A PVS Graph Theory Library - Semantic Scholar

NASA

/ TM-1998-206923

A PVS Graph Ricky

W.

Langley A

Air

Force

National

Library

Butler Research

Jon

Theory

Center,

Hampton,

Virginia

Sjogren Office

Aeronautics

of Scientific

and

Space Administration Langley Research Center Hampton, Virginia 23681-2199

February

1998

Research,

Washington,

DC

Available

from the following:

NASA Center for AeroSpace 800 Elkridge Landing Road Linthicum Heights, (301) 621-0390

Information

MD 21090-2934

(CASI)

National 5285 Port

Technical Royal

Information

Road

Springfield, VA 22161-2171 (703) 487-4650

Service

(NTIS)

Abstract This

pal)or

documents

the NASA

Langley

PVS graph

theory

library.

provides fimdanmntal definitions fi)r graphs, subgraphs, walks, paths, ated by walks, trees, cycles, degree, separating sets, aim four notions Theorems notiolis

provided include connectedness.

of

Ramsey's

and

Menger's

and

The

library

subgraphs gen(wof commcl_,dness.

tim equivalence

of all fimr

Contents 2

1

Introduction

2

Definition

3

Graph

Operations

5

4

Graph

Degree

6

5

Subgraphs

6

Walks

7

Connected

8

Circuits

14

9

Trees

15

2

of a Graph

7 and

8

Paths

13

Graphs

10 Ramsey's

Theorem

15

11 Menger's

Theorem

16

12 PVS

18

Theories

13 Concluding A

20

Remarks

APPENDIX:

Other

A.1

Gral)h

Inductions

A.2

Subgrat)hs

A.3 A.4

Maximmn Minimmn

A.5

Abstra(:t

Supporting

Theories

2O

.................................

G(_nerated

From

Walks

........................

Sul)grat)hs ............................... Walks .................................. Min an(t

Max

Thoori(_s

2O

.........................

21 21 22 23

1

Introduction

This the

paper

(l 0

connected?(del_vert(G,v)))

size(G)

bool

= NOT empty?(G) AND (FORALL (x,y: (vert(G))): (EXISTS (w: Walk(G)):

seq(w)(0)

= x AND

seq(w)(l(w)-l)

piece_connected?(G):

bool

= NOT empty?(G) AND (FORALL HI,H2: G = union(HI,H2) NOT IMPLIES

complected?(G):

l)rovides

path_connected,

bool

N0T

empty?(Hl)

AND

AND N0T

(x:

empty?(H2)

empty?(intersection(vert(Hl), vert(H2))))

= IF isolated?(G) THEN singleton?(G) ELSIF (EXISTS (v: (vert(G))): deg(v,G) (EXISTS

= y))

(vert(G))):

deg(x,G)

= i) THEN

= 1 AND

connected?(del_vert(G,x))) ELSE (EXISTS

(e: (edges(G))):

connected?(del_edge(G,e))) ENDIF

These

definitions

equivaleilce

aro

graph_connected[T:

are located

located in the

in the theory

graph_conn_defs graph_connected:

TYPE] : THEORY

13

theory.

The

following

lemmas

about

BEGIN

G:

VAR graph[T]

conn_eq_path

:THEOREM

8

= path_connected?(G)

path_eq_piece:

THEOREM

path_connected?(G)

piece_eq_conn:

THEOREM

piece_connected?(G)

conn_eq_complected:

END

connected?(G)

THEOREM

= piece_connected?(G)

= connected?(G)

connected?(G)

= complected?(G)

graph_connected

Circuits

A slightly

non-traditional

in tile same

place

(i.e.

definition

of circuit

a pre_circuit)

an(l

is used. is cyclically

reducible?(G: graph[T], w: Seq(G)): bool l(w) - 1 AND w(k-l) = w(k+l))

reduced?(G:

graph[T],

w:

cyclically_reduced?(G: reduced?(G,w)

pre_circuit?(G:

Seq(G)):

graph[T], AND w(1)

graph[T],

A circuit

bool

w:

=

reduced

(i.e.

(EXISTS

= NOT

Seq(G)):

graph[T],

that

starts

and

ends

cyclically_reduced?).

(k: posnat):

k
2 AND

/= w(l(w)-2)

w: prewalk):

bool

= walk?(G,w) w(0)

circuitT(G:

is a walk

w: Seq(G)):

bool

AND

= w(l(w)-l)

= walk?(G,w)

AND

cyclically_reduced?(G,w)

AND

pre_circuit?(G,w)

Tile

following cir_deg_G

prol)erties

are

: LEMMA

proved (EXISTS

in the

circuit_deg

(a,b:

(vert(G))):

a /= b AND deg(z,G)

circuit_deg

: LEMMA

FORALL

theory:

edge?(G)(a,z) >=

below(l(w))):

deg(w(i),G

14

AND

AND

edge?(G)(b,z)

2

(w: Walk(G),i: IMPLIES

vert(G)(z)

circuit?(G,w)

from(G,w))

>= 2

) IMPLIES

9

Trees

Trees

are

defin(,(I

G: VAR

r(,('ursiv(qy

as

follows:

graph[T]

tree?(G):

RECURSIVE

bool

= card [T] (vert (G) ) =

(EXISTS MEASURE

and

the

Tree

Tree:

type

TYPE

(v: (vert(G))): tree? (del_vert

size(G)

is defined

as follows:

= {G: graph[T]

I tree?(G)}

Theflmdammltall)ropertythattre(,s tree_no_circuits:

havenocircuitsist)rovt,din

THEOREM

(FDRALL

(w: N0T

10

Ramsey's

This

work

entitled Basin

builds "The

and

i, j: VAR n, p, q,

Walk(G)):

tree_circ

tree?(G)

=>

circuit?(G,w))

upon

a w_rification Prover

Kauflnann

of this and

theorem

Nuprl:

by

An

Natarajan

Experimental

Shankar Comt)arison"

2.

T ii: VAR

nat

g: VAR graph[T] G: VAR Graph[T] V: VAR

% nonempty

finite_set[T]

contains_clique(g, n): bool = (EXISTS (C: finite_set[T]): subset?(C,vert(g)) (FORALL

i,]:

i/=]

AND AND

card(C) C(i)

>= n AND AND

C(j)

contains_indep(g, n): bool = (EXISTS (D: finite_set[T]): "CLI

Technical

theory.

Theorem

Boycr-Moore

Matt

I 0R deg(v,G) = I AND [T] (G, v) ) )

1Report 58, .luly 17, 1990. 15

IMPLIES

edge?(g)(i,j)))

and

the

paper

by

David

subset?(D, (FORALL

vert(g)) i, j:

subgraph_clique:

AND

i/=j

card(D)

AND D(i)

(FORALL

LEMMA

>= n AND

AND

D(j)

IMPLIES

N0T

IMPLIES : LEMMA

(FORALL

contains_clique(g,

IMPLIES THEOREM

(EXISTS

contains_indep(g,

(G: Graph[T]):

IMPLIES

To

Menger's state

menger's

COml)licated G:

theoreln

one

We

size(G)

>= n

(contains_clique(G,

Ii)

contains_indep(G,

12))))

OR

graph[T]

=

first

define

begin

with

minimum the

(_oncept

separating

sets.

of a set)arating

This

is fairh"

set:

graph[T] VAR

e:

VAR

doubleton[T]

V:

VAR

finite_set[T]

T

del_verts(G,V): (# vert

:= difference[T](vert(G),V),

edges

:= {e

separates(G,V,s,t):

bool N0T

minimuln

nmst

system.

v,s,t:

In other

p))

Theorem

in a formal

VAR

p)

V),

(n: posnat):

(FORALL

11

p)

V), p))

(V: set[T]):

contains_indep(subgraph(g,

ramseys_theorem:

j)))

(V: set[T]):

contains_clique(subgraph(g,

subgraph_indep

edge?(g)(i,

words

theory.

The

(lesired

properties

net

result

min_sep_set(G,s,t):

v:

= N0T

V(s)

s and

(w:

we end

t when up

finite_set[T]

V(v)

AND

IMPLIES

N0T

prewalk):

set, we use an abstract is that

AND

(FDRALL

(EXISTS

V separates

separating

I edges(G)(e)

e(v))}

#)

AND

walk_from?(del_verts(G,V),w,s,t))

its removal minimum

with

V(t)

N0T

disconnects function

a function

s and

defined

min_sep_set

t.

in the with

(v:

(LAMBDA

(v: seps(G,s,t)):

16

seps(G,s,t)):

card(v)), true)]

the

abstract_min

all of the

= min[seps(G,s,t), (LAMBDA

To define

following

separable?(G,s,t):

bool

= (s /= t AND

min_sep_set_edge:

LEMMA

NOT

min_sep_set_card:

LEMMA

FORALL

NOT

separable?(G,s,t) IMPLIES min sep_set(G,s,t) = vert(G)

(s,t:

(vert(G))):

IMPLIES

min_sep_set_seps:

LEMMA

edge?(G)(s,t))

separates(G,V,s,t)

card(min_sep_set(G,s,t))

separable?(G,s,t)

IMPLIES

separates(G,min_sep

min_sep_set_vert:

LEMMA

separable?(G,s,t) IMPLIES

ends_not_in_min_sep_set:

\%'(, t,hen

Noxt,

w( _ define

nat

a prrdicato

independent?(wl,w2:

AND min_sep_set(G,s,t)(v) vert(G)(v)

separable?(G,s,t) AND min_sep_set(G, IMPLIES v /= s AND v /= t

independent? prewalk):

that

bool

(tofin(,s

of a set

(i,j: nat):

of ind(_t)(ul(hult

set_of_paths(G,s,t):

TYPE

ind_path_set?(G,s,t,(pset:

t)aths

i < l(wl)

- I AND

j > 0 AND

j < l(w2)

- I IMPLIES

/= seq(w2)(]))

is (h'fine(l

as follows:

(pl,p2:

in(h,t)('n(lent. easy_menger:

a set

\V(, can LEMMA

of paths now stat(, FORALL

is an

AND pset(p2)

ind_path_set?

M(ulg(w's (ips:

bool

tlmor(un

AND

pl

if all

in I)oth

pairs

0 AND

set_of_paths(G,s,t))):

(FORALL

In

wlmn

=

seq(wl)(i)

('on(:et)t

s, t)(v)

= card(min_sep_set(G,s,t))

(FORALL

The

set(G,s,t),s,t)

as f'olhnv,_:

(|(,fine sep_num

sep_num(G,s,t):

LEMMA



Max

1)o instantiated

T : TYPE size:

and

c()ntexts.

sul)grat)h that

t h('ori(_s min_walks

that

ill many

theories

in the

need

it is us('ful ways

that

satisfies

a t)arti('ular

a minimal

walk

to d(weh)l)

abstract

t() 1)rovid(_

(tifferent

flom inin

bv

()l).j(_('t for

flm('tion that

ol)j('('t

whi('h

t)y whi(:h the

rain

a min flmction

ol)jects flm(;tion

are

compared

must

hay('

abstract_min[T:

TYPE,

size:

[T -> nat],

P: pred[T]]:

abstract_max[T:

TYPE,

size:

[T -> nat] , P: pred[T]]

THEORY

alia

23

: THEORY

satis[v

is ime(t('d

s t() t rain

and

and lnaX

To simplify

the

detail.

abstract_max

The

In order exists

that

following

discussion, theory

for a minimum satisfies

the

tile

abstract_rain

is conceptually

only

identical.

function

property.

Thus,

to t)e defined, the

theory

theory

it is necessary contains

will be elat)orated that

at least

the following

one

assuming

in

ol)ject claus(,

ASSUMING T_ne: ASSUMPTION EXISTS

(t: T): P(t)

ENDASSUMING User's of this theory are required to prove that PVS's TCC generatioll mechanism). A flmction minimal? (S : T) is then (lefined minimal?(S):

this

assunlption

hol(ts

for their

tyl)e

T (via

as follows:

bool = P(S) AND (FORALL (SS: T): P(SS) IMPLIES size(S)