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Relational Setting Gabriel A. Baum1 and Marcelo F. Frias2

1 Universidad Nacional de La Plata, LIFIA, Departamento de Informatica.

Calle 50 y 115 - 1er. piso (1900), La Plata, Argentina. Tel./FAX: +54 - 221 - 422 - 8252, e{mail: gbaum@sol.info.unlp.edu.ar

2 Universidad de Buenos Aires, Departamento de Computacion, and

Universidad Nacional de La Plata, LIFIA, Departamento de Informatica. e{mail: mfrias@sol.info.unlp.edu.ar

Abstract. In [1] we presented the logicP=PML, a formalism suitable

for the specication and construction of Real{Time systems. The main algebraic result, namely, the interpretability of P=PML into an

equa-tional calculus based on!-closure fork algebras (which allows to reason

about Real{Time systems in an equational calculus) was stated but not proved because of the lack of space.

In this paper we present a detailed proof of the interpretability theorem, as well as the proof of the representation theorem for!-closure fork

alge-bras which provides a very natural semantics based on binary relations for the equational calculus.

1 Introduction 1.1 Motivations

The motivation for this work is the need to describe industrial processes as part of a project for a telecommunications company. We want to be able to give formal descriptions of such processes so as to be able to analyze such descriptions. For example, we want to be able to calculate critical paths for tasks in processes, throughput times of processes, etc. We also want to demonstrate correctness of process descriptions in relation to their specications (where this is appropriate), derive implementations of process specications in terms of the available concrete apparatus in the factory, validate (using formal techniques) an implementation against its abstract description, and so on. Available languages for describing processes are unsuitable for various reasons, most having to do with the nature of the formalization of such processes being used in the project.

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as a model, in the sense of logic, of the product. We may see products as being characterized by data types in rst order logic, for example.

The distinguishing characteristic of products is that they exist `indepen-dently' at an instant in time, where time is used here in its normal scientic sense. (Independence here means that a product is dened without recourse to any other referent or only in terms of other (sub)products. Products of the former kind are called atomic products.) In fact, all products have a time at-tribute whose value in a product instance indicates the time instant at which the values of the attributes were (co)determined, presumably by some appropriate measurement procedures. On the other hand, processes are distinguished enti-ties which do not exist at a time instant, but which have time duration. Further, processes are not independently denable, but are dened in terms of their input and output products.

Processes also model entities of the real world and again are dened in terms of attributes. The method imposes a very restrictive notion of process, namely one in which all processes have a single input and a single output. (The rea-sons for this restriction need not detain us here, except to say that they are methodologically very well motivated. The restriction clearly will have a pro-found inuence on the nature of the language we dene below.) Distinguished attributes of a process include the transfer function(s) `computed' by the pro-cess (i.e., how the input product is transformed into the output product), upper and lower bounds on the time taken for the process to execute, a ag indicat-ing whether the process is `enabled', and so on. The transfer function may be described in terms of an underlying state machine used to organize phases of the process being dened and to `sense' important external state information re-quired to control the execution of the process. Like products, processes may be dened in terms of `sub-processes' and we now turn to this language of processes. We can see an analogy between inputs/outputs and products and between programs and processes. Both programs and processes are intended to model entities that dene families of executions on the machine used to execute the program/process. This is exactly how we want to understand processes, i.e., as dening a class of potential executions over some (abstract) machine. We do not envisage a single abstract machine which will underpin all potential processes. Rather, we assume that our abstract machine is provided by an object, in the sense of object oriented programming.

1.2 The Results

In [1], the logic P=PMLwas dened by extending rst order dynamic logic with a parallel combinator and the ability to express real time constraints. The seman-tics of dynamic logic uses a notion of transition system that is used to represent the underlying abstract machine capable of executing the atomic processes. The logic was extended with variables over processes so that we can specify abstractly the processes we are interested in building. There is a notion of renement associ-ated with such specications, allowing us to demonstrate that a process satises its specication.

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algebras (!CFA). These algebras are extensions of relation algebras [17] with three new operators, a pairing operator called fork [7][6], a choice operator [15] and the Kleene star. A consequence preserving function mapping formulas of the logic to equations in the language of !CFAwill be dened. The use of the mapping enables the use of equational inference tools in the process of systems construction. Even though having the possibility of using an equational calculus is interesting by itself, this calculus has the particularity of being complete with respect to a very insightful and clean semantics based on binary relations. The last is guaranteed by the proof of a representation theorem for the class!CFA.

In order to simplify the reading of the paper, the logic and the classes of algebras (already presented in [1]) are presented here once more. The reader in-terested in more comprehensive motivations and examples is strongly encouraged to read the paper [1].

The paper is organized as follows: in Section 2 we will present a rst order formalization of objects. In Section 3 will be presented the logic we propose for specifying and reasoning about the properties of processes. In Section 4 we introduce the class of omega closure fork algebras and prove the representation theorem. In Section 5 we present the algebraization and prove the interpretability theorem. Finally, in Section 6 we present our conclusions about this work.

2 Objects

The rst problem we confront when trying to formalize the previous concepts is that of characterizing the `abstract machine' over which our processes will be dened. These processes are meant to use the underlying capabilities of the or-ganization, as represented by the behaviors displayed by individual components within the organization. (Such individual components may be people, groups, manufacturing machines, etc.) These behaviors are organized (at least in some abstract sense) into a joint behavior which IS our `abstract machine'. We will assume as given some object (which may be very complex and built as a system from less complex components [4][5]), which represents the potential behaviors of the organization as an abstract machine. The denitions below give a somewhat non standard account of objects in terms of the underlying transition system dening the object's allowed behaviors. However, the standard parts of such descriptions (i.e., methods, state variables, etc) are easily distinguishable.

Denition1. An object signature is a pair hA;iin which =hS;F;Pi is a many-sorted rst-order signature with set of sortsS, set of function symbols F and set of predicate symbolsP. Among the sorts, we will single out one sort called the time sort, denoted byT.Ais a set of action symbols. To eacha2Ais associated a pairhs

1 ;s

2 i2(S

) 2

called its arity. We will denote the input arity ofaby ia(a) and the output arity ofaby oa(a).

Denition2. Given an object signature S =hA;hS;F;Pii, an object struc-ture for S is a structure A= hS;A;F;Pi in which Sis an S-indexed family of nonempty sets, where the set T is the T-th element in S. In general, the set corresponding to sort s will be denoted by s. A is an A-indexed family of binary relations satisfying the typing constraints of symbols from A, i.e., if ia(a) =s

1

:::sm 2S

and oa( a) =s

0 1

:::s 0

n 2S , then

a

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thea-th element fromA) is contained in (s 1

sm)(s 0 1

s 0

n). To eachf :s

1

:::sk !sin F is associated a functionf A :

s 1

sk !s2F. To eachpof aritys

1

:::sk inP is associated a relationp A

s 1

sk2P. Regarding the domainTassociated to the time sortT, we will not deepen on the dierent possibilities for modeling time, but will rather choose some adequate (with respect to the application we have in mind) representation, as for instance the elds of rational or real numbers, extended with a maximum element1. We will distinguish some constants, as 0,, etc.

3 The Logic, the Relational Variables and the Time

In this section we will present the Product/Process Modeling Logic (P=PML). In order to achieve this goal we will extend a standard notation for specifying and reasoning about programs, namely dynamic logic.

An important aspect of P=PML is the real{time aspect. We adapt a real{ time logic developed in [2] which presents an extension of the logic presented in [4]. Each basic action is supplemented with a specication of lower and up-per time bounds for occurrences of that action. These bounds may have various interpretations, amongst which we have the following: the lower bound is inter-preted as the minimum time that must pass before which the action's eects are committed to happen and the upper bound gives a maximum time by which the action's eects are committed to happen. Specications of processes will also have associated lower and upper bounds, and renements will be expected to provably meet these bounds.

Consider the formula '(x) := [xAx](x) where A is an action term (a bi-nary orn-ary relation) and the notation [xAx] means that \all executions of actionAestablish the property". According to our previous discussion about processes and products, we read ' as stating that is a truth of the system A, then proving the truth of ' can be seen as the verication of the property in the system described by A. Opposed to the previous view, is the notion of an implicit specication of a system, in which A is not a ground term, but rather may contain some relational variables that represent subsystems not yet fully determined. In what follows we will denote by RelVar the set of relational variablesfR ;S;T;:::g.

Denition3. Given an object signature S =hA;hS;F;Pii, the sets of rela-tional terms and formulas onS are the smallest setsR T(S) and For(S) such that

1. a2R T(S) for alla2A[RelVar[f1,t:t2S

g. 2. If r 2 R T(S) and ia(r) = oa(r), then r

2 R T(S). We dene ia(r ) = oa(r

) = ia( r).

3. If r;s 2 R T(S), ia(r) = ia(s) and oa(r) = oa(s), then r+s 2 R T(S) and rs 2 R T(S). We dene ia(r+s) = ia(rs) = ia(r) and oa(r+s) = oa(rs) = oa(r).

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5. If 2For(S) is quantier free and has free variablesx 1

;:::;xn withxi of sortsi, then ?2R T(S) and ia(?) = oa(?) =s

1 :::sn.

6. The set of rst-order atomic formulas on the signature is contained in For(S).

7. If;2For(S), then:2For(S) and_ 2For(S)..

8. If 2 For(S) andx is an individual variable of sort s, then (9x:s) 2 For(S).

9. If 2 For(S), t 2 R T(S) with ia(t) = s 1

:::sm and oa(t) = s 0

1 :::s

0 n, !

x= x

1

;:::;xmwithxi of sortsi, !

y= y

1

;:::;yn withyi of sorts 0

i andl,u are variables of sortT, then

D ! x ltu

! y

E

2For(S). Denition4. Let R 2R T(S) with ia(R) =s

1

:::sm and oa(R) =s 0

1 :::s

0 n, !

x= x 1

;:::;xm with xi of sortsi, !

y= y

1

;:::;yn with yi of sorts 0

i, and l, u variables of sortT. An expression of the form

! x l

Ru !

y is called a timed action term.

We will assume that a lower and an upper bound are assigned to atomic actions, namely la 2Tand ua 2Tfor each actiona2A. From the bounds of the atomic actions it is possible to dene bounds for complex actions in a quite natural way.

Denition5. LetSbe an object signature. The functionslandufromR T(S)[ For(S) toTare dened as follows

1: 1. Ifa2A, thenl(a) =la andu(a) =ua.

2. IfR=X 2RelVar, thenl(X) = 0 andu(X) =1. 3. If R= 1,t, witht2S

, then

l(R) = 0 andu(R) =( being a constant of sortT).

4. IfR=S , then

l(R) = 0 andu(R) =1.

5. IfR=S+T, thenl(R) = minfl(S);l(T)gandu(R) = maxfu(S);u(T)g. 6. IfR=ST, then l(R) = maxfl(S);l(T)gandu(R) = maxfu(S);u(T)g. 7. IfR=S;T, then l(R) =l(S) andu(R) =u(S) +u(T).

8. IfR=? with2For(S) quantier free and with free variables ! x,

l(R) = l() andu(R) =u().

9. If=p(t 1

;:::;tk), thenl() =lp2Tand u() =up2T, withlpup. 10. If=:, thenl() =l() andu() =u().

11. If=op withop2f_;^;!g, thenl() = minfl();l()gandu() = maxfu();u()g.

12. If= D

! x lRu

! y

E

, thenl() =l(R) andu() =u(R) +u(). Given a set of sorts S = fs

1

;:::;skg and domains S = fs 1

;:::;skg for these sorts, by a valuation of the individual variables of sort si we refer to a function : IndVars

i

!si. A valuation of the relational variables is a function : RelVar !P(S

S

).

1 We will only consider quantier-free formulas, since these are the ones used for

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Denition6. Given a valuation of the individual variables and an array of variables!

x= x

1 ;:::;x

n, by (

!

x) we denote the tuple h(x

1)

;:::;(x n)

i. LetA be an object structure and a valuation of the relational variables. Given valuations of the individual variables and

0 and a timed action term ! x l R u ! y, by

! x l R u ! y

0 we denote the fact that: D ( ! x) ; 0( ! y) E 2 R A (the denotation of the relational term R, formally dened in Def. 7), for every variablez not occurring in

! y,

0(

z) =(z), and, (l)l(R) and(u)u(R). The semantics of formulas is now dened relative to valuations of individ-ual variables and relational variables. In the following denition, the notation Aj=

P=PML

[][], is to be read \The formula is satised in the object struc-tureAby the valuations and".

Denition7. Let us have an object signatureS=hA;hS;F;Piiand an object structure A=hS;A;F;Pi. Let be a valuation of individual variables and a valuation of relational variables. Then:

1. Ifa2Athena A

is the element with index

ain A. 2. IfR2RelVar, thenR

A

=

(R). 3. IfR= 1,

t with t=s

1 :::s

k, R

A

=

fh ha 1

;:::;a k

i;ha 1

;:::;a k

i i:a i

2s i

g. 4. If R =S

, with

S 2 R T(S), then R A

is the reexive-transitive closure of the binary relationS

A .

5. IfR=S+T, withS;T 2R T(S), thenR A = S A [T A . 6. IfR=ST, with S;T 2R T(S), thenR

A = S A \T A . 7. If R=S;T, with S;T 2R T(S), then R

A

is the composition of the binary relationsS A and T A .

8. If R = ? with 2 For(S) quantier free and with free variables !

x= x

1 ;:::;x

n, then R A = nD ( ! x) ;( ! x) E

:Aj= P=PML

[][] o

. 9. If'=p(t

1 ;:::;t

n) with

p2P,Aj= P=PML

'[][] if

t 1

A

;:::;t n A 2p A. 10. If'=:, thenAj=

P=PML

'[][] ifA6j= P=PML

[][]. 11. If'=_,Aj=

P=PML

'[][] ifAj= P=PML

[][] orAj= P=PML

[][]. 12. If ' = (9x : s), then A j=

P=PML

'[][] if there exists a 2 s such that Aj=

P=PML [ a x][ ] ( a

x, as usual, denotes the valuation that agrees with in all variables butx, and satises

a x(

x) =a). 13. If '=

D ! x l R u ! y E

, then Aj= P=PML

'[][] if there exists a valuation 0 such that ! x l R u ! y 0 and Aj=

P=PML [

0][ ].

4 Omega Closure Fork Algebras

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In this section we present the omega calculus for closure fork algebras (!CCFA), an extension of the calculus of relations (CR) and of the calculus of relations with fork [6]. Because of the non enumerability of the theory of dynamic logic, an in-nitary equational inference rule will be required in !CCFA. From the calculus we dene the class!CFA of the omega closure fork algebras and a representa-tion theorem is presented, showing that the Kleene star as axiomatized, indeed characterizes reexive-transitive closure.

In the following paragraphs we will introduce the Omega Calculus for Closure Fork Algebras (!CCFA).

Denition8. Given a set of relation symbolsR, the set of!CCFAterms onRis the smallest set T!CCFA(R) satisfying:R[RelVar[f0;1;1,gT!CCFA(R). Ifx2T!CCFA(R),thenfx; x

;x

gT!CCFA(R). Ifx;y2T!CCFA(R),then fx+y;xy;x;y;xrygT!CCFA(R).

The symbol denotes a choice function (see [15,

x3]), which is necessary in order to prove Thm. 2.

Denition9. Given a set of relation symbolsR, the set of!CCFAformulas on R is the set of identitiest

1= t

2, with t

1 ;t

2

2T!CCFA(R).

Denition10. Given termsx;y;z;w2T!CCFA(R), the identities dened by the following conditions are axioms:

Identities axiomatizing the relational calculus [17], The following three axioms for the fork operator:

xry= (x; (1,r1))(y; (1r1,)); (Ax. 1) (xry) ;(zrw) = (x;z)(y; w); (Ax. 2)

(1,r1)r(1r1,)1,: (Ax. 3)

The following three axioms for the choice operator, taken from [15, p. 324]:

x ;1;

x

1,; (Ax. 4)

x

;1; x

1,; (Ax. 5)

1; (xx

) ;1 = 1;

x;1: (Ax. 6)

The following two axioms for the Kleene star:

x

= 1, + x;x

; (Ax. 7)

x ;

y y + x ; (

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Let us denote by 1,U the partial identity

R an 1r1

. Then, the following axiom is added

1;1,U;1 = 1

; (Ax. 9)

which states the existence of a nonempty set of non splitting elements (that we will call urelements).

Theorem 1.

The following properties are derivable in the calculus!CCFA. 1. is a monotonic operator, i.e., if

xy, thenx

y . 2. Ifx is reexive and transitive, then x

= x. 3. Ify is reexive and transitive and xy, then x

y.

The proof of Thm. 1 requires simple equational manipulations. Notice that from Thm. 1, x

is the smallest reexive and transitive rela-tion that includesx. The rules of inference for the calculus!CCFA are those of equational logic adding the following inference rule2:

`1, y x

i

y`x i+1

y `x

y

Denition 11.

We dene the class of the omega closure fork algebras (!CFA) as the models of the identities provable in!CCFA.

The standard models of the !CCFA are the Proper Closure Fork Algebras (PCFAfor short). In order to dene the classPCFA, we will rst dene the class PCFA.

Denition 12.

Let E be a binary relation on a set U, and let R be a set of binary relations. A PCFA is a two sorted structure with domains R and U hR ;U;[;\;{;;;E;;;Id;;r;

;

;?isuch that 1. S

RE,

2. ?:U U !U is an injective function when its domain is restricted to the setE,

3. If we denote byIdthe identity relation on the setU, then;,EandIdbelong toR,

4. Ris closed under set choice operator dened by the condition:

x

x and jx

j= 1 () x6=;:

5. R is closed under set union ([), intersection (\), complement relative to E ({), composition of binary relations (;), converse (), reexive-transitive closure () and fork, the last being dened by the formula

SrT =fhx;?(y;z)i:xSy andxTzg: Note thatx

denotes an arbitrary pair in

x. That is whyx

is called a choice operator. We will call the setUin Def. 12 the eld of the algebra, and will denote the eld of an algebra

A

byU

A. We will also denote the set of urelements of

A

by UrelA.

2

Giveni>0,byx i

wedenotetherelationinductivelydenedasfollows:x 1

=x,and x

i+1 =x;x

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Denition 13.

We dene the classPCFAas

Rd

PCFAwhere

Rd

takes reducts to structures of the formhR ;[;\;{;;;E;;;Id;;r;

;

i.

Notice that given

A

2PCFA, the terms (1,r1) and (1r1,) denote respec-tively the binary relationsfh a?b;ai:a;b2Agandfha?b;bi:a;b2Ag. Thus, they behave as projections with respect to the injection?. We will denote these terms by and, respectively.

From the operator fork we dene xy = (;x)r(;y). The operator (cross), when interpreted in an proper closure fork algebra behaves as a parallel product:xy=fh a?b;c?di:h a;ci2x ^ hb;di2yg.

A relation R is constant if it satises: R;R 1,, 1;R =R, and R;1 = 1. Constant relations are alike constant functions, i.e., they relate every element from the domain to a single object3. We will denote the constant whose image is the valueabyC

a.

Denition 14.

We denote by FullPCFAthe subclass ofPCFAin which the re-lation E equals UU for some setU and R is the set of all binary relations contained inE.

Similarly to the relation algebraic case, where every proper relation algebra (PRA)

A

belongs to

4

ISP

Full

PRA, it is easy to show that every PCFAbelongs to

ISP

FullPCFA. We nally present the representation theorem for!CFA.

Theorem 2.

Given

A

2!CFA, there exists

B

2PCFA such that

A

is isomor-phic to

B

.

Proof. Let us consider the fork algebra reduct

A

0 of the algebra

A

. By the representation theorem for fork algebras [7] there exists a proper fork algebra

C

0 such that

A

0 is isomorphic to

C

0 (we will denote by

r the fork operation in

C

0). Let

h :

A

0

!

C

0 be an isomorphism. Let us consider the structure

C

=h

C

0

;

;

i, where

and are dened by the conditions: x

= h

h

1( x)

andx

= h

h

1( x)

for eachx2

C

0

: It is easy to check thath:

A

!

C

is an !CFA isomorphism. It is also easy to check using the innitary rule that forx2

A

, x

=

lub

x i :

i2! . Thus, using the isomorphismh, we can prove that for eachx2

C

,

x

=

lub

x i:

i2! : (1)

Notice that it is not necessarily the case that for x 2

C

, x

equals the reexive-transitive closure of x, since innite sums may not correspond to the innite union of the binary relations5.

3 This comment is in general a little strong and applies to simple algebras, but is

nevertheless useful as an intuitive aid for the non specialist.

4 By

I,SandPwe denote the closure of an algebraic class under isomorphic copies,

subalgebras and direct products, respectively.

5 Examples of proper relation algebras in which

lub Adoes not agree with S

A (for

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Let D be C RA reduct. Since D is point-dense (see [15][16] for details on point-density), by [15, Thm. 8] D has a complete representation D

0. Let g : D!D

0 be an isomorphism satisfying g P i2I x i =S i2I g(x

i). LetB=hD

0 ;r;

;

i, where r,

and are dened as follows: xry=g g

1( x)rg

1( y) ; x = g g 1( x) ; x = g g 1( x) :

Then, g : C ! B is also an isomorphism. That r as dened is a fork operation on B follows from the fact the relationsg() andg() are a pair of quasi-projections [15][19] inB. Since

satises Ax. 4{Ax. 6, it is easy to prove that it is a choice operator. Finally, let us check thatx

is the reexive-transitive closure ofx, for eachx2B.

x = g g 1( x)

(by Def. x ) =g 0 @ X i0 g 1( x) i 1

A (by (1))

=[ i0 g g 1( x) i

(g complete representation)

=[ i0 g g 1( x) i

(g homomorphism)

=[

i0 x

i

: (g one-to-one)

The last union computes the reexive-transitive closure of the relationx, as was to be proved.

Finally, the mappingf :A!B,f(x) =g(h(x)), is an isomorphism between the!CFAAand thePCFA B.

5 Interpretability of P=PML in !CCFA

In this section we will show how theories onP=PMLcan be interpreted as equa-tional theories in!CCFA. This is very useful because allows to reason equation-ally in a logic with variables over two dierent sorts (individuals and relations).

Denition15. LetS,F andP be sets consisting of sort, function and relation symbols, respectively. By!CCFA

+

(S;A;F;P) we denote the extension of!CCFA obtained by adding the following equations as axioms.

1. For each s;s 0

2 S (s6= s

0), the equations 1,

s+1,U = 1,U and 1,s 1,

s 0 = 0 (elements from types do not split, and dierent types are disjoint).

2. For each a 2 A with ia(a) = s 1

:::s k and

oa(a) = s 0 1

:::s 0

n, the equation (1,s1

1, sk) ;

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3. For eachf :s1:::sk

!s2F, f;f + 1,s= 1,sand (1,s 1

1,s

k) ;f =f, stating thatf is a functional relation of the right sorts.

4. For each p of arity s1:::sk in P, the equation (1,s 1

1,s

k) ;p;1 = p, stating thatpis a right-ideal relation expecting inputs of the right sorts.

Denition16. A model for the calculus!CCFA

+(S;A;F;P) is a structure A=

A;S

A;AA;FA;PA

;m

where: A 2 !CFA. S

A is a set of disjoint partial identities, one for each sort symbol inS.AAis a set of binary relations, one for each action symbola2A. Besides, if ia(a) =s

1:::sk and oa(a) =s 0 1:::s

0 n, then

aAsatises the condition in item 2 of Def. 15.FAis a set of functional relations, one for each function symbol inF. Besides, iff :s1:::sk

!s, thenf

Asatises the conditions in item 3 of Def. 15. PAis a set of right ideal relations, one for each predicate symbol p2 P. Besides, if p has aritys

1:::sk, thenp

A satises the conditions in item 4 of Def. 15.m: RelVar !A.

Notice that the mappingm in a!CCFA +

(S;A;F;P) model extends homo-morphically to arbitrary relational terms. For the sake of simplicity, we will use the same name for both.

In the following paragraphs we will dene a function mapping formulas from P=PML(S;A;F;P) to!CCFA

+(S;A;F;P) formulas. In the next denitions,is a sequence of numbers increasingly ordered. Intuitively, the sequencecontains indices of those individual variables that appear free in the formula (or term) being translated. ByOrd(n;) we will denote the position of the indexnin the sequence, by [n] we denote the extension of the sequence with the index

n, and by(k) we denote the element in thek-th position of. In what follows,

t;n is an abbreviation fort;

;t(ntimes). For the sake of completeness,t ;0 is dened as 1,. We will denote by IndTerm(F) the set of terms from P=PMLbuilt from the set of constant and function symbolsF. By RelDes(K) we denote the set of terms from!CCFAthat are built from the set of relation constantsK. Denition17. The function: IndTerm(F)!RelDes(F), mapping individ-ual terms into relation designations, is dened inductively by the conditions:

1. (vi) =

;Ord(i;) 1;ifiis not the last index in;

;Length() 1 ifiis the last index in: 2. (f(t1;::: ;tm)) = ((t1)

rr(tm));f for eachf 2F.

Given a sequence such thatLength() =land an indexn (n < !) such that vn has sorts, we dene the term;n(n < !) by the condition6

;n= 8 < :

(v(1))

rr(v (k 1))

r1sr(v (k+1))

rr(v (l)) ifk=Ord(n;[n])< l;

(v(1))

rr(v (l 1))

r1s ifOrd(n;[n]) =l: Notation1 Let be a sequence of indices of individual variables of length

n. Let !

x= hx

1;::: ;xk

i be a vector of variables whose indices occur in . We will denote by ;!

x the relation that given a tuple of values for the vari-ables whose indices appear in, projects the values corresponding to the vari-ables appearing in !

x. For example, given = h2;5;7;9i and !

x= hv 2;v7

i, 6

By1 s

wedenotetherelation1;1 , s

(12)

; ! x =

fha

1? a2? a3? a4;a1? a3 i:a

1;a2;a3;a4

2Ag. Similarly, Arrange ;

! x denotes the relation that, given two tuples of values (one for the variables with indices in and the other for the variables in!

x), produces a new tuple of values for the variables with indices in updating the old values with the values in the second tuple. For the previously dened and !

x, we have Arrange; !

x = fh(a

1? a2? a3? a4)?(b1? b2);b1? a2? b2? a4 i:a

1;a2;a3;a4;b1;b2

2Ag. Note that these two relations can be easily dened using the projections and

previously dened.

Denition 18.

The mappings M : RT(S) !RelDes(A) and T

: For( S) ! RelDes(A[F[P) are mutually dened by

M(a) =afor eacha2A[RelVar; M(1, s

1 :::s

k) = 1, s

1

1, s

k;

M(R) =M(R)

; M

(R+S) =M(R)+M(S); M(RS) =M(R)M(S); M(R;S) =M(R);M(S);

M(?) =T ()

1,;

T(p(t1;::: ;tk)) = ((t1)

rr

(tk)) ;p; T(

:) =T ();

T(( 9v

n:s)) =;n;T

[n](); T

(

_) =T

()+T();

T D

!

x lR u

!

yE

= 0

@ 1

, r

; ! x

;M(R ) 1 A

;Arrange ;

! y

;T() (vl;)C l(R)

;1 (vu;)C u(R)

;1.

Notation 2

We will denote by:

{

`

!CCFA the provability relation in the calculus!CCFA.

{

j=

Fullthe validity relation on the class of full!CCFA models.

{

j=

!CCFA the validity relation on the class of!CCFAmodels.

Notation 3

Given an object structure A = h

S

;

A

;

F

;

P

i, a valuation of the individual variables ,

A

2 PCFA such that Urel

A

S

S

, and a sequence of indices, bys; we denote the elementa1?

? a i?

? a n

2U

A such that: 1. n=Length(),

2. ai =(v

(i)) for all i, 1

in. In case=hi,s

; denotes an arbitrary element fromUA. Given a formula or term, bywe denote the sequence of indices of variables with free occurrences in , sorted in increasing order.

Throughout the next theorems we will assumeS=hA;hS;F;Piiis a xed but arbitrary object signature.

Theorem 3.

Let 2 For(S), let = , let

A be an object structure for S and let be a valuation of the relational variables. Then there exists a model B=

B

;SB;AB;FB;PB

;m0

for !CCFA+

(S;A;F;P) such that

Aj=

P=PML[][]

() s

;

2dom(m 0(T

())): Proof. AssumeA=h

S

;

A

;

F

;

P

i. Let us deneBas follows.

1. Let

B

be the FullPCFAwith set of urelementsS

(13)

2. For each sorts2S, 1,s=fhx;xi:x2sg, 3. aB =

ha

1?

? am;b 1?

? bni:hha

1;::: ;am i;hb

1;::: ;bn ii2a

A , for eacha2A,

4. fB=

ha 1?

? an;bi:f A(a

1;::: ;an) =b , for eachf 2F, 5. pB=

ha

1?

? an;bi:p A(a

1;::: ;an) andb 2U

B , for eachp 2P, 6. m0(R) =

ha

1?

? am;b 1?

? bni:hha

1;::: ;am i;hb

1;::: ;bn i i2R

A for allR2RelVar.

The proof follows by simultaneous induction on the structure of terms from

RT(S) and formulas fromFor(S).

Corollary 1.

Let2For(S) without free variables over individuals, let A be an object structure forS, and letbe a valuation of the relational variables. Then there exists a full modelB=

B

;SB;AB;FB;PB

;m

for!CCFA+(S;A;F;P) such that

Aj=

P=PML[]

() m T

hi()

= 1:

Notation 4

Given

A

2PCFA,s=a 1?

?ak (ai2Urel

A for alli, 1

ik) and a sequence of indices increasingly sorted and of length k, by s; we denote the set of valuations of individual variables satisfying(v(i)) =ai. In case = hi, for each s 2 U

A s; denotes the set of all valuations. Given an array of variables !

x of length n whose indices occur in , (a1?

? ak); !

x =

b1?

? bn wherebi=aOrd

(j;), withj the index of the individual variablesxi. Bytuple(a1?

? ak) we denote the tupleha

1;::: ;ak i.

Theorem 4.

Let2For(S),=andA=

A

;SA;AA;FA;PA

;m be a full !CCFA+(S;A;F;P) model. Then there exists an object structure

Band a valuation of relational variables such that

s2dom(m(T())) () Bj=

P=PML[][] for all 2s;: Proof. Since

A

2FullPCFA, for each sorts, lets=fa2Urel

A:a 21,sg. For eacha2A, dene

aB= fhha

1;::: ;am i;hb

1;::: ;bn ii:ha

1?

? am;b 1?

? bni2m(a)g: For eachf :s1:::sk

!s2F,

fB(a

1;::: ;ak) =b i ha

1?

? ak;bi2f A: For eachpof aritys1:::sk inP,

ha

1;::: ;ak i2p

B i a

1?

? ak 2dom p A

:

For eachR2RelVar with arityhs

1:::sm;s 0 1:::s

0 ni,

(R) =fhha

1;::: ;am i;hb

1;::: ;bn ii:ha

1?

? am;b 1?

(14)

Corollary 2.

Let 2 For(S) without free variables over individuals and let A =

A

;SA;AA;FA;PA

;m

be a full !CCFA

+(S;A;F;P) model. Then there exists an object structure B and a valuation of the relational variables such that

Bj=

P=PML[]

() m T

hi()

= 1:

Theorem 5.

Let [f'gbe a set of P=PML(S;A;F;P) formulas without vari-ables over individuals. Then,

j=

P=PML'

()

Thi() = 1 :

2 j=

FullThi(') = 1: Proof. In order to prove the theorem, it suces to show that

6j=

P=PML'

()

Thi() = 1 :

2 6j= FullT

hi(') = 1: (2) Formula (2) follows from Cors. 1 and 2.

Theorem 6.

LetV be the variety generated by FullPCFA. Then,V =!CFA. Proof. From Thm. 2 and the factPCFA=

ISP

FullPCFA,!CFA=

ISP

FullPCFA.

The next theorem states the interpretability of theories from P=PML as equational theories in!CCFA.

Theorem 7.

Let [f'gbe a set of P=PML formulas without free individual variables. Then, j=

P=PML' ()

Thi() = 1 :

2 `

!CCFA Thi(') = 1. Proof. By Thm. 5,

j=

P=PML'

()

Thi() = 1 :

2 j=

FullT

hi(') = 1: (3) By Thm. 6,

Thi() = 1 :

2 j=

FullT

hi(') = 1 ()

Thi() = 1 :

2 j=

!CCFA T

hi(') = 1: (4) By the denition of the class!CFAand the denition ofj=

!CCFA,

Thi() = 1 :

2 j=

!CCFA T

hi(') = 1 ()

Thi() = 1 :

2 `

!CCFA T

hi(') = 1: (5) Finally, by (3), (4) and (5),

j=

P=PML'

()

Thi() = 1 :

2 `

!CCFA Thi(') = 1: 6 Conclusions

(15)

1. Baum, G.A., Frias, M.F. and Maibaum, T.S.E., A Logic for Real{Time Systems Specication, Its Algebraic Semantics and Equational Calculus, in Proceedings of AMAST'98, Springer{Verlag, LNCS 1548, pp. 91{105, 1999.

2. Carvalho, S.E.R., Fiadeiro, J.L. and Haeusler, E.H.,A Formal Approach to Real{ Time Object Oriented Software. In Proceedings of the Workshop on Real{Time Programming, pp. 91{96, sept/1997, Lyon, France, IFAP/IFIP.

3. Fenton, N. E., Software Metrics. A Rigorous Approach, International Thomson Computer Press, 1995.

4. Fiadeiro, J. L. L. and Maibaum, T. S. E., Temporal Theories as Modularisation Units for Concurrent System Specications, Formal Aspects of Computing, Vol. 4, No. 3, (1992), 239{272.

5. Fiadeiro, J. L. L. and Maibaum, T. S. E.,A Mathematical Toolbox for the Soft-ware Architect, in Proc. 8th International Workshop on Software Specication and Design, J. Kramer and A. Wolf, eds., (1995) (IEEE Press), 46{55.

6. Frias M. F., Baum G. A. and Haeberer A. M.,Fork Algebras in Algebra, Logic and Computer Science, Fundamenta Informaticae Vol. 32 (1997), pp. 1{25.

7. Frias, M. F., Haeberer, A. M. and Veloso, P. A. S., A Finite Axiomatization for Fork Algebras, Logic Journal of the IGPL, Vol. 5, No. 3, 311{319, 1997.

8. Frias, M. F. and Orlowska, E.,A Proof System for Fork Algebras and its Applica-tions to Reasoning in Logics Based on Intuitionism, Logique et Analyse, vol. 150{ 151{152, pp. 239{284, 1995.

9. Frias, M. F. and Orlowska, E.,Equational Reasoning in Non{Classical Logics, Jour-nal of Applied Non Classical Logic, Vol. 8, No. 1{2, 1998.

10. Gries, D.,Equational logic as a tool, LNCS 936, Springer{Verlag, 1995, pp. 1-17. 11. Gries, D. and Schneider, F. B., A Logical Approach to Discrete Math., Springer{

Verlag, 1993.

12. Kaposi, A. and Myers, M., Systems, Models and Measures, Springer{Verlag Lon-don, Formal Approaches to Computing and Information Technology, 1994. 13. Lyndon, R., The Representation of Relational Algebras, Annals of Mathematics

(Series 2) vol.51 (1950), 707{729.

14. Maddux, R.D.,Topics in Relation Algebras, Doctoral Dissertation, Univ. Califor-nia, Berkeley, 1978, pp. iii+241.

15. Maddux, R.D.,Finitary Algebraic Logic, Zeitschr. f. math. Logik und Grundlagen d. Math. vol. 35, pp. 321{332, 1989.

16. Maddux, R.D.,Pair-Dense Relation Algebras, Transactions of the A.M.S., vol. 328, No. 1, pp. 83{131, 1991.

17. Maddux, R.D.,Relation Algebras, Chapter 2 of Relational Methods in Computer Science, Springer Wien New York, 1997.

18. Roberts, F. S.,Measurement Theory, with Applications to Decision{Making, Utility and the Social Sciences, Addison{Wesley, 1979.

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