topical media & game development

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object-oriented programming

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Behavioral refinement

Ultimately, types are meant to specify behavior in an abstract way. To capture behavioral properties, we will generalize our notion of types as constraints to include behavioral descriptions in the form of logical assertions.

Behavioral refinement

types as behavior
verification
abstraction and representation
behavioral compositions

Additional keywords and phrases: behavioral subtypes, state transformers, correctness formulae, assertion logic, transition systems, invariants, formal specification

slide: Behavioral refinement

In this chapter we will explore the notion of behavioral (sub)types. First we characterize the trade-offs between statically imposed (typing) constraints and dynamic constraints resulting from the specification of behavioral properties. We will provide a brief introduction to the assertion logic underlying the verification of behavioral constraints. Also, we look at how we may characterize the behavior of object-based systems in a mathematical way. Then we will describe the duality between abstraction and representation in defining behavioral subtypes that define concrete realizations of abstract specifications. In particular, we specify the correspondence requirements for behavioral subtypes. We will conclude this chapter by discussing the problems involved in specifying behavioral compositions, and explore what specification techniques are available to model the behavior of object-based systems.

object-oriented programming

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Types as behavior

In the previous chapter we have developed a formal definition of types and the subtyping relation. However, we have restricted ourselves to (syntactic) signatures only, omitting (semantic) behavioral properties associated with function and object types.

Subtype requirements -- signature and behavior

preservation of behavioral properties

Safety properties -- nothing bad

invariant properties -- true of all states
history properties -- true of all execution sequences

slide: Subtyping and behavior

From a behavioral perspective, the subtype requirements (implied by the substitutability property) may be stated abstractly as the preservation of behavioral properties. According to [Liskov93], behavioral properties encompass safety properties (which express that nothing bad will happen) and liveness properties (which express that eventually something good will happen). For safety properties we may further make a distinction between invariant properties (which must be satisfied in all possible states) and history properties (which hold for all possible execution sequences). See slide 10-properties. Behavioral properties (which are generally not captured by the signature only) may be important for the correct execution of a program. For example, when we replace a stack by a queue (which both have the same signature if we rename push and insert into put, and pop and retrieve into get) then we will get incorrect results when our program depends upon the LIFO (last-in first-out) behavior of the stack. As another example, consider the relation between a type FatSet (which supports the methods insert, select and size) and a type IntSet (which supports the methods insert, delete, select and size). See slide 10-history.

Example -- $IntSet \not<= FatSet$

FatSet -- insert, select, size
IntSet -- insert, delete, select, size

History property -- not satisfied by $IntSet$

$\A s : FatSet. \A %f, %q : State. %f < %q /\ s \e dom (%q).$
\hspace{2cm} $\A x : Int. x \e s_{%f} => x \e s_{%q}$

slide: History properties -- example

With respect to its signature, IntSet merely extends FatSet with a delete method and hence could be regarded as a subtype of FatSet. However, consider the history property stated above, which says that for any (FatSet) s, when an integer x is an element of s in state

%f

then x will also be an element of s in any state

%q

that comes after

%f

. This property holds since instances of FatSet do not have a method delete by which elements can be removed. Now if we take this property into account, IntSet may not be regarded as a subtype of FatSet, since instances of IntSet may grow and shrink and hence do not respect the FatSet history property. This observation raises two questions. Firstly, how can we characterize the behavior of an object or function and, more importantly, how can we extend our notion of types to include a behavioral description? And secondly, assuming that we have the means to characterize the behavior of a function or object type, how can we verify that a subtype respects the behavioral constraints imposed by the supertype?

The answer to the first question is suggested by the observation that we may also express the constraints imposed by the signature by means of logical formulae that state the constraints as assertions which must be satisfied.

Types as behavior -- constraints

$x : 9..11$ $<=>$ $x : Int /\ 9 <= x <= 11$

Behavioral constraints -- signature versus assertions

$f(x:9..11) : 3..5 { ... }$


    int f(int x) {
       require( 9 <= x  && x <= 11 );
       ...
       promise( 3 <= result && result <= 5);
       return result;
       }

slide: Types and behavioral constraints

For example, we may express the requirement imposed by typing a variable as an element of an integer subrange also by stating that the variable is an integer variable that respects the bounds of the subrange. Similarly, we can express the typing constraints on the domain and range of a function by means of pre- and post-conditions asserting these constraints. See slide 10-constraints. More generally, we may characterize the behavior of a function type by means of pre- and post-conditions and the behavior of an object type by means of pre- and post-conditions for its methods and an invariant clause expressing the invariant properties of its state and behavior. Recall that this is precisely what is captured in our notion of contract, as discussed in section contracts. With regard to the second question, to verify behavioral properties (expressed as assertions) we need an assertion logic in the style of Hoare. Such a logic will be discussed in the next section. In addition, we need a way in which to verify that (an instance of) a subtype respects the behavioral properties of its supertype. In section correspondence we will give precise guidelines for a programmer to check the behavioral correspondence between two types.

object-oriented programming

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Verifying behavioral properties

The concern with program correctness stems from a period when projects were haunted by what was called the software crisis. Projects delivered software that contained numerous bugs and large programs seemed to become unmanageable, that is never error-free. One of the most radical ideas proposed to counteract the software crisis was to require that programs should formally be proven correct before acceptance. The charm of the idea, I find personally, is that programming in a way becomes imbued with the flavor of mathematics, which may in itself be one of the reasons that the method never became very popular.

State transformers

Proving the correctness of (imperative) programs is based on the notion of states and the interpretation of programs as state transformers. A state, in a mathematical sense, is simply a function that records a value for each variable in the program. For example, having a program S in which the (integer) variable i occurs, and a state

%f

, we may have

%f(i) = 3

. States may be modified by actions that result from executing the program, such as by an assignment of a value to a variable. We employ substitutions to modify a state. As before, substitutions may be defined by an equation, as given in slide 10-subst.

Substitutions


  \%f [x:=v](y)  ==
         v             if  $x = y$ 
         \%f (y)        otherwise

slide: Substitution

A substitution

%f[x:=v](y)

states that modifying

%f

by assigning the value v to the variable x then, for a variable y, the state

%f

will deliver v whenever y is identical to x and

%f(y)

otherwise.

When we have, for example, an assignment $i = 5$ then we have as the corresponding transition $%f -{i=5}-> %f'$ where $%f' = %f[i:=5]$ , that is $%f'$ is like $%f$ except for the variable i for which the value 5 will now be delivered. Whenever we have a sequence of actions $a_1,...,a_n$ then, starting from a state $%f_0$ we have corresponding state transformations resulting in states $%f_1,...,%f_{n-1}$ as intermediary states and $%f_n$ as the final state. Often the states $%f_0$ and $%f_n$ are referred to as respectively the input and output state and the program that results in the actions $a_1,...,a_n$ as the state transformer modifying $%f_0$ into $%f_n$ .

Program state -- $%f$

$%f \e %S : Var -> Value$

State transformations -- operations $a_1,a_2,...,a_n$

$%f_{0} -{a_1}-> %f_{1} ... -{a_n}-> %f_n$

Correctness formulae -- Hoare logic

${ P } S { Q }$

Verification

$%f_i -{a}-> %f_j /\ %f_i |= P => %f_j |= Q$

slide: The verification of state transformations

To characterize the actions that result from executing a program, we need an operational semantics that relates the programming constructs to the dynamic behavior of a program. We will study such a semantics in section \ref{10:behavior}. The requirements a program (fragment) has to meet may be expressed by using predicates characterizing certain properties of a program state. Then, all we need to do is check whether the final state of a computation satisfies these requirements. Predicates characterizing the properties of a state before and after executing a program (fragment) may be conveniently stated by correctness formulae of the form

{ P } S { Q }

where S denotes a program (fragment) and P and Q respectively the pre-condition and post-condition associated with S. A formula of the form

{ P } S { Q }

is true if, for every initial state

%f

that satisfies P and for which the computation characterized by S terminates, the final state

%f'

satisfies Q. This interpretation of

{ P } S { Q }

characterizes partial correctness, partial since the truth of the formula is dependent on the termination of S (which may, for example, for a while statement, not always be guaranteed). When termination can be guaranteed, then we may use the stronger notion of total correctness, which makes the truth of

{ P } S { Q }

no longer dependent on the termination of S. Pre- and post-conditions may also be used to check invariance properties. As an example, consider the following correctness formula:

{ s = i*(i+1)/2 } i = i + 1; s = s + i; { s = i*(i+1)/2 }

It states that the begin and end state of the computation characterized by

i = i + 1; s = s + i

is invariant with respect to the condition

s = i*(i+1)/2

. As an exercise, try to establish the correctness of this formula! To verify whether for a particular program fragment S and (initial) state

%f_i

satisfying P the correctness formula

{ P } S { Q }

holds, we need to compute the (final) state

%f_j

and check that Q is true for

%f_j

. In general, for example in the case of non-deterministic programs, there may be multiple (final) states resulting from the execution of S. For each of these states we have to establish that it satisfies (the post-condition) Q. We call the collection of possible computation sequences of a program fragment S the traces of S. Traces characterize the (operational) behavior of a program.

Assertion logic

Reasoning about program states based on the traces of a program may be quite cumbersome. Moreover, a disadvantage is that it relies to a great extent on our operational intuition of the effect of a program on a state. Instead, [Hoare69] has proposed using an axiomatic characterization of the correctness properties of programming constructs. An axiomatic definition allows us to prove the correctness of a program with respect to the conditions stating its requirements by applying the appropriate inference rules.

Axioms

assignment -- ${ Q [x:=e] } x = e { Q }$
composition -- ${ P } S1 { R } /\ { R } S2 { Q } => { P } S1;S2 { Q }$
conditional -- ${ P /\ b } S { Q } => { P } if (b) S { Q }$
iteration -- ${ I /\ b } S { I } => { I } while (b) S { I /\ \neg b }$

Consequence rules

$P -> R /\ { R } S { Q } => { P } S { Q }$
$R -> Q /\ { P } S { R } => { P } S { Q }$

Procedural abstraction

$m(x) |-> S(x) /\ { P } S(e) { Q } => { P } m(e) { Q }$

slide: The correctness calculus

In slide 10-correctness correctness axioms have been given for assignment, sequential composition, conditional statements and iteration. These axioms rely on the side-effect free nature of expressions in the programming language. Also, they assume convertibility between programming language expressions and the expressions used in the assertion language. The assignment axiom states that for any post-condition Q we can derive the (weakest) pre-condition by substituting the value e assigned to the variable x for x in Q. This axiom is related to the weakest pre-condition calculus introduced by [Dijkstra76]. It is perhaps the most basic axiom in the correctness calculus for imperative programs. As an example, consider the assignment

x = 3

and the requirement

{ P }   x = 3   { y = x }

. Applying the assignment axiom we have

{ y = 3 }   x = 3   { y = x }

. Consequently, when we are able to prove that P implies

y = 3

, we have, by virtue of the first consequence rule, proved that

{ P }   x = 3   { x = y }

. The next rule, for sequential composition, allows us to break a program (fragment) into parts. For convenience, the correctness formulae for multiple program fragments that are composed in sequential order are often organized as a so-called proof outline of the form

{ P }   S1   { R }   S2   { Q }

. When sufficiently detailed, proof outlines may be regarded as a proof. For example, the proof outline

{ s = i*(i+1)/2 }   i = i + 1;   { s + i = i * (i+1)/2 }   s = s + i;   { s = i*(i+1)/2 }

constitutes a proof for the invariance property discussed earlier. Clearly, the correctness formula for the two individual components can be proved by applying the assignment axiom. Using the sequential composition rule, these components can now be easily glued together. As a third rule, we have a rule for conditional statements of the form

\! if \! (b)   S

. As an example, consider the correctness formula

{ true }   if (x > y )   z = x;   { z > y }

. All we need to prove, by virtue of the inference rule for conditional statements, is that

{ x > y }   z = x   { z > y }

which (again) immediately follows from the assignment axiom. As the last rule for proving correctness, we present here the inference rule for iterative (while) statements. The rule states that whenever we can prove that a certain invariant I is maintained when executing the body of the while statement (provided that the condition b is satisfied) then, when terminating the loop, we know that both I and

\neg b

hold. As an example, the formula

{ true }

while ( $i>0$ ) i--;

{ i <= 0 }

trivially follows from the while rule by taking I to be true. Actually, the while rule plays a crucial role in constructing verifiable algorithms in a structured way. The central idea, advocated among others by [Gries], is to develop the algorithm around a well-chosen invariant. Several heuristics may be applied to find the proper invariant starting from the requirements expressed in the (output) predicate stating the post-condition. In addition to the assignment axiom and the basic inference rules related to the major constructs of imperative programming languages, we may use so-called structural rules to facilitate the actual proof of a correctness formula. The first structural (consequence) rule states that we may replace a particular pre-condition for which we can prove a correctness formula (pertaining to a program fragment S) by any pre-condition of which the original pre-condition is a consequence, in other words which is stronger than the original pre-condition. Similarly, we may replace a post-condition for which we know a correctness formula to hold by any post-condition that is weaker than the original post-condition. As an example, suppose that we have proved the formula

{ x >= 0 }   S   { x < 0 }

then we may, by simultaneously applying the two consequence rules, derive the formula

{ x > 0 }   S   { x <= 0 }

which amounts to strengthening the pre-condition and weakening the post-condition. The intuition justifying this derivation is that we can safely promise less and require more, as it were. Finally, the rule most important to us in the present context is the inference rule characterizing correctness under procedural abstraction. Assuming that we have a function m with formal parameter x (for convenience we assume we have only one parameter, but this can easily be generalized to multiple parameters), of which the (function) body consists of

S(x)

. Now, moreover, assume that we can prove for an arbitrary expression e the correctness formula

{ P }   S(e)   { Q }

, with e substituted for the formal parameter x in both the conditions and the function body, then we also have that

{ P }   m(e)   { Q }

, provided that P and Q do not contain references to local variables of the function m. In other words, we may abstract from a complex program fragment by defining a function or procedure and use the original (local) correctness proof by properly substituting actual parameters for formal parameters. The procedural abstraction rule, which allows us to employ functions to perform correct operations, may be regarded as the basic construct needed to verify that an object embodies a (client/server) contract.

object-oriented programming

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On the notion of behavior

The assertion logic presented in the previous section allows us to reason about the behavior of a system without explicitly generating the possible sequences of states resulting from the execution of the program. However, underlying the inference rules of our assertion logic we need a mathematical model for the operational behavior of a system. An operational model is needed to prove the soundness of the inference rules. Further, an operational model may aid in understanding the meaning of particular language constructs and their associated correctness rules. In the following we will sketch the construction of a transition system modeling the behavior of an object-based program. Studying the formal semantics is relevant to understanding object orientation only in so far as it provides a means with which to characterize the desired behavior of object creation and message passing in an unambiguous manner.

Transition system

A transition system for a program is a collection of rules that collectively describe the effect of executing the statements of the program. A labeled transition system is one that enables us to label the transition from one state to another by some label indicating the observable behavior of a program step. In the transition system defined below, we will employ states

%f

, which may be decorated by object identifiers

%a

, as in

%f^{%a}

. Object identifiers are created when creating a new instance of an object type

%t

. We assume newly created object identifiers to be unique. We assume that each object type

%t

has a constructor (which is a, possibly empty, statement that we write as

S_{%t}

) and an arbitrary number of methods m. Each method m is assumed to be defined by some statement, which we write as

S_m (e)

, for method calls of the form

m(e)

. Also we allow an object

%a

of type

%t

to have attributes or instance variables v that may be accessed (read-only) as

%a.v

for an object identifier

%a

x.v

for an object variable x (which must have

%a

as its value). To determine the visible behavior of a program, we will employ labels of the form

%a

(to denote the creation of an object

%a

) and

m_{%a}

(to indicate the invocation of a method m for object

%a

). We allow transitions to be labeled by sequences of labels that we write as

%l

and which are concatenated in the usual way. We will define a transition system for a simple language of which the syntax is defined in slide 10-syntax.

Expressions

syntax

$e ::= v | x.v$

Elementary statements

$s ::= v = e | x = new %t | x.m(e)$

Compound statement

$S ::= %e | s | S1;S2 | \! if (b) S | while (b) S$

slide: The syntax of a simple OO language

Expressions are either local variables v or object instance variables that we write as

x.v

, where x is an object variable. As elementary statements we have

v = e

(indicating the assignment of (the value of) an expression e to a local variable v),

x = new   %t

(which stands for the creation of a new object of type

%t

), and

x.m(e)

(which calls a method m with arguments e for object x). The object variable x is associated with an object identifier

%a

by the state

%f

in which the statement in which x occurs is executed. As compound statements we have an empty statement

%e

(which is needed for technical reasons), an elementary statement s (as defined above), a sequential composition statement, a conditional statement and an iteration statement, similar to that in the previous section. The transition rules for elementary statements are given in slide 10-rules.

Assignment

rules

$<< v = e, %f^{%a} >> -> << %e, %f [ %a.v := e_{%f} ] >>$

Object creation

\ffrac{ $<< S_{%t}, %f^{%a} >> \apijl{%l} << %e, %f' >>$ }{ $<< x \! = \! new %t, %f >> \! \apijl{ %a \. %l } \! << %e , %f'[ x \! := %a \! ]>>$ }

Method call

\ffrac{ $<< S_m(e), %f^{%a} >> \apijl{%l} << %e, %f' >>$ }{ $<< x.m(e), %f >> \apijl{ m_{%a} \. %l } << %e, %f' >>$ }

slide: Transition system -- rules

The assignment rule states that the assignment of (the value of) an expression e to a (local) variable v in state

%f

decorated by an object identifier

%a

results in the empty statement and the state

%f

modified by assigning the value of e in

%f

(which is written as

e_{%f}

) to the instance variable v of object

%a

. Hence, decorations allow us to work in the local environment of the object indicated by the decoration. The object creation rule states that if we assume a transition

<< S_{%t}, %f^{%a} >> \apijl{%l} << %e, %f' >>

(which states that the constructor for type

%t

executed in state

%f

decorated by

%a

results in the state

%f'

with behavior

%l

) then we may interpret the creation of a new

%t

object to result in behavior

%a \. %l

(where

%a

is the newly created object identifier) and state

%f'

in which the object variable x has the value

%a

. Finally, in a similar way, the method call rule states that if we assume a transition

<< S_m(e), %f^{%a} >> \apijl{%l} << %e, %f' >>

(which states that executing the statement

S_m(e)

, that is the code associated with method m and arguments e, for object

%a

in state

%f

, results in behavior

%l

and state

%f'

) then we may interpret the method call

x.m(e)

%f

as a transition to state

%f'

displaying behavior

m_{%a} \. %l

. The rules for object creation and method call already indicate that transition rules may be used to construct a complex transition from elementary steps. In other words, a transition system defines a collection of proof rules that allow us to derive (state) transitions and to characterize the behavior that may be observed. To obtain a full derivation, we need in addition to the rules for elementary statements the rules for compound statements listed in slide 10-compound.

Composition

compound

\ffrac{ $<< S_1, %f >> \apijl{ %l_1 } << %e, %f' >>$ \ifsli{\hspace{-0.5cm}}{\hspace{1cm}} $<< S_2, %f' >> \apijl{ %l_2 } << %e, %f'' >>$ }{ $<< S_1;S_2, %f >> \apijl{ %l_1 \. %l_2 } << %e, %f'' >>$ }

Conditional

$<< \! if (b) S, %f>> -> << %e, %f>>$ if $%f(b) == false$
\ffrac{ $<< S, %f >> \apijl{%l} << %e, %f' >>$ }{ $<< \! if (b) S, %f >> \apijl{%l} << %e, %f' >>$ } if \ifsli{ $%f(b)$ }{ $%f(b) == true$ }

Iteration

$<< while \; (b) S, %f>> -> << %e, %f>>$ if $%f(b) == false$
\ffrac{ $<< S, %f >> \apijl{%l} << %e, %f' >>$ }{ $<< while (b) S, %f >> \apijl{%l} << while (b) S, %f' >>$ } \ifsli{}{ if $%f(b) == true$ }

slide: Transition system -- compound statement

The composition rule states that if a statement

S_1

transforms

%f

into

%f'

with behavior

%l_1

and

S_2

transforms

%f'

into

%f''

with behavior

%l_2

then the compound statement

S_1;S_2

transforms

%f

into

%f''

with behavior

%l_1 \. %l_2

. The conditional rules state that, dependent on the value of the boolean b, the statement

if   (b)   S

has either no effect or results in a state

%f'

assuming that S transforms

%f

into

%f'

with behavior

%l

. The iteration rules state that dependent on the value of the boolean b the statement

while   (b)   S

has either no effect or results in a state

%f'

assuming that S transforms

%f

into

%f'

with behavior

%l

. In contrast to the conditional, an iteration statement is repeated when b is true, in accordance with our operational understanding of iteration.

Example

In our rules we have made a distinction between unadorned states

%f

and states

%f^{%a}

decorated with an object identifier

%a

. This reflects the distinction between the execution of a program fragment in a global context and a local context, within the confines of a particular object

%a

. Assume, for example, that we have defined a counter type ctr with a method inc that adds one to an instance variable n. In slide 10-t-example, a derivation is given of the behavior resulting from a program fragment consisting of the creation of an instance of ctr, a method call to inc and the assignment of the value of the attribute n of the counter to a variable v.

Program

$x = new ctr; x.inc(); v = x.n$

Transitions

$<< x = new ctr, %f_1 >> \apijl{ctr_1} << %e, %f_1[ x := ctr_1 ]>>$

$[1]$

$<< n = n + 1, %f_2^{%a} >> -> <<%e, %f_2[%a.n = %a.n + 1]>>$

$[2]$

$<< x.inc(), %f_2 >>$ $\apijl{inc_{%a}$ $<< %e, %f_2[%f_2(x).n := %f_2(x).n ]>>$

$[2']$

$<< v = x.n, %f_3>> -> << %e, %f_3[v := %f_3(x).n] >>$

$[3]$

Trace

$%f \apijl{%l} %f'$ with $%f = %f_1$ , $%f' = %f_3$ and $%l = ctr_1 \. inc_{ %a }$

slide: Transitions -- example

To derive the transition

%f \apijl{%l} %f'

corresponding with the program fragment

x = new   ctr; x.inc(); v = x.n

we must dissect the fragment and construct transitions for each of its (elementary) statements as shown in

[1]

[2]

and

[3]

. The second statement, the method call

x.inc()

, needs two transitions

[2]

and

[2']

, of which the first represents the execution of the body of inc in the local context of the object created in

[1]

and the second represents the effect of the method call from a global perspective. For the first statement, we have assumed that the constructor for ctr is empty and may hence be omitted. Notice that the object identifier

%a

(introduced in

[1]

) is assumed in

[2]

to effect the appropriate local changes to n. After constructing the transitions for the individual statements we may compose these transitions by applying the composition rule and, in this case, identifying

%f_2

with

%f_1[x := %a ]

and

%f_3

with

%f_2[ %a.n := %a.n + 1]

. As observable behavior we obtain

ctr_1 \. inc_{%a}

(where

ctr_1 = %a

), which represents the creation of a counter and its subsequent modification by inc.

Discussion

Transition systems, such as the one given above, were originally introduced as a means to model the behavior of CSP. They have been extensively used to model the operational semantics of programming languages, including concurrent and object-oriented languages. See, for example, [ABKR89] and [Eliens92]. In [AptO], transition systems have been used as a model to prove the soundness and completeness of correctness rules for concurrent programming constructs. Also in [AmBo93], transition systems are used to demonstrate the validity of a proof system for a parallel object-oriented programming language. The interested reader is invited to explore the sources mentioned for further study.

object-oriented programming

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Objects as behavioral types

subsections:

Abstraction and representation
Correspondence

A syntax-directed correctness calculus as presented in section \ref{10:verification} provides, in principle, excellent support for a problem-oriented approach to program development, provided that the requirements a program has to meet can be made explicit in a mathematical, logical framework. When specifying requirements, we are primarily interested in the abstract properties of a program, as may be expressed in some mathematical domain. However, when actually implementing the program (and verifying its correctness) we mostly need to take recourse to details we do not wish to bother with when reasoning on an abstract level. In this section we will discuss how we may verify that an abstract type is correctly implemented by a behavioral (implementation) subtype, following [Am90]. Also, we will define precise guidelines for determining whether two (behavioral) types satisfy the (behavioral) subtype relation, following [Liskov93].

Abstraction and representation

In [Am90] a proposal is sketched to define the functionality of objects by means of behavioral types. Behavioral types characterize the behavioral properties of objects in terms of (possible) modifications of an abstract state. So as to be able to ignore the details of an implementation when reasoning about the properties of a particular program, we may employ a representation abstraction function which maps the concrete data structures and operations to their counterparts in the abstract domain.

Abstract data types -- representation function

Representation function -- abstraction

$%a(%c)( %a(%f)) = %a(%f')$ $<=>$ $%c (%f) = %f'$

slide: Abstraction and representation

The diagram in slide 10-ADT pictures the reasoning involved in proving that a particular implementation is correct with respect to a specification in some abstract mathematical domain. Assume that we have, in the concrete domain, an action a that corresponds with a state transformation function

%c

. Now assume that we have a similar operation in the abstract domain, that we will write as

%a(a)

, with a corresponding state transformation function

%a(%c)

. To prove that the concrete operation a correctly implements the abstract operation

%a(a)

, we must prove that the concrete state modification

%c

resulting from a corresponds with the modification that occurs in the abstract domain. Technically speaking, we must prove that the diagram above commutes, that is, that

%c(%f) = %f' <=> %a(%c)(%a(%f)) = %a(%f')

whenever we have that

%f -{a}-> %f'

. To prove that a particular implementation a respects the abstract operation

%a(a)

, for which we assume that it has abstract pre- and post-conditions

%a(P)

and

%a(Q)

, we must find a representation invariant I and (concrete) pre- and post-conditions P and Q for which we can prove that

%a(P) /\ I => P

and that

%a(Q) /\ I => Q

. Furthermore, the representation invariant I must hold before and after the concrete operation a. The proof strategy outlined above is of particular relevance for object-oriented program development, since the behavior of objects may, as we have already seen, be adequately captured by contracts. As an additional advantage, however, the method outlined enables us to specify the behavior of an object in a more abstract way than allowed by contracts as supported by Eiffel.

Realization

As an example, consider the specification of a generic stack as given in slide 10-specification. The specification of the stack is based on the (mathematically) well-known notion of sequences. We distinguish between empty sequences, that we write as

<< >>

, and non-empty (finite) sequences, that we write as

< >

. Further, we assume to have a concatenation operator for which we define

s \. <<>> = <<>> \. s = s

and

< > \. < > = < >

. A sequence is employed to represent the state of the stack.

Sequences -- abstract domain

empty sequence -- $<< >>$
concatenation -- $< > \. < >$ $=$ $< >$

Specification


  type stack T {
   $s : seq   T$ ;
  axioms:
   ${ true } push(t:T) { s' = s \. < > }$ 
   ${ s \neq << >> } pop() { s = { s' } \. < > }$ 
  };

slide: The specification of a stack

The operations push and pop may conveniently be defined with reference to the sequence representing the (abstract) state of the stack. We use s and

s'

to represent the state respectively before and after the operation. The operations themselves are completely specified by their respective pre- and post-conditions. Pushing an element e results in concatenating the one-element sequence

< >

to the stacks state. For the operation pop we require that the state of the stack must be non-empty. The post-condition specifies that the resulting state

s'

is a prefix of the original state, that is the original state with the last element (which is returned as a result) removed. To prove that a particular implementation of the stack is conformant with the type definition given above we must prove that

{ I /\ pre(%a(m(e))) } m(e) { I' /\ post(%a(m(e))) }

for both methods push and pop. These proofs involve both an abstraction function

%a

and a representation invariant I, relating the abstract state of the stack to the concrete state of the implementation. Now consider an implementation of the generic stack in C++, as given in slide 10-realization.


  template    implementation  
  class as {
  int t;
  T a[MAX];
  public:
  as() { t = 0; }
  void push(T e) { 
         require(t0); return a[--t]; }
  
  invariant:
  	0 <= t && t < MAX;
  };

slide: The realization of a stack

To prove that this implementation may be regarded as an element of the (abstract) type stack, we must find a representation (abstraction) function to map the concrete implementation to the abstract domain, and further we must specify a representation invariant that allows us to relate the abstract properties to the properties of the implementation. For the implementation in slide 10-realization, the abstraction function

%a

simply creates the sequence of length t, with elements

a[0],...,a[t-1]

. The representation invariant, moreover, gives an explicit definition of this relation. See slide 10-absf.

Abstraction function

$%a(a,t) =$

Representation invariant

$I(a,t,s) == t = length(s) /\ t >= 0 /\ s = %a(a,t)$

slide: Abstraction function and representation invariant

In order to verify that our implementation of the abstract data type stack is correct (that is as long as the bound MAX is not exceeded), we must show, given that the representation invariant holds, that the pre-conditions of the concrete operations imply the pre-conditions of the corresponding abstract operations, and, similarly, that the post-conditions of the abstract operations imply the post-conditions of the concrete operations. First, we show that for the operation push the post-condition of the abstract type specification is indeed stronger than the (implicit) post-condition of the implementation. This is expressed by the following formula.


   $s' = s \. /\  I(a',t',s') => t' = t + 1 /\ a'[t'] = e$

Since we know that

I(a',t',s')

, we may derive that

t' = t + 1

and

a'[t'] = e

To establish the correctness of the operation pop, we must prove that the pre-condition specified for the abstract operation is indeed stronger than the pre-condition specified for the concrete operation, as expressed by the formula $I(a,t,s) /\ s \not= <> => t > 0$ It is easy to see that $t > 0$ immediately follows from the requirement that the sequence is non-empty. Finally, to prove that the operator pop leaves the stack in a correct state, we must prove that $s = s' \. /\ I(a',t',s') => result = a'[t] /\ t' = t - 1$ which is done in a similar manner as for push.

The correspondence relation

Behavioral refinement is not restricted to the realization of abstract specifications. We will now look at a definition of behavioral refinement, following [Liskov93], that may serve as a guideline for programmers to define behavioral subtypes, both abstract and concrete, including subtypes extending the behavioral repertoire of their supertypes. In [Liskov93] the relation between behavioral types is explained by means of a so-called correspondence mapping, that relates a subtype to its (abstract) supertype.

Correspondence

$<< %a, %r, %x >>$

$%a$ abstraction -- maps $%s$ values to $%t$ values
$%r$ renaming -- maps subtype to supertype methods
$%x$ extension -- explains effects of extra methods

slide: The subtype correspondence mapping

A correspondence mapping is a triple consisting of an abstraction function

%a

(that projects the values of the subtype on the value domain of the supertype), a renaming

%r

(that defines the relation between methods defined in both types) and an extension map

%x

(that defines the meaning of additional methods). See slide 10-correspondence. Technically, the function

%a

must be onto, that is each value of the supertype domain must be representable by one or more values of the subtype domain. Generally, when applying the abstraction function, we loose information (which is irrelevant from the perspective of the supertype), for example the specific ordering of items in a container.

Signature

$%s <= %t$

$dom( m_{%t} ) <= dom( m_{%s} )$
$ran( m_{%s} ) <= ran( m_{%t} )$

Behavior

$pre( m_{%t} )[ x_{%t} := %a( x_{%s} )] => pre( m_{ %s } )$
$post( m_{ %s } ) => post( m_{ %t } )[ x_{ %t } := %a( x_{ %s } ) ]$
$invariant( %s )$ $=>$ $invariant( %t )[ x_{ %t } := %a( x_{ %s } ) ]$

Extension -- diamond rule

$%x(x.m(a)) = %p_{m}$

$%f \apijl{ x.m(a) } %f' /\ %f \apijl{ %p_{m} } %Q$
$=>$ $%a( x_{ %f } ) = %a( x_{ %Q } )$

slide: Behavioral subtyping constraints

To determine whether a type

%s

is a (behavioral) subtype of a type

%t

, one has to define a correspondence mapping

<< %a, %r, %x >>

and check the issues listed in slide 10-subtyping. First, syntactically, we must check that the signature of

%s

and

%t

satisfy the (signature) subtyping relation defined in the previous chapter. In other words, for each method m associated with the object type

%t

(which we call

m_{%t}

), and corresponding method

m_{%s}

(which is determined by applying the renaming

%r

) we must check the (contravariant) function subtyping rule, that is

dom( m_{%t} ) <= dom( m_{ %s } )

and

ran( m_{%s} ) <= ran( m_{ %t } )

, where ran is the range or result type of m. Secondly, we must check that the behavioral properties of

%s

respect those of

%t

. In other words, for each method m occurring in

%t

we must check that

pre( m_{%t} )[ x_{%t} := %a( x_{%s} )] => pre( m_{%s} )

and that

post( m_{ %s } ) => post( m_{ %t } )[ x_{ %t } := %a( x_{ %s } ) ]

. Moreover, the invariant characterizing

%s

must respect the invariant characterizing

%t

, that is

invariant( %s ) => invariant( %t )[ x_{ %t } := %a( x_{ %s } ) ]

. The substitutions

[ x_{ %t } := %a( x_{ %s } )]

occurring in the behavioral rules are meant to indicate that each variable of type

%t

must be replaced by a corresponding variable of type

%s

to which the abstraction function is applied (in order to obtain a value in the (abstract) domain of

%t

). And thirdly, in the final place, it must be shown that the extension map

%x

is well-defined. The extension map must be defined in such a way that each method call for an object x of type

%s

, which we write as

x.m(a)

where a represents the arguments to the call, is mapped to a program

%p_{m}

in which only calls appear to methods shared by

%s

and

%t

(modulo renaming) or external function or method calls. In addition the diamond rule must be satisfied, which means that the states

%f'

and

%q

resulting from applying respectively

x.m(a)

and

%p_{m}

in state

%f

must deliver identical values for x from the perspective of

%t

, that is after applying the abstraction function. In other words, extension maps allow us to understand the effect of adding new methods and to establish whether they endanger behavioral compatibility. In [Liskov93] a distinction is made between constructors (by which objects are created), mutators (that modify the state or value of an object) and observers (that leave the state of an object unaffected). Extension maps are only needed for mutator methods. Clearly, for observer methods the result of

%x

is empty, and constructors are taken care of by the abstraction function.

Behavioral subtypes

The behavioral subtyping rules defined above are applicable to arbitrary (sub)types, and not only to (sub)types defined by inheritance. As an example, we will sketch (still following [Liskov93]) that a stack may be defined as a behavioral subtype of the type bag. Recall that a bag is a set allowing duplicates. See slide 10-ex-subtype.

Behavioral subtypes

$stack <= bag$

bag -- put, get
stack -- push, pop, settop

Representation

bag -- $<< elems, bound >>$ \zline{multiset}
stack -- $<< items, limit >>$ \zline{sequence}

Behavior -- put/push


   $put(i):$ 
     $require   size( b.elems ) < b.bound$ 
     $promise   b' = << b. elems \uplus { i }, b.bound >>$ 
  
   $push(i):$ 
     $require   length( s.items ) < s.limit$ 
     $promise   s' = << s.items \. i, s.limit >>$

slide: Behavioral subtypes -- example

Let the type bag support the methods

put(i:Int)

and

get():Int

and assume that the type stack supports the methods

push(i:Int)

pop():Int

and in addition a method

settop(i:Int)

that replaces the top element of the stack with its argument. Now, assume that a bag is represented by a pair

< >

, where elems is a multiset (which is a set which may contain multiple elements of the same value) and bound is an integer indicating the maximal number of elements that may be in the bag. Further, we assume that a stack is represented as a pair

<< items, limit >>

, where items is a sequence and limit is the maximal length of the sequence. For example

<< { 1,2,7,1 }, 12 >>

is a legal value of bag and

<< 1 \. 2 \. 7 \. 1, 12 >>

is a legal value of stack. The behavioral constraints for respectively the method put for bag and push for stack are given as pre- and post-conditions in slide 10-ex-subtype. To apply put, we require that the size of the multiset is strictly smaller than the bound and we ensure that the element i is inserted when that pre-condition is satisfied. The multi-set union operator

\uplus

is employed to add the new element to the bag. Similarly, for push we require the length of the sequence to be smaller than the limit of the stack and we then ensure that the element is appended to the sequence. As before, we use the primed variables

b'

and

s'

to denote the value of respectively the bag b and the stack s after applying the operations, respectively put and push. Proceeding from the characterization of bag and stack we may define the correspondence mapping

<< %a, %r, %x >>

as in slide 10-ex-correspondence.

Correspondence

$stack -> bag$

%a(< >) =

<< mk_set( items ), limit >>

where


   $mk_set($ \%e ) = \0
   $mk_set( e \. s ) = mk_set(s) \uplus { e }$

$%r( push ) = put$ , $%r( pop ) = get$
$%x(s.settop(i))$ $=$ $s.pop(); s.push(i);$

slide: Behavioral subtypes -- correspondence

To map the representation of a stack to the bag representation we use the function

mk_set

(which is inductively defined to map the empty sequence to the empty set and to transform a non-empty sequence into the union of the one-element multiset of its first element and the result of applying

mk_set

to the remaining part). The stack limit is left unchanged, since it directly corresponds with the bound of the bag. The renaming function

%r

maps push to put and pop to get, straightforwardly. And, the extension map describes the result of

settop(i)

as the application of (subsequently)

pop()

and

push(i)

Proof obligations -- push/put

$size( %a(s). elems ) < %a(s).bound$
$=>$ $length( s.items ) < s.limit$
$s' = << s.items \. i, s.limit >>$
$=>$ $%a(s')$ $=$ $<< %a(s).elems \uplus { i }, %a(s).bound >>$

slide: Behavioral subtypes -- proof obligations

With respect to the behavioral definitions given for push and put we have to verify that

pre( put(i) )[ b := %a(s) ] => pre( push(i) )

and that

post( push(i) ) => post( put(i) )[ b := %a(s) ]

. These conditions, written out fully in slide 10-ex-proof, are easy to verify. Generally, a formal proof is not really necessary to check that two types satisfy the behavioral subtype relation. As argued in [Liskov93], the definition of the appropriate behavioral constraints and the formulation of a correspondence mapping is already a significant step towards verifying that the types have the desired behavioral properties.

object-oriented programming

[] readme course preface 1 2 3 4 5 6 7 8 9 10 11 12 appendix lectures resources

Specifying behavioral compositions

\label{Cooperation} The notion of behavioral types may be regarded as the formal underpinning of the notion of contracts specifying the interaction between a client and server (object); cf. [Meyer93]. Due to the limited power of the (boolean) assertion language, contracts as supported by Eiffel are more limited in what may be specified than (a general notion of) behavioral types. However, some of the limitations are due, not to limitations on the assertion language, but to the local nature of specifying object behavior by means of contracts. See also [Meyer93]. To conclude this chapter, we will look at an example illustrating the need to specify global invariants. Further we will briefly look at alternative formalisms for specifying the behavior of collections of objects, and in particular we will explore the interpretation of contracts as behavioral compositions.

Global invariants

Invariants specify the constraints on the state of a system that must be met for the system to be consistent. Clearly, as elementary logic teaches us, an inconsistent system is totally unreliable. Some inconsistencies cannot be detected locally, within the scope of an object, since they may be caused by actions that do not involve the object directly. An example of a situation in which an externally caused inconsistent object state may occur is given in slide 10-invariants. (The example is taken from [Meyer93], but rephrased in C++.)

Problem -- dynamic aliasing


  class A {
  public:
  A() { forward = 0; }
  attach(B* b) { forward = b; b->attach(this); }
  bool invariant() {
     return !forward || forward->backward == this;
     }
  private:
  B* forward;
  };
  
  class B {
  public:
  B() { backward = 0; }
  attach(A* a) { backward = a; }
  bool invariant() {
     return !backward || backward->forward == this;
     }
  private:
  A* backward;
  };

slide: Establishing global invariants

When creating an instance of A, the forward pointer to an instance of B is still empty. Hence, after creation, the invariant of the object is satisfied. Similarly when, after creating an instance of B, this instance is attached to the forward pointer, and as a consequence the object itself is attached to the backward pointer of the instance of B. After this, the invariant is still satisfied. However, when a second instance of A is created, for which the same instance of B is attached to the forward pointer, the invariant for this object will hold, but as a result the invariance for the first instance of A will become violated. See below.


  A a1, a2; B b;
  a1.attach(b);
  a2.attach(b); // violates invariant a1

This violation cannot be detected by the object itself, since it is not involved in any activity. Of course, it is possible to check externally for the objects not directly involved whether their invariants are still satisfied. However, the cost of exhaustive checking will in general be prohibitive. Selective checking is feasible only when guided by an adequate specification of the possible interferences between object states.

Specifying interaction

Elementary logic and set-theory provide a powerful vehicle for specifying the behavior of a system, including the interaction between its components. However, taking into account that many software developers prefer a more operational mode of thinking when dealing with the intricacies of complex interactions, we will briefly look at formalisms that allow a more explicit specification of the operational aspects of interaction and communication, yet support to some extent to reason about such specifications. See slide 10-interaction.

Contracts -- behavioral compositions

interaction

specification, refinement, conformance declarations

Scripts -- cooperation by enrollment

roles, initialization/termination protocols, critical role set

Multiparty interactions -- communication primitive

frozen state, fault-tolerance, weakening synchrony

Joint action systems -- action-oriented

state charts, refinement, superposition

slide: Specifying interactions

In [HHG90], a notion of behavioral contracts is introduced that allows for characterizing the behavior of compositions of objects. Behavioral contracts fit quite naturally in the object oriented paradigm, since they allow both refinement and (type) conformance declarations. See below. Somewhat unclear, yet, is what specification language the behavioral contracts formalism is intended to support. On the other hand, from an implementation perspective the interactions captured by behavioral contracts seem to be expressible also within the confines of a class system supporting generic classes and inheritance. A similar criticism seems to be applicable to the formalism of (role) scripts as proposed in [Francez]. Role scripts allow the developer to specify the behavior of a system as a set of roles and the interaction between objects as subscribing to a role. In contrast to behavioral contracts, the script formalism may also be applied to describe the behavior of concurrently active objects. In particular, the script formalism allows for the specification of predefined initialization and termination policies and for the designation of a so-called critical role set, specifying the number and kind of participants minimally required for a successful computation. Also directed towards the specification of concurrent systems is the multi-party interactions formalism proposed in [Evangelist], which is centered around a (synchronous) communication primitive allowing multiple objects to interact simultaneously. The notion of frozen state (which may be understood as an invariance requirement that holds during the interaction) may be useful in particular for the specification of fault-tolerant systems. An interesting research issue in this respect is to what extent the assumption of synchrony may be weakened in favor of efficiency. A rather different orientation towards specifying the interaction between collections of concurrently active objects is embodied by the joint action systems approach described in [Kurki]. Instead of relying on the direct communication between objects, joint action systems proceed from the assumption that there exists some global decision procedure that decides which actions (and interactions) are appropriate.

Joint action systems


  action  $service()$  by client c; server s is
  when c.requesting && s.free do
  	<body>

slide: Specifying actions -- example

An example of an action specification is given in slide 10-action. Whether the service is performed depends upon the state of both the client and the server object selected by the action manager. [Kurki] characterize their approach as action-oriented to stress the importance of specifying actions in an independent manner (as entities separate from classes and objects). An interesting feature of the joint action systems approach is that the behavior of individual objects is specified by means of state charts, a visual specification formalism based on [Harel87]. The specification formalism adopted gives rise to interesting variants on the object-oriented repertoire, such as inheritance and refinement by superposition. From a pragmatic viewpoint, the assumption of a global manager seems to impose high demands on system resources. Yet, as a specification technique, the concept of actions may turn out to be surprisingly powerful. In summary, this brief survey of specification formalisms demonstrates that there is a wide variety of potentially useful constructs that all bear some relevance to object-oriented modeling, and as such may enrich the repertoire of (object-oriented) system developers.

Contracts as protocols of interaction

Contracts as supported by Eiffel and Annotated C++ are a very powerful means of characterizing the interaction between a server object and a client object. However, with software becoming increasingly complex, what we need is a mechanism to characterize the behavior of collections or compositions of objects as embodied in the notion of behavioral contracts as introduced in [HHG90]. A contract (in the extended sense) lists the objects that participate in the task and characterizes the dependencies and constraints imposed on their mutual interaction. For example, the contract model-view, shown below (in a slightly different notation than the original presentation in [HHG90]), introduces the object model and a collection of view objects. Also, it characterizes the minimal assumptions with respect to the functionality these objects must support and it gives an abstract characterization of the effect of each of the supported operations.


  contract model-view< V > {    MV(C)  
  subject : model supports [
   state : V;
   value( val : V ) $|->$ [state = val]; notify(); 
   notify() $|->$ $\forall v \e   $views $\bl$ v.update(); 
   attach( v : view ) $|->$ v $\e$ views;
   detach( v : view ) $|->$ v $\not\e$ views;
   ]
  views : set where view supports [
   update() $|->$ [view reflects state];
   subject( m : model ) $|->$ subject = m;
   ]
  invariant:
   $\forall$ v $\e$ views $\bl$ [v reflects subject.state]
  instantiation:
   $\forall$ v $\e$ views $\bl$ subject.attach(v) & v.subject(subject);
   subject.notify();
  }

slide: The Model-View contract

To indicate the type of variables, the notation $v : type$ is used expressing that variable v is typed as type. The object subject of type model has an instance variable state of type V that represents (in an abstract fashion) the value of the model object. Methods are defined using the notation

method $|->$ action

Actions may consist either of other method calls or conditions that are considered to be satisfied after calling the method. Quantification as for example in

$\A$ v $\e$ views $\bl$ v.update()

is used to express that the method $update()$ is to be called for all elements in views. The model-view contract specifies in more formal terms the MV part of the MVC paradigm discussed in section MVC. Recall, that the idea of a model-view pair is to distinguish between the actual information (which is contained in the model object) and the presentation of that information, which is taken care of by possibly multiple view objects. The actual protocol of interaction between a model and its view objects is quite straightforward. Each view object may be considered as a handler that must minimally have a method to install a model and a method update which is invoked, as the result of the model object calling notify, whenever the information contained in the model changes. The effect of calling $notify()$ is abstractly characterized as a universal quantification over the collection of view object. Calling $notify()$ for subject results in calling $update()$ for each view. The meaning of $update()$ is abstractly represented as

update() $|->$ [view reflects state];

which tells us that the state of the subject is adequately reflected by the view object. The invariant clause of the model-view contract states that every change of the (state of the) model will be reflected by each view. The instantiation clause describes, in a rather operational way, how to initialize each object participating in the contract. In order to instantiate such a contract, we need to define appropriate classes realizing the abstract entities participating in the contract, and further we need to define how these classes are related to their abstract counterparts in the contract by means of what we may call, following [HHG90], conformance declarations. Conformance declarations specify, in other words, how concrete classes embody an abstract role, in the same sense as in in the realization of a partial type by means of inheritance.

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