Computer Science 1JC3: Introduction to Computational Thinking

Instructor: Bill Farmer

Lecture on Logic and Computational Expressions in Haskell

Importance of Boolean Expressions

• Boolean expressions are essential for decision-making in programming.
• They allow programs to skip code, making them flexible.

Conditional Expressions

• Definition: An expression that depends on the value of a Boolean expression.
• In Haskell, written as if-then-else.
  • Syntax: if c then e1 else e2
  • e1 and e2 must have the same type.
  • Example: Absolute value function for integers.
    • Conditional expression determines if n is non-negative or negative to return n or -n.

Guarded Function Definitions

• Provides a compact way to define functions without multiple if-then-else statements.
• Uses guards (conditions) to determine the output expression, ei, for given inputs.
• Example: Defining an absolute value function using guards for conditions n >= 0 and n < 0.

Case Statements

• A generalization of conditional statements in Haskell.
• Syntax: case expression of pattern -> result
• Matches expression E with patterns and returns corresponding results.
• Example: Converts a boolean to an int, mapping False to 0 and True to 1.

Alonzo Church

• American logician, mathematician, and computer scientist.
• Known for:
  • Showing undecidable problems exist (Church's Theorem).
  • Developing Lambda Calculus for function abstraction.
  • Church-Turing Thesis: Equivalence of Lambda Calculus and Turing Machines as models of computation.
  • Church-Rosser Theorem.

Lambda Calculus

• Originated by Alonzo Church.
• Provides a way to define anonymous functions using lambda notation.
• Example: A lambda expression for squaring a real number.
  • λx. x * x denotes squaring function.*

Lambda Expressions in Haskell

• Notation: \x -> expression
• Does not require explicit type declaration.
• Example: Defining anonymous function for squaring using Haskell syntax.

Curried Functions in Haskell

• Allows partial application of functions.
• Example discussed: Function f that takes x and y to return sqrt(x - y).
  • f 7 returns a new function \y -> 7 - y.

Summary

• Covered conditional expressions, guarded function definitions, and case statements.
• Discussed Alonzo Church's contributions to computation and lambda calculus.
• Demonstrated application of lambda expressions and curried functions in Haskell.

Thank you for attending! See you next time.