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Formal Software Development Methods

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Rice's theorem states that no algorithm can determine all non-trivial properties about the language recognized by a Turing machine (or any Turing-complete program), meaning it's impossible to know for any given program if it possesses an interesting behavioral characteristic, like whether it halts or accepts specific inputs, without running it (which might not even finish)

Set

image-20260122125518904

Cartesian product 笛卡尔积：如果 A={1， 2}且 B={x， y}，则 A×B = {（1， x）， （1， y）， （2， x）， （2， y）}

Relation

Property Definition
Reflexive
Symmetric
Transitive
Antisymmetric
  • Equivalence: reflexive+symetric+transitive
  • Partial order: reflexive+antisymmetric+transitive
  • Preorder: reflexive+transitive

Function

A particular input has a single output

Computations not a function?

Language

Alphabet, language, words

Regular expression

syntax+semantics

Simple imperative Programming language

Syntax->Graph

Graph(V, E)

Parse tree

Flow graph

DAG(Directed A~ Graph)

CFG(control flow graph)

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