These notes are based on the following page:
- RELATIONS::= See http://cse.csusb.edu/dick/maths/logic_40_Relations.html.
First time readers might like to see
- INTRODUCTION::= See http://cse.csusb.edu/dick/maths/intro_relation.html,
before looking at RELATIONS above and the rest of this page.
There is special listing of the special kinds of homogeneous relations
(transitive, reflexive, etc) in
- STANDARD_KINDS::= See http://www.csci.csusb.edu/dick/maths/math_11_STANDARD.html#Kinds of relations
Homogeneous relations have the same type for domain and codomain. As a result, for any type
the homogeneous relations for that type form a complex algebra. They are
closed under union, intersection and complement. A homogeneous relation
can be composed with itself, giving another relation of the same type -
hence powers and series of relations. In other words we have a boolean algebra
and a monoid combined together.
- For T:Types.
- For A:@T, A^2::= A><A.
- (-1)|-for A, A^2---A^@---A^(0..1).
- For X:@T, x,y:variables, W:expression(@), rel [x,y:T](W) ::= {(x,y):X><X ||(W)},
- null_relation::@(T,T)= rel [x,y:T](false).
- (-1)|- (nr): null_relation= {}.
- For n:Int, R:@(X,X), pow(R, n)::@(X,X)= if n=0 then Id(X) else if n=1 then R else if n=-1 else /R else if n<0 then (/R)^(-n) else if n>1 then R;( R^(n-1)).
- For n:Int, R:@(X,X), R^n::= pow(R,n).
- (above)|-for n,m:Int, R^(n+m)=R^n ; R^m
- For N:@Int, R:@(X,X), R^N::= |{ R^n || n:N}.
So for example:
- (above)|-R^(0..1) = Id | R.
- (above)|-R^(1..*) = R^1 | R^2 | R^3 | ....
.RoadWorks_Ahead
Conversion to a homogeneous relation and
Blocks.
- For R:@(X, X><Z ), {R}::@(X,X)= R;map[x:X, z:Z] (x).
Note -- the above defines {...} to act like a block in C/C++/Java/PHP etc.
by removing any variables (Z) added to R inside the braces.
This is a way to handle iterative (and recursive) programs that dates back to
Whitehead and Russel's work. The idea is simple: If some property is not
changed by applying a relation, then iterating many times also leaves it
unchanged. So, iterating a relation any
number of times does not change those things that can not be varied by a
single use of the relation. Hence we define the invariants of a relation.
Floyd(and hence Hoare) used this to assign meanings to programs and prove
their properties.
- invariants::= map[ R:@(X,X)] {A:@X || for x:A, y:X ( if x R y then y in A)},
- Invariants of R=sets that are closed or invariant under R.
- inv::=invariants.
- iterate::= map [R:@(X,X)] ( rel [x,y]( for all A:inv(R) (if x in A then y in A))`,
- any number of Rs = transitive, reflexive closure of R.
- do::=iterate.
Note: do is short hand. For documents that are read by people who are
new to MATHS then iterate should be used.
This implies that do(R) is the limit of the Kleene series:
- Id | R | R^2 | R^3 | ...
Informally,
do(R) is the smallest upper bound of the above series of sets.
- (pow)|- (Nat_do): R^Nat0 = R^(0..*) = do(R).
- For R:@(X,X), R^*::=do(R).
- (above)|- (inv_do): for all A:inv(R), a:A, b:a.(do(R)) ( b in A ).
- (above)|- (no_do): for x,y ( not x do(R) y) iff for some A:inv(R) ( x in A and y not in A ).
- (above)|- (inv_id1): for R, do(Id(dom(R)))=Id(dom(R))==>do(R).
- (above)|- (inv_id2): for R, R.dom.Id.do=R.dom.Id ==>R.do.
- (above)|- (do1): for R, R==>do(R).
- (above)|- (do2): if P==>do(R) then (P;R) ==> do(R).
- -- if P is a power of R so is P;R
- (above)|- (do3): if P=R;do(S;R) then P=do(R;S);R.
- (above)|- (do4): R;do(R)=do(R);R.
- (above)|- (do5): /do(R) = do(/R),
- (above)|- (do6): do(R) = Id(dom(R))|R; do(R).
- (above)|- (do7): for R, all n:Nat0, (R^n ==> do(R)).
- (above)|- (fund_do): for R, do(R) = |[n:0..](R^n).
- (above)|- (reg_rel): RegularAlgebra(@(X,X), |, {}, (;), Id(X), do).
[ math_45_Three_Operators.html ]
Given a set of relations B:@@(X,X), then the closure of B|{Id} with respect
to union(|), composition(;) is also closed with respect to iteration (do).
The smallest regular algebra which contains B is defined to be the set of
programs on the operations B:
- For B:@@(X,X), Programs(B)::@@(X,X)::=smallest{R:@@(X,X)|| B==>R and RegularAlgebra(R, |, {}, (;), Id(X), do)}.
This definitions as an algebra has some deep implications -- for example,
if P:Programs(B) then Q defined by Q::=E(Q) for some programs(B|{Q}) is
also a program.
For a set of strings B which represent a set of relations in @(X,X), the
set of simple programs on base B, SP(B) is the set of meanings of the
finite regular expression of items in B as expressions in @(X,X). To set
this up we first define a homomorphism from regular expressions into
relations and then apply it to the set of finite regular expressions:
- m::=meaning.
- For B:@Character, m:B->@(X,X), E:=regular_expression(B), m::= the [m:E->@(X,X)] ( for all A,B:E( m(A|B)=m(A)|m(B) and m(A;B)=M(A);m(B) and ...) ).
- simple_programs(B)::=img(m).
- SP(B)::=simple_programs(B).
- (above)|-SP(B)=>>Programs(B)
(while): For A:@T,while(A) ::= [R:@(T,T)](do(A;R);(T~A)), [Botting 87]
(for): For A:@T,for(I,J,K) ::= [R:@(T,T)](I;while(J)({R};K)), [Kernighan & Ritchie]
(do-od): For A:@T,do F+>f [] G+>g od::= do{F;f|G;g};@(T,T)~(F|G), [Dijkstra 76]
- Enigma::=following,
Net
- n':Nat;
- while(n>1)(2*n'=n | 2*n'=3*n+1)},
Source: Lagarias 85, J.C. Lagarias, "The 3x+1 problem and its generalizations," American Math Monthly, V92, Jan 1985, pp33-22.
(End of Net
Enigma)
Notice that when the relation is a function f:X->X that has a fixed point p, then f(p)=p and so p f p and so {p} in inv(f). Thus fix(f) ==> inv(f). However any cycle of f (where f^r(x)=x) also defines another invariant of f.
[ math_15_Unary_Algebra.html ]
Given a set S, a value x,
- For S:@T, x:T, x in S::= test for x in S.
- For S:@T, x:T, x|:S::= S'=S|{x}, Put an x in S.
- For S:@T, x:T, x:~:S::= (x' in S and S'=S~{x'}),remove an x from S.
- For S:@T, x:T, X~:S::= (X ==> S and S'=S~X), -- accept and remove any members of X in S.
If S is ordered then the minimum values are input first - whatever
sequence they are output. So a poset is MATHS's model to COBOL's SORT
verb or a library sort.
- For S:@T, (<=) :order(S), x:T, x:~:S::= (x' in min(S) and S'=S~{x'}),
- For S:@T, X:@T, X|:S::= S'=S|X, -- put members of X into S.
By giving S different structures then many standard data storage systems
can be modeled: RAM, Queues, stacks, Bags,... .
Given a homogeneous relation on a set and a map into that set then there is an equivalent relation in the domain in the map.
This is the paradigmatic example of a powerful technique - using inverse
mappings to transfer structure from codomain to domain:
Notation:
- For f:X->Y, x1, x2:X, R:@(Y,Y), x1 R mod f x2::@= f(x1) R f(x2)
or equivalently:
- (above)|- (mod): For R:@(Y,Y), f:X->Y, R mod f = f;R;/f.
Example:
- R:= <=
- X:=People
- Y:=Money
- f:=wages
- x <= mod wages y iff x earns less than y
Results:
- (above)|- (mod_pow): (R mod f)^n = (R^n)mod f.
- (above)|- (do_mod): do (R mod f) = (do(R))mod f.
- For X: @T, Equivalence_relations(X)= {E:@(X,X)|| I ==> E = (E ; E) = /E}.
Equivalence relations partition their set into non-overlapping parts. This
is done by associating with each element in X the set of elements that
are equivalent to it:
/ :: X><Equivalence_relations(X) -> @@X.
- For X: @T, E: Equivalence_relations(X), x:X, x/E={y:X||x E y}.
The family of all these sets then forms a partition.
- For X, E, x, A:@X, A/E={a/E||a:A}.
- (above)|- (eqr_part1): X>==X/E.
- (above)|- (eqr_part2): X/E in partition(X).
[ Partitions in logic_31_Families_of_Sets ]
- For all f:X->Y, (Id(X)/f) in Equivalence_relations(X).
.Dangerous_Bend
- X./f is not X/f.
Homogeneous relations provide a handy way to talk about changing systems.
If the relation R holds between the current state and a future state
then the powers: R^2, R^3, ... describe multi-step changes and do(R)
the long term possibilities.
A continuous time system with sates in S
based on a set of durations T can be
modelled as a map R from T into @(S,S), if:
Net
- (T, +, 0, -): group.
- R(t1+t2) = R(t1); R(t2).
- R(0)=Id.
(End of Net)
THe best way to describe the structure of these systems (and
any homogeneous relation) is to use the language of Directed
Graphs
- DIGRAPH::= See http://www.csci.csusb.edu/dick/maths/math_22_graphs.html#DIGRAPH.
There has been much research on the long term properties of
such systems. There are 4 properties that are commonly studied:
- (EF): It is possible to get to a particular set of states.
- (AG): The system always remains in a given set of states.
- (EG): The system can follow a path that lies in a given set.
- (AF): Whatever path the system follows it must go into a given set.
[ logic_9_Modalities.html ]
Two of the above(EF and AG) are easily stated by using do(R):
- For Type T, t:T, S:@T, R:@(T,T).
- EF(t,R,S)::= some ( t.do(R) & S ).
- AG(t,R,S)::= (t.do(R) ==> S).
The other two(EG and AF) need a model of paths or trajectories:
- paths::@(T,T)-> @(T, #T), paths relate elements to strings of elements.
- For R:@(T,T), t:T, t.paths(R)={ s:#T. head(s)=t and for all (x,y) in s ( x R y).
- trajectories(R,t)::=t.paths(R).
Given the above we can express EG and AF as:
- EG(t,R,S)::= some( t.paths(R) & #S ).
- AF(t,R,S)::= (t.paths(R) ==> #T S #T).
Notice the use of regular expressions to describe sets of paths.
We can show:
- (above)|- (path1): AG(t,R,S) iff t.do(R) ==> S iff t.paths(R)==>#S.
Thus the language of paths and regular expressions is a way to talk about
the behaviors of complex systems. There is no reason not to use grammars to express
irregular patterns as well.