... value^3.1

A VDM operation may also access and change a system state, but we will delay discussion of this idea until later in the chapter.

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... Databases^3.2

In C.J.Date's `An Introduction to Database Systems' [Date 90], an Integrity Constraint is defined as `a condition that all correct states of the database are required to satisfy' - which is essentially the meaning of a data type invariant

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... limit^4.1

Although this is a well-worn example, the stack provides a simple introduction to the use of the Sequence, and also to Algebraic Specifications in chapter 8

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... Intelligence^6.1

Refer to [Rich and Knight 91] for a good introduction to Planning.

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... chosen^6.2

Note that our interpretation of Goal and State are different. A state is interpreted with the implicit assumption that anything not asserted in it is assumed to be false. A goal, on the other hand, specifies a set of states - exactly all those which contain all the goal's literals.

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... implementation^7.1

In fact, systematically generating instances of soln from complete(soln, I(pp)) in a planner corresponds to very inefficient search strategies such as `breadth-first search'.

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... Axis.^8.1

©Hewlett-Packard 1988 [Coleman, Dollin, Gallimore, Arnold and Rush 88]

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... Miranda^9.1

©Miranda is a trademark of Research Software Ltd

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... algebra.^10.1

For completeness, and the mathematically curious, we append a formal definition of initiality. An algebra ${\cal I}$ is initial in a category C of algebras over a presentation if and only if ${\cal I}$ is a member of C and for every algebra $\cal{A}$ which belongs to C, a unique homomorphism from ${\cal I}$ to $\cal{A}$ exists. (A category of algebras with respect to a given presentation is a set of algebras denoted by the presentation together with a number of homomorphisms between these algebras. For our purposes, the category C is the variety over a presentation together with all possible homomorphisms between the algebras of the variety).

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... algebras.^10.2

An algebra $\cal F$ is final in a category $\cal C$ of algebras over a presentation if and only if $\cal F \in \cal C$and for each algebra $\cal A \in \cal C$, a unique homomorphism in $\cal C$ from $\cal{A}$ to $\cal F$ does exist.

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... Axis^11.1

©Hewlett-Packard Ltd. 1988

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... Axis^13.1

©Hewlett-Packard 1988

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... function^14.1

We can express the output n_i as

n_i = H(w_i1 n₁ + w_i2 n₂ + w_i3 n₃ - T_i)

where H(x - a) is the Heaviside step function defined by

$$H(x - a) = \left\{ \begin{array}{rl} 1 & \mbox{if $x \ge a$} \\ 0 & \mbox{otherwise} \end{array} \right.$$

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... assumption^15.1

Exactly the same idea of default persistence underpins the action representation which we built up in the Planning case study of chapter 6.

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... specifications^15.2

This is precisely the `referential transparency' characteristic of pure functional programming languages.

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