A

Dialectica

Model

of

State

Marcelo

da

Silva

Corr

ea

e-mail

correa@inf.puc-rio.br

Edward

Hermann

Haeusler

e-mail

hermann@inf.puc-rio.br

Valeria

Correa

Vaz

de

Paiva

Computer

Laboratory

University

of

Cambridge

New

Museums

Site

Pembroke

Street

Cambridge

U.K.

Computer

Laboratory

University

of

Cambridge

e-mail

Valeria.Paiva@cl.cam.ac.uk

PUC-Rio

Abstract

Reddy

-LSB-

-RSB-

introduced

an

extended

intuitionistic

linear

calculus

to

model

some

features

of

state-manipulation

His

calculus

LLMS

-LCB-

for

Linear

Logic

Model

of

State

-LCB-

includes

the

connective

before

and

an

its

associated

modality

y.

De

Paiva

-LSB-

-RSB-

presents

a

-LRB-

collection

of

-RRB-

dialectica

categorical

models

for

Classical

Linear

Logic

the

categories

GC

These

categories

contain

an

extra

tensor

product

functor

and

a

comonad

structure

corresponding

to

a

modality

related

to

it

It

is

surprising

that

these

works

arising

from

completely

different

motivations

can

be

related

in

a

meaningful

way

In

this

paper

we

present

a

dialectica

category

G

which

models

the

sequent

calculus

LLMSc

the

commutative

version

of

Reddy

's

LLMS

Furthemore

we

present

a

computational

characterization

of

the

fragment

f

g

of

LLMSc

in

the

framework

of

CSP

-LSB-

-RSB-

We

draw

some

preliminary

conclusions

and

point

out

extensions

and

future

work

Resumo

Reddy

-LSB-

-RSB-

apresenta

um

c?alculo

linear

intuicionista

para

modelar

alguns

aspectos

de

manipula?c

ao

de

estados

Seu

c?alculo

chamado

Linear

Logic

Model

of

State

-LRB-

LLMS

-RRB-

inclui

o

conectivo

before

e

uma

modalidade

y

associada

a

este

conectivo

De

Paiva

-LSB-

-RSB-

apresenta

uma

cole?c

ao

de

modelos

categ?oricos

dial?eticos

para

a

L?ogica

Linear

Cl?assica

em

termos

das

categorias

GC

Estas

categorias

incluem

al?em

do

produto

tensorial

e

a

conjun?c

ao

padr

ao

um

funtor

produto

e

uma

estrutura

de

comonada

para

uma

modalidade

relativa

a

este

produto

Surpreendentemente

estes

trabalhos

que

t

em

motiva?c

oes

completamente

diferentes

podem

ser

relacionados

de

um

modo

bastante

interessante

Neste

trabalho

n?os

apresentamos

uma

categoria

dial?etica

G

que

determina

um

modelo

categ?orico

para

o

c?alculo

de

sequentes

LLMSc

uma

vers

ao

comutativa

do

c?alculo

LLMS

Al?em

disso

n?os

apresentamos

uma

caracteriza?c

ao

computacional

do

fragmento

f

g

de

LLMSc

em

CSP

-LSB-

-RSB-

N?os

conclu

mos

o

trabalho

destacando

algumas

extens

oes

e

trabalhos

futuros

Palavras-Chave

L?ogica

Linear

Intuicionista

Teoria

da

Prova

Modelos

Categ?oricos

e

Concorr

encia

Dept.

de

An?alise

Univ.

Federal

Fluminense

Introduction

The

concept

of

state-manipulation

plays

an

important

role

in

computation

There

are

many

attempts

to

formalise

this

concept

especially

in

the

framework

of

Girard

's

Linear

Logic

-LRB-

-LSB-

-RSB-

-RRB-

Some

aspects

of

state-manipulation

like

sequencing

and

regenerative

values

are

captured

in

an

interesting

way

by

Reddy

's

work

-LSB-

-RSB-

Reddy

presents

an

extended

intuitionistic

linear

calculus

for

modelling

state

manipulation

His

calculus

LLMS

-LRB-

for

Linear

Logic

Model

of

State

-RRB-

adds

two

extra

connectives

to

those

of

Intuitionistic

Linear

Logic

a

sequencing

binary

operator

called

before

and

a

regenerative

storage

operator

y

a

modality

associated

with

before

The

connective

before

is

motivated

as

the

denotation

of

sequential

composition

of

components

while

the

regenerative

storage

operator

allows

us

to

build

sequentially

reusable

storage

objects

These

two

constructors

are

used

to

express

dynamic

values

and

imperative

programs

within

the

framework

of

linear

logic

while

Girard

's

of

course

modality

is

used

to

express

static

values

The

approach

is

shown

to

work

by

embbeding

a

higher-order

Algol-like

language

in

the

system

LLMS

Reddy

's

intuitive

idea

is

that

the

connective

before

captures

the

computational

feature

of

composition

with

possible

one

way

-LRB-

left

to

right

-RRB-

communication

Thus

this

connective

must

be

noncommutative

If

one

considers

a

variant

of

this

connective

which

is

commutative

one

obtains

an

operator

denoting

non-ordered

-LRB-

or

aleatory

-RRB-

combination

of

elements

Non-ordered

combination

does

not

mean

here

communication

both

ways

but

it

can

be

said

to

represent

concurrent

execution

without

interaction

or

synchronization

In

this

paper

we

follow

this

idea

we

introduce

a

new

connective

which

is

just

the

commutative

version

of

Reddy

's

and

we

modify

the

system

LLMS

accordingly

Hence

the

presentation

of

the

system

of

LLMSc

follows

strictly

the

presentation

of

LLMS

Reddy

also

presents

a

categorical

model

for

his

calculus

using

a

variant

of

coherent

spaces

and

linear

maps

However

if

we

do

consider

the

commutative

version

of

the

connective

before

which

is

related

to

interleaving

as

we

suggest

in

section

we

have

realised

that

we

end

up

with

a

calculus

-LRB-

dubbed

LLMSc

-RRB-

for

which

we

had

already

a

dialectica

categorical

model

Dialectica

models

were

introduced

by

de

Paiva

-LSB-

-RSB-

as

a

model

for

the

intuitionistic

fragment

of

Linear

Logic

-LRB-

the

motivation

for

the

model

comes

from

G?odel

's

Dialectica

Interpretation

hence

the

name

-RRB-

Later

de

Paiva

introduced

the

categories

GC

-LSB-

-RSB-

a

different

kind

of

dialectica

model

which

following

a

suggestion

of

Girard

model

the

whole

of

Linear

Logic

Actually

the

categories

GC

have

some

extra

structure

-LRB-

an

extra

tensor

product

and

an

extra

modality

associated

with

this

tensor

product

-RRB-

and

hence

can

be

seen

as

a

model

of

LLMSc

The

extra

tensor

product

denoted

here

by

can

be

seen

as

representing

the

aleatory

combination

of

elements

The

extra

-LRB-

monoidal

-RRB-

comonad

can

be

seen

as

an

extra

modality

which

we

think

of

as

a

commutative

version

of

the

regenerative

storage

operator

In

the

section

we

present

the

proof

system

LLMSc

obtained

by

modifying

LLMS

to

make

the

connective

before

commutative

In

the

section

we

describe

one

specific

dialectica

category

-LRB-

built

over

Sets

-RRB-

and

show

that

it

is

a

sound

model

for

LLMSc

In

the

section

we

present

a

characterization

of

the

fragment

of

LLMSc

consisting

of

cartesian

product

the

tensor

product

and

the

extra

tensor

product

in

CSP

-LSB-

-RSB-

Detailed

proofs

of

the

results

stated

here

can

be

found

in

-LSB-

-RSB-

Finally

we

conclude

pointing

out

some

future

work

The

system

LLMSc

The

presentation

of

the

system

of

LLMSc

follows

strictly

the

presentation

of

LLMS

-LSB-

-RSB-

The

left

context

of

a

sequent

in

LLMSc

has

the

following

syntax

ffljAj

where

ffl

-LRB-

empty

context

-RRB-

is

interpreted

as

and

the

contexts

and

are

interpreted

as

and

respectively

A

context

which

does

not

contain

any

occurrences

of

the

connective

is

called

an

independent

context

In

order

to

provide

a

pomset

characterization

to

a

context

the

following

objects

are

associated

with

each

context

ffl

j?j

the

set

of

formula

occurrences

in

and

ffl

a

partial

order

on

j?j

such

that

the

pair

-LRB-

j?j

-RRB-

is

a

pomset

defined

inductively

as

follows

jfflj

fg

jAj

fAg

A

f

-LRB-

A

A

-RRB-

g

j

j

j

-LSB-

j

j

j

-LSB-

-LSB-

-LRB-

j

j

-RRB-

-LSB-

-LRB-

j

j

-RRB-

The

Rules

of

LLMSc

look

like

those

of

LLMS

-LSB-

-RSB-

Structural

Rule

A

A

Ser

if

j?j

j

and

Identity

Rules

A

A

Id

A

-LSB-

A

-RSB-

B

-LSB-

-RSB-

B

Cut

Multiplicative

Rules

A

B

A

B

R

-LSB-

A

B

-RSB-

C

-LSB-

A

B

-RSB-

C

L

A

B

A

B

R

-LSB-

A

B

-RSB-

C

-LSB-

A

B

-RSB-

C

L

-LSB-

ffl

-RSB-

C

-LSB-

-RSB-

C

A

B

A?ffiB

ffiR

A

-LSB-

B

-RSB-

C

-LSB-

A?ffiB

-RSB-

C

ffiL

Additive

Rules

R

A

B

A&B

R

-LSB-

A

-RSB-

C

-LSB-

A&B

-RSB-

C

L

-LSB-

B

-RSB-

C

-LSB-

A&B

-RSB-

C

L

ordered

multiset

Modalities

-LSB-

ffl

-RSB-

C

-LSB-

yA

-RSB-

C

yWeak

-LSB-

A

-RSB-

C

-LSB-

yA

-RSB-

C

yDer

-LSB-

yA

yA

-RSB-

C

-LSB-

yA

-RSB-

C

yThread

A

yA

-LRB-

if

is

an

independent

context

with

only

or

y

formulas

-RRB-

-LSB-

yA

-RSB-

C

-LSB-

A

-RSB-

C

Ser

-LSB-

A

A

-RSB-

C

-LSB-

A

-RSB-

C

Contr

A

A

-LRB-

if

is

an

independent

context

with

only

formulas

-RRB-

Next

we

present

some

inferences

which

are

enclosed

by

the

structural

rule

and

which

determine

interesting

properties

of

the

connectives

and

Proposition

The

following

inferences

are

allowed

by

the

Structural

Rule

-LRB-

Ser

-RRB-

-LRB-

i

-RRB-

-LSB-

A

B

-RSB-

C

-LSB-

A

B

-RSB-

C

-LRB-

ii

-RRB-

-LSB-

B

A

-RSB-

C

-LSB-

A

B

-RSB-

C

-LRB-

iv

-RRB-

-LSB-

-LRB-

A

B

-RRB-

C

-RSB-

D

-LSB-

A

-LRB-

B

C

-RRB-

-RSB-

D

-LRB-

vi

-RRB-

-LSB-

-LRB-

A

B

-RRB-

C

-RSB-

D

-LSB-

A

-LRB-

B

C

-RRB-

-RSB-

D

-LRB-

viii

-RRB-

-LSB-

-LRB-

A

B

-RRB-

C

-RSB-

D

-LSB-

A

-LRB-

B

C

-RRB-

-RSB-

D

-LRB-

iii

-RRB-

-LSB-

B

A

-RSB-

C

-LSB-

A

B

-RSB-

C

-LRB-

v

-RRB-

-LSB-

A

-LRB-

B

C

-RRB-

-RSB-

D

-LSB-

-LRB-

A

B

-RRB-

C

-RSB-

D

-LRB-

vii

-RRB-

-LSB-

A

-LRB-

B

C

-RRB-

-RSB-

D

-LSB-

-LRB-

A

B

-RRB-

C

-RSB-

D

-LRB-

ix

-RRB-

-LSB-

A

-LRB-

B

C

-RRB-

-RSB-

D

-LSB-

-LRB-

A

B

-RRB-

C

-RSB-

D

-LRB-

x

-RRB-

-LSB-

-LRB-

A

C

-RRB-

-LRB-

B

D

-RRB-

-RSB-

E

-LSB-

-LRB-

A

B

-RRB-

-LRB-

C

D

-RRB-

-RSB-

E

Proof

Regarding

that

-LSB-

-RSB-

-LSB-

-RSB-

is

the

common

form

of

each

inference

considered

here

we

must

proof

in

each

case

that

j

-LSB-

-RSB-

j

j

-LSB-

-RSB-

j

and

-LSB-

-RSB-

-LSB-

-RSB-

This

can

be

done

by

means

of

a

structural

induction

on

-LSB-

-RSB-

where

the

base

step

is

the

case

in

which

-LSB-

-RSB-

is

like

-LRB-

and

so

-LSB-

-RSB-

is

like

-RRB-

Base

Step

We

have

that

j?j

j

for

each

case

-LRB-

i

-RRB-

to

-LRB-

x

-RRB-

since

and

have

the

same

occurrences

of

formulas

Then

it

remains

to

prove

that

-LRB-

i

-RRB-

A

B

A

-LSB-

B

f

-LRB-

A

A

-RRB-

-LRB-

B

B

-RRB-

g

A

B

A

-LSB-

B

-LSB-

-LRB-

jAj

jBj

-RRB-

-LSB-

-LRB-

jBj

jAj

-RRB-

f

-LRB-

A

A

-RRB-

-LRB-

B

B

-RRB-

-LRB-

A

B

-RRB-

-LRB-

B

A

-RRB-

g

Then

A

B

A

B

The

proofs

of

the

cases

-LRB-

ii

-RRB-

and

-LRB-

iii

-RRB-

are

based

on

the

commutativity

of

-LSB-

and

the

proofs

of

the

cases

-LRB-

iv

-RRB-

and

-LRB-

v

-RRB-

are

based

on

the

associativity

of

-LSB-

A

B

f

-LRB-

A

A

-RRB-

-LRB-

B

B

-RRB-

g

B

A

A

B

f

-LRB-

A

A

-RRB-

-LRB-

B

B

-RRB-

-LRB-

A

B

-RRB-

-LRB-

B

A

-RRB-

g

B

A

A

-LRB-

B

C

-RRB-

f

-LRB-

A

A

-RRB-

-LRB-

B

B

-RRB-

-LRB-

C

C

-RRB-

g

-LRB-

A

B

-RRB-

C

-LRB-

vi

-RRB-

and

-LRB-

vii

-RRB-

A

-LRB-

B

C

-RRB-

A

-LSB-

B

C

-LSB-

-LRB-

jAj

jB

Cj

-RRB-

-LSB-

-LRB-

jB

Cj

jAj

-RRB-

A

f

-LRB-

A

A

-RRB-

g

B

C

B

-LSB-

C

-LSB-

-LRB-

jBj

jCj

-RRB-

-LSB-

-LRB-

jCj

jBj

-RRB-

f

-LRB-

B

B

-RRB-

-LRB-

C

C

-RRB-

-LRB-

B

C

-RRB-

-LRB-

C

B

-RRB-

g

So

A

-LRB-

B

C

-RRB-

f

-LRB-

A

A

-RRB-

-LRB-

B

B

-RRB-

-LRB-

C

C

-RRB-

-LRB-

B

C

-RRB-

-LRB-

C

B

-RRB-

-LRB-

A

B

-RRB-

-LRB-

A

C

-RRB-

-LRB-

B

A

-RRB-

-LRB-

C

A

-RRB-

g

-LRB-

A

B

-RRB-

C

A

B

-LSB-

C

-LSB-

-LRB-

jA

Bj

jCj

-RRB-

-LSB-

-LRB-

jCj

jA

Bj

-RRB-

A

B

f

-LRB-

A

A

-RRB-

-LRB-

B

B

-RRB-

-LRB-

A

B

-RRB-

-LRB-

B

A

-RRB-

g

So

-LRB-

A

B

-RRB-

C

f

-LRB-

A

A

-RRB-

-LRB-

B

B

-RRB-

-LRB-

A

B

-RRB-

-LRB-

B

A

-RRB-

-LRB-

C

C

-RRB-

-LRB-

A

C

-RRB-

-LRB-

B

C

-RRB-

-LRB-

C

A

-RRB-

-LRB-

C

B

-RRB-

g

Then

A

-LRB-

B

C

-RRB-

-LRB-

A

B

-RRB-

C

-LRB-

viii

-RRB-

A

-LRB-

B

C

-RRB-

A

-LSB-

B

C

f

-LRB-

A

A

-RRB-

-LRB-

B

B

-RRB-

-LRB-

C

C

-RRB-

-LRB-

B

C

-RRB-

-LRB-

C

B

-RRB-

g

-LRB-

A

B

-RRB-

C

A

B

-LSB-

C

-LSB-

-LRB-

jA

Bj

jCj

-RRB-

-LSB-

-LRB-

jCj

jA

Bj

-RRB-

f

-LRB-

A

A

-RRB-

-LRB-

B

B

-RRB-

-LRB-

C

C

-RRB-

-LRB-

A

C

-RRB-

-LRB-

B

C

-RRB-

-LRB-

C

A

-RRB-

-LRB-

C

B

-RRB-

g

Then

A

-LRB-

B

C

-RRB-

-LRB-

A

B

-RRB-

C

-LRB-

ix

-RRB-

Analogous

to

the

previous

case

-LRB-

x

-RRB-

-LRB-

A

B

-RRB-

-LRB-

C

;D

-RRB-

A

B

-LSB-

C

;D

f

-LRB-

A

A

-RRB-

-LRB-

B

B

-RRB-

-LRB-

A

B

-RRB-

-LRB-

B

A

-RRB-

-LRB-

C

C

-RRB-

-LRB-

D

D

-RRB-

-LRB-

C

D

-RRB-

-LRB-

D

C

-RRB-

g

-LRB-

A

C

-RRB-

-LRB-

B

;D

-RRB-

A

C

-LSB-

B

;D

-LSB-

-LRB-

jA

Cj

jB

Dj

-RRB-

-LSB-

-LRB-

jB

Dj

jA

Cj

-RRB-

A

C

-LSB-

B

;D

f

-LRB-

A

A

-RRB-

-LRB-

C

C

-RRB-

-LRB-

B

B

-RRB-

-LRB-

D

D

-RRB-

g

jA

Cj

jB

Dj

f

-LRB-

A

B

-RRB-

-LRB-

A

D

-RRB-

-LRB-

C

B

-RRB-

-LRB-

C

D

-RRB-

g

jB

Dj

jA

Cj

f

-LRB-

B

A

-RRB-

-LRB-

B

C

-RRB-

-LRB-

D

A

-RRB-

-LRB-

D

C

-RRB-

g

Then

-LRB-

A

B

-RRB-

-LRB-

C

;D

-RRB-

-LRB-

A

C

-RRB-

-LRB-

B

;D

-RRB-

Induction

Step

We

must

consider

structural

constructions

done

on

the

basic

contexts

analised

above

In

order

to

obtain

a

general

treatment

those

pairs

of

basic

contexts

will

be

represented

by

its

common

form

and

respectively

Suppose

that

j?j

j

and

where

contexts

and

are

constructed

on

and

respectively

There

are

a

few

cases

to

consider

Let

be

a

formula

of

LLMSc

-LRB-

a

-RRB-

j

j

j?j

j

j

j

j

j

j

j

fi

fi

since

fi

-LSB-

and

fi

-LSB-

-LRB-

b

-RRB-

The

proof

is

analogous

to

a

pair

of

contexts

and

-LRB-

c

-RRB-

j

j

j?j

j

j

j

j

j

j

j

-LSB-

-LSB-

-LRB-

j?j

j

j

-RRB-

-LSB-

-LRB-

j

j

j?j

-RRB-

fi

-LSB-

-LSB-

-LRB-

j

j

j

-RRB-

-LSB-

-LRB-

j

j

j

-RRB-

Then

by

induction

hypothesis

-LRB-

d

-RRB-

The

proof

is

analogous

to

a

pair

of

contexts

and

We

can

obtain

a

Gentzen

's

style

proof

of

the

cut-elimination

for

LLMSc

Theorem

-LRB-

Cut

Elimination

-RRB-

If

a

sequent

is

provable

in

LLMSc

then

it

is

provable

in

LLMSc

without

an

application

of

the

cut

rule

The

proof

of

this

theorem

is

a

consequence

of

the

following

lemma

and

it

is

made

by

induction

on

the

number

of

applications

of

the

cut

rule

in

a

proof

of

a

sequent

in

LLMSc

Lemma

If

ss

is

a

proof

of

a

sequent

S

in

LLMSc

which

contains

only

one

application

of

the

cut

rule

occurring

as

last

inference

then

there

is

a

cut-free

proof

of

S

in

LLMSc

Proof

The

degree

g

-LRB-

A

-RRB-

of

a

formula

A

by

g

-LRB-

A

-RRB-

-LRB-

for

A

atomic

-RRB-

g

-LRB-

A

B

-RRB-

max

-LRB-

g

-LRB-

A

-RRB-

g

-LRB-

B

-RRB-

-RRB-

g

-LRB-

-RCB-

A

-RRB-

g

-LRB-

A

-RRB-

The

rank

of

a

proof

ss

-LRB-

ended

by

a

cut

rule

-RRB-

is

defined

as

rank

-LRB-

ss

-RRB-

rankl

-LRB-

ss

-RRB-

rankr

-LRB-

ss

-RRB-

where

rankl

-LRB-

ss

-RRB-

-LRB-

rankr

-LRB-

ss

-RRB-

-RRB-

is

the

number

of

consecutive

sequents

counting

upward

from

the

left

-LRB-

right

-RRB-

upper

sequent

of

the

end

sequent

that

contain

the

cut

formula

Notice

that

rank

-LRB-

ss

-RRB-

Considering

ss

be

a

such

proof

of

the

sequent

S.

Let

-LRB-

ss

-RRB-

hn

mi

be

a

scale

for

measuring

the

complexity

of

ss

where

n

g

-LRB-

A

-RRB-

-LRB-

A

is

the

cut

formula

-RRB-

and

m

rank

-LRB-

ss

-RRB-

Lets

consider

that

hn

mi

iff

either

n

or

-LRB-

n

and

m

-RRB-

The

proof

will

be

by

induction

on

-LRB-

ss

-RRB-

of

a

proof

ss

so

we

will

be

focusing

in

the

application

of

the

cut

rule

and

its

left

and

right

upper

sequents

all

the

time

Special

Cases

The

left

or

right

upper

sequent

is

an

initial

sequent

-LRB-

the

cut

formula

A

is

atomic

g

-LRB-

A

-RRB-

-RRB-

It

includes

the

cases

in

which

the

rank

is

equal

or

greater

then

A

A

-LSB-

A

-RSB-

B

-LSB-

A

-RSB-

B

A

A

A

A

To

these

proofs

there

are

the

following

cut-free

proofs

respectively

-LSB-

A

-RSB-

B

A

Case

the

degree

of

the

cut

formula

is

greater

than

a

-RRB-

Lets

consider

a

proof

ss

such

that

g

-LRB-

ss

-RRB-

and

rank

-LRB-

ss

-RRB-

Induction

Hypothesis

For

every

proof

of

a

sequent

S

which

contains

a

cut

rule

only

as

the

last

inference

and

-LRB-

-RRB-

-LRB-

ss

-RRB-

there

is

a

cut-free

proof

of

S

a.

-RRB-

The

cut

formula

is

A

B

ss

A

B

A

B

-LSB-

A

B

-RSB-

C

-LSB-

A

B

-RSB-

C

-LSB-

-RSB-

C

Consider

the

following

proof

A

-LSB-

A

B

-RSB-

C

-LSB-

B

-RSB-

C

-LRB-

-RRB-

-LRB-

ss

-RRB-

since

g

-LRB-

B

-RRB-

g

-LRB-

A

B

-RRB-

So

by

induction

hypothesis

there

is

a

cut-free

proof

of

the

sequent

-LSB-

B

-RSB-

C.

Then

the

following

proof

also

has

less

complexity

than

ss

B

-LSB-

B

-RSB-

C

-LSB-

-RSB-

C

Therefore

by

induction

hypothesis

there

is

a

cut-free

proof

of

-LSB-

-RSB-

C.

a.

-RRB-

The

cases

in

which

the

cut

formula

is

either

A?B

A?ffiB

or

A&B

are

analogous

a.

-RRB-

The

cut

formula

is

A

ss

A

A

-LSB-

A

-RSB-

C

-LSB-

yA

-RSB-

C

-LSB-

A

-RSB-

C

-LSB-

-RSB-

C

Then

lets

consider

the

following

proof

A

-LSB-

A

-RSB-

C

-LSB-

-RSB-

C

Its

complexity

is

less

than

the

complexity

of

ss

Then

by

induction

hypothesis

there

is

a

cut-free

proof

of

-LSB-

-RSB-

C

We

have

been

considered

the

case

in

which

it

is

applied

the

rule

yDer

Instead

of

this

if

it

is

applied

the

rule

-LSB-

ffl

-RSB-

C

-LSB-

yA

-RSB-

C

y

Weak

we

could

obtain

a

direct

proof

of

-LSB-

-RSB-

C

from

-LSB-

ffl

-RSB-

C

by

some

applications

of

the

rules

yWeak

and

Ser

since

contains

only

formulas

a.

-RRB-

The

case

in

which

yA

is

the

cut

formula

is

analogous

b

-RRB-

Lets

consider

a

proof

ss

such

that

g

-LRB-

ss

-RRB-

and

rank

-LRB-

ss

-RRB-

So

rankl

-LRB-

ss

-RRB-

and/or

rankr

-LRB-

ss

-RRB-

Induction

Hypothesis

For

every

proof

of

a

sequent

S

which

contains

a

cut

rule

only

as

the

last

inference

and

-LRB-

-RRB-

-LRB-

ss

-RRB-

there

is

a

cut-free

proof

of

S

b.

-RRB-

Suppose

that

rankl

-LRB-

ss

-RRB-

b.

-RRB-

The

left

upper

sequent

A

is

a

lower

sequent

of

the

structural

rule

ss

A

A

-LSB-

A

-RSB-

C

-LSB-

-RSB-

C

Consider

the

following

proof

of

-LSB-

-RSB-

C.

A

-LSB-

A

-RSB-

C

-LSB-

-RSB-

C

-LRB-

-RRB-

-LRB-

ss

-RRB-

since

que

rankl

-LRB-

-RRB-

rankl

-LRB-

ss

-RRB-

So

by

induction

hypotesis

there

is

a

cut-free

proof

of

-LSB-

-RSB-

C.

Moreover

j

-LSB-

-RSB-

j

j

-LSB-

-RSB-

j

and

-LSB-

-RSB-

-LSB-

-RSB-

since

and

were

used

in

an

application

of

the

structural

rule

Then

the

following

proof

whose

its

last

inference

is

the

structural

rule

-LSB-

-RSB-

C

-LSB-

-RSB-

C

is

a

cut-free

proof

of

-LSB-

-RSB-

C.

b.

-RRB-

The

left

upper

sequent

is

a

lower

sequent

of

an

application

of

a

rule

whose

principal

formula

is

not

the

cut

formula

ss

-LSB-

-RSB-

A

-LSB-

-RSB-

A

-LSB-

A

-RSB-

C

-LSB-

-LSB-

-RSB-

-RSB-

C

Consider

the

following

proof

of

-LSB-

-LSB-

-RSB-

-RSB-

C.

-LSB-

-RSB-

A

-LSB-

A

-RSB-

C

-LSB-

-LSB-

-RSB-

-RSB-

C

Analogously

to

the

previous

case

has

less

complexity

than

ss

So

by

induction

hypotesis

there

is

a

cut-free

proof

of

-LSB-

-LSB-

-RSB-

-RSB-

C.

Then

the

following

proof

whose

its

last

inference

is

fiL

-LSB-

-LSB-

-RSB-

-RSB-

C

-LSB-

-LSB-

-RSB-

-RSB-

C

is

a

cut-free

proof

of

-LSB-

-LSB-

-RSB-

-RSB-

C

The

cases

in

which

the

rule

applied

is

either

L

ffiL

L

L

yWeak

yDer

yThread

Ser

or

Contr

are

analogous

b.

-RRB-

Suppose

that

rankr

-LRB-

ss

-RRB-

b.

-RRB-

The

right

upper

sequent

is

a

lower

sequent

of

the

structural

rule

There

are

many

cases

to

consider

especially

those

proved

in

the

Proposition

-LSB-

-RSB-

C

-LSB-

-RSB-

C

-LSB-

-RSB-

C

-LSB-

-RSB-

C

-LSB-

-RSB-

C

-LSB-

-RSB-

C

-LSB-

-LRB-

-RRB-

-RSB-

C

-LSB-

-LRB-

-RRB-

-RSB-

C

-LSB-

-LRB-

-RRB-

-RSB-

C

-LSB-

-LRB-

-RRB-

-RSB-

C

-LSB-

-LRB-

-RRB-

-RSB-

C

-LSB-

-LRB-

-RRB-

-RSB-

C

-LSB-

-LRB-

-RRB-

-RSB-

C

-LSB-

-LRB-

-RRB-

-RSB-

C

-LSB-

-LRB-

-RRB-

-RSB-

C

-LSB-

-LRB-

-RRB-

-RSB-

C

-LSB-

-LRB-

-RRB-

-RSB-

C

-LSB-

-LRB-

-RRB-

-RSB-

C

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-RSB-

C

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-RSB-

C

where

the

cut

formula

can

be

any

Bi

We

will

prove

only

the

first

one

and

the

others

can

be

proved

in

an

analogous

way

ss

A

-LSB-

A

B

-RSB-

C

-LSB-

A

B

-RSB-

C

-LSB-

B

-RSB-

C

Consider

the

following

proof

of

-LSB-

B

-RSB-

C.

A

-LSB-

A

B

-RSB-

C

-LSB-

B

-RSB-

C

has

less

complexity

than

ss

So

by

induction

hypotesis

there

is

a

cut-free

proof

of

-LSB-

B

-RSB-

C.

Moreover

j

-LSB-

B

-RSB-

j

j

-LSB-

B

-RSB-

j

and

-LSB-

B

-RSB-

-LSB-

B

-RSB-

since

-LSB-

A

B

-RSB-

and

-LSB-

A

B

-RSB-

were

used

in

an

application

of

the

structural

rule

Then

the

following

proof

whose

its

last

inference

is

the

structural

rule

-LSB-

B

-RSB-

C

-LSB-

B

-RSB-

C

is

a

cut-free

proof

of

-LSB-

-RSB-

C.

b.

-RRB-

The

right

upper

sequent

is

a

lower

sequent

of

a

rule

whose

principal

formula

is

not

the

cut

formula

The

cases

enclosed

here

are

analogous

to

the

case

proved

in

-LRB-

b.

-RRB-

b.

-RRB-

The

right

upper

sequent

is

a

lower

sequent

of

a

rule

whose

principal

formula

is

the

cut

one

b.

-RRB-

Cut

formula

A

ss

A

A

-LSB-

A

A

-RSB-

C

-LSB-

A

-RSB-

C

-LSB-

-RSB-

C

where

is

an

independent

context

with

only

formulas

The

complexity

of

the

following

proof

is

less

than

the

complexity

of

ss

A

A

-LSB-

A

A

-RSB-

C

-LSB-

A

-RSB-

C

Then

there

is

cut-free

proof

of

-LSB-

A

-RSB-

C.

So

the

following

proof

also

has

less

complexity

than

ss

A

A

-LSB-

A

-RSB-

C

-LSB-

-RSB-

C

Then

there

is

a

cut-free

proof

of

-LSB-

-RSB-

C

In

the

other

hand

has

only

occurrences

of

operator

and

formulas

So

there

is

a

cut-free

proof

of

-LSB-

-RSB-

C

from

by

some

permutations

allowed

by

the

Structural

Rule

-LRB-

Proposition

-RRB-

-LSB-

-RSB-

C

-LSB-

-RSB-

C

and

contractions

-LRB-

Contr

-RRB-

-LSB-

-RSB-

C

-LRB-

and

Contr

-RRB-

-LSB-

-RSB-

C

b.

-RRB-

Cut

formula

yA

A

yA

-LSB-

yA

yA

-RSB-

C

-LSB-

yA

-RSB-

C

-LSB-

-RSB-

C

Analogously

to

the

previous

case

there

is

a

cut-free

proof

of

-LSB-

-RSB-

C.

Although

has

only

occurrences

of

operator

and

or

y

formulas

Lets

consider

-LRB-

i

-RRB-

-LRB-

y

i

y

n

-RRB-

where

every

j

and

y

k

are

occurrences

of

formulas

in

j

i

and

i

k

n

and

the

following

inferences

allowed

by

the

Strucutural

Rule

-LRB-

Proposition

-RRB-

in

addition

to

-LSB-

-RSB-

C

-LSB-

-RSB-

C

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-RSB-

C

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-RSB-

C

and

contractions

-LRB-

Contr

-RRB-

Then

there

is

the

following

cut-free

proof

of

-LSB-

-RSB-

C

-LSB-

-RSB-

C

-LRB-

-RRB-

-LSB-

-RSB-

C

-LRB-

-RRB-

-LSB-

-LRB-

-RRB-

-LRB-

i

i

-RRB-

-LRB-

y

i

y

i

-RRB-

-LRB-

y

n

y

n

-RRB-

-RSB-

C

-LRB-

yThread

-RRB-

-LSB-

-LRB-

-RRB-

-LRB-

i

i

-RRB-

-RRB-

y

i

y

n

-RSB-

C

-LRB-

-RRB-

-LSB-

-LRB-

-RRB-

-LRB-

i

i

-RRB-

y

i

y

n

-RSB-

C

-LRB-

Contr

-RRB-

-LSB-

i

y

i

y

n

-RSB-

C

-LRB-

-RRB-

-LSB-

-RSB-

C

b.

-RRB-

Cases

in

which

the

formula

has

two

different

occurrences

in

the

right

upper

sequent

as

the

cut

formula

and

as

the

principal

formula

ss

A

B

-LRB-

-LSB-

A

B

-RSB-

-RRB-

-LSB-

A

B

-RSB-

C

-LRB-

-LSB-

A

B

-RSB-

-RRB-

-LSB-

A

B

-RSB-

C

-LRB-

-LSB-

-RSB-

-RRB-

-LSB-

A

B

-RSB-

C

Consider

the

following

proof

of

-LRB-

-LSB-

-RSB-

-RRB-

-LSB-

A

B

-RSB-

-RSB-

C.

A

B

-LRB-

-LSB-

A

B

-RSB-

-RRB-

-LSB-

A

B

-RSB-

C

-LRB-

-LSB-

-RSB-

-RRB-

-LSB-

A

B

-RSB-

C

has

less

complexity

than

ss

So

by

induction

hypotesis

there

is

a

cut-free

proof

of

-LRB-

-LSB-

-RSB-

-RRB-

-LSB-

A

B

-RSB-

C.

Then

the

following

proof

is

a

cut-free

proof

of

-LRB-

-LSB-

-RRB-

-LSB-

A

B

-RSB-

C.

-LRB-

-LSB-

-RSB-

-RRB-

-LSB-

A

B

-RSB-

C

-LRB-

-LSB-

-RSB-

-RRB-

-LSB-

A

B

-RSB-

C

L

The

cases

in

which

the

rule

applied

is

either

L

ffiL

L

L

yWeak

yDer

yThread

Ser

or

Contr

are

analogous

Notice

that

there

are

a

few

cases

in

which

the

principal

formula

occurs

in

the

right

side

of

the

right

upper

sequent

but

they

can

be

reduced

to

initial

sequent

A

Categorical

Model

for

LLMSc

To

present

a

categorical

characterization

of

LLMSc

we

consider

an

instantiation

G

of

the

-LRB-

symmetric

monoidal

closed

-RRB-

categories

GC

developed

in

-LSB-

-RSB-

As

usual

the

interpretation

of

the

formulae

is

given

by

objects

of

the

category

G

and

proofs

of

LLMSc

are

interpreted

by

morphisms

of

G

In

the

most

general

case

a

GC-like

category

is

obtained

from

a

symmetric

monoidal

closed

category

C

with

products

-LRB-

and

satisfying

some

other

conditions

-LSB-

-RSB-

-RRB-

but

for

purposes

of

this

paper

our

base

category

will

be

Sets

Readers

interested

in

the

-LRB-

several

already

known

-RRB-

generalizations

of

this

work

should

consult

-LSB-

-RSB-

Recall

that

the

partial-order

with

two

elements

where

one

should

think

of

as

false

and

as

true

is

a

closed-poset

It

has

a

tensor

product

-LRB-

-RRB-

and

an

internal-hom

-LRB-

ffi

-RRB-

such

that

a

b

c

iff

a

b?ffic

Actually

is

the

usual

conjunction

and

the

internal-hom

linear

is

the

usual

Heyting

implication

these

operations

can

be

described

by

the

usual

truth-tables

Definition

The

category

G

is

described

by

ffl

Objects

are

relations

on

Sets

that

is

maps

U

X

written

as

-LRB-

U

-LRB-

-RRB-

X

-RRB-

ffl

A

morphism

from

-LRB-

U

-LRB-

-RRB-

X

-RRB-

to

-LRB-

V

-LRB-

-RRB-

Y

-RRB-

consists

a

pair

of

maps

f

U

V

and

F

Y

X

such

that

the

following

condition

is

satisfied

-LRB-

u

Fy

-RRB-

-LRB-

fu

y

-RRB-

where

is

the

usual

order

in

Thus

there

is

a

morphism

-LRB-

f

F

-RRB-

from

to

iff

for

all

u

in

U

and

all

y

in

Y

if

uffF

-LRB-

y

-RRB-

then

f

-LRB-

u

-RRB-

y

This

could

represented

by

the

following

diagram

U

Y

F

i

U

X

f

idY

j

jj

j

jj

V

Y

We

usually

depict

a

morphism

-LRB-

f

F

-RRB-

in

G

as

follows

U

X

f

j

jj

jj

j

F

V

Y

The

next

-LRB-

big

-RRB-

definition

defines

the

-LRB-

multiplicative

and

additive

-RRB-

structure

of

the

category

G

we

are

interested

in

Definition

Let

A

-LRB-

U

-LRB-

-RRB-

X

-RRB-

and

B

-LRB-

V

-LRB-

-RRB-

Y

-RRB-

be

objects

of

G.

ffl

The

tensor

bifunctor

G

G

G

is

given

by

A

B

-LRB-

U

V

-LRB-

-RRB-

XV

Y

U

-RRB-

where

-LRB-

u

v

-RRB-

-LRB-

f

g

-RRB-

iff

ufff

-LRB-

v

-RRB-

and

v

g

-LRB-

u

-RRB-

Its

unit

is

the

object

I

-LRB-

-LRB-

-RRB-

-RRB-

where

is

the

identity

relation

on

ffl

The

tensor

bifunctor

G

G

G

is

given

by

A

B

-LRB-

U

V

-LRB-

-RRB-

X

Y

-RRB-

where

-LRB-

u

v

-RRB-

-LRB-

x

y

-RRB-

iff

uffx

and

v

y.

Its

unit

is

the

same

object

I

above

ffl

The

internal-hom

bifunctor

-LSB-

-RSB-

G

Gop

G

G

is

given

by

-LSB-

A

B

-RSB-

G

-LRB-

V

U

XY

-LRB-

-RRB-

U

Y

-RRB-

where

-LRB-

f

F

-RRB-

-LRB-

u

y

-RRB-

iff

uffx

implies

v

y

The

relation

is

given

by

the

composition

V

U

XY

U

Y

eval

evali

U

Y

V

X

ffi

ffl

The

product

bifunctor

G

G

G

is

given

by

A&B

-LRB-

U

V

-LRB-

-RRB-

X

Y

-RRB-

where

-LRB-

u

v

-RRB-

x

y

iff

either

uffx

or

v

y.

Its

unit

is

the

object

T

-LRB-

e

-LRB-

-RRB-

-RRB-

We

could

also

describe

coproducts

-LRB-

-RRB-

and

a

par

connective

-LRB-

-RRB-

but

these

will

not

play

an

important

role

in

the

following

discussion

Lemma

For

any

A

and

B

objects

of

G

there

are

natural

morphisms

-LRB-

i

-RRB-

o

-LRB-

A

B

-RRB-

A

B

A

-LRB-

ii

-RRB-

weak-distributivity

w

-LRB-

A

B

C

-RRB-

A

-LRB-

B

C

-RRB-

-LRB-

A

B

-RRB-

C

-LRB-

iii

-RRB-

v

-LRB-

A

B

C

;D

-RRB-

-LRB-

A

B

-RRB-

-LRB-

C

D

-RRB-

-LRB-

A

C

-RRB-

-LRB-

B

D

-RRB-

Proof

-LRB-

i

-RRB-

The

natural

morphism

o

-LRB-

A

B

-RRB-

is

given

by

-LRB-

idU?V

-LRB-

-RRB-

-RRB-

U

V

XV

Y

U

idU?V

j

jj

jj

j

-LRB-

-RRB-

U

V

X

Y

where

-LRB-

-RRB-

X?Y

XV

Y

U

consists

of

constant

maps

-LCB-

or

exponential

transposes

of

the

projections

-LCB-

in

each

coordinate

is

another

natural

morphism

oeA

B

-LRB-

A

B

-RRB-

-LRB-

-RRB-

which

shall

not

concern

us

here

-LRB-

ii

-RRB-

The

natural

morphism

w

-LRB-

A

B

C

-RRB-

corresponding

to

the

first

weak-distributivity

is

given

by

-LRB-

ff

-LRB-

-RRB-

-RRB-

U

-LRB-

V

W

-RRB-

-LRB-

-RRB-

XV?W

-LRB-

Y

Z

-RRB-

U

j

jj

jj

j

-LRB-

-RRB-

-LRB-

U

V

-RRB-

W

-LRB-

-RRB-

-LRB-

XV

Y

U

-RRB-

Z

where

U

-LRB-

V

W

-RRB-

-LRB-

U

V

-RRB-

W

is

the

associativity

isomorphism

XV!XV?W

-LRB-

XV

-RRB-

W

is

the

constant

map

Y

U

Y

U

is

the

identity

map

and

Z

ZU

is

the

constant

map

again

-LRB-

note

that

we

need

more

associativity

and

the

isomorphism

-LRB-

Y

Z

-RRB-

U

Y

U

ZU

here

-RRB-

-LRB-

iii

-RRB-

The

natural

morphism

v

-LRB-

A

B

C

;D

-RRB-

is

given

by

-LRB-

-LRB-

-RRB-

-RRB-

-LRB-

U

V

-RRB-

-LRB-

W

P

-RRB-

-LRB-

-RRB-

-LRB-

ae

-RRB-

-LRB-

X

Y

-RRB-

W?P

-LRB-

Z

Q

-RRB-

U?V

j

jj

jj

j

-LRB-

-RRB-

-LRB-

U

W

-RRB-

-LRB-

V

P

-RRB-

-LRB-

-RRB-

-LRB-

ae

-RRB-

-LRB-

XW

ZU

-RRB-

-LRB-

Y

P

QV

-RRB-

where

-LRB-

U

V

-RRB-

-LRB-

W

P

-RRB-

-LRB-

U

W

-RRB-

-LRB-

V

P

-RRB-

is

the

commutativity

isomorphism

and

XW

XW?P

ZU

Z

-LRB-

U?V

-RRB-

Y

P

Y

W?P

and

QV

QU?V

are

constant

maps

and

we

use

some

isomorphisms

to

get

the

parenthesis

in

the

right

places

Recall

that

Sets

does

have

free

-LRB-

commutative

-RRB-

monoid

structures

ffl

There

is

an

adjunction

F

a

U

Monc

Sets

where

Monc

is

the

category

of

commutative

monoids

U

is

the

forgetful

functor

and

in

F

-LRB-

X

-RRB-

-LRB-

X

eX

mX

-RRB-

X

can

be

viewed

as

finite

sequences

-LRB-

up

to

permutation

-RRB-

of

elements

of

X

eX

X

is

the

empty

sequence

of

elements

of

X

and

mX

X

X

X

means

concatenation

of

sequences

ffl

There

is

a

monad

in

Sets

-LRB-

-RRB-

where

U

ffi

F

Sets

Sets

and

we

write

f

for

f

applied

to

each

element

of

the

sequence

This

monad

's

unit

is

given

by

the

singleton

sequence

-LRB-

-RRB-

X

X

X

and

its

multiplication

-LRB-

-RRB-

X

X

X

by

forgetting

parenthesis

We

will

need

to

consider

another

monad

structure

in

Sets

Consider

the

monad

-LRB-

TU

TU

TU

-RRB-

where

for

a

given

U

in

Sets

TU

Sets

Sets

is

the

endofunctor

which

takes

X

XU

Y

Y

U

and

if

g

X

Y

f

XU

is

taken

to

f

g

Y

U

Its

unit

is

given

by

the

transpose

of

the

projection

that

is

the

constant

map

-LRB-

TU

-RRB-

X

X

XU

and

its

multiplication

-LRB-

TU

-RRB-

X

XU?U

XU

is

simply

precomposition

with

the

diagonal

map

on

U

We

define

two

endofuctors

on

G

to

model

the

modalities

of

LLMSc

Definition

Consider

the

following

endofunctors

on

G

The

functor

y

on

G

given

on

objects

by

y

-LRB-

U

-LRB-

-RRB-

X

-RRB-

-LRB-

U

y

-LRB-

-RRB-

X

-RRB-

where

the

relation

y

-LRB-

-RRB-

is

defined

by

u

-LRB-

y

-RRB-

-LRB-

xk

-RRB-

iff

and

and

uffxk

and

y

acts

on

maps

as

y

-LRB-

f

F

-RRB-

-LRB-

f

F

-RRB-

The

functor

G

on

G

given

by

G

-LRB-

U

-LRB-

-RRB-

X

-RRB-

-LRB-

U

Gff

-LRB-

-RRB-

XU

-RRB-

where

the

relation

Gff

is

given

by

u

-LRB-

Gff

-RRB-

f

iff

ufff

-LRB-

u

-RRB-

The

functor

G

applied

to

a

map

-LRB-

f

F

-RRB-

A

B

is

-LRB-

f

F

-LRB-

-RRB-

f

-RRB-

GA

GB

Formally

the

relation

y

is

given

by

the

map

U

X

y

which

is

the

composition

U

X

X

-LRB-

U

X

-RRB-

-LRB-

the

map

U

X

U

X

-LRB-

U

X

-RRB-

takes

a

tuple

hu

xki

hhu

hu

xkii

and

using

the

tensor

product

in

-RRB-

and

the

relation

Gff

is

given

by

the

composition

U

XU

evali

U

X

Lemma

The

functors

y

and

G

have

natural

comonad

structures

-LRB-

y

ffly

ffiy

-RRB-

and

-LRB-

G

fflG

ffiG

-RRB-

induced

by

the

monad

structure

of

and

the

monads

TU

respectively

ffl

The

counit

-LRB-

ffly

-RRB-

A

yA

A

is

given

by

identity

on

U

and

the

singleton

map

X

-LRB-

-RRB-

X

X

and

the

comultiplication

-LRB-

ffiy

-RRB-

A

yA

is

given

by

identity

on

U

and

forgetting

brackets

X

-LRB-

-RRB-

X

X

in

the

second

coordinate

as

in

the

diagram

U

y

X

idU

j

jj

jj

j

-LRB-

-RRB-

X

U

X

U

y

X

idU

j

jj

jj

j

-LRB-

-RRB-

X

U

X

Moreover

this

comonad

is

monoidal

with

respect

to

the

tensor

structure

that

is

there

is

a

natural

morphism

yA

yB

y

-LRB-

A

B

-RRB-

given

by

U

V

y

y

X

Y

idU?V

j

jj

jj

j

r

-LRB-

X

Y

-RRB-

U

V

y

-LRB-

-RRB-

-LRB-

X

Y

-RRB-

where

r

-LRB-

X

Y

-RRB-

-LRB-

X

Y

-RRB-

X

Y

transforms

a

sequence

of

pairs

hxi

yii

into

the

the

pair

of

sequences

hxii

and

hyii

ffl

The

counit

-LRB-

fflG

-RRB-

A

GA

A

is

given

by

identity

on

U

and

the

natural

constant

map

-LRB-

TU

-RRB-

X

X

XU

in

the

second

coordinate

the

comultiplication

-LRB-

ffiG

-RRB-

A

GA

is

given

by

identity

on

U

and

restriction

to

the

diagonal

-LRB-

TU

-RRB-

X

XU?U

XU

in

the

second

coordinate

U

Gff

XU

idU

j

jj

jj

j

-LRB-

TU

-RRB-

X

U

X

U

Gff

XU

idU

j

jj

jj

j

-LRB-

TU

-RRB-

X

U

XU?U

This

comonad

is

monoidal

with

respect

to

the

tensor

structure

as

there

is

a

natural

morphism

B

GA?GB

G

-LRB-

A

B

-RRB-

given

by

U

V

Gff

Gfi

-LRB-

XU

-RRB-

V

-LRB-

Y

V

-RRB-

U

j

jj

jj

j

U

V

G

-LRB-

-RRB-

-LRB-

XV

Y

U

-RRB-

U?V

where

XU?V?V

Y

U?U?V

XU?V

Y

U?V

is

just

restriction

to

the

respective

diagonal

Now

there

is

a

distributive

law

relating

our

comonads

y

and

G.

Then

by

abstract

nonsense

-LSB-

-RSB-

we

have

that

these

comonads

compose

and

that

G

lifts

to

the

category

of

coalgebras

for

y

The

next

lemma

collects

some

facts

about

G

and

y.

Lemma

ffl

There

is

a

distributivity

law

of

comonads

A

G

-LRB-

yA

-RRB-

y

-LRB-

GA

-RRB-

given

by

U

G

-LRB-

y

-RRB-

-LRB-

X

-RRB-

U

idU

j

jj

jj

j

X

U

y

-LRB-

Gff

-RRB-

-LRB-

XU

-RRB-

where

each

component

X

-LRB-

XU

-RRB-

-LRB-

X

-RRB-

U

of

the

natural

transformation

is

given

by

the

exponential

transpose

of

the

following

composition

-LRB-

XU

-RRB-

U

XU

U

-LRB-

XU

U

-RRB-

ev

X

ffl

There

is

a

natural

morphism

yA

yB

y

-LRB-

A

B

-RRB-

given

by

-LRB-

idU?V

-LRB-

-RRB-

ffi

r

-RRB-

as

the

diagram

shows

U

V

y

-LRB-

-RRB-

y

-LRB-

-RRB-

-LRB-

X

-RRB-

V

-LRB-

Y

-RRB-

U

idU?V

j

jj

jj

j

-LRB-

-RRB-

ffi

r

U

V

y

-LRB-

-RRB-

-LRB-

XV

Y

U

-RRB-

where

intuitively

the

map

r

-LRB-

XV

Y

U

-RRB-

-LRB-

XV

-RRB-

-LRB-

Y

U

-RRB-

transforms

a

sequence

fk

gki

into

pairs

of

sequences

-LRB-

fki

gki

-RRB-

and

the

maps

-LRB-

XV

-RRB-

-LRB-

X

-RRB-

V

and

-LRB-

Y

U

-RRB-

-LRB-

Y

-RRB-

U

are

components

of

natural

transformations

which

satisfies

the

conditions

for

a

distributive

law

of

monads

ffl

There

is

a

comonad

morphism

y

where

A

A

yA

is

given

by

U

-LRB-

X

-RRB-

U

idU

j

jj

jj

j

X

U

y

X

where

X

X

-LRB-

X

-RRB-

U

is

given

by

-LRB-

TU

-RRB-

applied

to

the

object

X

This

morphism

is

similar

to

Reddy

's

Ser

but

it

is

not

a

monomorphism

in

G.

Definition

Consider

the

composite

comonad

-LRB-

G

y

-RRB-

and

call

it

Its

functor

part

acts

on

objects

as

-LRB-

U

X

-RRB-

-LRB-

U

-LRB-

X

-RRB-

U

-RRB-

and

on

maps

as

-LRB-

f

F

-RRB-

-LRB-

f

F

-LRB-

-RRB-

f

-RRB-

We

have

-LRB-

ffl

-RRB-

A

A

A

and

-LRB-

ffi

-RRB-

A

A

A

as

follows

U

-LRB-

X

-RRB-

U

idU

j

jj

jj

j

X

U

X

U

-LRB-

X

-RRB-

U

idU

j

jj

jj

j

X

U

-LRB-

-LRB-

-LRB-

X

-RRB-

U

-RRB-

-RRB-

U

where

X

is

given

by

X

-LRB-

-RRB-

X

X

-LRB-

TU

-RRB-

X

-LRB-

X

-RRB-

U

and

X

is

given

by

the

exponential

transpose

of

the

long

composition

-LRB-

-LRB-

-LRB-

X

-RRB-

U

-RRB-

-RRB-

U

U

hev

ssi

-LRB-

-LRB-

X

-RRB-

U

-RRB-

U

U

-LRB-

X

-RRB-

U

U

ev

X

-LRB-

-RRB-

X

X

We

emphasize

some

properties

which

are

used

in

the

proof

of

soundness

of

the

model

Proposition

For

any

G-object

A

A

has

a

comonoid

structure

-LRB-

A

dA

eA

-RRB-

with

respect

to

The

comonoid

counit

eA

A

I

is

given

by

the

terminal

map

on

U

and

the

map

-LRB-

X

-RRB-

U

The

comultiplication

dA

A

A

A

is

given

by

the

diagonal

U

U

U

and

the

map

-LRB-

X

-RRB-

U?U

-LRB-

X

-RRB-

U?U

-LRB-

X

-RRB-

U

The

map

is

given

by

taking

a

pair

of

functions

-LRB-

OE

-RRB-

each

of

them

of

the

form

U

U

X

to

the

following

map

U

U

U

OE

X

X

mX

X

where

mX

X

X

X

is

concatenation

of

sequences

This

comonoid

structure

determines

a

functor

R

G

Comon?G

which

takes

A

to

-LRB-

A

dA

eA

-RRB-

For

any

G-object

A

yA

has

a

natural

comonoid

structure

with

respect

to

There

is

a

functor

Ry

G

Comon

G

which

takes

A

to

-LRB-

yA

A

A

-RRB-

Moreover

Ry

is

right-adjoint

to

the

forgetful

functor

U

Comon

G

G.

There

are

natural

isomorphisms

for

all

objects

A

and

B

in

G

-LRB-

A&B

-RRB-

A

B

Lemma

If

there

is

a

morphism

from

-LSB-

-LSB-

-LSB-

B

-RSB-

-RSB-

-RSB-

to

-LSB-

-LSB-

C

-RSB-

-RSB-

and

a

morphism

from

-LSB-

-LSB-

A

-RSB-

-RSB-

to

-LSB-

-LSB-

B

-RSB-

-RSB-

then

there

is

a

morphism

from

-LSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

-RSB-

to

-LSB-

-LSB-

C

-RSB-

-RSB-

Proof

Suppose

that

there

are

morphisms

-LRB-

f

F

-RRB-

-LSB-

-LSB-

-LSB-

B

-RSB-

-RSB-

-RSB-

-LSB-

-LSB-

C

-RSB-

-RSB-

and

-LRB-

h

H

-RRB-

-LSB-

-LSB-

A

-RSB-

-RSB-

-LSB-

-LSB-

B

-RSB-

-RSB-

Let

us

consider

the

following

cases

-LRB-

i

-RRB-

-LSB-

B

-RSB-

has

only

the

formula

B

Then

the

morphism

is

-LRB-

f

F

-RRB-

ffi

-LRB-

h

H

-RRB-

-LRB-

ii

-RRB-

-LSB-

B

-RSB-

has

the

form

-LRB-

B

-RRB-

Then

-LSB-

-LSB-

-LSB-

B

-RSB-

-RSB-

-RSB-

-LRB-

-LSB-

-LSB-

B

-RSB-

-RSB-

-LSB-

-LSB-

-RSB-

-RSB-

-RRB-

-LSB-

-LSB-

-RSB-

-RSB-

So

there

is

a

morphism

-LRB-

f

F

-RRB-

ffi

-LRB-

-LRB-

h

H

-RRB-

ID

-LSB-

-LSB-

-RSB-

-RSB-

-RRB-

ID

-LSB-

-LSB-

-RSB-

-RSB-

from

-LSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

-RSB-

to

-LSB-

-LSB-

C

-RSB-

-RSB-

The

permutations

of

this

possibility

are

proved

analogously

-LRB-

iii

-RRB-

-LSB-

B

-RSB-

has

the

form

-LRB-

B

-RRB-

So

-LSB-

-LSB-

-LSB-

B

-RSB-

-RSB-

-RSB-

-LRB-

-LSB-

-LSB-

B

-RSB-

-RSB-

-LSB-

-LSB-

-RSB-

-RSB-

-RRB-

-LSB-

-LSB-

-RSB-

-RSB-

So

there

is

a

morphism

-LRB-

f

F

-RRB-

ffi

-LRB-

-LRB-

h

H

-RRB-

ID

-LSB-

-LSB-

-RSB-

-RSB-

-RRB-

ID

-LSB-

-LSB-

-RSB-

-RSB-

from

-LSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

-RSB-

to

-LSB-

-LSB-

C

-RSB-

-RSB-

The

permutations

of

this

possibility

are

proved

analogously

-LRB-

iv

-RRB-

The

case

in

which

-LSB-

B

-RSB-

has

either

the

form

-LRB-

B

-RRB-

or

the

form

-LRB-

B

-RRB-

and

all

permutations

of

these

possibilities

are

analogous

to

the

second

and

third

cases

-LRB-

v

-RRB-

The

cases

in

which

-LSB-

B

-RSB-

contains

two

occurrences

of

formulas

are

particular

cases

of

the

above

ones

Theorem

-LRB-

Soundness

-RRB-

The

symmetric

monoidal

closed

category

G

with

bifunctors

internal

hom

-LSB-

-RSB-

G

tensor

product

product

cartesian

product

and

the

comonads

and

y

is

a

model

for

LLMSc

Thus

each

entailment

LLMSc

A

corresponds

to

the

existence

of

a

morphism

in

G

-LRB-

f

F

-RRB-

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

Proof

We

have

only

to

check

that

the

rules

of

LLMSc

are

preserved

by

the

interpretation

First

lets

consider

that

to

each

context

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

-RSB-

-RSB-

where

is

a

formula

obtained

from

by

substituting

any

ocurrences

of

the

operators

and

by

the

connectives

and

respectively

-LRB-

i

-RRB-

Structural

Rule

The

proofs

of

the

main

inferences

allowed

by

the

Structural

Rule

-LRB-

Proposition

-RRB-

are

based

on

the

Lemma

the

Lemma

and

on

the

facts

that

the

functors

and

are

commutative

and

associative

-LRB-

ii

-RRB-

Identity

Rules

ffl

To

each

initial

sequent

inferred

by

the

rule

Id

corresponds

an

identity

morphism

in

G.

ffl

The

proof

of

the

Cut

rule

can

be

obtained

directly

from

the

Lemma

-LRB-

iii

-RRB-

Multiplicative

Rules

ffl

The

rules

R

and

fiR

only

say

that

the

tensor

product

and

the

product

are

bifunctors

If

-LSB-

-LSB-

-RSB-

-RSB-

-LRB-

f

F

-RRB-

-LSB-

-LSB-

A

-RSB-

-RSB-

and

-LSB-

-LSB-

-RSB-

-RSB-

-LRB-

h

H

-RRB-

-LSB-

-LSB-

B

-RSB-

-RSB-

then

-LRB-

f

F

-RRB-

-LRB-

h

H

-RRB-

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

A

B

-RSB-

-RSB-

If

-LSB-

-LSB-

-RSB-

-RSB-

-LRB-

f

F

-RRB-

-LSB-

-LSB-

A

-RSB-

-RSB-

and

-LSB-

-LSB-

-RSB-

-RSB-

-LRB-

h

H

-RRB-

-LSB-

-LSB-

B

-RSB-

-RSB-

then

-LRB-

f

F

-RRB-

-LRB-

h

H

-RRB-

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

A

B

-RSB-

-RSB-

ffl

The

proofs

of

the

rules

L

and

fiL

follow

immediately

by

the

Lemma

since

-LSB-

-LSB-

A

B

-RSB-

-RSB-

-LSB-

-LSB-

A

B

-RSB-

-RSB-

and

-LSB-

-LSB-

A

B

-RSB-

-RSB-

-LSB-

-LSB-

A

B

-RSB-

-RSB-

ffl

The

proofs

of

the

rules

and

are

based

on

the

fact

that

-LSB-

-LSB-

-RSB-

-RSB-

I

is

the

unit

of

the

bifunctors

and

ffl

The

rules

ffiR

and

ffiL

reflect

the

monoidal

closed

structure

Thus

If

-LSB-

-LSB-

A

-RSB-

-RSB-

-LRB-

f

F

-RRB-

-LSB-

-LSB-

B

-RSB-

-RSB-

using

the

adjunction

we

get

-LSB-

-LSB-

-RSB-

-RSB-

-LRB-

f

F

-RRB-

-LSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

-LSB-

-LSB-

B

-RSB-

-RSB-

-RSB-

G

where

-LSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

-LSB-

-LSB-

B

-RSB-

-RSB-

-RSB-

G

-LSB-

-LSB-

A?ffiB

-RSB-

-RSB-

and

-LSB-

-LSB-

A

-RSB-

-RSB-

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

Further

if

-LSB-

-LSB-

-RSB-

-RSB-

-LRB-

f

F

-RRB-

-LSB-

-LSB-

A

-RSB-

-RSB-

then

we

have

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

A?ffiB

-RSB-

-RSB-

-LRB-

f

F

-RRB-

Id

-LSB-

-LSB-

A

-RSB-

-RSB-

-LSB-

-LSB-

A?ffiB

-RSB-

-RSB-

ev

-LSB-

-LSB-

B

-RSB-

-RSB-

So

by

the

Lemma

if

there

is

a

morphism

from

-LSB-

-LSB-

-RSB-

-RSB-

to

-LSB-

-LSB-

A

-RSB-

-RSB-

and

there

is

a

morphism

from

-LSB-

-LSB-

-LSB-

B

-RSB-

-RSB-

-RSB-

to

-LSB-

-LSB-

C

-RSB-

-RSB-

then

there

is

a

morphism

from

-LSB-

-LSB-

-LSB-

A?ffiB

-RSB-

-RSB-

-RSB-

to

-LSB-

-LSB-

C

-RSB-

-RSB-

-LRB-

iv

-RRB-

Additive

Rules

The

additive

rules

characterize

the

properties

of

a

cartesian

product

where

-LSB-

-LSB-

A&B

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

-LSB-

-LSB-

B

-RSB-

-RSB-

Thus

ffl

-LSB-

-LSB-

-RSB-

-RSB-

-LRB-

e

-LRB-

-RRB-

-RRB-

is

the

unit

of

the

bifunctor

ffl

If

-LSB-

-LSB-

-RSB-

-RSB-

-LRB-

f

F

-RRB-

-LSB-

-LSB-

A

-RSB-

-RSB-

and

-LSB-

-LSB-

-RSB-

-RSB-

-LRB-

h

H

-RRB-

-LSB-

-LSB-

B

-RSB-

-RSB-

then

-LRB-

-LRB-

f

h

-RRB-

F

H

-RRB-

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

A&B

-RSB-

-RSB-

ffl

The

rules

L

and

L

correspond

to

the

existence

of

projections

in

G

-LSB-

-LSB-

A

-RSB-

-RSB-

-LSB-

-LSB-

B

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

and

-LSB-

-LSB-

A

-RSB-

-RSB-

-LSB-

-LSB-

B

-RSB-

-RSB-

-LSB-

-LSB-

B

-RSB-

-RSB-

given

by

-LRB-

-RRB-

and

-LRB-

-RRB-

where

U

V

U

and

U

V

V

are

canonical

projections

and

X

X

Y

and

Y

X

Y

are

canonical

injections

-LRB-

v

-RRB-

Modalities

The

proofs

of

the

rules

of

the

modalities

y

and

are

based

on

its

comonad

properties

described

in

the

Lemma

Lemma

Proposition

and

using

the

Lemma

where

-LSB-

-LSB-

yA

-RSB-

-RSB-

y

-LSB-

-LSB-

A

-RSB-

-RSB-

and

-LSB-

-LSB-

A

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

ffl

To

prove

the

rules

yWeak

yDer

yThread

it

is

enough

to

consider

the

maps

-LSB-

-LSB-

A

-RSB-

-RSB-

y

-LSB-

-LSB-

A

-RSB-

-RSB-

I

-LRB-

ffly

-RRB-

-LSB-

-LSB-

A

-RSB-

-RSB-

y

-LSB-

-LSB-

A

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

and

-LSB-

-LSB-

A

-RSB-

-RSB-

y

-LSB-

-LSB-

A

-RSB-

-RSB-

y

-LSB-

-LSB-

A

-RSB-

-RSB-

y

-LSB-

-LSB-

A

-RSB-

-RSB-

ffl

To

prove

the

rules

Contr

and

Ser

it

is

enough

to

consider

the

maps

d

-LSB-

-LSB-

A

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

and

-LSB-

A

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

y

-LSB-

-LSB-

A

-RSB-

-RSB-

ffl

To

prove

the

rule

A

yA

-LRB-

if

is

an

independent

context

with

only

or

y

formulas

-RRB-

suppose

that

there

is

a

morphism

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

First

notice

that

corresponds

to

i

y

i

y

n.

For

any

G-object

A

there

are

the

following

maps

A

-LRB-

ffi

-RRB-

A

A

A

y!A

and

y

A

-LRB-

ffiy

-RRB-

A

y

y

A

So

there

is

the

following

map

-LSB-

-LSB-

-RSB-

-RSB-

y

-LSB-

-LSB-

-RSB-

-RSB-

y

-LSB-

-LSB-

i

-RSB-

-RSB-

y

y

-LSB-

-LSB-

i

-RSB-

-RSB-

y

y

-LSB-

-LSB-

n

-RSB-

-RSB-

Using

-LRB-

a

generalisation

of

-RRB-

the

maps

mA

B

of

Lemma

we

obtain

a

map

y

-LSB-

-LSB-

-RSB-

-RSB-

y

-LSB-

-LSB-

i

-RSB-

-RSB-

y

y

-LSB-

-LSB-

i

-RSB-

-RSB-

y

y

-LSB-

-LSB-

n

-RSB-

-RSB-

y

-LSB-

-LSB-

-RSB-

-RSB-

there

is

a

map

y

-LSB-

-LSB-

-RSB-

-RSB-

y

-LSB-

-LSB-

A

-RSB-

-RSB-

Therefore

there

is

a

map

ffi

-LRB-

ffi

-RRB-

from

-LSB-

-LSB-

-RSB-

-RSB-

to

y

-LSB-

-LSB-

A

-RSB-

-RSB-

-LSB-

-LSB-

yA

-RSB-

-RSB-

ffl

To

prove

the

rule

A

A

-LRB-

if

is

an

independent

context

with

only

formulas

-RRB-

we

first

recall

that

corresponds

to

n

As

we

have

seen

in

the

proposition

above

there

is

a

isomorphism

-LSB-

-LSB-

n

-RSB-

-RSB-

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

n

-RSB-

-RSB-

-LRB-

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

n

-RSB-

-RSB-

-RRB-

H

Given

any

map

H

-LSB-

-LSB-

A

-RSB-

-RSB-

we

can

obtain

a

map

H

ffi

H

-LRB-

f

F

-RRB-

-LSB-

-LSB-

A

-RSB-

-RSB-

A

fragment

of

LLMSc

in

CSP

In

this

section

we

present

an

interpretation

of

a

fragment

of

LLMSc

including

the

operators

tensor

product

cartesian

product

and

the

operator

in

a

formalism

which

gives

an

interesting

characterization

of

computational

processes

so-called

CSP

-LRB-

Communicating

Sequential

Process

-LSB-

-RSB-

-RRB-

In

CSP

the

word

process

stands

for

the

behaviour

pattern

of

objects

This

concept

comprehends

sets

of

kinds

of

events

or

actions

adequately

chosen

in

order

to

give

an

description

of

such

a

behaviour

pattern

The

set

of

events

associated

to

an

object

is

called

its

alphabet

Initially

we

present

intuitively

descriptions

of

some

concepts

which

we

will

use

here

We

follow

the

definitions

given

by

Hoare

-LSB-

-RSB-

ffl

The

alphabet

associated

to

an

object

whose

behaviour

pattern

is

described

by

a

process

P

is

denotated

ffP

ffl

STOPA

is

the

process

which

never

engages

in

any

of

the

events

of

its

alphabet

A.

ffl

-LRB-

x

P

-RRB-

is

the

process

which

first

engages

in

the

event

x

and

so

behaves

as

described

by

P

provided

that

x

ffP

ffl

A

trace

of

the

behaviour

of

a

process

is

a

finite

sequence

of

symbols

recording

the

events

in

which

the

process

has

engaged

up

to

some

moment

in

time

ffl

The

most

important

operation

on

traces

is

catenation

The

catenation

of

a

trace

s

with

a

trace

t

denoted

by

s

t

determines

a

trace

by

simply

putting

s

and

t

together

in

this

order

ffl

A

trace

s

is

a

prefix

of

a

trace

t

-LRB-

s

t

-RRB-

if

s

is

a

copy

of

an

initial

subsequence

of

t

i.e.

there

is

an

extension

u

of

s

such

that

the

sequence

constructed

by

the

events

of

s

followed

by

the

events

of

u

is

the

same

as

t

So

hi

t

for

any

trace

t

where

hi

is

the

empty

trace

ffl

The

sequential

composition

of

the

processes

P

and

Q

-LRB-

P

Q

-RRB-

is

the

process

which

first

behaves

like

P

but

when

P

terminates

sucessfully

it

continues

by

behaving

like

Q.

ffl

The

concurrency

operator

k

concerns

on

the

characterization

of

the

interactions

occurring

between

processes

when

they

run

concurrently

So

when

it

is

applied

to

processes

which

have

the

same

alphabet

it

determines

they

interact

with

each

other

in

a

kind

of

a

lock-step

synchronization

When

the

operator

k

is

applied

to

processes

P

and

Q

which

have

different

alphabets

the

events

that

are

in

both

their

alphabets

require

simultaneous

participation

of

both

P

and

Q

the

events

that

are

in

ffP

but

are

not

in

ffQ

are

no

concern

to

Q

thus

P

may

engage

in

them

independently

of

Q.

Similarly

Q

may

engage

alone

in

events

which

are

in

ffQ

but

are

not

in

ffP

ffl

On

the

other

hand

the

interleaving

operator

jjj

concerns

on

the

characterization

of

the

concurrently

combination

of

processes

with

the

same

alphabet

without

directly

interacting

or

synchronizing

with

each

other

So

each

event

of

a

trace

of

the

interleaving

of

two

processes

is

an

event

which

one

of

them

can

engage

in

it

however

if

both

processes

can

engage

in

a

same

event

in

a

certain

moment

of

time

the

choice

of

which

of

them

engage

in

such

an

event

is

nondeterministic

Each

trace

s

of

the

interleaving

of

two

processes

P

and

Q

can

be

constructed

from

a

trace

of

P

and

a

trace

of

Q

by

mixing

their

events

in

a

nondeterministic

way

So

we

say

that

s

interleaves

and

ffl

We

must

emphasize

that

processes

with

differents

alphabets

can

be

transformed

in

order

to

have

the

same

alphabet

without

change

the

behaviour

patterns

that

they

describe

For

instance

ffP

can

be

extended

by

including

symbols

which

are

only

in

ffQ

and

we

can

obtain

the

process

P

-LRB-

ffQ?ffP

-RRB-

-LRB-

P

k

STOP

-LRB-

ffQ?ffP

-RRB-

-RRB-

which

has

exactly

the

same

traces

of

P

but

ffP

-LRB-

ffQ?ffP

-RRB-

ffP

-LSB-

ffQ

Lets

denote

P

-LRB-

ffQ?ffP

-RRB-

as

PQ

and

Q

-LRB-

ffP?ffQ

-RRB-

as

QP

The

denotation

of

the

formulas

in

general

will

be

given

as

processes

in

CSP

In

order

to

give

an

interpretation

to

the

linear

tensor

product

and

the

operator

we

must

take

into

account

that

the

first

one

is

usually

considered

as

representing

the

absence

of

communication

and

is

more

restrict

than

the

second

one

The

concept

of

communication

of

processes

is

represented

in

CSP

by

means

of

the

concurrency

operator

k

but

two

processes

might

communicate

only

if

their

alphabets

have

a

common

event

So

we

could

obtain

a

denotation

for

the

tensor

product

by

means

of

the

concurrency

operator

applied

to

processes

which

have

disjoint

alphabets

Thus

we

must

transform

the

original

processes

into

processes

with

different

alphabets

This

can

be

done

in

CSP

labelling

each

process

by

a

different

name

such

that

each

event

of

a

labelled

process

is

also

labelled

by

that

name

In

order

to

characterize

perfectly

the

tensor

product

which

drives

two

premisses

into

only

one

we

make

the

labelled

processes

run

concurrently

until

one

of

then

stops

In

CSP

a

process

P

labelled

by

l

is

denoted

by

l

P

and

an

event

of

the

alphabet

of

l

P

is

denoted

by

pairs

l

a

where

a

is

an

event

of

the

alphabet

of

P

So

we

define

the

following

operator

in

order

to

give

a

denotation

for

the

tensor

product

Definition

Let

P

and

Q

be

processes

and

distinct

labels

and

d

a

non-labelled

event

such

that

d

ffP

-LSB-

ffQ

Then

the

operator

V

defined

by

P

V

Q

-LRB-

P

d

STOP

k

Q

d

STOP

-RRB-

Now

lets

recall

that

the

tensor

product

is

more

restrict

than

the

operator

then

it

seems

that

the

denotation

for

the

operator

could

be

given

by

means

of

the

interleaving

operator

Both

of

them

are

commutative

and

associative

moreover

the

interleaving

operator

captures

very

well

the

unrestricted

combination

of

elements

determined

by

the

categorical

interpretation

of

On

the

other

hand

the

cartesian

product

is

usually

considered

as

representing

the

intersection

of

elements

It

will

be

denoted

here

by

the

interaction

of

processes

by

means

of

the

concurrency

operator

regarding

that

we

must

transform

the

original

processes

into

processes

with

the

same

alphabet

We

will

compare

the

denotations

of

the

formulas

by

means

of

a

kind

of

relation

of

inclusion

defined

in

terms

of

traces

of

processes

However

such

a

relation

will

be

little

more

complex

than

would

be

expected

since

the

traces

of

the

denotation

of

the

tensor

product

are

particularly

differents

of

the

others

they

have

labelled

events

So

we

may

have

to

deal

with

a

trace

s

containing

some

events

labelled

by

different

labels

ln

It

will

not

be

a

problem

here

since

we

can

eliminate

those

labels

obtaining

a

labels-free

sequence

denoted

s

lng

like

it

is

recursively

defined

below

Definition

Let

s

be

a

trace

and

L

the

set

of

labels

which

occur

in

s.

-LRB-

i

-RRB-

hi

L

hi

-LRB-

ii

-RRB-

-LRB-

hl

ai

-RRB-

L

-LRB-

hai

-RRB-

L

if

s

hl

ai

-LRB-

iii

-RRB-

-LRB-

hai

-RRB-

L

hai

-LRB-

L

-RRB-

if

a

is

a

non-labelled

event

and

s

hai

Now

we

are

able

to

present

more

strictly

the

interpretation

discussed

above

Definition

Let

LLMS

c

be

a

fragment

of

LLMSc

which

contains

only

the

connectives

and

Let

-LSB-

-LSB-

-RSB-

-RSB-

be

a

mapping

such

that

ffl

-LSB-

-LSB-

A

-RSB-

-RSB-

is

a

process

if

A

is

atomic

ffl

-LSB-

-LSB-

A

B

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

V

-LSB-

-LSB-

B

-RSB-

-RSB-

ffl

-LSB-

-LSB-

A&B

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

-LSB-

-LSB-

B

-RSB-

-RSB-

k

-LSB-

-LSB-

B

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

ffl

-LSB-

-LSB-

A

B

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

-LSB-

-LSB-

B

-RSB-

-RSB-

jjj

-LSB-

-LSB-

B

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

ffl

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

-RSB-

-RSB-

where

the

formula

is

obtained

interpreting

each

occurrence

of

the

operators

and

in

the

context

as

and

respectively

ffl

C

is

interpreted

as

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

C

-RSB-

-RSB-

where

the

relation

P

Q

reads

P

is

subsumed

by

Q

is

defined

by

P

Q

iff

for

each

trace

of

P

there

is

a

trace

of

Q

such

that

where

-LRB-

-RRB-

is

the

set

of

labels

which

occur

in

-LRB-

respectively

-RRB-

We

have

hence

the

following

Theorem

-LRB-

Soundness

-RRB-

The

mapping

-LSB-

-LSB-

-RSB-

-RSB-

preserves

the

sequent

rules

of

LLMS

c.

Proof

For

the

sake

of

simplicity

when

we

will

be

dealing

with

the

relation

instead

of

considering

the

original

traces

of

the

related

denotations

we

will

consider

directly

the

labels-free

sequences

obtained

from

those

as

representing

such

traces

It

does

not

entail

any

lack

of

generality

since

we

can

recover

the

original

traces

in

a

reverse

way

that

we

follow

in

order

to

elimiminate

its

occurrences

of

labels

-LRB-

i

-RRB-

R

Suppose

that

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

and

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

B

-RSB-

-RSB-

Lets

consider

that

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

ffi

-RSB-

-RSB-

and

is

a

trace

of

-LSB-

-LSB-

ffi

-RSB-

-RSB-

We

must

consider

two

cases

-LRB-

a

-RRB-

The

trace

is

obtained

by

a

successful

termination

of

the

process

-LSB-

-LSB-

-RSB-

-RSB-

such

that

it

includes

some

events

which

constitute

a

prefix

of

a

trace

of

-LSB-

-LSB-

ffi

-RSB-

-RSB-

So

let

-LSB-

-LSB-

-RSB-

-RSB-

be

a

trace

of

determined

by

such

a

successful

termination

of

-LSB-

-LSB-

-RSB-

-RSB-

and

let

-LSB-

-LSB-

ffi

-RSB-

-RSB-

be

such

a

trace

of

-LSB-

-LSB-

ffi

-RSB-

-RSB-

Thus

by

hypothesis

there

is

a

trace

of

-LSB-

-LSB-

A

-RSB-

-RSB-

such

that

-LSB-

-LSB-

-RSB-

-RSB-

and

there

is

a

trace

of

-LSB-

-LSB-

B

-RSB-

-RSB-

such

that

-LSB-

-LSB-

ffi

-RSB-

-RSB-

So

let

be

a

sequence

of

events

of

-LRB-

-LSB-

-LSB-

A

B

-RSB-

-RSB-

-RRB-

which

can

be

obtained

from

and

in

such

a

way

that

ffl

ffl

contains

all

events

of

such

that

it

ends

with

the

last

one

and

ffl

includes

only

that

events

of

which

also

are

in

Then

is

a

trace

of

-LSB-

-LSB-

A

B

-RSB-

-RSB-

obtained

by

a

successful

termination

of

-LSB-

-LSB-

A

-RSB-

-RSB-

whose

trace

is

such

that

it

includes

some

events

which

constitute

a

prefix

of

the

trace

of

-LSB-

-LSB-

B

-RSB-

-RSB-

Therefore

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

A

B

-RSB-

-RSB-

-LRB-

b

-RRB-

is

obtained

by

a

successful

termination

of

-LSB-

-LSB-

ffi

-RSB-

-RSB-

such

that

it

includes

some

events

which

constitute

a

prefix

of

a

trace

of

-LSB-

-LSB-

-RSB-

-RSB-

This

case

is

analogous

to

the

previous

one

-LRB-

ii

-RRB-

fiR

Suppose

that

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

and

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

B

-RSB-

-RSB-

Lets

consider

that

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

ffi

-RSB-

-RSB-

and

is

a

trace

of

-LSB-

-LSB-

ffi

-RSB-

-RSB-

interleaves

a

trace

of

-LSB-

-LSB-

-RSB-

-RSB-

and

a

trace

of

-LSB-

-LSB-

ffi

-RSB-

-RSB-

So

let

-LSB-

-LSB-

-RSB-

-RSB-

be

such

a

trace

of

-LSB-

-LSB-

-RSB-

-RSB-

and

-LSB-

-LSB-

ffi

-RSB-

-RSB-

such

a

trace

of

-LSB-

-LSB-

ffi

-RSB-

-RSB-

Thus

by

hypothesis

there

is

a

trace

of

-LSB-

-LSB-

A

-RSB-

-RSB-

such

that

-LSB-

-LSB-

-RSB-

-RSB-

and

there

is

a

trace

of

-LSB-

-LSB-

B

-RSB-

-RSB-

such

that

-LSB-

-LSB-

ffi

-RSB-

-RSB-

Then

let

be

a

sequence

of

events

of

-LSB-

-LSB-

A

B

-RSB-

-RSB-

which

interleaves

and

in

a

such

way

that

So

there

is

a

trace

of

-LSB-

-LSB-

A

B

-RSB-

-RSB-

such

that

Therefore

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

A

B

-RSB-

-RSB-

-LRB-

iii

-RRB-

fiL

and

L

These

cases

are

directly

proved

from

the

semantic

definition

since

-LSB-

-LSB-

-LSB-

A

B

-RSB-

-RSB-

-RSB-

-LSB-

-LSB-

-LSB-

A

B

-RSB-

-RSB-

-RSB-

and

-LSB-

-LSB-

-LSB-

A

B

-RSB-

-RSB-

-RSB-

-LSB-

-LSB-

-LSB-

A

B

-RSB-

-RSB-

-RSB-

-LRB-

iv

-RRB-

R

Suppose

that

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

and

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

B

-RSB-

-RSB-

Let

be

a

trace

of

-LSB-

-LSB-

-RSB-

-RSB-

Thus

by

hypothesis

ffl

there

is

a

trace

of

-LSB-

-LSB-

A

-RSB-

-RSB-

such

that

and

ffl

there

is

a

trace

of

-LSB-

-LSB-

B

-RSB-

-RSB-

such

that

So

as

-LSB-

-LSB-

A&B

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

-LSB-

-LSB-

B

-RSB-

-RSB-

k

-LSB-

-LSB-

B

-RSB-

-RSB-

-LSB-

-LSB-

A

-RSB-

-RSB-

there

is

a

trace

of

-LSB-

-LSB-

A&B

-RSB-

-RSB-

such

that

Therefore

-LSB-

-LSB-

-RSB-

-RSB-

-LSB-

-LSB-

A&B

-RSB-

-RSB-

-LRB-

v

-RRB-

L

and

L

The

proofs

of

these

cases

are

based

on

the

facts

that

every

trace

of

-LSB-

-LSB-

A&B

-RSB-

-RSB-

is

a

prefix

of

a

trace

of

-LSB-

-LSB-

A

-RSB-

-RSB-

and

a

trace

of

-LSB-

-LSB-

B

-RSB-

-RSB-

-LRB-

vi

-RRB-

Structural

Rule

The

main

inferences

enclosed

by

the

structural

are

-LSB-

A

B

-RSB-

C

-LSB-

A

B

-RSB-

C

The

proof

of

this

case

follows

of

the

proof

that

-LSB-

-LSB-

A

B

-RSB-

-RSB-

-LSB-

-LSB-

A

B

-RSB-

-RSB-

-LSB-

-LSB-

A

B

-RSB-

-RSB-

-LSB-

-LSB-

A

B

-RSB-

-RSB-

So

lets

consider

a

trace

of

-LSB-

-LSB-

A

B

-RSB-

-RSB-

We

must

consider

two

cases

-LRB-

a

-RRB-

The

trace

is

obtained

by

a

successful

termination

of

the

process

-LSB-

-LSB-

A

-RSB-

-RSB-

such

that

it

includes

some

events

which

constitute

a

prefix

of

a

trace

of

-LSB-

-LSB-

B

-RSB-

-RSB-

So

let

-LSB-

-LSB-

A

-RSB-

-RSB-

be

a

trace

of

-LSB-

-LSB-

A

-RSB-

-RSB-

determined

by

such

a

successful

termination

of

-LSB-

-LSB-

A

-RSB-

-RSB-

and

let

-LSB-

-LSB-

B

-RSB-

-RSB-

be

such

a

trace

of

-LSB-

-LSB-

B

-RSB-

-RSB-

So

let

be

a

sequence

of

events

which

can

be

obtained

from

-LSB-

-LSB-

A

-RSB-

-RSB-

and

-LSB-

-LSB-

B

-RSB-

-RSB-

in

a

such

way

that

interleaves

-LSB-

-LSB-

A

-RSB-

-RSB-

and

-LSB-

-LSB-

B

-RSB-

-RSB-

and

Then

there

is

a

trace

of

-LSB-

-LSB-

A

B

-RSB-

-RSB-

such

that

Therefore

-LSB-

-LSB-

A

B

-RSB-

-RSB-

-LSB-

-LSB-

A

B

-RSB-

-RSB-

-LRB-

b

-RRB-

is

obtained

by

a

successful

termination

of

-LSB-

-LSB-

B

-RSB-

-RSB-

such

that

it

includes

some

events

which

constitute

a

prefix

of

a

trace

of

-LSB-

-LSB-

A

-RSB-

-RSB-

This

case

is

analogous

to

the

previous

one

-LSB-

B

A

-RSB-

C

-LSB-

A

B

-RSB-

C

-LSB-

B

A

-RSB-

C

-LSB-

A

B

-RSB-

C

The

proofs

of

these

cases

are

based

on

the

facts

that

the

operators

k

and

jjj

are

commutative

-LSB-

-LRB-

A

B

-RRB-

C

-RSB-

D

-LSB-

A

-LRB-

B

C

-RRB-

-RSB-

D

-LSB-

A

-LRB-

B

C

-RRB-

-RSB-

D

-LSB-

-LRB-

A

B

-RRB-

C

-RSB-

D

-LSB-

-LRB-

A

B

-RRB-

C

-RSB-

D

-LSB-

A

-LRB-

B

C

-RRB-

-RSB-

D

-LSB-

A

-LRB-

B

C

-RRB-

-RSB-

D

-LSB-

-LRB-

A

B

-RRB-

C

-RSB-

D

The

proofs

of

these

cases

are

based

on

the

facts

that

the

operators

k

and

jjj

are

associative

-LRB-

-RRB-

-LSB-

-LRB-

A

B

-RRB-

C

-RSB-

D

-LSB-

A

-LRB-

B

C

-RRB-

-RSB-

D

-LRB-

-RRB-

-LSB-

A

-LRB-

B

C

-RRB-

-RSB-

D

-LSB-

-LRB-

A

B

-RRB-

C

-RSB-

D

It

is

enough

to

prove

that

-LSB-

-LSB-

A

-LRB-

B

C

-RRB-

-RSB-

-RSB-

-LSB-

-LSB-

A

-LRB-

B

C

-RRB-

-RSB-

-RSB-

-LSB-

-LSB-

-LRB-

A

B

-RRB-

C

-RSB-

-RSB-

-LSB-

-LSB-

-LRB-

A

B

-RRB-

C

-RSB-

-RSB-

So

lets

consider

a

trace

of

-LSB-

-LSB-

A

-LRB-

B

C

-RRB-

-RSB-

-RSB-

We

must

consider

two

cases

-LRB-

a

-RRB-

The

trace

is

obtained

by

a

successful

termination

of

the

process

-LSB-

-LSB-

A

-RSB-

-RSB-

such

that

it

includes

some

events

which

constitute

a

prefix

of

a

trace

of

-LSB-

-LSB-

B

C

-RSB-

-RSB-

So

let

-LSB-

-LSB-

A

-RSB-

-RSB-

be

a

trace

determined

by

such

a

successful

termination

of

-LSB-

-LSB-

A

-RSB-

-RSB-

and

let

-LSB-

-LSB-

B

C

-RSB-

-RSB-

be

such

a

trace

of

-LSB-

-LSB-

B

C

-RSB-

-RSB-

On

the

other

hand

-LSB-

-LSB-

B

C

-RSB-

-RSB-

interleaves

a

trace

of

-LSB-

-LSB-

B

-RSB-

-RSB-

and

a

trace

of

-LSB-

-LSB-

C

-RSB-

-RSB-

So

let

-LSB-

-LSB-

B

-RSB-

-RSB-

such

a

trace

of

-LSB-

-LSB-

B

-RSB-

-RSB-

and

-LSB-

-LSB-

C

-RSB-

-RSB-

such

a

trace

of

-LSB-

-LSB-

C

-RSB-

-RSB-

Then

let

be

a

sequence

which

interleaves

a

trace

of

-LSB-

-LSB-

A

B

-RSB-

-RSB-

and

-LSB-

-LSB-

C

-RSB-

-RSB-

such

that

and

which

can

be

obtained

from

-LSB-

-LSB-

A

-RSB-

-RSB-

and

-LSB-

-LSB-

B

-RSB-

-RSB-

in

such

a

way

that

ffl

contains

all

events

of

-LSB-

-LSB-

A

-RSB-

-RSB-

such

that

it

ends

with

the

last

one

and

ffl

includes

only

that

events

of

-LSB-

-LSB-

B

-RSB-

-RSB-

which

also

are

in

Then

there

is

a

trace

of

-LSB-

-LSB-

-LRB-

A

B

-RRB-

C

-RSB-

-RSB-

such

that

Therefore

-LSB-

-LSB-

A

-LRB-

B

C

-RRB-

-RSB-

-RSB-

-LSB-

-LSB-

-LRB-

A

B

-RRB-

C

-RSB-

-RSB-

-LRB-

b

-RRB-

is

obtained

by

a

successful

termination

of

-LSB-

-LSB-

B

C

-RSB-

-RSB-

such

that

it

includes

some

events

which

constitute

a

prefix

of

a

trace

of

-LSB-

-LSB-

A

-RSB-

-RSB-

This

case

is

analogous

to

the

previous

one

Analogous

to

the

previous

case

-LSB-

-LRB-

A

C

-RRB-

-LRB-

B

D

-RRB-

-RSB-

E

-LSB-

-LRB-

A

B

-RRB-

-LRB-

C

D

-RRB-

-RSB-

E

It

is

enough

to

prove

that

-LSB-

-LSB-

-LRB-

A

B

-RRB-

-LRB-

C

D

-RRB-

-RSB-

-RSB-

-LSB-

-LSB-

-LRB-

A

B

-RRB-

-LRB-

C

D

-RRB-

-RSB-

-RSB-

-LSB-

-LSB-

-LRB-

A

C

-RRB-

-LRB-

B

D

-RRB-

-RSB-

-RSB-

-LSB-

-LSB-

-LRB-

A

C

-RRB-

-LRB-

B

D

-RRB-

-RSB-

-RSB-

So

lets

consider

a

trace

of

-LSB-

-LSB-

-LRB-

A

B

-RRB-

-LRB-

C

D

-RRB-

-RSB-

-RSB-

We

must

consider

two

cases

-LRB-

a

-RRB-

The

trace

is

obtained

by

a

successful

termination

of

the

process

-LSB-

-LSB-

A

B

-RSB-

-RSB-

such

that

it

includes

some

events

which

constitute

a

prefix

of

a

trace

of

-LSB-

-LSB-

C

D

-RSB-

-RSB-

So

let

-LSB-

-LSB-

A

B

-RSB-

-RSB-

be

a

trace

of

-LSB-

-LSB-

A

B

-RSB-

-RSB-

determined

by

such

a

successful

termination

of

-LSB-

-LSB-

A

B

-RSB-

-RSB-

and

let

-LSB-

-LSB-

C

D

-RSB-

-RSB-

be

such

a

trace

of

-LSB-

-LSB-

C

D

-RSB-

-RSB-

Then

-LSB-

-LSB-

A

B

-RSB-

-RSB-

interleaves

a

trace

of

-LSB-

-LSB-

A

-RSB-

-RSB-

and

a

trace

of

-LSB-

-LSB-

B

-RSB-

-RSB-

and

-LSB-

-LSB-

C

D

-RSB-

-RSB-

interleaves

a

trace

of

-LSB-

-LSB-

C

-RSB-

-RSB-

and

a

trace

of

-LSB-

-LSB-

D

-RSB-

-RSB-

So

let

-LSB-

-LSB-

A

-RSB-

-RSB-

such

a

trace

of

-LSB-

-LSB-

A

-RSB-

-RSB-

and

-LSB-

-LSB-

B

-RSB-

-RSB-

such

a

trace

of

-LSB-

-LSB-

B

-RSB-

-RSB-

let

-LSB-

-LSB-

C

-RSB-

-RSB-

such

a

trace

of

-LSB-

-LSB-

C

-RSB-

-RSB-

and

-LSB-

-LSB-

D

-RSB-

-RSB-

such

a

trace

of

-LSB-

-LSB-

D

-RSB-

-RSB-

Then

let

be

a

sequence

which

interleaves

a

trace

of

-LSB-

-LSB-

A

C

-RSB-

-RSB-

and

a

trace

of

-LSB-

-LSB-

B

D

-RSB-

-RSB-

such

that

where

which

can

be

obtained

from

-LSB-

-LSB-

A

-RSB-

-RSB-

and

-LSB-

-LSB-

C

-RSB-

-RSB-

in

such

a

way

that

ffl

contains

all

events

of

-LSB-

-LSB-

A

-RSB-

-RSB-

such

that

it

ends

with

the

last

one

and

ffl

includes

only

that

events

of

-LSB-

-LSB-

C

-RSB-

-RSB-

which

also

are

in

which

can

be

obtained

from

-LSB-

-LSB-

B

-RSB-

-RSB-

and

-LSB-

-LSB-

D

-RSB-

-RSB-

in

such

a

way

that

ffl

contains

all

events

of

-LSB-

-LSB-

B

-RSB-

-RSB-

such

that

it

ends

with

the

last

one

and

ffl

includes

only

that

events

of

-LSB-

-LSB-

D

-RSB-

-RSB-

which

also

are

in

Then

there

is

a

trace

of

-LSB-

-LSB-

-LRB-

A

C

-RRB-

-LRB-

B

D

-RRB-

-RSB-

-RSB-

such

that

Therefore

-LSB-

-LSB-

-LRB-

A

B

-RRB-

-LRB-

C

D

-RRB-

-RSB-

-RSB-

-LSB-

-LSB-

-LRB-

A

C

-RRB-

-LRB-

B

D

-RRB-

-RSB-

-RSB-

-LRB-

b

-RRB-

is

obtained

by

a

successful

termination

of

-LSB-

-LSB-

C

D

-RSB-

-RSB-

such

that

it

includes

some

events

which

constitute

a

prefix

of

a

trace

of

-LSB-

-LSB-

A

B

-RSB-

-RSB-

This

case

is

analogous

to

the

previous

one

Conclusions

The

motivation

for

this

work

came

from

O'Hearn

's

and

Reddy

's

use

of

Linear

Logic

to

deal

with

the

semantics

of

Algol-like

programming

languages

Earlier

on

we

realised

that

we

could

use

Reddy

's

ideas

to

give

a

syntactical

characterization

to

some

of

the

-LRB-

somewhat

mysterious

-RRB-

constructions

of

de

Paiva

's

dialectica

categorical

models

GC

for

Classical

Linear

Logic

In

particular

it

was

interesting

to

find

an

intuitive

explanation

for

the

comonad

y

in

terms

of

the

extra

tensor

product

Summing

up

the

work

so

far

we

adapted

the

system

LLMS

providing

it

with

a

commutative

version

of

the

connective

before

and

an

associated

modality

A

dialectica

category

G

was

constructed

on

Sets

and

shown

to

be

a

sound

model

of

LLMSc

Furthermore

we

gave

a

computational

characterization

of

the

operator

by

means

of

the

interleave

operator

of

CSP

while

the

tensor

product

and

the

connective

were

interpreted

as

restricted

assynchronous

concurrency

and

the

interaction

of

processes

respectively

But

much

work

remains

to

be

done

Firstly

going

back

to

our

original

motivation

more

work

is

needed

to

relate

the

system

LLMSc

to

O'Hearn

's

and

Reddy

's

recent

work

-LSB-

-RSB-

Secondly

from

a

logical

pespective

we

might

want

to

use

this

system

with

two

modalities

as

a

blueprint

for

work

on

relating

modalities

Thirdly

the

interpretation

of

a

fragment

of

LLMSc

in

terms

of

CSP

operators

is

useful

as

it

gives

a

very

tangible

explanation

of

our

plethora

of

tensor

products

But

it

would

be

more

convincing

if

we

could

also

explain

the

modalities

in

CSP

terms

The

domain-theoretical

interpretation

of

the

addressed

fragment

of

LLMSc

can

be

built

by

using

the

standard

event

structure

based

semantics

for

CSP

Extending

this

to

the

whole

of

LLMSc

is

also

a

topic

for

future

work

References

-LSB-

-RSB-

J.

Beck

Distributive

laws

In

Seminar

on

Triples

and

Categorical

Homoly

Theory

LNM

pages

-LCB-

Springer-Verlag

-LSB-

-RSB-

M.

da

S.

Corr

ea

E.

H.

Haeusler

and

V.

C.

V.

de

Paiva

A

dialectica

model

of

state

Technical

Report

PUC-Rio

Dept.

de

Inform?atica

Pontif

cia

Univ.

Cat?olica

PUC-Rio

August

-LSB-

-RSB-

V.

C.

V

de

Paiva

Dialectica

categories

In

Categories

in

Computer

Science

and

Logic

Boulder

Colorado

June

American

Mathematical

Society

-LRB-

Contemporary

Mathematics

Vol

-RRB-

-LSB-

-RSB-

V.

C.

V

de

Paiva

Dialectica

Categories

PhD

thesis

Lucy

Cavendish

College

Cambridge

November

Printed

as

Technical

Report

No.

University

of

Cambridge

Computer

Laboratory

January

-LSB-

-RSB-

V.

C.

V.

de

Paiva

A

dialectica-like

model

of

linear

logic

In

Category

Theory

in

Computer

Science

volume

of

LNCS

pages

-LCB-

September

-LSB-

-RSB-

V.

C.

V.

de

Paiva

A

category

of

multirelations

Technical

Report

University

of

Cambridge

Computer

Laboratory

April

-LSB-

-RSB-

V.

C.

V.

de

Paiva

and

J.

M.

E.

Hyland

Generalising

dialectica

categories

-LRB-

Revised

version

of

a

talk

in

the

Montreal

Category

Theory

Meeting

-RRB-

-LSB-

-RSB-

J.-Y

Girard

Towards

a

geometry

of

interaction

In

Categories

in

Computer

Science

and

Logic

pages

-LCB-

Boulder

Colorado

June

American

Mathematical

Society

-LRB-

Contemporary

Mathematics

Vol

-RRB-

-LSB-

-RSB-

C.

A.

R.

Hoare

Communicating

Sequential

Processes

Prentice-Hall

International

London

-LSB-

-RSB-

Y.

Lafont

The

linear

abstract

machine

Theoretical

Computer

Science

-LCB-

-LSB-

-RSB-

P.

W.

O'Hearn

Linear

logic

and

interference

control

In

Categories

in

Computer

Science

and

Logic

volume

of

LNCS

pages

-LCB-

Paris

France

September

Springer-Verlag

Berlin

-LSB-

-RSB-

P.

W.

O'Hearn

A.

J.

Power

M.

Takeyama

and

R.

D.

Tennent

Syntactic

control

of

interference

revisited

In

MFPS

XI

volume

of

Eletronic

Notes

in

Theoretical

Computer

Science

New

Orleans

March

-LSB-

-RSB-

U.

S.

Reddy

A

linear

logic

model

of

state

Eletronic

manuscript

-LRB-

ftp

-RRB-

csc.uiuc.edu

directory

pub/reddy/papers

October

-LSB-

-RSB-

U.

S.

Reddy

Passivity

and

independence

In

Annual

IEEE

Symposium

on

Logic

in

Computer

Science

pages

-LCB-

Paris

France

IEEE

Computer

Society

Press

Los

Alamitos

California

-LSB-

-RSB-

P.

Wadler

Linear

types

can

change

the

world

In

Programming

Concepts

and

Methods

Amsterdam

North-Holland

-LRB-

Proc

IFIP

TC

Working

Conf

Sea

of

Galilee

Israel

-RRB-

-LSB-

-RSB-

P.

Wadler

Is

there

a

use

for

linear

logic

In

Proc

Symp

on

Partial

Evaluation

and

Semantics-Based

Program

Manipulation

ACM

pages

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SIGPLAN

Notices

Sep.

-RRB-

categorical models and concurrency
proof theory