Foundations of Programming Languages: A Comprehensive Overview

Required Textbook

• Practical Foundations for Programming Languages (2nd Ed.) by Robert Harper (free PDF from CMU).
• Types and Programming Languages by Benjamin C. Pierce (supplemental).

Module 2: Simply Typed Lambda Calculus (STLC)

The λ-calculus is powerful but unsafe (you can apply True to 3). Adding types fixes this. You learn the typing judgment: Γ ⊢ e : τ (In context Γ, expression e has type τ). You also meet the Type Safety Theorem, which proves:

• Progress: A well-typed term is not stuck (it’s either a value or can take a step).
• Preservation: If a term steps, the new term has the same type.

Sample Homework Exercise

Problem: Define a small language of Boolean expressions with and (short-circuiting) and or.

1. Give the abstract syntax.
2. Write small-step operational semantics (SOS) for call-by-name evaluation.
3. Write typing rules.
4. Prove that if e has type bool and e → e', then e' has type bool (Preservation).

Solution Sketch:

• Syntax: ( e ::= \texttrue \mid \textfalse \mid \textand(e_1, e_2) \mid \textor(e_1, e_2) )
• Typing: ( \Gamma \vdash \texttrue : \textbool ), similarly false. For and/or: both subterms must be bool.
• Semantics (short-circuit and): [ \frac{}\textand(\textfalse, e_2) \rightarrow \textfalse \quad \frac{}\textand(\texttrue, e_2) \rightarrow e_2 ]

2. Debugging Complex Errors

That cryptic "Mismatched types" error in TypeScript or MyPy is not a bug—it’s a proof failure. 15-312 teaches you to read inference rules. You learn to ask: Which typing rule did I violate?

1. Orthogonality

• A language is orthogonal if its features can be combined in a consistent and predictable way.

15312 Foundations - Of Programming Languages ((link))

Foundations of Programming Languages: A Comprehensive Overview

Required Textbook

• Practical Foundations for Programming Languages (2nd Ed.) by Robert Harper (free PDF from CMU).
• Types and Programming Languages by Benjamin C. Pierce (supplemental).

Module 2: Simply Typed Lambda Calculus (STLC)

The λ-calculus is powerful but unsafe (you can apply True to 3). Adding types fixes this. You learn the typing judgment: Γ ⊢ e : τ (In context Γ, expression e has type τ). You also meet the Type Safety Theorem, which proves:

• Progress: A well-typed term is not stuck (it’s either a value or can take a step).
• Preservation: If a term steps, the new term has the same type.

Sample Homework Exercise

Problem: Define a small language of Boolean expressions with and (short-circuiting) and or. 15312 foundations of programming languages

1. Give the abstract syntax.
2. Write small-step operational semantics (SOS) for call-by-name evaluation.
3. Write typing rules.
4. Prove that if e has type bool and e → e', then e' has type bool (Preservation).

Solution Sketch:

• Syntax: ( e ::= \texttrue \mid \textfalse \mid \textand(e_1, e_2) \mid \textor(e_1, e_2) )
• Typing: ( \Gamma \vdash \texttrue : \textbool ), similarly false. For and/or: both subterms must be bool.
• Semantics (short-circuit and): [ \frac{}\textand(\textfalse, e_2) \rightarrow \textfalse \quad \frac{}\textand(\texttrue, e_2) \rightarrow e_2 ]

2. Debugging Complex Errors

That cryptic "Mismatched types" error in TypeScript or MyPy is not a bug—it’s a proof failure. 15-312 teaches you to read inference rules. You learn to ask: Which typing rule did I violate? Practical Foundations for Programming Languages (2nd Ed

1. Orthogonality

• A language is orthogonal if its features can be combined in a consistent and predictable way.