Looking back...

• Functional programming - "declarative" programming (recursion over persistent data structures)
  • higher-order functions
  • polymorphism

• Logic, the mathematical basis for software engineering:

                 logic                        calculus
          --------------------   =   ----------------------------
          software engineering       mechanical/civil engineering
  

  • inductively defined sets and relations
  • inductive proofs
  • proof objects

• Coq, an industrial-strength proof assistant
  • functional core language
  • core tactics
  • automation

• Foundations of programming languages
  • notations and definitional techniques for precisely specifying
    • abstract syntax
    • operational semantics
      • big-step style
      • small-step style
    • type systems
  • program equivalence
  • Hoare logic
  • fundamental metatheory of type systems
    • progress and preservation
  • theory of subtyping

Looking forward...

Some good places to go for more...

• Several optional chapters of Software Foundations
• Cutting-edge conferences on programming languages and formal verification:
  • POPL
  • PLDI
  • OOPSLA
  • ICFP
  • CAV
  • (and many others)
• More on functional programming
  • Learn You a Haskell for Great Good, by Miran Lipovaca (ebook)
  • and many other texts on Haskell, OCaml, Scheme, Scala, ...
• More on Hoare logic and program verification
  • The Formal Semantics of Programming Languages: An Introduction, by Glynn Winskel. MIT Press, 1993.
  • Many practical verification tools, e.g. Microsoft's Boogie system, Java Extended Static Checking, etc.
• More on the foundations of programming languages:
  • Types and Programming Languages, by Benjamin C. Pierce. MIT Press, 2002.
  • Practical Foundations for Programming Languages, by Robert Harper. Forthcoming from MIT Press. Manuscript available from his web page.
  • Foundations for Programming Languages, by John C. Mitchell. MIT Press, 1996.
• More on Coq:
  • Certified Programming with Dependent Types, by Adam Chlipala. A draft textbook on practical proof engineering with Coq, available from his web page.
  • Interactive Theorem Proving and Program Development: Coq'Art: The Calculus of Inductive Constructions, by Yves Bertot and Pierre Casteran. Springer-Verlag, 2004.
  • Iron Lambda (http://iron.ouroborus.net/) is a collection of ​Coq formalisations for functional languages of increasing complexity. It fills part of the gap between the end of the​ Software Foundations course and what appears in current research papers. The collection has at least Progress and Preservation theorems for a number of variants of STLC and the polymorphic lambda-calculus (System F)