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λ Programming Language Theory

Type Theory • Lambda Calculus • Semantics • Compilers • Formal Verification • Category Theory

Master the mathematical foundations of programming — from lambda calculus and type systems to denotational semantics, program logics, and compiler design.

45+

Programming Language Theory Concepts & Formalisms

PLT Basics

Syntax, semantics, pragmatics, formal languages.

Formal Languages

Strings, grammars, Chomsky hierarchy.

Lambda Calculus

α-conversion, β-reduction, η-conversion, combinators.

Simply Typed Lambda

Types, type inference, λ→, STLC.

System F (Polymorphism)

Parametric polymorphism, type abstraction.

Dependent Types

Π-types, Σ-types, theorem proving.

Type Systems

Static/dynamic, strong/weak, inference.

Hindley-Milner

Type inference, let-polymorphism, unification.

Algebraic Data Types

Sum, product, recursion, pattern matching.

Linear Types

Resource management, substructural logic.

Cubical Type Theory

Homotopy type theory, univalence.

Operational Semantics

Small-step, big-step, evaluation.

Denotational Semantics

Domains, functions, continuity.

Axiomatic Semantics

Hoare logic, Floyd-Hoare, weakest precondition.

Game Semantics

Dialogical, strategic, compositionality.

Category Theory

Functors, natural transformations, adjunctions.

Monads & Effects

Moggi, monad transformers, algebraic effects.

Cartesian Closed Categories

CCC, λ-calculus as internal language.

Hoare Logic

Partial/total correctness, loop invariants.

Separation Logic

Pointer programs, frame rule, ownership.

Dependent Verification

Coq, Agda, proof assistants.

Theorem Proving

Compiler Design

Lexical analysis, parsing, IR, code generation.

Parsing Theory

LL(k), LR(k), ambiguity, parser generators.

Type Checking

Algorithm W, subtyping, bidirectional.

Program Optimization

Dataflow, SSA, loop optimizations.

Process Calculi

CCS, CSP, π-calculus, mobility.

Session Types

Communication protocols, linearity.

Model Checking

LTL, CTL, spin, symbolic checking.

Interactive Theorem Proving

Coq, Lean, Isabelle, tactics.

Domain-Specific Languages

Internal/external DSLs, language workbenches.

Embedded DSLs

Shallow/deep embedding, combinators.

Functional Paradigm

Immutability, higher-order, purity.

OOP Theory

Classes, inheritance, subtyping, F-bounded.

Logic Programming Theory

Horn clauses, resolution, SLD.

Language Design

Syntax design, trade-offs, evolution.

Effect Systems

Algebraic effects, handlers, row types.

Gradual Typing

Dynamic + static, cast insertion.

Lambda Cube

λ→, λ2, λP, λω, CoC.

Constructive Logic

BHK interpretation, Curry-Howard, intuitionism.

λ Programming Language Theory Mastery Roadmap

🔰 Foundation

Lambda calculus, type systems basics, operational semantics

📶 Core

Hindley-Milner, ADTs, denotational semantics, compilers

🔥 Deep Dive

Dependent types, category theory, monads, process calculi

🏆 Frontier

HoTT, cubical type theory, theorem proving, effect systems

λ Formalize, Prove, Compile — Master Programming Language Theory

From lambda calculus to theorem proving, explore the mathematical underpinnings of computation.

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