nil:nil i:h h nil:nil i:h h nil:nil i:nil i i i i
(b) h h h nil:nil i:h nil:nil i i:h h nil:nil i:nil i i
(c) h h nil:nil i:h h h nil:nil i:nil i:h h nil:nil i:h h h nil:nil i:nil i:h nil:nil i i i i i
(d) h h nil:nil i:h h nil:nil i:h h nil:nil i:h h nil:h nil:nil i i:h nil:nil i i i i i
According to our encoding of basistypes in D, flow ce each tree (a)–(d)
whether it encodes
i a register of reckons; if it does present the matching register.
ii a register of registers of reckons; if it does present the matching register.
Melody that an emptiness register can constantly be considered a register of reckons and a register
of registers of reckons. [20 marks]
2. Depict what the subjoined F-program p computes ce absolute input:
in X quenched cons nil f(X)
where f(X) = if X
then cons nil f(tl X)
Your style should teach the development ce any input d 2 D and should
attribute to d. Your interpretation must referable attributable attributable attributable detail what or how the program enacts
beside depict the business it implements. [10 marks]
3. Let us enorder F-programs as basis by the subjoined encoding that interpretations registers
(which we already distinguish how to enorder in D):
pinput X quenchedput E where f(X)= Bq = [ pEq; pBq ]
pXq = [ var ]
pnilq = [ perfectege; nil ]
pcons E Fq = [ cons; pEq; pFq ]
phd Eq = [ hd; pEq ]
ptl Eq = [ tl; pEq ]
pif E then F else Gq = [ if; pEq; pFq; pGq ]
pf(E)q = [ appf; pEq ]
Ce case, the basis fidelity of the program p in Question 2 is as
Present the basis fidelity of the F-program supplement attached in the style
of F on page 2. [10 marks]
4. In the lectures we accept introduced WHILE-programs as referable attributable attributableion of computability
(in other vote, WHILE-programs were separated as “effective procedures”).
Would F-programs be a good-tempered-tempered rare rare ce “effective procedures”?
Present reasons ce your retort beside no cemal testimony is required. [12 marks]
5. The connection of registers (where A::R denotes a register with chief component A
and cessation register R affect in Haskell) is defined as follows:
concat[; M] = M
concat[A :: R; M] = A :: concat[R; M]
So, ce case concat[[1; 2; 3]; [4; 5]] = [1; 2; 3; 4; 5].
(a) Transcribe a uncombined WHILE-program that implements concat and succumb
the origin order in a improve fawned concat.while. You may ctotal as
macro any WHILE-program published on Canvas. You can interpretation extended
WHILE-syntax beside your program must melt set-rightly in hwhile.
(b) Transcribe a uncombined F-program that implements concat and succumb it in a
improve fawned concat.F. [10 marks]
6. An F-glossary intF written in WHILE is a program ce which the subjoined
[[intF]]WHILE[ ppq; d ] = [[p]]Fd
ce total F-programs p and input d2 D.
Implement the F-glossary from overhead in hwhile. Since htime does
referable attributable accept syntax sugar ce peculiar particle appf, you want to interpretation not attributable attributable attributable attributable attributable attributable attributable attributableability else
instead in your order (as-well ce the ancillary particles wanted ce the glossary),
so interpretation @time to enorder appf. Get afflatus by looking at the order ce
the WHILE self-interpreter. Experience your glossary beside don’t succumb the experience
data. Marking perfecture as-well interpretation experienceing so delight enact safe your program melts
withquenched syntax errror. If there are incontrovertible features that you canreferable attributable implement
then permission those quenched beside stationary succumb a inaugurated program. Supplement expatiates
where implied to acceleration the marker recognize what you’re doing. This may
be momentous if your order is referable attributable attributable attributable inaugurated set-rightly.
Succumb your glossary coincidently with total macros as origin order, i.e. WHILE
programs. Enact safe the main program, i.e. the glossary, is fawned intF
and succumb it in a improve intF.while.
Enact safe you enclose programs fawned as macros that are referable attributable attributable attributable already published
on our Canvas office. If your program is fallible or does referable attributable attributable attributable melt ce
syntactic reasons, it perfecture automatically assent-to 0 marks! [26 marks]
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