The document discusses the theoretical underpinnings of functional programming including lambda calculus, semantics of lambda calculus using alpha, beta, and eta conversions, and how lambda calculus has the same computational strength as a Turing machine by representing numbers, booleans, and recursive functions. It also summarizes the basic characteristics of functional languages as having only functions and immutable variables without statements, and emphasizes composability through higher-order functions, currying, and pattern matching without side effects.

2. Theoretical Underpinnings
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 2 / 10

3. Lambda Calculus
e→x variable
e1 e2 application
λx.e abstraction
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 3 / 10

4. Semantics of Lambda Calculus
λxi .e ⇒ λxj .[xj /xi ]e iff xj is not free in e (α-conversion, renaming)
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 4 / 10

5. Semantics of Lambda Calculus
λxi .e ⇒ λxj .[xj /xi ]e iff xj is not free in e (α-conversion, renaming)
(λx.e1 )e2 ⇒ [e2 /x]e1 (β-conversion, application)
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 4 / 10

6. Semantics of Lambda Calculus
λxi .e ⇒ λxj .[xj /xi ]e iff xj is not free in e (α-conversion, renaming)
(λx.e1 )e2 ⇒ [e2 /x]e1 (β-conversion, application)
λx.(e x) ⇒ e iff x is not free in e (η-conversion)
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 4 / 10

7. Computational Strength of Lambda Calculus
Lambda calculus has the same strength as Turing machine. In particular,
you can define functions that represent
numbers,
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 5 / 10

8. Computational Strength of Lambda Calculus
Lambda calculus has the same strength as Turing machine. In particular,
you can define functions that represent
numbers,
booleans and conditions and
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 5 / 10

9. Computational Strength of Lambda Calculus
Lambda calculus has the same strength as Turing machine. In particular,
you can define functions that represent
numbers,
booleans and conditions and
recursive computations.
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 5 / 10

10. Functional Languages
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 6 / 10

11. Basic Characteristics
No statements, just functions.
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 7 / 10

12. Basic Characteristics
No statements, just functions.
Immutable variables.
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 7 / 10

13. Basic Characteristics
No statements, just functions.
Immutable variables.
No implicit state.
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 7 / 10

14. Composability
No side effects.
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 8 / 10

15. Composability
No side effects.
Higher-order functions.
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 8 / 10

16. Composability
No side effects.
Higher-order functions.
Currying.
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 8 / 10

17. Composability
No side effects.
Higher-order functions.
Currying.
Pattern matching.
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 8 / 10

18. Evaluation Order
Non-strict (lazy) evaluation, possibly infinite data structures.
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 9 / 10

19. Evaluation Order
Non-strict (lazy) evaluation, possibly infinite data structures.
Parallel execution.
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 9 / 10

20. Evaluation Order
Non-strict (lazy) evaluation, possibly infinite data structures.
Parallel execution.
Memoization.
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 9 / 10

21. See
Hudak, P. Conception, evolution, and application of functional
programming languages. ACM Computing Surveys 21, 3 (September
1989), 359–411. http://doi.acm.org/10.1145/72551.72554
Michal P´se (CTU in Prague)
ıˇ Object Prgrmmng L. 3: Functional Concepts October 12, 2010 10 / 10