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Software Engineering 3: Domains, Requirements, and Software Design

Dines Bjørner

Chapter 6

On Defining and on Definitionss - all with Video Answers

Educators

Chapter Questions

Problem 1

For each of these three exercises keep repeating your solution until it, in your own considered opinion, comes close enough to ours.

Definition, Definiens and Definiendum. Write down in your mother tongue a characterisation of what a definition consists of and name its parts and characterise those parts.

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01:13

Problem 2

For each of these three exercises keep repeating your solution until it, in your own considered opinion, comes close enough to ours.
Basis Clause, Inductive Clause and Extremal Clause. What role do the basis, the inductive and the extremal clauses serve in definitions?

Kashif Qureshi
Kashif Qureshi
Numerade Educator
01:06

Problem 3

For each of these three exercises keep repeating your solution until it, in your own considered opinion, comes close enough to ours.
What Is Art? How many characterisations were given of the concept of art? Try reformulate the essence of each of these characterisations.

Ernest Castorena
Ernest Castorena
Numerade Educator
04:44

Problem 4

Mostly make sense when using this book in a formal course over this volume. The exercises all illustrate the issue of proper definitions, also formally. Anyway, we have hinted at some definitions in our formulations of the exercises. When using this book in an informal course over this volume try reformulate the definitions along the lines of: $A$ tree consists of a root and a finite set of zero, one or more subtrees. Subtrees are trees - albeit inserting root and branch labels as appropriate. We otherwise, for Exercises 6.4-6.11, refer to Sect. 6.4 , especially its subsection, Sect. 6.4 .2 , on uniqueness and identification.
Finite Root-Labelled Trees. Define a concept of trees, informally, as in Example 6.7, but for which all roots are labelled - and such that no two 'immediate', but otherwise 'distinct' subtrees of a tree have 'identically labelled' roots.
Suggest what we might mean by immediate, distinct and identical labels.
Figure 6.3 gives a snapshot of the kind of labelled trees referred to in Example 6.7 and in Exercises 6.4-6.6.
(FIGURE CANT COPY).Fig. 6.3. Unlabelled and labelled trees

R M
R M
Numerade Educator

Problem 5

Mostly make sense when using this book in a formal course over this volume. The exercises all illustrate the issue of proper definitions, also formally. Anyway, we have hinted at some definitions in our formulations of the exercises. When using this book in an informal course over this volume try reformulate the definitions along the lines of: $A$ tree consists of a root and a finite set of zero, one or more subtrees. Subtrees are trees - albeit inserting root and branch labels as appropriate. We otherwise, for Exercises 6.4-6.11, refer to Sect. 6.4 , especially its subsection, Sect. 6.4 .2 , on uniqueness and identification.

Finite Branch-Labelled Trees. Define a concept of trees, informally, as in Example 6.7, but for which all branches (the things that "connect" a root of a tree with the roots of its immediate subtrees) are labelled - and such that no two branches of a tree are identically labelled.

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Problem 6

Mostly make sense when using this book in a formal course over this volume. The exercises all illustrate the issue of proper definitions, also formally. Anyway, we have hinted at some definitions in our formulations of the exercises. When using this book in an informal course over this volume try reformulate the definitions along the lines of: $A$ tree consists of a root and a finite set of zero, one or more subtrees. Subtrees are trees - albeit inserting root and branch labels as appropriate. We otherwise, for Exercises 6.4-6.11, refer to Sect. 6.4 , especially its subsection, Sect. 6.4 .2 , on uniqueness and identification.
Finite Root- and Branch-Labelled Trees. Define a concept of trees, informally, as in Example

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Problem 7

Mostly make sense when using this book in a formal course over this volume. The exercises all illustrate the issue of proper definitions, also formally. Anyway, we have hinted at some definitions in our formulations of the exercises. When using this book in an informal course over this volume try reformulate the definitions along the lines of: $A$ tree consists of a root and a finite set of zero, one or more subtrees. Subtrees are trees - albeit inserting root and branch labels as appropriate. We otherwise, for Exercises 6.4-6.11, refer to Sect. 6.4 , especially its subsection, Sect. 6.4 .2 , on uniqueness and identification.
Distinctly Labelled Trees. As for Exercise 6.6, only now it is required that no two root labels are the same, that no two branch labels are the same, and that root and branch labels also differ.

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00:43

Problem 8

Mostly make sense when using this book in a formal course over this volume. The exercises all illustrate the issue of proper definitions, also formally. Anyway, we have hinted at some definitions in our formulations of the exercises. When using this book in an informal course over this volume try reformulate the definitions along the lines of: $A$ tree consists of a root and a finite set of zero, one or more subtrees. Subtrees are trees - albeit inserting root and branch labels as appropriate. We otherwise, for Exercises 6.4-6.11, refer to Sect. 6.4 , especially its subsection, Sect. 6.4 .2 , on uniqueness and identification.
Forest of Trees. Based on Exercises 6.4-6.7, define a concept of forest as consisting of trees, unlabelled, or labelled one way or another, but such that no two labels of any two somehow labelled trees are identical.
Does Figure 6.3 "portray" such a forest? Motivate the answer.

Joanna Quigley
Joanna Quigley
Numerade Educator
07:46

Problem 9

Mostly make sense when using this book in a formal course over this volume. The exercises all illustrate the issue of proper definitions, also formally. Anyway, we have hinted at some definitions in our formulations of the exercises. When using this book in an informal course over this volume try reformulate the definitions along the lines of: $A$ tree consists of a root and a finite set of zero, one or more subtrees. Subtrees are trees - albeit inserting root and branch labels as appropriate. We otherwise, for Exercises 6.4-6.11, refer to Sect. 6.4 , especially its subsection, Sect. 6.4 .2 , on uniqueness and identification.
Finite Node-Labelled Graphs. Define a concept of graphs, informally, as in Example 6.8, but for which all nodes are distinctly labelled.

Chris Trentman
Chris Trentman
Numerade Educator
07:46

Problem 10

Mostly make sense when using this book in a formal course over this volume. The exercises all illustrate the issue of proper definitions, also formally. Anyway, we have hinted at some definitions in our formulations of the exercises. When using this book in an informal course over this volume try reformulate the definitions along the lines of: $A$ tree consists of a root and a finite set of zero, one or more subtrees. Subtrees are trees - albeit inserting root and branch labels as appropriate. We otherwise, for Exercises 6.4-6.11, refer to Sect. 6.4 , especially its subsection, Sect. 6.4 .2 , on uniqueness and identification.
Finite Edge-Labelled Graphs. Define a concept of graphs, informally, as in Example 6.8, but for which all edges between any given pair of (in this exercise unlabelled) nodes are distinctly labelled.

Chris Trentman
Chris Trentman
Numerade Educator

Problem 11

Mostly make sense when using this book in a formal course over this volume. The exercises all illustrate the issue of proper definitions, also formally. Anyway, we have hinted at some definitions in our formulations of the exercises. When using this book in an informal course over this volume try reformulate the definitions along the lines of: $A$ tree consists of a root and a finite set of zero, one or more subtrees. Subtrees are trees - albeit inserting root and branch labels as appropriate. We otherwise, for Exercises 6.4-6.11, refer to Sect. 6.4 , especially its subsection, Sect. 6.4 .2 , on uniqueness and identification.
Finite Node- and Edge-Labelled Graphs. Define a concept of graphs, informally, as in Example 6.8, but for which all nodes are distinctly labelled, and for which all edges between any given pair of (in this exercise now labelled) nodes are distinctly labelled.

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