CS7120 (Prasad)L13-1-Lambda-Opt1 Typed Lambda Calculus Adapted from Lectures by Profs Aiken and Necula of Univ. of California at Berkeley

CS7120 (Prasad)L13-1-Lambda-Opt2 Lecture Outline Lambda Calculus –Review and examples Simple types Type checking Comparison with OO type systems

CS7120 (Prasad)L13-1-Lambda-Opt3 Lambda Calculus Review Grammar: E ::= x variables | E 1 E 2 function application | x. E function creation Evaluation: ( x. e) e’  [e’/x]e

CS7120 (Prasad)L13-1-Lambda-Opt4 Example 1 ( x.x) y.y  y.y ( x.x) y.y  [ y.y/x] x = y.y

CS7120 (Prasad)L13-1-Lambda-Opt5 Example 2 ( x. y.x) ( z.z) e  * z.z (( x. y.x) ( z.z)) e  ([ z.z/x] y.x) e  ( y. z.z) e  [e/y] z.z = z.z

CS7120 (Prasad)L13-1-Lambda-Opt6 Example 3 ( x. x x) ( x. x x)  ( x. x x) ( x. x x) ... ( x. x x) ( x. x x)  [( x. x x)/x](x x) = ( x. x x)

CS7120 (Prasad)L13-1-Lambda-Opt7 Question What programming errors can occur in lambda calculus?

CS7120 (Prasad)L13-1-Lambda-Opt8 Notes We can encode anything we like in lambda calculus –Booleans, integers, objects, if-then-else, recursion,... But don’t forget that these are encodings –Akin to programming directly with 0’s and 1’s

CS7120 (Prasad)L13-1-Lambda-Opt9 Extension Grammar: E ::= x variables | E 1 E 2 function application | x. E function creation | 0,1,2,… integers | + addition Evaluation: ( x. e) e’  [e’/x]e + i j  k where k = i + j

CS7120 (Prasad)L13-1-Lambda-Opt10 Digression There is nothing magic about adding integers and + as constants to the lambda calculus We could add any data types and operations we like They can be encoded directly in lambda calculus anyway

CS7120 (Prasad)L13-1-Lambda-Opt11 Examples ( x. + x 1) ( x. + x x) ( x. + x 1) 3  4 ( x. + x 1) ( y.y)  ?

CS7120 (Prasad)L13-1-Lambda-Opt12 What Happens? ( x. + x 1) ( y.y)  + ( y.y) 1  ? Answer: It depends. A runtime error; we can’t add to a function Or no error: If + and 1 are encoded as lambda terms, computation will continue!

CS7120 (Prasad)L13-1-Lambda-Opt13 Notes Assume 1, + are encoded as lambda terms Nothing guarantees the encoding of 1 is used as an integer –E.g., (1 x.x) –Evaluation doesn’t know what the encodings are supposed to represent

CS7120 (Prasad)L13-1-Lambda-Opt14 Back to the Future We need to be able to restrict uses of values to appropriate operations We need a type system! Note that we’d like a type system whether or not + ( y.y) 1 causes a runtime error –Catching these errors before program execution is better

CS7120 (Prasad)L13-1-Lambda-Opt15 Typed Lambda Calculus Grammar: E ::= x variables | E 1 E 2 function application | x: . E function creation | 0,1,2,… integers | + addition Evaluation: ( x: . e) e’  [e’/x]e + i j  k where k = i + j

CS7120 (Prasad)L13-1-Lambda-Opt16 The Types We have only two kinds of values –Integers –Functions Type grammar:  := int |   

CS7120 (Prasad)L13-1-Lambda-Opt17 Examples of Types int int  int (int  int)  int int  (int  int)

CS7120 (Prasad)L13-1-Lambda-Opt18 Type Judgments Type environment Type Expression “it is provable that” “has type”

CS7120 (Prasad)L13-1-Lambda-Opt19 Examples

CS7120 (Prasad)L13-1-Lambda-Opt20 More Examples

CS7120 (Prasad)L13-1-Lambda-Opt21 Type Rule: Variables A variable has the type assumed in the type environment. [Var]

CS7120 (Prasad)L13-1-Lambda-Opt22 Abstraction A function has type  1   2 if the function body has type  2 when we assume the argument has type  1. [Abs]

CS7120 (Prasad)L13-1-Lambda-Opt23 Application Applying a function of type  1   2 to an argument of type  1 gives a result of type  2 [App]

CS7120 (Prasad)L13-1-Lambda-Opt24 Integers An integer has type int [Int]

CS7120 (Prasad)L13-1-Lambda-Opt25 Addition Adding two int’s produces an int [Add]

CS7120 (Prasad)L13-1-Lambda-Opt26 Example 1

CS7120 (Prasad)L13-1-Lambda-Opt27 Example 2

CS7120 (Prasad)L13-1-Lambda-Opt28 Type Checking One recursive descent traversal. Top-down: Add type declarations of formal parameters to environments [Abs]

CS7120 (Prasad)L13-1-Lambda-Opt29 Type Checking (Cont.) Bottom-up: Check that types match in applications [App]

CS7120 (Prasad)L13-1-Lambda-Opt30 Structural Equality “Types match” means “types are equal” Equality is recursively defined int = int a  b = c  d  a = c  b = d

CS7120 (Prasad)L13-1-Lambda-Opt31 Notes In typed lambda calculus: –Types of all variables are declared –Types are structured (e.g., int  int) –Types are checked for equality Many typed languages in this class –Nearly all typed languages before 1980 –FORTRAN, ALGOL, Pascal, C –Captures C typing except for casts & coercions

CS7120 (Prasad)L13-1-Lambda-Opt32 Typed OO Languages In many typed object-oriented languages –Types of all variables are declared –Types are non-structural (just names) Declare all types and type relationships –Types are checked for subtyping relationships What if type declarations are omitted? –A language with type inference

CS7120 (Prasad)L13-1-Lambda-Opt33 Discussion What about structural types + subtyping? –Area of current research –Currently no consensus on the right way to combine C-like type systems with typical OO-like type systems