SE 504 (Formal Methods and Models)
Spring 2024

Syllabus

Homework Assignments

• Homework #1: Predicate Strength/Weakness   Sample Solutions
• Homework #2: Proofs of Simple Programs
• Homework #3: Catenation, Selection, Repetition
• Homework #4: Repetition, including Strengthening the Loop Invariant
• Homework #5: Tail Recursion
• Homework #6: Array Assignment
• Homework #7: Majorities; Array Assignment

Programming Assignments

• Prog. Assg. #1: Longest Common Subsequence

Dafny

• The Dafny Programming and Verification Language (home page)
• Getting Started with Dafny: A Guide (web page)
• Getting Started with Dafny: A Guide (original paper?)
• Dafny Reference Manual

Electronic Handouts

From SE 500:
• Gries/Schneider Theorems as presented by Warford
• On Proofs Involving the Replacement of A by B, where A implies B
• Developing Predicates from Informal Statements: A Checklist
• On the Strength/Weakness Relationship between Predicates
• Axioms for the (family of) if functions

New to SE 504:
• Properties/Laws of Hoare triples and the wp Predicate Transformer
• Law of Assumptions and Textual Substitution
• Program Verification Summary
• Correctness proofs of some simple programs:
  • A skip command
  • An assignment command   Another one
  • Calculating the right-hand side of an assignment command
  • A catenation (or sequential composition)
  • Proof of Correctness of a Selection Command
  • Two Examples of Deriving Selection Commands
  • A proof of a catenation of a selection and an assignment
• Repetition (Looping)
  • Notes on Loop Invariants
  • Correctness proof of a program that computes 1+2+...+n
  • Development of a linear search program
  • When a selection command is nested inside a loop
  • Strengthening the Loop Invariant:
    • Development of a program to calculate Fibonacci numbers
    • Development of a program to compute N³ without multiplication or exponentation
    • Development of a program that does negative-positive pair counting in an array
    • Development of a program for the Maximum Segment Sum Problem
• Arrays
  • Reasoning about arrays
  • Irrelevant Array Element Axiom
  • Development of a program for the Prefix Sums Problem
  • A solution to the Array Decomposition Problem
  • A solution to the 2-color version of Dutch National Flag Problem
  • A solution to the Length of Longest Common Subsequence problem
• Tail Recursion
  • Developing programs to compute tail recursive functions
• Search by Elimination (including the Celebrity Problem and Saddleback Search)
• Development of a program for determining the majority of a bag
• Longest Increasing Subsequence (i.e., Upsequence): Gries   Manber

External Links

• The Correctness-by-Construction Approach to Programming, by D.G. Kourie and Bruce W. Watson.
• Wikipedia entry on Design by Contract

Work by Bertrand Meyer on Program Correctness

• Bertrand Meyer's technology+ blog
• Loop invariants: analysis, classification, and examples
• Links to Publications
• Applying "Design by Contract"
• A Framework for Proving Contract-equipped Classes
• Towards Practical Proofs of Class Correctness
• Proving Pointer Program Properties Part 1: Context and Overview
• Proving Pointer Program Properties Part 2: The Overall Object Structure