![Algebraic specification
In the algebraic specification technique an
object class or type is specified in terms
of relationships existing between the
operations defined on that type. It was
first brought into prominence by Guttag
[1980, 1985] in specification of abstract
data types.](https://image.slidesharecdn.com/module3-l7se-181201151913/85/Software-Engineering-Module-3-Lesson7-2-320.jpg)
Software Engineering - Module 3: Lesson7
- 1. Module 3 Requirements Analysis and Specification Lesson 7: Algebraic Specification
- 2. Algebraic specification In the algebraic specification technique an object class or type is specified in terms of relationships existing between the operations defined on that type. It was first brought into prominence by Guttag [1980, 1985] in specification of abstract data types.
- 3. Representation of algebraic specification Essentially, algebraic specifications define a system as a heterogeneous algebra. A heterogeneous algebra is a collection of different sets on which several operations are defined. Traditional algebras are homogeneous. A homogeneous algebra consists of a single set and several operations; {I, +, -, *, /}.
- 4. Types section: • Exceptions section This section gives the names of the exceptional conditions that might occur when different operations are carried out. These exception conditions are used in the later sections of an algebraic specification. • Syntax section This section defines the signatures of the interface procedures. The collection of sets that form input domain of an operator and the sort where the output is produced are called the signature of the operator. For example: append : queue x element → queue
- 5. • Equations section This section gives a set of rewrite rules (or equations) defining the meaning of the interface procedures in terms of each other. In general, this section is allowed to contain conditional expressions. For example: isempty(create()) = true Set Of Required Operations. 1. Basic construction operators. These operators are used to create or modify entities of a type. The basic construction operators are essential to generate all possible element of the type being specified.
- 6. 2. Extra construction operators. These are the construction operators other than the basic construction operators. 3. Basic inspection operators. These operators evaluate attributes of a type without modifying them, e.g., eval, get, etc. Let S be the set of operators whose range is not the data type being specified. The set of the basic operators S1 is a subset of S, such that each operator from S-S1 can be expressed in terms of the operators from S1.
- 7. “ 4. Extra inspection operators. These are the inspection operators that are not basic inspectors. Develop algebraic specification of simple problems A simple way to determine whether an operator is a constructor (basic or extra) or an inspector (basic or extra) is to check the syntax expression for the operator. If the type being specified appears on the right hand side of the expression then it is a constructor, otherwise it is an inspection operator.
- 8. Example: Let us specify a FIFO queue supporting the operations create, append, remove, first, and isempty where the operations have their usual meaning. Types: defines queue uses boolean, integer Exceptions: underflow, novalue Syntax: 1. create : φ → queue 2. append : queue x element → queue 3. remove : queue → queue + {underflow} 4. first : queue → element + {novalue} 5. isempty : queue → boolean
- 9. Equations: 1. isempty(create()) = true 2. isempty((append(q,e)) = false 3. first(create()) = novalue 4. first(append(q,e)) = is isempty(q) then e else first(q) 5. remove(create()) = underflow 6. remove(append(q,e)) = if isempty(q) then create() else append(remove(q),e) Properties of algebraic specifications • Completeness: This property ensures that using the equations, it should be possible to reduce any arbitrary sequence of operations on the interface procedures. There is no simple procedure to ensure that an algebraic specification is complete.
- 10. • Finite termination property: This property essentially addresses the following question: Do applications of the rewrite rules to arbitrary expressions involving the interface procedures always terminate? For arbitrary algebraic equations, convergence (finite termination) is undecidable. But, if the right hand side of each rewrite rule has fewer terms than the left, then the rewrite process must terminate. • Unique termination property: This property indicates whether application of rewrite rules in different orders always result in the same answer.
- 11. Structured specification • Incremental specification. The idea behind incremental specification is to first develop the specifications of the simple types and then specify more complex types by using the specifications of the simple types. • Specification instantiation. This involves taking an existing specification which has been developed using a generic parameter and instantiating it with some other sort.
- 12. Advantages and disadvantages of algebraic specifications Algebraic specifications have a strong mathematical basis and can be viewed as heterogeneous algebra. Therefore, they are unambiguous and precise. Using an algebraic specification, the effect of any arbitrary sequence of operations involving the interface procedures can automatically be studied.
- 13. Executable specification language (4GLs). If the specification of a system is expressed formally or by using a programming language, then it becomes possible to directly execute the specification. However, executable specifications are usually slow and inefficient, 4GLs3 (4th Generation Languages) are examples of executable specification languages. 4GLs are successful because there is a lot of commonality across data processing applications. 4GLs rely on software reuse, where the common abstractions have been identified and parameterized. Example of a 4GL is Structured Query Language (SQL).