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Imperative Program with Input
http://cis.k.hosei.ac.jp/~yukita/
Imperative Programs
Let be an alphabet and consider functors from free monoid on
to Sets.
Such a functor can be considered as a collection of functions
f a : X X (a ) for a fixed set X ( the state space) with
the property f ba f b f a for all pairs a, b .
If | X | , we call such a functor
a deterministic finite automaton or deterministic finite machine.
2
Def. Finite State Recognizer
A finite state recognizer is a finite state automaton with
one specified start state J and a finite set K1 , K 2 , of
specified end states.
A recognizer defines a subset U of * as follows :
an an 1 a1 U
if
f an f a2 f a1 ( J ) K i
for some end state K i .
3
Theorem.
The set U * is recognized by a finite state recognizer
if and only if U is a regular language.
Proof. A recognizer is a special type of regular grammar. ...
Conversely , given a grammar X (without ), we show how
to construct an automaton which recognizes the same language
as that generated by the grammar.
Take the states of the automaton to be 2 obj X .
Put an arrow a : Y1 Y2
if Y2 { y2 | y1 Y1 ; a : y1 y2 in the grammar}.
Take the start state to be {J }. Take the end states to be all
subsets containing at least one end object of the grammar.
4
Imperative Program with Input
Def. Given an aphabet , an imperative program with input from
is a functor * Sets, constructe d out of given functions using the
operations available in a distributi ve category.
Giving such a functor amounts to giving a set X (the state space)
and functions f a : X X (a ) constructe d from given functions
in the appropriat e way.
A behavior of an implerativ e program consists of the effect of a sequence
of inputs on an initial state.
a3
a2
a1
x2
x1
x0
f
f
f
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Ex. 37. Exponential
The state space is X ( I I ).
The input alphabet is {clock } .
Notice | | .
Let ( p, d , t ) be a typical element of X .
Rough idea :
clock
t 0 calculatio n takes place
t 1
ignore
number
ignore
number input
6
Ex. 37 continued
f clock : ( I I ) ( I I )
2
2
( p d , d 1,0)
( p, d ,0)
( p, d ,1)
if d 1
if d 1
( p, d ,1) ( p, d ,1).
fn : (I I ) (I I )
2
2
( p, d ,0) ( p, d ,0)
( p, d ,1) (1, n,0).
7
Ex. 38. Product and Sum
We start with the following two programs.
f a : X X with 1 {a}.
f b : X X with 2 {b}.
1 2
New program 1 :
g a f a 1Y : X Y X Y ,
g b 1X f b : X Y X Y .
New program 2 :
ha f a 1Y : X Y X Y ,
hb 1X f b : X Y X Y .
8
Ex. 38. continued
Program 1 allows the two users to work independen tly.
Program 2 put the two users in total conflict.
If the initial state belongs to the first component of X Y
then every input from the first user has effect; no input of
the second user has effect.
If the initial state belongs to the second component then
the roles are reversed.
9
Basic Idea behind Functional
Languages
• The move towards programming by
specification of goals or functions,
• as distinct from imperative programming.
• In many cases, the specification actually
contains the means for computation.
10
Ex. 39 f(x)=2x
Graph Data has two objects I and N and two arrows
0 : I N (zero) ,
s : N N (successor ).
Graph Function contains Data and one extra arrow
f : N N.
The set of equations between paths Equation consists of
fs ssf ,
f 0 0.
11
Def. Functional Specification
A functional specification consists of two graphs,
Data Function,
and a set, Equation, of equations between paths in Function.
Data has three specified objects, I, J, K ; Function has a
specified arrow f : J K .
A funtional specificat ion determines a relation
f : PathsData ( I , J ) PathsData ( I , K )
as follows : f ( path1) path2 if f ( path1) path2 is
deducible in Function from the equations in Equation.
12
Ex. 40. Calculation in Ex. 39
We let J K N .
f : Paths( I , N ) Paths( I , N )
s n 0 s m 0,
if fs n 0 s m 0 is deducible from fs ssf and f 0 0.
It is clear that
f : s n 0 s 2n 0.
13
Note.
• A functional specification may give rise to
a partial function instead of a (full) function,
• or even a multi-valued function.
• Robbie Gates showed that the partial
functions specifiable from to are
precisely the partial recursive functions.
14
Ex. 41. Length of a list
Let Data has three objects I, L, and N and arrows
0 : I N,
o : I L,
s:N N
a1 , a2 , a3 , : L L.
Think of o as the empty list, and each letter a as appending
a letter to the list.
Function consists of Data together with one extra arrow
length : L N .
Equation consists of two equations :
length ai s length
(ai ),
length o 0.
15
Ex. 41. continued
length : Paths( I , L) Paths( I , N ) gives the length of a list.
A typical calculatio n goes like this :
length a1 a3 a3 a2 o s length a3 a3 a2 o
s s length a3 a2 o
s s s length a2 o
s s s s length o
s s s s 0 s 4 0.
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Ex. 42. Sort
The alphabet is ordered as a1 a2 a3 .
Data consists of two objects I and L and arrows
o : I L, a1 , a2 , a3 , : L L.
Function has two extra arrows :
transfer : L L and sort : L L.
Arrow transfer is intended to alter a list in the following
way : it takes the leftmost element of a list and moves it
to the left of the first element wh ich is larger or the same.
17
Ex. 42. continued
Equation consists of
sort o o,
sort a transfer a sort
(a ),
transfer o o,
transfer a o a o,
transfer ai a j ai a j
(ai a j ),
transfer ai a j a j transfer ai
(ai a j ).
sort : Paths( I , L) Paths( I , L) sorts a list into
ascending order.
18
Ex. 42. A typical calculation
sort a2 a1 a3 a1 o
transfer a2 sort a1 a3 a1 o
transfer a2 transfer a1 sort a3 a1 o
transfer a2 transfer a1 transfer a3 sort a1 o
transfer a2 transfer a1 transfer a3 transfer a1 sort o
transfer a2 transfer a1 transfer a3 transfer a1 o
transfer a2 transfer a1 transfer a3 a1 o
transfer a2 transfer a1 a1 transfer a3 o
transfer a2 transfer a1 a1 a3 o
transfer a2 a1 a1 a3 o
a1 transfer a2 a1 a3 o
a1 a1 transfer a2 a3 o
a1 a1 a2 a3 o
19
