Delay-insensitive directed trace structures satisfy the foam rubber wrapper postulate

Delay-insensitive directed trace structures satisfy the foam

rubber wrapper postulate

Citation for published version (APA):

Verhoeff, T., & Schols, H. M. J. L. (1985). Delay-insensitive directed trace structures satisfy the foam rubber wrapper postulate. (Computing science notes; Vol. 8504). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1985

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.... =

ARD

81

CSN

Eindhoven University of Technology

Department of Mathematics and Computing Science Computing Science Section

Computing Science Notes

I

belay-insensitive Directed Trace Structures

I

Satisfy the Foam Rubber Wrapper Postulate

! , by Tom Verhoeff Huub M.J.L. Schols 85/04 l'j.'

1

I

7

COMPUTING SCIENCE NOTES

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Eindhoven University of Technology

Department of Mathematics and Computing Science P.O. Box 513

5600 MB EINDHOVEN The Netherlands All rights reserved editor: M.L. Potters

1

Delay-insensitive Directed Trace Structures Satisfy the

Foam Rubber Wrapper Postulate

o

Abstract

In [JTU] Udding defines C4• the class of delay-insensitive directed trace

struc-tures. Schols defines the foam rubber wrapper postulate in [HS]. This postulate is a formalization of the foam rubber wrapper principle defined by Molnar. Fang. and Rosenberger in [MFR]. In this paper we prove that a directed trace stucture that is a C4 • satisfies the foam rubber wrapper postulate and has absence of

danger of transmission inference (the reverse is proven in [HS]). Furthermore we show that absence of danger of transmission interference. which is explicitly required in the definition of C4• is superft.uous in order to prove that a directed

1

D

-2-1 Notations

We explain the notation. that we use for variable-binding constructs. Universal quantification is denoted by

(Al:D :E)

where A is the quantifier. l denotes a list of bound variables, D denotes a predi-cate. and E denotes the quantified expression. D and E contain in general -variables from l. D indicates the domain of the bound variables. E is quantified for variable values that satisfy D. Existential quantification is denoted analo-gously. using quantifier E instead of A. By

P:D:E~

we denote the set of all values of E obtained by substituting for all variables in l

values that satisfy D. By

(Sl :D:A)

we denote the sum of all elements of

II :

D : A

J,

where A denotes the quantified arithmetic expression. In all notations the domain D is omitted when obvious from the context.

For expressions E and G an expression of the form E =::> G is often proved in a number of steps by the introduction of intermediate expressions. For instance. we can prove E

=>

G by proving E

=

F and F

=>

G for some expression F. In

order to prevent that the reader has to perform a string comparison to establish the (for the argument essential) sameness of the two occurrences of F. we represent proofs like this as follows

E

=

l

hint why E = F

J

F

=>

l

hint why F=> G

J

G

1 I ~ P 2 Trace Theory 2.0 Introduction

We present an introduction to trace theory. which is sufficient for our purposes.

An extended description can be found in [MR]. [RSU]. and [JvdS]. 2.0.0 Traces and directed trace structures

An alphabet is a finite set of symbols. For an alphabet A, A* denotes the set of all

finite-length sequences of elements of A. including the empty sequence. which is denoted by E. A trace is a finite-length sequence of symbols. A directed trace structure T is a triple <iTloT,tT

>.

where iT is the input alphabet of T, oT is tbe output alphabet of T. and tT is the trace set of T. iT and oT are disjoint. We denote iTuoT by aT. the alphabet of T. tT is a subset of (aT)*. Elements of iT

are called input symbols of T. or symbols of type input. Elements of oT are called output symbols of T. or symbols of type output. Elements of tT are called traces of T.

Note

Unless stated otherwise. small and capital letters near the end of the Latin alphabet denote traces and directed trace structures respectively. Small and capital letters near the beginning of the Latin alphabet denote symbols and alphabets respectively.

end of note

2.0.1 Directed traces and partially directed traces

We may postfix symbols with an exclamation point or a question mark. Symbols

a. aI, and a? are three distinct symbols. For alphabet A. A! denotes the set

~ a: ae:A: al ~ and A? denotes ~ a: a e:A: a?~. A. AI. and A? are three disjoint

sets. Elements of (A!uA?)* are called directed traces and elements of (AuAIUA? )* are called partially directed traces. Elements of A* are referred to

as traces. Notice that all traces and all directed traces are partially directed traces. For directed trace structure T, T!? denotes {oT )!U (iT)?

i

n

-4-2.0.2 Operations

Definition 2.0 (prefix)

For partially directed trace

t

the set of all prefixes of

t.

denoted by pref

(t ).

is the trace set

~ u.w ; t

=

uw; u

I

end of definition

(Concatenation is denoted by juxtaposition). We extend this operator to trace sets.

Definition 2.1 (prefix-closure)

For trace set T the prefix-closure of T. denoted by pref ( T). is the trace set ~

t .

u ;

t

E: T /\ u E: pref (

t ) ;

u ~

end of definition

We denote the length of a partially directed trace

t

by

I(

t ) :

Definition 2.2 (length)

(i) I( £)

=

0

(ii) for partially directed trace

t

and symbol a. I (

ta. )

=

1 (

t )

+

1

end of definition

The projection of a partially directed trace

t

on an alphabet A is denoted by

t ~A:

Definition 2.3 (projection) For alphabet A

(i) E ~A

=

E

(U) for partially directed trace

t

and symbol a.-.such that a. E:A.

(ta. )

~ A

=

(t

~ A )a.

h

(iii) for partially directed trace

t

and symbol a. such that a ~ A.

(ta

HA

=t ~A.

end of definition

For a partially directed trace t and a symbol

a

we denote 1 (t ~la

D

by

#11.

t. We define the function direct, denoted by dir. which maps partially directed traces on partially directed traces:

Definition 2.4 (direct)

For alphabet A and partially directed trace u we define dir(A,u) recursively:

(i) dir{A,e)

=

e

(li) for partially directed trace

t

and symbol a, such that a E: A,

dir(A,ta)

=

dir(A,t )a!a?

(Ui) for partially directed trace

t

and symbol a, such that a~A,

dir(A,ta )

=

dir(A.t )a

end of definition 2.0.3 Undirecting

Definition 2.5 (immediate undirect)

For partially directed traces t and u,

t

is an immediate undirect of u, denoted by tiundirectu, iff (Ea,x,Yo,Yt,Z :(YoYt>~la? ~=t; : ( (t

=

x Yo a Y 1 Z ) 1\ ( u

=

x a! Yo Y 1 a? Z ) ) V ( ( t

=

x Yo a Y

d

1\ ( U

=

x a! Yo Y 1 ) ) ) end of definition

The reflexive and transitive closure of immediate undired is a partial order called undirect. We denote this partial order among partially directed traces

t

b

-6-2.1 Notions related to the FRlf-postulate

The notions introduced in this section have been adopted from [HS]. Note

In the remainder of this section S and T denote directed trace structures, such that is equals oT and oS equals iT.

end of note

Definition 2.6 (absence of deadlock)

Traces sand

t,

such that s E: tS and

t

E:

tT,

have absence of deadlock. denoted by

s no deadlock

t .

iff

end of definition

Traces s of S and

t

of T have absence of deadlock if and only if for all symbols a

in iT. which equals oS, and b in is. which equals oTt and for any natural numbers i and j, such that l~i~#(Jt and l~j~#bs, in s the j-th input of b is preceded by the i-th output of a or in

t

the i-th input of a is preceded by the

j-th output of b.

Definition 2.7 (composable)

Traces s and

t.

such that s E: tS and t E: tT, are composable. denoted by c (s.t ).

i.f!

end of definition

A trace of S and a trace of T can be seen as observations of the same communi-cation. if each input occurring in one of them, occurs in the other as output and they have absence of deadlock. Such traces we call composable.

,

7

Definition 2.8 (directed resultant)

For a directed trace x and traces s and t, such that SEtS, t E tT, and c (s ,t ). x is a directed resultant of s and t. denoted by x dres (s . t ). iff

«x

=t)l\(s =t)/\(t=t»

V (Ea.so.xo: (s =soa )I\xodres(so,t ): (aE oS ~ (x = xoa!» 1\( aEiS ~(x =xoa?

»)

V (Ea,to.xo:

(t

=

toa )I\xodres(s .to): (aE oT ~ (x

=

xoaq) 1\( aEiT ~ (x =xoa?

»)

end of definition

In a directed resultant x we use a! or a? to indicate that this occurrence of a in

x originates from the output alphabet of a directed trace structure or the input

alphabet of a directed trace structure respectively. Notice that for each pair of composable traces a set of directed resultants is defined.

Property 2.0

For directed traces x and y and composable traces s and

t.

such that (ry )dres( s.t ). and for symbol a

end of property

Each input in a directed resultant is preceded by its corresponding output.

Definition 2.9 (resultant)

For traces s, t, and z. such that sEtS, tE tT. Z E( as'vaT)*, and c(s ,t ).

z is a resultant of sand

t.

denoted by z res (

s.t ).

iff

(Ex: x dres ( s •

t ):

Z undirectx )

end of definition

Notice that since the alphabets of S and T are equal. as'vaT equals as' and aT.

For .each pair of composable traces a set of resultants is defined. A resultant is a minimal element with respect to the partial order undirect , since a resultant is an element of ( as'v aT) *,

DE

-8-Definition 2.10 (composite)

The composite of two directed trace structures Sand T, such that

is

= oT and oS

=

iT, which is denoted by S @T, is the directed trace structure

<iSn oT,oSniT,prer

U

S ,t ,Z : sEtS At E

tT

Ac(

s,t

)I\zres(s ,t ): z

n>

end of definition

For a directed trace structure T, <iT,oT,tT>,

T

denotes its complement. i.e.

<oT,iT,tT>. Property 2.1 (1')

=

T end of property Property 2.2 tTct(T@T) end of property

Definition 2.11 (foam rubber wrapper postulate)

A directed trace structure T does justice to the foam rubber wrapper principle, it!

T=T@T

end of definition

For an explanation of our notion of the foam rubber wrapper principle we refer to

[HS]

and [MFR]. Our notion "foam rubber wrapper principle" equals the notion "FRW-postulate", that Molnar, Fang, and Rosenberger use in [MFR].

I

I

l

I

f:

?

2.2 Notions related to C₄

The notions introduced in this section have been adopted from [JTU). C. is the class of delay-insensitive directed trace structures.

Definition 2.12 (C_{4 )}

A directed trace structure T is an element of C₄ if it satisfies the requirements Ro through R5:

(Ro) iTu oT=aT

(RI ) iT is prefix-closed and nonempty

(R2 ) for trace s and symbol a E: aT

saa ~ tT

(Ra) for traces s and

t.

and for symbols a E: aT and bE: aT of the same type ( sabt € tT )

= (

sbat E: t T )

(R4) for traces s and

t.

and for symbols aE: aT. bE: aT. and e € aT with b of another type than a and e

(sabte E: tT I\sbat € tT) ~ (sbate E: tT)

( R5 ) for trace s and symbols a E: aT and b E: aT of ditTerent types

(sa E: tT I\sb E: tT) ~ (sab E: tT) end of definition

Definition 2.13 (from)

For directed trace structure T. and for composable traces

t

€ tT and U E: tr we define from

(t

,u ) as

f

x : x E: { 0 Tn iT)·1\ (Aa : a E: 0 Tn iT: #~ x

=

#a

t -

#Il

U ) : x ~ end of definition

Since from(t,u) is nonempty and the lengths of the traces in from(t,u) are equal, we define I ( from (

t

,U ) ) as the length of the traces in from (

t .

u ).

Definition 2.14 (mismatches)

For directed trace structure T, and for composable traces

t

€ tT and u € tr. we define

mm(t ,u)

=

l{from(

t,u»+

l(from{u,t}) end of definition

t

10

-Property 2.3

For directed trace structure T. traces to. t l' uo. and u1• such that toE: tT,

tlE:tT. uoe:tf. uIE:tf. c(to.uo}. and

C(t 1

.Ul). and directed traces x and y.

such that x dres (t o. uo) and y <ires (

t

1-u 1 ).

(Aa: ae:( T!? u T!? ) :#ISX =#ISY) ~ (mm( to.uo)

=

mm( t 1.u1

»

end of property

Udding uses a definition of composability of traces that differs from our definition. cf. [JTU]. In [HS) is proven that these definitions are equivalent.

I

I

I

_b

11

-3 Directed trace structures that satisfy the foam rubber wrapper postulate

Determining whether a directed trace structure satisfies the foam rubber wrapper postulate. comes to checking whether the resultants of its traces are elements of its trace set. Resultants are obtained by applying directed resultant and immediate undirect operators. In section 3.0 some tools are presented which are used in lhe subsequent sections. We deal with directed resultant and immediate undirecl operations in section 3.1. In section 3.2 mathematical induction on the number of immediate undirect operations is applied. Theorems 3.0 and 3.1 in section 3.3 are conclusions drawn from the results of the previous sections.

Note

For the remainder of this chapter U denotes a directed trace structure that satisfies Ro. R1• Rs. R4 • and R:5.

end of note 3.0 Preparation

In order to indicate the type of symbols with respect to a directed trace struc-ture we introduce the notion postfix type. Notice that for a symbol a. a! and a?

are symbols. not concatenations of symbols and an exclamation point or a ques-tion mark respectively.

Definition 3.0 (postfix type)

For directed trace structure T and trace

t.

such that

t

E: tT. the trace denoted by postf( T.t). in which the symbols in tare postftxed by their type with respect to T. is defined by:

(i) postf (T.E) = E

(ii) for trace u and symbol a. such that uaE: tTand aE: oT

postf (T.ua) = postf (T,u )a!

(iii) for trace u and symbol a, such that uaE:tT and aE: iT

postf( T,ua)

=

postf (T,u )a?

end of definition

b

12

-Property 3.0 is derived from the definition of directed resultant, composability, and postfix type.

Property 3.0

For directed trace structure T , composable traces t and u, such that t E tT and uEtr, and directed trace x, such that XE{ T!? IJ T!? )*,

xdres{t ,u)

=

«x ~T!?

=

posU( T,t» /\(x ~t!?

=

postl( r,u»

1\ ( Aa, y :y E pref ( x );

#a

,y ~ #a.? y )

) end of property

In lemma 3.0 we deal with absence of danger of computation interference, Le. all symbols on their way between composable traces can be received. For a definition of these notions we refer to [JTU]. Our notion "absence of danger of computation interference" equals the notion "absence of computation interfer-ence" that Udding uses in [JTU].

Lemma 3.0

For directed trace structure T, _{that satisfies R o, R}1, R3 , R4 , and R:;, and for

compos able traces

t

and u, such that tEtT and uEtr,

(Aa: aEoT

niT /\(

#at >#a.u) :ua Etf) Proof

This is a theorem which is proved by Udding [JTU, pA5 and ppA9-58]; T and

T

satisfy aU conditions of connectabie directed trace structures except Hz; Udding does not use R2 to prove that theorem.

end of lemma

Note

Udding refers to R 2, [JTU, p.50]. He uses R2 to prove absence of (danger of) transmission interference only, not to prove absence of danger of computation interference.

I

_\

I

b

3.1 Interchanging adjacent symbols in directed resultants

Lemmata 3.1 and 3.2 deal with interchanging adjacent symbols in directed resul-tants. In order to derive them we present some properties and lemmata. Pro-perties 3.1. 0 and 3.1.1 are derived from property 3.0. Property 3.1.2 is derived from property 3.0 and the definitions of composability, directed resultant, and postfix type.

The occurrences of the symbols interchanged in properties 3.1.0 and 3.1.1 ori-ginate from distinct directed trace structures.

Property 3.1.0

For traces t and u, such that te:tU, ue:tV, and c(t,u), directed traces x and

y, and symbols a and b of distinct types (i) (xa!b !y )dres (t ,u )

=

(xb !a!y )dres (t ,u )

(it) (xa?b?y )dres( t ,u )

=

(xb?a?y )dres(

t

,u)

end of property Property 3.1.1

Fortracest andu, such that te:tU,ue:tV , andc{t,u), directed traces x and

y, and distinct symbols a and b of the same type

(xa!b?y )dres(

t

,u)

=

(xb?a!y )dres(

t

,u)

end of property

Property 3.1.2 expresses that prefixes of directed resultants of composable traces be directed resultants of composable prefixes of those traces ..

Property 3.1.2

For traces t and u, such that te:tU, ue:tV, and c(t,u), and directed trace x,

such that xdres{

t

,u),

(Axo:xoe:pref{x)

)

: (Eto,uo: toe:pref( t )/\uo e:pref( u)

)

/\(xo~U!? =postf( U,to»/\{xo~V!? =postf( U,uo»

: c{ to,uo) I\xodres{ to,uo)

w

l

-

14-In lemmata 3.1.0 and 3.1.1 we deal with interchanging occurrences. that ori-ginate from the same directed trace structure. of symbols of the same type. In lemma 3.1.0 we treat occurrences. that originate from the directed trace struc-ture in which the symbols are outputs.

Lemma 3.1.0

For directed traces x and y. such that xy E ( U!? u U!? )*. and symbols a and b

of the same type

(Eto.Ua: toE tU /\UoE tV /\ c (ta.uo): (xa!b!y )dres (to.uo»

=

(Et.u:tEtV/\uEtU/\c(t.u):(xb!a!Y)dres(t.u»

Proof

Given directed traces x and y. such that xy E ( V!? u V!? )*, and symbols a and

b, such that aEoU and bEOU. We derive:

(Eta,uo: toE tU /\UoE tU /\c( to.uo): (xa!b!y )dres( to.uo»

=

l

property 3.0. definition of directed resultant. and calculus

J

(Eto,uo: toE tU /\UoEtU

)

: «xa!b!y HU!?

=

postr ( U,fo» I\«xa!b!y) ~ U!?

=

postr (U,uo»

I\{Ac ,Z : Z E pref (xa!b!y): #c!z ~#c? Z )

=

l

calculus, a!E(oU)!, and b!E(OV)! ~

{Eto.t ltt2.uo: toE tU /\UoE

tV /\

(to

=

t₁abt_{2 ) / \} (x ~ U!?

=

postr ( U,td)

)

: «xa!b!y) ~U!?

=

posU( V,t1abt2 ) )I\«xy} ~ V!?

=

postl( V.uo»

I\(Ac,z: zEpref(xb!a!y) :#c!Z ~#c?Z)

=

~ U satisfies Rs. a!E(OU)!, b!E(OU}!. definition of composability, and definition of postfix type

J

(Et_l,t₂,UO:(t₁bat₂)EtU /\UoEtV /\(x ~V!?

=

postf ( U.tl

»

)

: «xb !a!y nUl?

=

postr ( U,t Ibat₂

»

1\( (xb !a!y HU!?

=

postf ( Cl.uo))

I\(Ac ,Z: zEpref(xb!a!y): #c!z '??#c?z )

=

l

calculus. property 3.0. and definition of directed resultant ~

(Et ,u: tE tU /\UEtU I\c(

t

,u ): (xb !a!y )dres(

t

,'ll»

For: symbols a en b, suctJ. that a~oV andb_E~V, ~he proof ~sanalogo~s.

I

1

1

I

_h

Lemma 3.1.1 is the counterpart of lemma 3.1.0: occurrences, that originate from the directed trace structure in which the symbols are input symbols, are treated.

Lemma 3.1.1

For directed traces x and y, such that :ryE ( U!? u U!? )*, and symbols a and b

of the same type

(Eto,uo: toEtU l\uoE tU Ac( to,uo): (xa?b?y )dres(to,uo»

=

(Et ,u: tE tU I\UE tU /\c( t ,u): (xb?a?y )dres( t ,u)

The proof of this lemma is analogous to the proof of lemma 3.1.0.

end of lemma

Lemma 3.1.2 deals with interchanging occurrences, that originate from the same directed trace structure, of symbols of distinct types in one way: output backward and input forward.

Lemma 3.1.2

For directed traces x and y, such that :ryE ( U!? u U!? )*, and symbols a and b

of distinct types

(Eto.uo: toEtU l\uoEtU /\c( to,uo): (xa!b?y )dres( to,uo»

=>

(E t ,u : t E t U 1\ u E t U A c ( t ,u ) : ( xb? a! y ) dres ( t ,u ) )

Proof

We prove this lemma by mathematical induction on the length of y. Given directed traces x and y. such that :ryE ( U!? u U!? )*, and symbols a and b,

such that aEOU and bEiU.

Induction hypothesis

\ ..

<;$;~?'

;.::.,: .

.-(~9~.;:l"( Yo)

<

1 ( y

t:.

i>;"'I" " ,0

. ~·~.:'l~~~'" 4';"'~' ~. 'IIo~ ... '~' • • ~~ n .. -; ~:

),(~to,Uo:

toEtu

/\~~'EtU

Ac(

to,uo)

:(;~!b?yo)dres(

tci,Uo»

~:~T:t$.':yC~:t,~

: t E tU /\UE:

tV

I\c

(t

,u):

(x~?a!yo

)dres(

t

,U ) t: )

Base : I ( Y )

=

0 We derive:

16

-(Eto,Uo: toE tU /\UoE tU /\ c( te.Ue): (xa!b?y )dres( te.Ue»

=

~ Y

=

t , since I( y)

=

0 ~

(Eto,uo: teE tU /\UoE tU /\c (te.Uo): (xa!b? )dres( te.uo»

=

~ calculus and property 3.0

i

(Eto.uo: teE tU /\UeE tU /\c( te.ue)

)

: «xa!b? ) ~U!?

=

postl( U,t c

»

I\«xa!b? HU!?

=

postf ( U,Uc» I\(xa!b? )dres(tc.uo)

=

~ calculus, a! E (oU )!, b? E (iU)? , and definition of composability

i

(Etc,t I,UC: tcE tU 1\( to = t lab) /\UcEtU I\c (t lab ,Uc)

)

: (x ~ U!?

=

postl( _{U,t l ) )/\(x} ~U!?

=

postl( D,uc»

I\(xa!b? )dres(tlab,uc)

=>

~ property 3.1.2 and calculus ~

(Et l,UC:

(t

lab )EtU /\UeEtD I\c(t lab ,Uc): c(

t

l'UC) I\xdres (tl.uc»

. =>

~ definition of composability. b EiU • and tU is prefix closed

l

(Et loue: (t la)E tU l\ucE tV: Db tl <#bueI\C (t IoUC) 1\ xdres(

t

I.UC»

=

~ lemma 3.0 using property 2.1, and calculus

i

(Et I,UO: (t la)E tU l\ucEtV: Db t I <#bueI\C (t 1.Ue) 1\( t lb )E tU I\x dres ( t l,ue»

=

~ b EiU • definition of composability, and definition of directed resultant

J

(Etl,Ue: (t la.)E tU l\ueE: tV: c(t lb ,uc) 1\(

t

lb )EtU I\(xb? )dres(

t

lb ,uc»

=>

~ U _{satisfies R o,} a E oU, bE i U, definition of compos ability,

and definition of directed resultant using property 2.0

(Etl,uc :ucE tU: c(t lba,uc) 1\ (t lba)e: tU 1\ (xb?a! )dres

(t

lba,ue»

=>

l

Y = t and calculus J

(Et ,U:

t

EtU l\uE tU 1\ c(

t

,u): (xb?a!y )dres( t,u»

&ep: I(y»O

We distinguish four cases: (0) (Eyc,c :CEOV:y =ycc!) (1) (EYe,c :CEiV:y=ycc?) (2) (Eyo,c :_CEOU:y=Yec!)

l

Case (0): {Eyo,c : c E:oV:y =Yoc!}

(Eto,uo: toE: tU l\ueE:

tV

1\ c( to,uo): (xa.!b?y )dres( to,uo» I\(Eyo,c :CE:oV:y =Yoc!)

=

~ calculus and property 3.0 ~ (Ec ,to,uo.Yo

)

: c E: oV 1\ toE: tU l\uoE: tV 1\ (y =Yoc! )

: (xa.!b?yoc! )dres( to,uo) I\c (to,uo)

A( (xa.!b?yoc!) tU!?

=

posU ( _{U,t o}» A( (xa.!b?yoc!) ~ V!?

=

postf ( V,uo» .

=

~ calculus J

(Ec ,to.tLO,UI'YO

)

: c E: oU AtoE: tU AUoE:tU 1\( uo=u IC ) A (y

=

Yoc!)

: (xa.!b?yoc ! )dres( tO,UI c ) A c (t O. u_{l C )}

A( (xa.!b?yo)t V!?

=

posU( V,t_o

»

A( (xa.!b?yo)

f

U!?

=

posU (U,u I»

=>

~ property 3.1. 2 and calculus J

(Ec .to,uI'YO: c E: oU I\toEtU A( ulC )E: tV I\(y =Yoc!)

: (xa.!b?yo)dres( to,udA( (xa.!b?yo) HI!?

=

posU ( V,ud) )

=>

~ induction hypothesis

J

(Ec ,t l,uI,u2,YO:

c

E

oU

AtlE: tU /\( ulC )E: tU l\u2E:

tV

A (y =Yoc!)

: (Zb?C1!Yo )dres (tloU2) A«xa.lb?yo) ~Vl?

=

postf( V,ud) )

=

~ property 3.0 and calculus

J

(Ec ,t IluI,u2,YO: C E

oU

At IE: tU /\ (UlC )E tV I\U2E: tU A (y =yoc I)

)

: (Zb?C1!yo)dres( t1,ua)A«xb?rzlyo) ~V!?

=

posU(

V,u2»

A«xa.!b?YoHU!?

=

posU ( V,UI»

=>

~ calculus, definition of postfix type, and (b?a.! ) ~ V!?

=

e ~

(Ec ,t I,UI,U2'YO: C E

oU

At IE: tu /\ (UlC )E tV _{l\u 2E:} tV A (y =Yoc!) : (xb?a.!yo)dres( tloUa) A( UI =U2)

)

=

~ calculus, definition of composability, and definition of directed resultant

J

(Ec

,t

I,UloYO: c E: oU At I EtU 1\( U IC }E:

tV /\

(y

=

yoc!) :C(tI,tL1,C )I\(zb?rz!yoc !)dres(t1.u1c)

)

=>

~ calculus

J

( E t ,tL : t E: t U 1\ U E: t U 1\ c (

t .

tL ) : ( zb? a.!y ) dres (

t ,U ) )

end of case

18

-Case (1) : (Eyo,c : c EiV: y =Yoc? )

(EtO'UO: toE tU /\UOE tU /\c( to,UO): (xa!b'?y )dres (to,UO» /\(EC,YO:CEiU:y =yOC?)

:: l

calculus and property 3.0 j

(Ec ,to,uo,yo: C EiU /\toE tV /\UoE tV /\( Y =yoc? )/\c( to,uo)

)

: (xa!b?yoc? )dres(to,uo)/\«xa!b?yoc? )~U!? =posU(V,to»

/\( (xa!b?yoc? ) ~ V!?

=

postf ( V,uo»

:: f

calculus and property 2.0 j

(Ec ,to,UO,UI,YO

)

:cEiU/\toEtU/\UoEtV/\(UO=UIC )/\(y::yoc?)

: #,A xa!b?yoc? )~#c? (xa!b?yoc? ) /\ (xa!b?yoc? )dres( to,u IC ) /\ c( to'u1c ) /\( (xa!b?yo) ~U!?

=

posU ( v,t o) )/\((xa!b?yo) ~ V!?

=

posU (V,UI»

=>

l

property 3.1.2 and calculus j

(Ec ,ta,UI,YO: c EiU /\taE tU /\( Ulc )EtV /\( y =yoc? )

)

: #cA

xa!b?Ya) >#c? (xa!b?yo) /\(xa!b?yo )dres(ta,ul )

/\({xa!b?yo) tV!? :: postf( V,ud)

=>

f

induction hypothesis and calculus j

(EC,t₁,u₁,U2,YO: C EiV /\tlEtU /\(u}c )EtU /\U2EtV /\(y =Yac? )

)

: #e

!(xb?a!yo)

>

#e? (xb?a!yo)/\ (xb?a!Ya )dres( tl,U2) 1\( (xa!b?yo)

H7!?

=

posU (D,uI

»

:: f

property 3.0 and calculus j (Ec ,t },uI,u2,yo

:cEiU/\t1E:tU/\(UIC )EtV/\U2EtV /\(y=Ya c?) : #e!(xb?a!yo)

>

#e? (xb?a!yo) /\ (xb?a!yo )dres( t l'UZ)

/\( (xb?a!yo) ~V!? :: posU( V,U2) )/\( (xa!b?yo) ~V!?

=

posU (V,ud)

)

=>

l

calculus, definition of postfix type, and (b?a! ) ~ V!?

=

~ ~

(Ec

,t

l'U l,u2,YO: C E:iV /\t 1 E tU /\( U1C )E tV /\ U2E _{tV /\ (y}

=

_{yoc? )}

: #e

tl

>

#eU2/\( xb?a!yo )dres( t 1,U2) /\( UI

=

U2)

)

:: f

calculus, definition of composability, and definition of directed resultant ~

(Ec,t1,UI,YO:CEiU/\t1ElU/\(UIC

)ElV-/\(y=yoc?)--: c(t IoUIC )/\ (xb?a!yoc! )dres( t IoUlc )

)

=>

f

calculus

J

end of case

Case (2): (Eyo,c :CE:.oU:y=Yoc!)

(Eto,uo: toE:. tU /\uoE: tU /\ c (to,uo): (xa!b?y }dres (to,uo» /\(Ec,yo:cE:.OU:y =Yoc!)

=

~ calculus and property 3.0 J

(Ec ,to,uo,yo: c E:. oU 1\ toE:. tU /\uoE:. tV /\ (y = Yoc!) /\c( to,uo)

)

: (xa!b?yoc! )dres(to,uo)/\( (xa!b?yoc!

H

U!? = postf ( U,t o» /\( (xa!b?yoc!) ~V!? =postf( V,uo»

=

~ calculus

J

(Ec ,to,t l,uO,YO: c E:. aU /\ toE:. tU /\( to

=

tIc )l\uoE:. tV /\ (y

=

yoc!) /\c (t IC ,Uo)

: (xa!b?yoc! )dres( t IC ,Uo) /\( (xa!b?yo) ~ U!?

=

posU ( U,t

1»

/\( (xa!b?yo) ~U= postf( V,uo»

)

=>

~ property 3.1.2 and calculus

J

(Ec,tl,uo,yo: c E:.oU 1\( tIc }E:.tU /\UoE:tV /\(y =yoc!)

)

: (xa!b?yo )dres( t l'UO ) / \ ( (xa!b?yo) ~ U!? = postf ( U,t d)

/\( (xa!b?yo) ~U!?

=

postf( V,uo»

==:»

~ induction hypothesis

J

(Ec ,t l.t 2,UO.u.,yo: c E:oU /\( tiC )E:. tU /\t2E:.tU l\uoE:. tU /\UI E: tV /\( Y = Yoc! ) : (xb?a!yo)dres( t 2,U I ) /\ «xa!b?yo) ~ U!? = postf( U,t

I»

/\«xa!b?yo)~V!? =postf(V,uo»

)

=

~ property 3.0 and calculus

J

{Ec ,t l,t 2,uO,UloYO

)

: c E:. aU I\(t IC )E:. tU /\ t2E:. tU /\uoE:. tV /\UIE:. tV /\ (y

=

Yec!)

: (xb?a!yo )dres( t2,UI)

/\({xb?a!yo HU!?

=

postf ( U,t_{2 )} )/\«xb?a!yo) ~V!? = postf (V,u l

»

/\({xa!b?yo)~U!?

₌

postf { u,tl»I\«xa!b?yo)~V!? =postf(V,uo»

=>

~ calculus, definition of postfix type, and (b?a! ) ~ U!? = s

J

(Ec ,t l,t 2,t3,t4,uo,uI,Ye

: c E:..oU /\{ t IC )E:. tU /\ t2E:. tU /\ UoE:. tV /\u1E:. tV /\ (y

=

Yoc!)

: (xb?a!yo )dres( t2,u d/\( t l=t3abt4)/\ (t2

=

t3bat4)/\ (ue = U I) )

-

20-(Ec ,t₃,t₄,uo,yo; c E oU A (t 3a.bt4c )E tU A (t3ba.t4}E: tU AUoEtU A (y = Yoc!)

: (zb?a.!yo )dres ( t 3ba14' uo)

)

~ ~ U satisfies R4 J

(Ec _{,t3,t4,uO,yo:} c E oU A (t3ba.t4C )E tU AUoE tU 1\( y =Yoc!)

: (zb?a.!yo )dres (t3ba.t4'UO) )

=

~ definition of composability, definition of directed resultant, and calculus

J

(Ec ,t 3,t4,uO,yo:c E oU A( t3ba.t4c )E tU AUoE tU 1\( y =yoc!) : c( t3ba.t4c ,U ) A (xb?a.!yoc! )dres( t 3ba14c ,U ) )

=> f

calculus J

(Et ,U: t E tU AuE: tV Ac(t ,u): (zb?a.!y )dres( t ,U

»

end of case

Case (3): (Eyo,c :cEiU:y =yoc?)

(Eto,Uo: toE tU AUoE tU Ac( to,Uo): (xa!b?y )dres( to,Uo» A(Ec,yo: cEiU:y =yoc? )

=

~ calculus and property 3.0 J

(Ec .to,uo,Yo

)

: c E iU AtoE tU AUoE tV A( y =Yoc? ) A c( to,Uo)

: (xa!b?yoc? )dres( to,uo)

A( (za!b?yoc?

H

U!?

=

postf ( U,t o

»

A( (xa.!b?yoc? ) ~ U!?

=

postf ( U,uo»

=

~ calculus and property 2.0

J

(Ec ,to,t l,uO,YO

)

:CEiUAtoEtUA(to=tlC )AuoEtVA(y=yoc? )AC(tIC,UO)

: #e!(xa.!b?yoc? )~#e? (xa.!b?yoc? ) A (xa!b?yoc? )dres( tIC ,Uo)

A( (xa!b?yo) ~ U!?

=

postf ( U,t l)} A «xa!b?yo) ~ V!?

=

postf ( U,Uo» ~ ~ property 3.1.2 and calculus

J

(Ec,tlouo,yo

)

:cEiUA(tlc )EtU/\UoEtUA(y =Yoc?)

: #e!(xa.!b?yo)

>

#e? (xa.!b?yo) A(xa!b?yo }dres(t l'UO)

A«xa!b?YoHU!?

=

postf( Uotd) A «xa!b?-YoHU!?

=

postf-{

u,uo-H

=>

~ induction hypothesis and calculus J

(Ec ,t l,t2,uO'U I'YO

)

:cEiU/\(tIC )EtU/\t 2EtU/\UoEtV/\U I EtU/\(y =yOC?)

: He!( xb?a!yo)

>

He? (xb?a!yo) /\(xb?a!yo )dres (t2'U

d

/\«xa!b?yo)~U!? =postf( U,tl»/\((xa!b?yo)~V!? =postf(V,Uo»

=

l

property 3.0 and calculus ~ (Ec ,t 1.t2,UO,u loYO

)

:cEiU/\(tlc )EtU/\t 2EtU/\UoEtV/\U I EtV/\(y =yoc?)

: He!( xb?a!yo)

>

He? (xb?a!yo )/\(xb?a!yo )dres(t2,ud

/\ «xb?a!yo) ~ U!? = postf ( _{U,t 2}» /\((xb?a!yo) ~ v!?

=

post! ( V.u l»

/\((xalb?yo) ~U!? =postf( U.td )/\( (xa!b?yo) ~V!?

=

post! ( V,uo»

~ l calculus, definition of postfix type. and (b?a!) ~ V!?

=

t

I

(Ec ,t l,t 2.t3,t 4,UO, U I,YO

:CEiU/\(tlc _{)EtU/\t2EtU/\UoEtU/\UlEtU/\(y =yoc?)}

: #eUl

>

_{He t 2/\ (xb?a!yo} )dres( _{t 2,Ul) /\(} tl = t 3abt4 ) /\( t2 = t 3bat4 ) /\( Uo = Ul)

)

=

l calculus and property 3.0

I

(Ec .t3.t4.UO.yO: c E iU /\ (t3bat4)E tU /\UoE

tV /\

(y

=

yoc? )

: Heuo

>

He (t3 bat 4) /\( xb?a!yo )dres (t 3bat4.uo)/\ C(t3 bat4.u O) )

=

llemma 3.0 using property 2.1. and definition of composability

I

(Ec ,t_S.t4,uo,yo: c E iU 1\ (t_{sbat 4c} )E: tU /\ UoE: tU /\ (y

=

yoc? )

: (xb?a!yo )dres( _{t 3bat 4,uo)} /\c( t3 bat 4C .uo) )

=

l definition of directed resultant

I

(Ec

,t

_3.t4,UO.YO: c E iU 1\ (t3bat4c )E tU /\UoE: tU /\ (y

=

YoC? )

: (xb?a!yoc? )dres( t3bat4c IUO) /\C(t3 bat4c .uo)

)

=>

l calculus

J

(Et.u: t EtU /\uE tV /\c( t ,u): (xb?a!y )dres( t.u»

end of case end of step

For symbols a and b, such that a E oV and bE: i U. the proof is analogous.

end of lemma

We combine the results obtained thus far into two lemmata. In lemma 3.1 we move output occurrences backward. Input occurrences are put forward in lemma 3.2.

- 22-Lemma 3.1

For symbol a. such that aE:aU. and partially directed traces Yo. Yl' Yz. and Y3. such that (YoYIY2Ya)E: (aVvV!? vO!? )* and (YIY2) ~!u.? J

=

t.

(Eto,'uo: toE: tV AUoE: to Ac( to.uo): (dir( aU.Yoa!y lYza?Y3) )dres( to.uo» ~ (Et.u: t E:tV AUE: tU Ac( t.u): (dir( aU.YaY la!Yzu.?Y3))dreS( t.'LL

»

Proof

Givena. Yo. Yl, Y2. andY3. such that aEaU. (YoYlY2Y3)E:(aUVV!?uf}!? )*, and

(YIY2) ~fa? ~

=

e. For shortness we introduce some abbreviations; xc. Xl. and X2 denote

dire

aU,yo).

dire

aU,Yl). and

dire

aV,Y2a?Ya) respectively. We proof this

lemma by mathematical induction on the length of Xl'

Induction hypothesis

(AWO.Wl; (WOWI =Xl )Al{ wo) <1{x 1)

)

: (Eto,uo: toE: tU AUoE: to Ac (to.uo): (xou. !WOWlx2 )dres{ to.uo»

=:;> (Et ,u: tE: tV Au E: tU AC( t,u): (xow lU.!w2 )dres( t,u»

Base : l(xt>

=

0 We derive:

(Eto.uo: toE: tU AUoE: tU Ac{ to.ue): (dir( aU.You.!YlY2a?Ya) )dres (to.uo»

=

f

definitions of direct. Xo. Xl. and x2. using a! ~ aU

J

{Eto.uo: toE: tU AUoE: tV Ac (to.uo): (xoa!x lx2 )dres{to.uo» = c,.... - - _: ___ If - \ - 0 1

( ;1;1 - c., .,u~..,c ~\"'l/- )

(Eto.uo: toE: tU AUeE: tU Ac{ to.ue): (XoXla!X2)dreS( to,uo»

Step : I( X

d

>

0

We derive:

(Eto.uo: toE: tU AUoE: tU Ac( to.uo): (dir( aU.you.!y IY2a?yS) )dres (te.u 0» A{Ew.b ; b E: ( U!? vV!? ) :Xl =wb )

=

~ definitions of direct. xo. x I, and x2, using a! ~ aU

J

(Eb ,to,uo,w: b E: ( U!? v U!? ) AtoE: tU AUoE: tV Ac( to.uo) A

(Xl

=

wb )

: (xoa!wbx₂ )dres( to,uo) )

::::::> ~ induction hypothesis J

{Eb ,t1,Ul.w: b E: (U!? v U!? ) AtIE: tU AUI E: tV Ac( t l,ud A (Xl

=

wb )

: (.z0'UJC1!bX2 )dres{ tl,ul)

We distinguish two cases: (0) aEoU (1) aE oU Case (0): aEoU We use case-analysis: (0.0) b E( oU)! (0.1) b E( oU)! (0.2) bE (iU)? /\ (b ~ a? ) (0.3) bE(iU)? /\(b ==a?) (0.4) bE(iU)? Case (0.0) : bE ( oD)!

(Eb ,e ,t l'u l'w: (b == e!) /\e E oD /\t IE tU /\u IE tU /\ c(

t

l'U I) /\(XI == wb )

: (xowa!e !x:a)dres(

t

l,ud )

=

~ property 3.1.0 (i) using a and e have distinct types, and calculusl

(Ec ,t1,Ul,W : e E oU /\tlE tU /\UIEtV /\c( tl,udA(XI == we!)

: (xowe !a!x:a )dres (t I,UI)

)

=

f

calculus

1

(Et ,U:

t

E tU /\u EtU /\ c(

t

,U): (XoXla!X:a)dres( t ,U»

end of case

Case (0.1): b E( oU)!

(Eb ,e ,tloul,W: (b :: e! )/\e E oU At IEtU /\UIEtU /\c( tloUI )/\(XI:: wb )

: (xowa!e IX:a)dres (tl,UI)

)

=

f

lemma 3.1.0 using a and e have the same type, and calculus ~

(Ee,t ,U,W: e EOU /\tEtU Au EtV Ac( t ,11.)/\ (Xl == we!)

: (xowe !alx:a)dres( t ,u)

)

=

~ calculus

J

(Et

,'ll: tE tU /\uE tV /\c(

t

,u): (XoXla1x2)dreS(

t

,u»

1

I

- 24-Case (0.2): bE(iV)? /\(b 7" a? ) (Eb,c,tl,UI'W ; (b

=

c? ) /\ e E i

V /\

(e 7" a ) /\ tiE t U /\ U lEt U /\ c (t I' U I ) /\ ( X 1

=

wb )

; (xowale?x2 )dres( t I,UI) )

=

t

property 3.1.1 using a and e have the same type, and calculus

1

(Ee ,t loUI,W ; c E i U /\ (c ;t a. ) /\ tiE tU /\u IE

tV /\

c ( t I,U I) /\ (x I

=

we? )

; (xowc?a!x2 )dres(tl,u I)

)

==

l

calculus

1

(Et ,U; t E tU /\UEtU /\c( t ,U): (xoX la!X2)dreS( t ,U

»

end of case

Case (0.3): bE(iU)? /\(b ==a?)

(Eb,t 1,UI,W:(b =a? )/\t IEtU/\U I EtUI\C(tl,Ul)/\(Xl=wb)

: (xowa.la?x2 )dres(t l'U 1 )

)

=

l

property 2.0 and calculus

1

(Et_{1,ul'w ; t l EtU} l\ulEtV /\c(t l,udl\(x 1 =wa? )

: (xowala.?x2 )dres( t I,UI) 1\#~!xO~#"? Xo

)

=

l Xl

=

dire

aU,y 1)'

#,,?

y 1

=

0, definitions of direct and prefix-closure

1

(Et1,UI'W; tlEtU /\UIEtV /\c( tl,UI )I\(XI =wa? )

: (xowa!a?x2 )dres (t I,U I) /\#,,!XO ~#,,? Xo /\ #"lX I ~ #o.? X I

)

=

t

calculus

1

(Et I.UI.W ; tIE tU I\UI E tV /\c( t l.ud 1\ (x l=wa? )

: (xowa!a?X2 )dres (t I,UI) I\#Q.!XOW

>

#,,?XOW

)

=

t

a!E (oU )!. a? E (iV)?, definitions of composability and directed resultant

1

(Et 1,ul'W : tiE tU I\Ul E

tV

/\c( t _{l'U I )} I\(X I =wa? )

: (XoWa.?a.!X2 )dres (t I,Ut ) )

- lcalculus

1

- __

(Et ,U: tE tU /\UEtU /\c( t ,U): (xoX lalx2)dres( t ,U»

Case (0.4) : bE (iU)?

(Eb,c,t1,UloW:(b =c? )ACEiUAt1EtVAuIEtUAC(tloUdA(XI=wb)

: (xowa!c?X2)dres( tl,UI )

)

=;> ~ lemma 3.1.1 using a and c have distinct types, and calculus J

(Ec ,t,u,w: c EiU AtEtU Au EtUAc(t ,U )I\(XI =wc? )

: (xowc?a!x2 )dres (t ,u ) )

=

~ calculus J

(Et ,u : tE tV AUEtU Ac( t ,u): (XoXla!X2)dreS( t ,u»

end of case end of case Case (1): aEoU

The proof of this case is analogous to the proof of case (0).

end of case end of lemma Lemma 3.2

For symbol a, such that aEaU, and partially directed traces Yo, YI' Y2, and Ys,

such that (YoY IY2YS)E (aUu U!? u U!? )* and (YIY2) ~~ a?

J

=

t,

(Eto,Uo: toE tU I\UoE

tV

I\c (to,Uo): (dir( aU,Yoa!YIY2a?yS) )dres( to,Uo»

=;> (Et ,U: t E tU AuE tU Ac( t ,U): (dir( aU,Yoa!y1a?Y2YS) )dres( t ,u»

Using lemma 3.1.1 instead of lemma 3.1.0 the proof of this lemma is analogous to the proof of lemma 3.1.

-

26-3.2 Undirecting directed resultants

In order to derive lemma 3.3 we define the notion D. It can be interpreted as the distance between two partially directed traces, that are ordered by the partial order undirect.

Definition 3.2.0 (D)

For partially directed traces x and y I such that yundirectx,

D (x ,y)

=

(Sa: : #alx - #alY )

end of definition Property 3.2.0 (i) D (x ,y

):?!

0

(ii) (D (x ,y )

=

0) ;;: (x

=

Y )

(iii) yundirectx ;;: (D (x ,Y ) = 1)

(iv) {zundirecty/\yundirectx);;: (D(x,y)+D(y,z )=D(x,z» end of property

Lemma 3.3

For traces

t

and u , such that tEtU, uEtV, c(t,u), and mm(t ,u) = 0, and par-tially directed traces x and y, such that xdres(

t

,U ) and yundirectx,

(Eto,Uo: toEtU /\UoE tV /\c( to,uo )/\( mm( te,ue)

=

0): (dir( aU,y ) )dres( to,ue»

Proof

Given

t

I U , and x, such that

t

EtU, uEtV, c(

t

,u). mm.(t ,u) = 0, and

xdres(t ,u). We proof this lemma for partially directed trace y. such that yundirectx, by mathematical induction on D (x,y). Given y, such that

yundirectx.

Induction hypothesis

(Az: zundirectx /\D (x,z) <D (x,y)

: (Eto,Uo: toE tU /\UoE tU /\c (tOIUO )/\( mm( to,'lLe) = 0): (dir( aU,z ) )dres( te,Ue» )

Base:D(x,y)=O We derive:

I

-

27-D(x,y)=O

=

l

property 3.2.0 (ii) and xdres(t ,u) J

(x =y) I\xdres( t ,u )

=

~ calculus J

ydres(

t

,u)

=

~ y

=

dir( aU,y ) due to the definitions of direct and directed resultant

J

(dir( aU,y» dres( t ,u)

=

f

calculus, using m.m(t ,u) = 0

J

(Eto,uo: toE tU I\UoEt

V

I\c( to,Ue) 1\ (m.m( to,Uo)

=

0): (dir( aU,y) )dres( te,ue» Step: D (x, y )

>

0

We derive:

D(x,y»O

=

t

definition of D, definition of immediate undirect, definition of undirect, x dres (t ,11.), and calculus

~

(E z : y iundirect z 1\ z undirectx : x dres (

t

,U ) )

=

~ property 3.2.0 (iii) and (iv), and definition of immediate undirect J

(E z : y iundirect z 1\ z undirectx 1\ D ( x ,z )

<

D ( x ,y ) : x dres (

t

,u ) )

=>

l

induction hypothesis J

(Eto,uo,z : teE tU /\uoE:tU /\c( te,uo) /\(mm.( to,uo) = 0 )I\yiundirectz

: (dir( aU,z ) )dres(te,uo)

)

=

t

definition of immediate undirect using m.m( te,ue)

=

0 ~

(Eto,uo,z

)

: toEtU l\uoE tV 1\ c{to,uo) 1\ (mm (to,uo) = 0) : (dir( aU,z) )dres( to,uo)

I\(Ea,yo'YhY2,Ya:(YlY2)~~a? ~=t:(Y =YoYlaY2Ya)I\(Z =Yea !YIY2a?Y3»

=>

t

lemmata 3.1 and 3.2, and calculus ~

(Ea,t ,to,u,ue.Yo'Yl,Y2,Ya

)

: t€ tU I\UEtV I\c(t ,u )l\teE tU l\ueE tU I\c( to,ue) 1\( m.m( te,ue)

=

0) 1\( (YIY2) ~~ a?

l

=

e)

: (dir( aU,Yoa!YlY2a?yS) )dres(to,uo )1\( dir( aU,YoYla!a?Y2Y3) )dres(t ,u)

I\(y =YoYla:y2Ya)

=>

~ property 2.3, and calculus ~

(Et ,u: tE tU /\uE tV I\c( t ,u) I\(m.m( t ,u)

=

0): (dir( aU,y) )dres( t ,u»

,~ ~ "r

1

- 28-3.3 Concluding theorems Theorem 3.0

Every directed trace structure U, that satisfies RI , Ra, R4 • and Ro satisfies the

foam rubber wrapper postulate.

Proof

Given a directed trace structure U, such that U satisfies R1, R a, R4 , and Ro,

traces t, u, and z, such that

t

E tU, uE tV, c( t ,u), ZE( aU )$, and z res (

t

,u).

We derive: true

= llemma 3.0, definition of mismatches, and zres(t,u)

!

(Eto,uo: toE tU I\UoEtU I\c(to,Uo )1\( mm(to,uo) = 0): (zres(to,uo» = l definition of resultant!

(Eto,uo,x: toEtU l\ u oEtU I\c(to,uo)l\(mm(to,uo) = 0) : zundirectx I\xdres(

t

_{I,U I )}

)

=>

llemma 3.3!

(Etl,UI: tIE tU I\UIE tU I\c( t1udl\(mm( t l,ud = 0): (dir( aU,z ) )dreS(tI,UI»

=>

f

property 3.0 !

(Etl : t1EtU: dir( aU,z )

N

aU )!? = postf{ U,t d)

= l z E (aU )$, and definitions of postfix type, direct, and projection! (Etl:tIEtU:Z=t l )

= l calculus

!

zEtU

We conclude, since tU is prefix-closed due to R I' that t( U @ V) c tU. Using pro-perty 2.2 we conclude that U satisfies the foam rubber wrapper postulate. end of theorem

Theorem 3.1

Every directed trace structure that is a C4, satisfies he foam rubber wrapper postulate and has absence of danger of transmission interference.

Proof

In the context of delay-insensitive directed trace structures absence of danger

- - _.

--of transmission interference equals H2. A directed trace structure, that is a C4,

satisfies the foam rubber wrapper postulate on account of theorem 3.0, and has absence of danger of transmission interference on account of R2'

end of theorem

References

[EWD] Edsger W. Dijkstra. Lecture notes "Predicate transformers" (Draft).

EWD835. 1982.

[HS] Huub M.J.L. Schols. Aformalisation of the foam rubber wrapper princi-ple, Master's Thesis, Department of Mathematics and Computing Science.

Eindhoven University of Technology, 1985.

[JvdS] Jan L.A. van de Snepscheut. Trace theory and VLSI design, Ph.D. Thesis,

Department of Mathematics and Computing Science. Eindhoven U niver-sity of Technology, 1983.

[JTU] Jan Tijmen Udding, Qassification and composition of delay-i:nsensitive circuits, Ph.D. Thesis. Department of Mathematics and Computing

Sci-ence, Eindhoven University of Technology, 1984.

[MFR] Charles E. Molnar, Ting-Pien Fang, and Frederick U. Rosenberger.

Syn-thesis

0/

/Jelay-insensitive Modules, in 1985 Chapel Hill Conference on VLSI, ed. Henry Fuchs, Computer Science Press, 1985, pp. 67-86.

[MR] Martin Rem. Concurrent Computations a.nd VLSI Circuits, in Control Flow and Data Flow: Concepts

0/

Distributed Programming, ed. M. Bray, Springer-Verlag Berlin Heidelberg, 1985, pp. 399-437.

[RSU] Martin Rem. Jan L.A van de Snepscheut. and Jan Tijmen Udding, Trace theory and the definition of hierarchical cO'T7l11onents, in Third Caltech Con/erence on ¥LSI, ed. Randal Bryant, Computer Science Press, 1983,

•

COMPUTING SCIENCE NOTES

In this ser~es appeared:

Nr. 85/01 85/02 85/03 85/04 Author{s) R.H. Mak W.M.C.J. van Overveld W.J.M. Lerrnnens T. Verhoeff H.M.J.L. Schols Title

The Formal Specification and Derivation of CMOS-circuits On arithmetic operations with M-out-of-N-codes

Use of a Computer for Evaluation of Flow Films

Delay-insensitive Directed Trace Structures Satisfy the Foam Rubber Wrapper Postulate