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A GOAL-ORIENTED GRAPH CALCULUS FOR RELATIONS

Paulo A. S. VELOSO

Systems & Computing Engin. Progr., COPPE-UFRJ; Rio de Janeiro, RJ, BRAZIL

Sheila R. M. VELOSO

Systems & Computing Dept., UERJ; Rio de Janeiro, RJ, BRAZIL

Graph calculus, algebras of relations, complement, refutation soundness, completeness

Research partly sponsored by CNPq & FAPERJ

1

OUTLINE

• 1. Introduction

graphical notation

• 2. Slices

graphical representation

• 3. Graphs

alternative slices

• 4. Conclusion

comments

2

1 Introduction

⊳ Diagrams & figures useful in science & everyday life Graphs & diagrams visualization

• Computing

automata, Petri nets, flowcharts

• Foundations of Mathematics

categories, allegories

• Engineering/Architecture

wiring diagrams, blueprints

• Metro journey

diagram of lines Venn diagrams heuristic appeal not proofs ∴ compile ♥ Graph manipulations precise syntax & semantics ∴ proof methods

3

Formulas traditionally written down on a single line Notations economy vs. visualization

• 1. Polish prefix (parenthesis-free)

→ ∧pq∨rs

• 2. usual (with parentheses)

(p∧q) → (r∨s)

• 3. two-dimensional

    p ∧ q     →     r ∨ s    

4

♥ Graph calculi 2-dimensional notation & nodes ⊳ Drawings for relations natural idea a related to b via relation R arc a R → b Operations on relations simple manipulations on arrows

• Bolean intersection ∩

parallel arcs

• Peircean transposal T

arrow reversal

• Peircean relative product (composition) |

consecutive arcs ∞ Reason about relations manipulate their representations ∠ visual appeal 2-dimensional manipulations

5

Overview

⊳ Goal-orientation P ⊆ Q iff P ∩ Q ⊆ / ♥ Reductions equivalent objects

• 1. Represent term P ∩ Q by slice S

∴ goal S ⊆ /

• 2. Convert slice S to graph G

∴ goal G ⊆ /

• 3. Expand graph G to H

∴ goal H ⊆ / ∞ (Correctness) Basic graph H: H ⊆ / iff H inconsistent

6

⊲ Relational terms generated   

from relation names by relational operations

⊲ Peircean operations 2-ary relations over set M (0) Constants (I I) Identity (diagonal) IM := {(a,b) ∈ M2 /a = b} (I D) Diversity IM:= {(a,b) ∈ M2 /a = b} (1) Unary operation (⌣) Transposition T (reversal) RT := {(a,b) ∈ M2 /(b,a) ∈ R} (2) Binary operations (;) Relative product | (composition) P | Q := {(a,b) ∈ M2 /∃c ∈ M[(a,c) ∈ P∧(c,b) ∈ Q]} (;) Relative sum | P |Q := {(a,b) ∈ M2 /∀c ∈ M[(a,c) ∈ P∨(c,b) ∈ Q]}

7

2 Slices

♥ Slice graphical representation

→x

y→ P

• Q
• term

    P ⊓ Q    

parallel arcs intersection ∩

⊲ Slice S            finite sets of    nodes labeled arcs 2 distinguished nodes: I/O →

8

Slice example Establish r⌣;r;s ⊑ s ⊳ Reduce r⌣;r;s ⊑ s

to

    r⌣;r;s ⊓ s     ⊑ I ⊥

(∩) Slice S0

parallel arcs ∩

→x

y→ r⌣;r;s

• s
• 9

( ) Eliminate double complement

s ≡ s S0

→x

y→ r⌣;r;s

• s
• ≡

→x

y→ r⌣;r;s

• s
• S1

10

(;) Eliminate relative product

consecutive arcs

new intermediate node

S1

→x

y→ r⌣;r;s

• s
• ≡

→x

y→ z r⌣

• r;s
• s
• S2

(a,b) ∈ P;Q ⇔ ∃c     (a,c) ∈ P ∧ (c,b) ∈ Q    

11

(⌣) Eliminate converse

invert arrow: xr⌣ →z ≡ x r ←z S2

→x

y→ z r⌣

• r;s
• s
• ≡

→x

y→ z r

• r;s
• s
• S3

(a,c) ∈ R⌣ ⇔ (c,a) ∈ R

12

(;) Eliminate complemented relative product

label: complemented slice S3

→x

y→ z r

• r;s
• s
• |||

S4

→x

y→ z r

• →z′ r

→ x′ s → y′ →

• s
• 13

(⊥) Slice S4

inconsistent Parallel paths: from z to y x y z r

• s
• y

z

→z′ r

→ x′ s → y′ →

• term r;s

term r;s r;s ∩ r;s = /

14

⊲ Slice concepts

• 1. Morphism

node mapping preserving arcs

• 2. Zero slice

parallel incompatible paths Slice S

with

embedded slice T w R

• t
• →x

P u s

• T

v Q y→ T := →x′ s w′ R w′′ t y′ →

15

Node mapping θ : NT NS preserves arcs

→x′

s

• θ
• w′

R

• θ
• w′′

t

• θ

y′ →

θ

• w

R

• t
• →x

P u r

• T

v Q y→ Parallel paths from u to v incompatible Slice S zero not satisfiable

16

3 Graphs

♥ Alternative slices

for ∪, /

⊲ Graph finite set of alternative slices

→x P⊔Q

− → y→ ≡     

→x P

→ y→,

→x Q

→ y→      2 alternative slices

→x I

⊥ → y→ ≡

• 0 alternative slices

17

Graph example Establish r;(s†t) ⊑ (r;s)†t ⊳ Reduce r;(s†t) ⊑ (r;s)†t

to

    r;(s†t) ⊓ (r;s)†t     ⊑ I ⊥

(∩) Difference slice DS(r;(s†t)\(r;s)†t)

parallel arcs ∩

→x

y→ r;(s†t)

• (r;s)†t
• 18

(⊲∗) Eliminate operations

but complement

→x

T1

• r

u T2

• v

t y→ Arc labels complemented slices T1 :=

→x1 r

→ u1 s → v1 → T2 :=

→u2 s

→ v2 t → y2 →

19

(∪) Expand

graph G: 2 alternative slices T+ & T− T+ T−

→x

T1

• r

u T2

• s
• v

t y→

→x

T1

• r

u T2

• s
• v

t y→ Equivalent alternative paths from u to v s ∪ s = M2

20

(⊥) Zero graph

inconsistent slices T+ & T− T+ Parallel paths from x to v: r;s r;s x

→x1 r

→ u1 s → v1 →

• r

u T2

• s
• v

t y T− Parallel paths from u to y: s;t s;t x T1

• r

u

→u2 s

→ v2 t → y2 →

• s
• v

t y

21

Gluing onto slice S slice & graph ♥ Gluing of slices addition of slice-label arc S + u T → v

→ xS

u S

← yS

v I I ← I I ← xT T yT

eliminate I I arcs

⊳ Eliminate arc wI Iz from slice S rename w to z w (or z to w) in S ∠ Gluing S u

vT ≡ S + u T

→ v pushout

22

Slices S & T S :=

→x

R → u s → v t → y→ T :=

↓↑

w z Q

• P
• Glued slice S u

vT

identify u,v to w S u

vT

=

→x

R w s

• P
• t

y→ z Q

• ⊲ Glued graph

glued slices S u

vH := {S u vT/T ∈ H}

23

4 Graph Calculus

⊲ Labels Labels generated           

from

   relation names slices & graphs

by relational operations

⊲ Basic objects Basic: labels    relation name or complement of basic slice

24

⊲ Label derivations Rules            Conversion    Operational relational operations Structural arc labels Expansion alternatives ∞ Valid label inclusion L ⊑ I ⊥ iff L ⊢ H H: zero graph ⊲ Normal linear derivation L Cnv∗ G Exp∗ H

• 1. Convert label L to graph G

e.g. basic form

• 2. Expand graph G to zero graph H

binary expansion

25

♥ Operational rules label ⊲ graph any context

• Meaning of operation but complement

given by graph

• Double complement

L ≡ L ∠ Operational rules

• eliminate relational operations but complement
• may introduce slices or graphs within labels

♥ Structural rules arc labels

• Addition of graph-label arc

glued graph

• Label vs. slice

L ≡ →x L → y→

• de Morgan laws

complement of ∪, complement of ∩ ⊲ ⊳ Conversion L ⊲∗ Lbs Every label L convertible to an equivalent basic graph Lbs

26

⊲ Expansion rule replace S by copies S u

vT & S + uTv

(Exp)

{S} {S u

vT, S + uTv}

(u,v) ∈ NS2 ♦ Label inclusion L ⊑ I ⊥ valid iff normal linear derivation L Cnv∗ G Exp∗ H zero graph ⊳

• 1. Convert label L to (basic) graph G

finite process

• 2. Expand graph G to zero graph H

unbounded search

27

5 Conclusion

• 1. Complement

(-) difficult to handle (+) goal-orientation

• 2. Goal-orientation

derive zero graph

• 3. Graph language

more expressive: labels (embedding)

• easy
• ne single object
• simple concepts

morphism

• Extension to hypotheses

erasing

• 4. Generalize: labels to n-labels

first-order predicate logic

⌣

28

A Details

• 1. Language

(a) Syntax

• bjects

(b) Semantics meaning (c) Concepts morhism, zero, basic (d) Constructions gluing, transformations (graph ↔ slice)

• 2. Calculus

(a) Conversion label to basic graph (b) Expansion (basic) slice to (basic) graph (c) Correctness sound & complete

• 3. Hypotheses

(a) Semantics consequence (b) Rule erase slice

29

⊳ Constants & operations square relational interpretation Arity Symbol Interpretation r arbitrary relation

• ver set M

I ⊥ empty relation / I ⊤ universal relation square M2 := M ×M I I identity relation diagonal IM I D diversity relation IM 1 Boolean complementation R:= M2 \R

⌣

Peircean transposition T 2 ⊓ Boolean intersection ∩ ⊔ Boolean union ∪ ; Peircean relative product | † Peircean relative sum |

30

∠ Constants & operations language interpretation (regular expressions)

Arity Symbol Interpretation r arbitrary language over alphabet A I ⊥ empty language / I ⊤ universal language A∗ I I null-word language Λ := {λ} I D non-null language A+ := {w ∈ A∗ /w = λ} 1 Language complementation L:= A∗ \L

⌣

Language reversal rv Lrv := {wrv ∈ A∗ /w ∈ L} 2 ⊓ Language intersection ∩ ⊔ Language union ∪ ;

• Lang. concatenation ·

P·Q := {u·v ∈ A∗ /u ∈ P∧v ∈ Q} †

• Lang. co-concatenation ·

P · Q := (P·Q )

31

A.1 Language

Denumerably infinite sets    Rn: relation names INd: nodes (alphabetical order: x,y,z,...) ⊲ Syntax mutual recursion

(L) Labels: generated from

  relation names slices, graphs   by relational operations

(a) Arc: triple uLv

u,v: nodes, L: label

(Σ) Sketch Σ = N,A

sets N: nodes & A: arcs

(D) Draft D = N,A

finite sketch

(S) Slice S = N,A : xS,yS

   underlying draft S = N,A input, output xS,yS: nodes

(G) Graph G: finite set of slices

32

(⊑) Label inclusion L⊑K

pair of labels

(\) Difference slice DS(L\K)

parallel arcs

→x

y→ L

• K
• ⊲ Morphism θ : Σ′ Σ′′

node mapping preserving arcs Nodes Arcs N′ uLv ∈ A′ ↓ θ ⇓ N′′ uθ Lvθ ∈ A′′ ⊳ Set of morphisms Mor[Σ′,Σ′′]

33

♥ Meaning labels,slices & graphs denote 2-ary relations arcs, sketches & drafts represent restrictions Pair (a,b) satisfies arc urv pair (a,b) in relation of r (of model) ∠ Semantics

• Model: relation name → 2-ary relation

M = M,(rM)r∈Rn

• Assignment g : N → M

w ∈ N → wg ∈ M ⊳ Model M relational term r → 2-ary relation rM ⊆ M2

34

⊲ Behavior mutual recursion

(L) Relation of label

[L]M ⊆ M2 concrete versions of operations

e.g. [r]M :=rM, [L⌣]M :=[L]M

T, [L;K]M :=[L]M | [K]M

(a) Satisfaction of arc u L

→ v (ug,vg) ∈ [L]M (with u,v ∈ N)

(Σ) Satisfaction of sketch Σ

g : Σ → M ⇔ g satisfies all arcs of Σ

(S) Extension of slice

[[S]]M values of I/O for assignments satisfying underlying draft [[S]]M := {(xS

g,yS g) ∈ M2 /g : S → M}

(G) Extension of graph

[[G]]M :=

S∈G [[S]]M 35

⊲ Label inclusion & equivalence

(M) holds

M | = L⊑K ⇔ [L]M ⊆[K]M

(| =) valid

| = L⊑K ⇔ M | = L⊑K (∀M)

(⊥) Null label L

⇔ L⊑I ⊥ valid

(≡) Equivalent labels

L≡K ⇔ L⊑K & K⊑L valid

36

Concepts ⊲ Zero sketch not satisfiable

→ xT

T

← yT

θ

• xTθ

yTθ T

• Σ

⊲ Zero slice & graph empty extension

(S) Slice T is zero

⇔ underlying draft T is zero sketch

(G) Graph H is zero

⇔ all its slices T ∈ H are zero slices

37

⊲ Basic labels, arcs, sketches, slices & graphs mutual recursion

(L) Label L is basic

⇔ L is    relation name or complement of basic slice (cf. below)

(a) Arc u L

→ v is basic ⇔

its label L is basic label

Basic sketches, slices & graphs

• nly basic arcs

(Σ) Sketch Σ = N,A is basic

⇔ all its arcs a ∈ A are basic arcs

(S) Slice S = S : xS,yS is basic

⇔ underlying draft S is basic sketch

(G) Graph G is basic

⇔ all its slices S ∈ G are basic slices

38

Constructions ⊲ Gluing slice onto draft: D u

vT

pushout D u

vT

D + {u,v} α ր σ ց

→x y←

PO

D u

vT

β ց τ ր T ⊲ Gluing onto slice S slice & graph

• 1. slice T

transfer I/O S u

vT:=S u vT : xTσ,yTσ

• 2. graph H

glued slices S u

vH := {S u vT/T ∈ H}

39

Slices S & T′, T′′ S :=

→x

R → u s → v t → y→ T′ :=

→w

P → z→ T′′ :=

↓↑

w z Q

• P
• Glued slice S u

vT′

identify w to u, z to v S u

vT′

=

→x

u v y→ R s

• P
• t

40

Glued slice S u

vT′′

identify u,v to w S u

vT′′

=

→x

R w s

• P
• t

y→ z Q

• Glued slice S x

yT′′

identify I/O S x

yT′′

=

→ ←w

u z v R

• s
• t
• P
• Q
• 41

⊲ Transformations graph ↔ slice

• Slice of graph Sl[G]:={x,y},{x S

→ y/S ∈ G} : x,y parallel arcs Graph {S,T} → slice →x y→ S

• T
• Graph of slice Gr(S)

1-arc complemented label slice, for each arc Slice →x t

• r

y→ s

• →

graph             

→x r

→ y→,

→x s

← y→,

→

t

• x y→

             ⊲ Slice is small ⇔ nodes: input, output ⋊ ⋉ Small slice S equivalence {S} ≡ Gr(S)

42

A.2 Calculus

⊲ Operational rules label ⊲ graph any context Constants Boolean I ⊥,I ⊤ & Peircean I I,I D

(I ⊥) I

⊥ ⊲

• I

⊥ ≡ empty graph (I ⊤) I

⊤ ⊲

• →x

y→

• I

⊤ ≡ 2-node arcless slice {x,y}, / 0 : x,y (I I) I

I ⊲

• →x→
• I

I ≡ 1-node arcless slice

→x→

(I D) I

D ⊲   

→x

y→

→x→

• 

 

I D≡I I (2-node 1-arc slice)

43

Boolean operations unary

& binary ⊓,⊔ ( ) L ⊲ L L ≡ L (⊓) L⊓K ⊲

             →x

y→ L

• K
• 

           

parallel arcs: L & K (⊔) L⊔K ⊲

  

→x L

→ y→,

→x K

→ y→   

alternative slices: L & K

44

Peircean operations unary ⌣ & binary ;,†

(⌣) L⌣ ⊲

• →x L

← y→

• reversed arrow

(;) L;K ⊲

• →x L

→ z K → y→

• consecutive arcs

L then K (†) L†K ⊲

  

→x

y→

→x L

→ z K → y→   

L†K ≡ L;K

45

Summary

• perations

arity: 0, 1

(I ⊥) I

⊥ ⊲

• empty graph

(I ⊤) I

⊤ ⊲ {{x,y}, / 0 : x,y}

2-node arcless slice

→x y→

(I I) I

I ⊲ {{x}, / 0 : x,x}

1-node arcless slice

→x→

(I D) I

D ⊲ {{x,y},{{x}, / 0 : x,x} : x,y}

2-node 1-arc slice

→x

y→

→x→

• ( ) L ⊲ L

replace L by L (⌣) L⌣ ⊲ {{x,y},{yLx} : x,y}

reversed-arc slice

→x L

← y→

46

Summary

• perations

arity: 2

(⊓) L⊓K ⊲ {{x,y},{xLy,xKy} : x,y}

parallel-arc slice:

→x

y→ L

• K
• (⊔) L⊔K ⊲

   {x,y},{xLy} : x,y, {x,y},{xKy} : x,y   

alternative slices:

→x L

→ y→

→x K

→ y→ (;) L;K ⊲ {Sl(L→K)}

consecutive-arc slice:

→x L

→ z K → y→ (†) L†K ⊲

{{x,y},{xSl(L→K)y} : x,y}

complemented label:

→x L

→ z K → y→

47

∠ Operational rules composite labels ⊲∗ graphs Term r;s⊓t graph G r;s⊓t

(;)

⊲

• →x r

→ z s⊓t → y→

• (⊓)

⊲             

→x

z r y→     

→x

y→ s

• t
• 

   

• 

           

• G

48

⊲ Structural rules arc labels

( ∪ →) {S + uHv} ⊲ S u

vH

replace graph label by glued slices (G) G ⊲ {Sl[G]} replace complemented graph by slice (S) small S: {S} ⊲ Gr(S) move inside small slice (r) r ⊲ →x r

→ y→

replace label r by complemented slice

⋊ ⋉ Derived structural rule replace compl. graph label by compl. slices

( ∪ →) {S + uHv} ⊲ {S + {uTv/T ∈ H}} replace u H → v by {u T → v/T ∈ H}

49

Graph G

(cont’d)

            

→x

z r y→     

→x

y→ s

• t
• 

   

• 

           

( ∪ →)

⊲   

→x r

→ z s → y→

→x r

→ z t → y→   

(r)

⊲                 

→x r

→ z

→x s

→ y→ y→ ,

→x r

→ z

→x t

→ y→ y→                 

50

⊳ Derived structural rule ( ∪ →) rules (∪) & ( ∪ →)

               S+ u v          T1 . . . Tn         

• 

             

(∪)

⊲                  S+ u v

→x

y→ . . . T1

• Tn
• 

               

( ∪ →)

⊲                S + u T1 − → v . . . u Tn − → v               

51

♥ Completeness

• 1. Family of zero slices

Z 0

• 2. Family of eventually zero slices:

S ∈ Z ∗ ⇔ S Exp∗ H ⊆ Z 0

• 3. Family of non-eventually zero slices:

S ∈ Z ∞ ⇔ S ∈ Z ∗ ⊳ If G ⊆ Z ∗ then G ≡ I ⊥ ⋊ ⋉ If S ∈ Z ∞ then S u

vT ∈ Z ∞

• r

S + uTv ∈ Z ∞ Chain of slices S ∈ Z ∞ S0 := S S0 ϕ0 → S1 ... ϕn → Sn+1

underlying drafts

Sn ∈ Z ∞ Co-limit sketch Σ counter-model C [[S]]C = / ⊲ ⊳ If G ⊆ Z ∗ then [[G]]C = /

52

⊲ Model M = M,(rM)r∈Rn natural for sketch Σ = NΣ,AΣ M = N rM = {(w,z) ∈ M2 /wrz ∈ A} ⋊ ⋉ Discrimination assignments & morphisms natural model C for sketch Σ Basic draft D with ES[D] ⊆ ES[Σ] g : D → C iff g : D Σ ⊳

Induction on rk(D) ∈ IN ⊲ Basic objects: rank and set of embedded slices structural measure (r) r ∈ Rn: rk(r) := 0 ES[r] := / ( ) compl. slice: rk(T) := rk(T)+1 ES[T] := ES[T]∪{T} (a) arc: rk(uLv) := rk(L) ES[uLv] := ES[L] (D) draft: rk(D) := ∑a∈AD rk(a) (Σ) sketch: ES[Σ] :=

a∈AΣ ES[a]

(S) slice: rk(S) := rk(S) ES[S] := ES[S]

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A.3 Hypotheses

r⊑s⇒r;t⊑s;t iff     r ⊓ s    ⊑I ⊥ ⇒     r;t ⊓ s;t    ⊑I ⊥

• 0. Hypothesis r⊑s

→

• diff. slice DS(r\s)

⊲∗ basic graph {S′} S′

→x

y→ r

• s
• ⊲∗

→x

y→ r

• →x′

s − → y′→

• 54

• 1. Difference slice DS(r;t\s;t) converts to basic slice:

S1 :=

→x

y→ z

→x1 s

→ z1 t → y1→

• r
• t
• 55

• 2. Expand {S1} to graph H = {S+,S−}

with T := →x s

→ y→ S+ :=

→x

y→ z

→x1 s

→ z1 t → y1→

• s
• r
• t
• S−

:=

→x

y→ z

→x1 s

→ z1 t → y1→

• →x′ s

→ y′ →

• r
• t
• 56

∠ Graph H = {S+,S−} {S′}-erasable

• Slice S+

parallel paths from x to y: terms s;t & s;t ∴ zero

• Slice S−

morphism θ : S′ S− with x → x,y → z ∴ erasable ∴ Graph H has empty extension in any model

where r⊓s has empty extension

• i. e.

hypothesis r ⊑ s holds

• [r]M ⊆[s]M

⇒ inclusion r;t ⊑ s;t holds

• [r;t]M ⊆[s;t]M

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⊲ Models & consequences

• 1. Inclusion L⊑K

Mod(L⊑K) models where [L]M ⊆[K]M

• 2. Set Λ of inclusions

Mod(Λ):=

L′⊑K′∈Λ Mod(L′⊑K′)

• 3. L⊑K follows from Λ

Λ | = L⊑K ⇔ Mod(Λ) ⊆ Mod(L⊑K) ⊲ Slice S is Γ-erasable ⇔ Mor[S′,S] = / 0 for some S′ ∈ Γ ⊲ Rule for hypothesis can erase Γ-erasable slice

(Hyp[Γ]) {S}

{ } if slice S is Γ-erasable ♦ Basic graph G, set Γ of basic slices Λ[Γ] := {S′ ⊑I ⊥/S′ ∈ Γ} Λ[Γ] | = G⊑I ⊥ G⊑I ⊥ follows from Λ[Γ] iff G ⊢(Exp∪Hyp[Γ]) H H : zero graph iff G ⊢(Exp) H′ H′ : zero or Γ-erasable graph

58