library Basic/Graphs

version 0.9

%% authors: M.Roggenbach, T.Mossakowski, L.Schr�der

%% Corresponding author: Till Mossakowski, till@tzi.de

%% date: 17.09.02

%left_assoc __intersection__ %[ Should go to StructuredDatatypes ]%

%left_assoc __+__

%prec({__isIn__, __subset__, __disjoint__,__ :: __ --> __}

< {__union__, __intersection__, __\\__, __*__,

__disjointUnion__} )%

%prec {__+__} < {__*__,__div__}

%%from Basic/StructuredDatatypes get List

%%from Basic/Numbers get Nat

logic HasCASL

spec Bool =

sort Bool

ops false, true : Bool;

end

spec Nat =

sort Nat

ops __+__,__*__,__div__ : Nat*Nat->Nat;

0,1,2 : Nat

pred __<__ : Nat*Nat;

end

spec List = {}

spec InternalLogic = {}

spec Extensionality = {}

%{ Typed sets are represented by predicates over the type.

Set membership is just holding of the predicate: x isIn s <=> s x

Note that for disjoint unions and products, the type changes. }%

spec Set = Bool and InternalLogic and Extensionality then

var S,T : Type

type Set S := S ->? Unit;

ops emptySet : Set S;

{__} : S -> Set S;

__isIn__ : S * Set S ->? Unit;

__subset__ :Pred( Set(S) * Set(S) );

__union__, __intersection__, __\\__ : Set S * Set S -> Set S;

__disjoint__ : Pred( Set(S) * Set(S) );

__*__ : Set S -> Set T -> Set (S*T);

__disjointUnion__ : Set S -> Set S -> Set (S*Bool);

injl,injr : S -> S*Bool;

forall x,x':S; y:T; s,s':Set S; t:Set(T)

. not (x isIn emptySet)

. x isIn {x'} <=> x=x'

. x isIn s <=> s x

. s subset s' <=> forall x:S . (x isIn s) => (x isIn s')

. (x isIn (s union s')) <=> ((x isIn s) \/ (x isIn s'))

. (x isIn (s intersection s')) <=> ((x isIn s) /\ (x isIn s'))

. x isIn s \\ s' <=> x isIn s /\ not (x isIn s')

. (s disjoint s') <=> ((s intersection s') = emptySet)

%% . ((x,y) isIn (s * t)) <=> ((x isIn s) /\ (y isIn t))

. (injl x) = (x,false)

. (injr x) = (x,true)

%%. (s disjointUnion s') = ((s * {false}) union (s' * {true}))

end

%{ Relations and partial equivalence relations (PERs) }%

spec Relation =

Set

then

var S : Type

ops reflexive, symmetric, transitive : Pred(Set(S*S))

forall r:Set(S*S)

. (reflexive r) <=> (forall x:S . r(x,x))

. (symmetric r) <=> forall x,y:S . (r(x,y)) => (r(y,x))

. (transitive r) <=> forall x,y,z:S . ((r(x,y)) /\ (r(y,z))) => (r(x,y))

type PER S := Set(S*S)

%% = { r : Set(S*S) . (symmetric r) /\ (transitive r) }

op dom : PER S -> Set S

%%forall x:S; r: PER S

%%. (x isIn (dom r)) <=> ((x,x) isIn r)

end

%{ Partial maps and endomaps }%

spec Map =

Set then

var S,T,U : Type

type Map S := S->?S

ops dom : (S->?T) -> Set S;

range : (S->?T) -> Set T;

image : (S->?T) -> Set S -> Set T;

emptyMap : (S->?T);

__ :: __ --> __ : Pred ( (S->?T) * Pred(S) * Pred(T) );

__ [__/__] : (S->?T) * S * T -> (S->?T);

__ - __ : (S->?T) * S -> (S->?T);

__o__ : (T->?U) * (S->?T) -> (S->?U);

__||__ : (S->?T) * Set S -> (S->?T);

undef__ : S ->?T;

ker : (S->?T) -> Pred (S*S);

injective : Pred(S->?T)

forall f,g:S->?T; h:T->?U; s,s':Set S; t:Set T; x,x':S; y:T

. x isIn dom f <=> def (f x)

. y isIn range f <=> exists x:S . f x = y

. y isIn image f s <=> exists x:S . x isIn s /\ f x = y

. not def (emptyMap x)

. f :: s --> t <=> (forall x:S . x isIn s => f x isIn t)

. f[x/y] x' = if x=x' then y else (f x)

. (f - x) x' = if x=x' then undef() else (f x)

. (s intersection s') x = if s x = s' x then s x else undef()

. def (s union s') <=> (forall x:S. def (s x) /\ def (s' x) => s x = s' x)

. def (s union s') => (s union s') x = if def (s x) then s x else (s' x)

. (h o g) x = h(g(x))

. (f || s) x = if x isIn s then f x else undef()

%% the kernel is a PER with same domain as the function, hence the existential equation

. (x,x') isIn ker f <=> f x =e= f x'

%% injectivity only for values in the domain, hence the existential equation

. injective f <=> (forall x,y:S . f x =e= f y => x=y)

end

%{ If we want to iterate the constructions product, disjoint union

and quotients on sets, we need to circumvent the problem that

the type of the sets changes with each construction.

This means that we need a specific type S which comes with

the possibility to represent the result of the above mentioned

operations again in S.

The following specification states the abstract requirement that

such representations exists for a given type S. }%

spec SetRepresentation =

Map and Relation

then

type S;

ops pair : S*S->S; %% represent S*S in S

inl,inr : S->S; %% represent S disjointUnion S in S, with left and right injection

%% represent quotient by a PER, yields the quotient map

%% the quotient is then the image of the quotient map

coeq: PER S -> Map S;

. injective pair

. injective inl

. injective inr

. (range inl) disjoint (range inr);

forall r:PER S

. ker (coeq r) = r

end

%{ Some sample set representation for the natural numbers ... }%

spec Nat_SetRepresentation =

Map and Nat and Relation

then

ops pair : Nat*Nat->Nat;

inl,inr : Nat->Nat;

coeq: PER Nat -> Map Nat;

min : Pred Nat ->? Nat

forall r:PER Nat; m,n:Nat

%% Use Cantor's diagonalization for pairs of natural numbers

. pair(m,n) = ((m+n)*(m+n+1)+2*m) div 2

%% Use even and odd numbers as two copies of the natural numbers

. inl m = 2*m

. inr m = 2*m+1

%% choose the minimal element as representative of an equivalence class

. min p = n <=> (p n /\ forall m:Nat . m<n => not p m)

. coeq r n = min (\ m:Nat . (m,n) isIn r)

end

%{ ... and the statement that this is indeed a set representation }%

view v : SetRepresentation to Nat_SetRepresentation = S |-> Nat

end

%{ Given a set representation, we now can define products, coproducts,

coequalizers and pushouts, while staying within the same type of sets.

Note that __*__ is now overloaded: for two given S-sets,

it delivers either an (S*S)-set, as defined in the specification

Set above, or an S-Set, as defined here.

We also specify the mediating morphisms that exist by the

respective (co)universal properties of the constructions. }%

spec AbstractSetConstructions[SetRepresentation] given Map = %def

ops __*__,__coproduct__ : Set S * Set S -> Set S;

%% the mediating morphisms for (co)products depend on two morphisms

%% going into the factors (out of the summands)

<__! __>, [__! __] : Map S -> Map S -> Map S;

%% the product projections (coproduct injections already come from SetRepresentation)

pi1,pi2 : S ->? S;

%% partial inverses of the coproduct injections

left, rigt : S ->? S;

%% given a PER, yield the resulting quotient set (=coequalizer)

%% then we have coeq r :: dom r --> factor r

factor : PER S -> Set S;

%% the mediating morphism for coequalizers

%% if we have another h :: dom r --> A, it factors through coeq r via the mediating morphism

mediate : PER S -> Map S ->? Map S

forall x,y:S; s,s1,s2,t:Set S; f,g,h : Map S; r:PER S

. pi1 (pair (x,y)) = x

. pi2 (pair (x,y)) = y

. s1 * s2 = image pair (s1*s2)

. <f!g> x = pair(f x, g x)

. s1 coproduct s2 = (image inl s1) union (image inr s2)

. [f!g] (inl x) = f x

. [f!g] (inr x) = g x

. def left x <=> x isIn range inl

. def right x <=> x isIn range inr

. left (inl x) = x

. right (inr x) = x

. factor r = range (coeq r)

. f :: dom r --> t /\ ker f subset r => mediate r f (coeq x) = f x

then %implies

. pi1 :: s1 * s2 --> s1

. pi2 :: s1 * s2 --> s2

. inl :: s1 --> s1 coproduct s2

. inr :: s2 --> s1 coproduct s2

. f :: s --> s1 /\ g :: s --> s2 =>

<f!g> : s --> s1*s2 /\

pi1 o <f!g> || s = f || s /\

pi2 o <f!g> || s = g || s

. f :: s1 --> s /\ g :: s2 --> s =>

[f!g] :: s1 coproduct s2 --> s /\

[f!g] o inl || s1 coproduct s2 = f || s1 coproduct s2 /\

[f!g] o inr || s1 coproduct s2 = g || s1 coproduct s2

. coeq r :: dom r --> factor r

. f :: dom r --> t /\ ker f subset r =>

mediate r f :: (factor r) --> t /\

mediate r f o coeq r || dom r = f || dom r

then %def

var S : Type

%% pushout mediators

%% the mediating morphism for pushouts depends on a cocone,

%% which is given by an object and two morphisms (ending in that object)

type Med Ob Mor = Ob * Mor * Mor ->? Mor

%% (co)product of maps

ops __*__,__coproduct__ : Map S * Map S -> Map S;

%% the pushout operation takes a diagram (three objects and two morphisms)

%% and yields a colimiting cocone (an object and two morphisms)

%% plus a mediator

pushout : Set S * Map S * Set S * Map S * Set S ->

(Set S * Map S * Map S * Med (Set S) (S->?S))

forall x,y:S; s,s1,s2,s3,t:Set S; f,g,h : Map S; r:Pred(S*S)

. f*g = <f o pi1 ! g o pi2>

. f copoduct g = [inl o f ! inr o g]

. def pushout (s1,f,s2,g,s3) <=> f :: s1 --> s2 /\ g :: s1 --> s3

%% pushouts are coequalizers of coproducts

. def pushout (s1,f,s2,g,s3) =>

pushout (s1,f,s2,g,s3) =

%[ first take the coproduct ... ]%

let s = s2 coproduct s3;

%[ r is the PER identifying those elements of s coming from

elements of s1 ]%

r = \(x,y:S) . exists z:S . z isIn s1 /\ inl (f z) =x /\ inr (g z) = y

as PER S

%[ ... and then the coequalizer ]%

q = coeq r

in

%[ the pushout object is the quotient, the cocone is the

composition of the cocones ]%

(factor r,q o inl,q o inr,

%[ the mediator first takes the coproduct mediator [h!k]

%% and then the coequalizer mediator (mediate r [h!k]) ]%

\(d,h,k:Med) (Set S) (S->?S) .

if h :: s2 --> d /\ k :: s3 --> d /\

h o f || s1 = k o g || s1

then mediate r [h!k] else undef())

then %implies

%% pushout properties

{}

end

%{ Directed graphs over some arbitrary but fixed node and edge types,

given by node set and source and target maps.

Note that the node set may be larger than the range of these maps

in case that there are isolated nodes. }%

spec DirectedGraph =

Set

then

var N,E : Type

type Graph N E = { (n,source,target) : Set N * (E->?N) * (E->?N) .

dom source=dom target

/\ (source : dom source --> n)

/\ (target : dom target --> n) }

end

%{ Some useful operations on directed graphs. }%

spec RichDirectedGraph =

DirectedGraph

then %def

var N,E : Type

ops emptyGraph : Graph N E;

nodes : Graph N E -> Set N;

edges : Graph N E -> Set E;

sourceMap, targetMap : Graph N E -> (E->?N);

addNode,removeNode : N -> Graph N E -> Graph N E;

addEdge : E * N * N -> Graph N E -> Graph N E;

removeEdge : E -> Graph N E -> Graph N E;

symmetric, transitive, loopFree, complete : Graph N E;

__subgraph__, __cliqueOf__, __maxCliqueOf__ : Pred(Graph N E * Graph N E);

__union__, __intersection__ : Graph N E * Graph N E -> Graph N E

forall n,n1,n2:N; e,e1,e2:E; g,g':Graph N E;

nn : Set N

. emtpyGraph = (emptySet,emptyMap,emptyMap) as Graph N E %(emtpyGraph)%

. nodes (nn,s,t) = nn

. edges (nn,source,target) = dom source %(edges)%

. sourceMap (nn,source,target) = source

. targetMap (nn,source,target) = target

. addNode n (nn,source,target) =

(nn union {n}, source, target) as Graph N E %(addNode)%

. removeNode n (nn,source,target) =

(nn \\ {n}, source, target) as Graph N E %(removeNode)%

. addEdge (e,n1,n2) (nn,source,target) =

(nn union {n1} union {n2},

source[e/n1], target[e/n2]) %(addEdge)%

. removeEdge e (nn,source,target) =

(nn, source-e, target-e) %(removeEdge)%

. symmetric g <=>

forall e:E .

e isIn edges g =>

exists e':E . e' isIn edges g

/\ sourceMapMap g e = targetMap g e'

/\ targetMap g e' = targetMap g e %(symmetric_def)%

. transitive g <=>

forall e1,e2:E .

e1 isIn egdes g /\ e2 isIn edges g /\

targetMap g e1 = sourceMap g e2 =>

exists e3:E . e3 isIn edges g

/\ sourceMap e3 g = sourceMap e1 g

/\ targetMap e3 g = targetMap e2 g %(transitive_def)%

. loopFree g <=>

not (exists e:E . e isIn edges g /\ sourceMap e g = targetMap e g)

%(loopFree_def)%

. g1 subgraph g2 <=>

(forall n:N . n isIn nodes g1 => n isIn nodes g2) /\

(forall e:E . e isIn edges g1 => e isIn edges g2) /\

sourceMap g1 = sourceMap g1 intersection sourceMap g2

targetMap g1 = targetMap g1 intersection targetMap g2

%(subgraph_def)%

. complete (nn,source,target) <=>

forall n1,n2:N . n1 isIn nn /\ n2 isIn nn =>

exists e:E . source e = n /\ target e = n %(complete)%

. g1 cliqueOf g2 <=>

g1 subgraph g2 /\ complete(g1) %(cliqueOf_def)%

. g1 maxCliqueOf g2 <=>

g1 cliqueOf g2

/\ forall g3:Graph N E .

g1 subgraph g3 /\ g3 cliqueOf g2 => g1=g3 %(max_cliqueOf_def)%

. g union g' =

(nodes g union nodes g',

sourceMap g union sourceMap g',

targetMap g union targetMap g') as Graph N E

. g intersection g' =

(nodes g intersection nodes g',

sourceMap g intersection sourceMap g',

targetMap g intersection targetMap g') as Graph N E

end

spec SimpleGraph =

RichDirectedGraph

then

type SimpleGraph N E = {(nn,source,target):Graph N E .

forall e1,e2:E . source e1 = source e2

/\ target e1 = target e2 => e1=e2 }

type LoopFreeGraph N E = {g:Graph N E . loopFree g}

end

spec GraphHomomorphism =

DirectedGraph

then

var N1, E1, N2, E2, N3, E3 : Type

type Hom N1 E1 N2 E2 =

{(g1,hn,he,g2) : Graph N1 E1 * (N1->?N2) * (E1->?E2) * Graph N2 E2 .

hn :: nodes g1 --> nodes g2 /\ he :: edges g1 --> edges g2

/\ forall e:E1 . e isIn edges e =>

( hn(source g1 e) = source g2 (he e)

/\ hn(target g1 e) = target g2 (he e) ) }

type HomHom E N = Hom E N E N

ops dom : Hom N1 E1 N2 E2 -> Graph N1 E1;

cod : Hom N1 E1 N2 E2 -> Graph N2 E2;

nodeMap : Hom N1 E1 N2 E2 -> (N1->?N2);

edgeMap : Hom N1 E1 N2 E2 -> (E1->?E2);

__o__ : Hom N2 E2 N3 E3 * Hom N1 E1 N2 E2 ->? Hom N1 E1 N3 E3

forall g1:Graph N1 E1; hn:N1->?N2; he:E1->?E2; g2:Graph N2 E2;

h1:Hom N1 E1 N2 E2; h2:Hom N2 E2 N3 E3

. dom ((g1,hn,he,g2) as Hom N1 E1 N2 E2) = g1

. cod ((g1,hn,he,g2) as Hom N1 E1 N2 E2) = g2

. nodeMap ((g1,hn,he,g2) as Hom N1 E1 N2 E2) = hn

. edgeMap ((g1,hn,he,g2) as Hom N1 E1 N2 E2) = he

. def h2 o h1 <=> cod h1 = dom h2

. def h2 o h1 =>

h2 o h1 =

(dom h1, nodemap h2 o nodemap h1, edgemap h2 o edgemap h1,cod h2)

as Hom N1 E1 N2 E2

end

%{ Constructions on directed graphs, defined in terms of the corresponding

operations on sets, usually acting component wise.

Note that we have to fix N and E here, since we need set representations for them

in order to get the needed constructions on sets. }%

spec DirectedGraphConstructions [SetRepresentation with S |-> N] [SetRepresentation with S |-> E] =

AbstractSetConstructions[SetRepresentation with S |-> N]

and

AbstractSetConstructions[SetRepresentation with S |-> E]

and GraphHomomorphism

then

type Congruence = {(g,rn,re) : Graph N E * PER N * PER E

. nodes g = dom rn /\ edges g = dom re /\

forall e1,e2:E . (e1,e2) isIn re =>

(sourceMap g e1,sourceMap g e2) isIn rn /\

(targetMap g e1,targetMap g e2) isIn rn }

ops graph : Congruence -> Graph N E;

nodeRel : Congruence -> PER N;

edgeRel : Congruence -> PER E;

factor : Congruence ->? Graph N E;

__coproduct__ : Graph N E * Graph N E -> Graph N E;

pushout : HomHom N E -> HomHom N E ->?

HomHom N E * HomHom N E * Med (Graph N E) (HomHom N E)

forall g,g1,g2: Graph N E; h1,h2:HomHom N E; r:Relation N E

. ((g,rn,re) in Congruence)

=> graph ((g,rn,re) as Congruence) = g

. ((g,rn,re) in Congruence)

=> nodeRel ((g,rn,re) as Congruence) = rn

. ((g,rn,re) in Congruence)

=> edgeRel ((g,rn,re) as Congruence) = re

. factor c =

(factor (graph c) (nodeRel c),

mediate (edgeRel c) (coeq (nodeRel c) o sourceMap (graph c)),

mediate (edgeRel c) (coeq (nodeRel c) o targetMap (graph c)) )

. g1 coproduct g2 =

(nodes g1 coproduct nodes g2,

sourceMap g1 coproduct sourceMap g2,

targetMap g1 coproduct targetMap g2)

. def pushout h1 h2 <=> dom h1 = dom h2

. def pushout h1 h2 =>

pushout h1 h2 =

let nm1 = nodeMap h1

nm2 = nodeMap h2

(np,ninl,ninr,nmed) = pushout(dom nm1,nm1,cod nm1,nm2,cod nm2)

em1 = edgeMap h1

em2 = edgeMap h2

(ep,einl,einr,emed) = pushout(dom em1,em1,cod em1,em2,cod em2)

s = nmed (ninl o (sourceMap (cod h1)),ninr o (sourceMap (cod h2)))

t = nmed (ninl o (targetMap (cod h1)),ninr o (targetMap (cod h2)))

pgraph = (np,s,t)

in

((cod h1,ninl,einl,pgraph) as HomHom N E,

(cod h2,ninr,einr,pgraph) as HomHom N E,

\(g:Graph N E, h:HomHom N E, k:HomHom N E) .

(pgraph,

nmed (nodeMap h,nodeMap k),

emed (edgeMap h,edgemap k),

g ) as HomHom N E )

end

%% still first order !!!!!

spec SymmetricClosure[sort NodeLabel][sort EdgeLabel] =

RichAbstractGraph[sort NodeLabel][sort EdgeLabel]

then

op sc : Graph -> Graph

forall n,n1,n2:NodeLabel; e:EdgeLabel; g:Graph

. n isIn sc(g) <=> n isIn g %(sc_def_1)%

. e::n1-->n2 isIn sc(g) <=>

(e::n1-->n2 isIn g \/ e::n2-->n1 isIn g) %(sc_def_2)%

end

spec TransitiveClosure[sort NodeLabel][sort EdgeLabel] =

RichAbstractGraph[sort NodeLabel][sort EdgeLabel]

and

List[sort EdgeLabel fit Elem |-> EdgeLabel]

and

SymmetricClosure[sort NodeLabel][sort List[EdgeLabel]]

with Graph |-> PathGraph

then

ops tc,stc : Graph -> PathGraph

preds __pathSubgraphOf__ : Graph * PathGraph;

pathTransitive : PathGraph

forall n,n1,n2:NodeLabel; e:EdgeLabel; g:Graph; g':PathGraph

. g pathSubgraphOf g' <=>

(forall n:NodeLabel . n isIn g <=> n isIn g') /\

(forall n1,n2:NodeLabel; e:EdgeLabel .

e::n1-->n2 isIn g <=> (e::[])::n1-->n2 isIn g')

%(pathSubgraphOf_def)%

. pathTransitive(g') <=>

forall n1,n2,n3:NodeLabel; e1,e2:List[EdgeLabel] .

e1::n1-->n2 isIn g' /\ e2::n2-->n3 isIn g' =>

(e1++e2)::n1-->n3 isIn g' %(pathTransitive_def)%

. pathTransitive(tc(g)) %(tc_def1)%

. g pathSubgraphOf tc(g) %(tc_def2)%

. g pathSubgraphOf g' /\ pathTransitive(g') =>

tc(g) subgraphOf g' %(tc_def3)%

. stc(g) = sc(tc(g)) %(stc_def)%

end

spec Minor[sort N1][sort E1][sort N2][sort E2] =

TransitiveClosure[sort N2][sort E2] with Graph |-> Graph2

and

GraphHomomorphism[sort N1][sort E1][sort N2][sort List[E2]]

with Graph2 |-> PathGraph

then

pred __minorOf__(g1:Graph1; g2:Graph2) <=>

exists h:Hom . source(h)=g1 /\ target(h)=stc(g2) %(minorOf_def)%

end

spec K5 =

free type Five ::= 1 | 2 | 3 | 4 | 5

then

SimpleAbstractGraph[sort Five] with Graph |-> K5

then

op k5 : K5

forall n,n1,n2 : Five

. n isIn k5 %(k5_def_1)%

. n1 --> n2 isIn k5 %(k5_def_2)%

end

spec K3_3 =

free type Three ::= 1 | 2 | 3

free type Three2 ::= left(Three) | right(Three)

then

SimpleAbstractGraph[sort Three2] with Graph |-> K3_3

then

op k3_3 : K3_3

forall n,n1,n2 : Three

. left(n) isIn k3_3 %(k3_3_def_1)%

. right(n) isIn k3_3 %(k3_3_def_2)%

. left(n1) --> right(n2) isIn k3_3 %(k3_3_def_3)%

end

%% planar graphs defined using the Kuratowski characterization

spec Planar[sort NodeLabel][sort EdgeLabel] =

K5 and K3_3

and

Minor[sort Five][sort Unit][sort NodeLabel][sort EdgeLabel]

with Graph1 |-> K5, Graph2 |-> Graph

and

Minor[sort Three2][sort Unit][sort NodeLabel][sort EdgeLabel]

with Graph1 |-> K3_3, Graph2 |-> Graph

then

pred planar(g:Graph) <=> not k5 minorOf g /\ not k3_3 minorOf g

%(planar_def)%

end

%%