CSc 520 Principles of Programming Languages. β-reduction Introductory Example. 22: Lambda Calculus Reductions

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1 Lambda Reductions CSc 520 Principles of Programming Languages 22: Lambda Calculus Reductions Christian Collberg To evaluate a lambda epression we reduce it until we can appl no more reduction rules. There are four principal reductions that we use: 1. α-reduction variable renaming to avoid name clashes in β-reductions. 2. β-reduction function application. 3. η-reduction formula simplification. 4. δ-reduction evaluation of predefined constants and functions. Department of Computer Science Universit of Arizona Copright c 2004 Christian Collberg [1] 520 [2] -reduction Introductor Eample β-reduction Introductor Eample Blindl appling β-reductions can lead to a problem known as variable capture. To prevent this we need a wa to change the name of a variable. Here s an eample of an α-reduction: ((λ.(λf.(f ))) ) α ((λz.(λf.(f ))) z) The equivalence of function application in a functional language is called β-reduction. Here s an eample of using β-reductions to evaluate a lambda epression: Twice (λf.(λ.(f (f )))) ((Twice (λn.(add n 1))) 5) (((λf.(λ.(f (f )))) (λn.(add n 1))) 5) β ((λ.((λn.(add n 1)) ((λn.(add n 1)) ))) 5) β ((λn.(add n 1)) ((λn.(add n 1)) 5)) β

2 reduction Introductor Eample... δ-reduction Introductor Eample ((Twice (λn.(add n 1))) 5) (((λf.(λ.(f (f )))) (λn.(add n 1))) 5) β ((λ.((λn.(add n 1)) ((λn.(add n 1)) ))) 5) β ((λn.(add n 1)) ((λn.(add n 1)) 5)) β (add ((λn.(add n 1)) 5) 1) β (add (add 5 1) 1) No further β-reductions can be made. A δ-reduction is used to evaluate non-pure lambda epression, i.e. those that contain predefined constants and functions (such as add, etc.). Without δ-reductions we couldn t evaluate the previous eample an further: ((Twice (λn.(add n 1))) 5) (add (add 5 1) 1) δ (add 6 1) δ 7 [5] 520 [6] Free Variable Substitution Free Variables To work with α-reductions we first need to define the concept of a variable substitution. The notation E[v M] means replace all free occurences of v with M in the epression E. Eample: (λv.(mul v))[ z] = (λv.(mul z v))

3 Free Variable Substitution... Free Variable Substitution A more traditional notation (used b Scott, for eample) for E[v M] is Eample: E[M\v] (λv.(mul v))[z\] = (λv.(mul z v)) Variable replacements aren t legal in E[v M] if a free variable in M becomes bound. For eample, the substitution (λ.(mul ))[ ] (λ.(mul )) is illegal because it causes a change in semantics: the new epression represents a squaring operation, the original doesn t. This is known as variable capture or a name clash. [9] 520 [10] Computing Free Variables Computing Free Variables Eample The free variables of an epression E, FV(E) is computed as 1. FV(c) = for an constant c. 2. FV() = {} for an variable. 3. FV((E 1 E 2 )) = FV(E 1 ) FV(E 2 ). 4. FV((λ.E)) = FV(E) {}. An epression that has no free variables (FV(E) = ) is called closed. FV((λ.( (λ.(( ) z))))) = FV(( (λ.(( ) z)))) {} = (FV() FV((λ.(( ) z)))) {} = ({} (FV((( ) z)) {})) {} = ({} ((FV(( )) FV(z)) {}) {} = ({} ((FV() FV()) {z}) {}) {} = ({} (({} {}) {z}) {}) {} = {, z}

4 Variable Substitution Algorithm Variable Substitution Eample a) v[v E 1 ] = E 1 for an variable v b) [v E 1 ] = for an variable v c) c[v E 1 ] = c for an constant c d) (E 1 E 2 )[v E 3 ] = (E 1 [v E 3 ] E 2 [v E 3 ]) e) (λv.e)[v E 1 ] = (λv.e) f) (λ.e)[v E 1 ] = (λ.e[v E 1 ]), when v and FV(E 1 ) g) (λ.e)[v E 1 ] = (λz.e[ z][v E 1 ]), when v and FV(E 1 ) where z v and z FV((E E 1 )) In g) the first substitution E[ z] replaces a bound variable b a new bound variable z. [13] 520 (λ.((λf.(f )) ))[ (f )] g) = (λz.((λf.(f )) )[ z][ (f )]) = (λz.((λf.(f )) z)[ (f )]) d) = (λz.((λf.(f ))[ (f )] z[ (f )])) b) = (λz.((λg.(g ))[ (f )] z)) d,b,a) = (λz.((λg.(g (f ))) z)) [14] Reductions Reductions The main rule for evaluating a lambda epression is called β-reduction. β-reduction is similar to function application in a functional language. To prevent variable capture (when a free variable in E 1 becomes bound during the subsitution E[v E 1 ]), we also need α-reductions. δ-reductions are used to evaluate predefined functions such as add. η-reductions are not strictl necessar but can be used to clean up mess epressions.

5 α-reductions β-reductions Scott calls this α-conversion. α-reduction sas that we can replace v b w in (λv.e) as long as w does not occur free in E. Formall, (λv.e) α (λw.e[v w]) where w does not occur free in E. Eample: Eample: (λ.((λf.(f )) )) α (λz.((λf.(f )) z)) (λz.((λf.(f )) z)) α (λz.((λg.(g )) z)) Let v be a variable and E and E 1 lambda epressions, then ((λv.e) E 1 ) β E[v E 1 ] provided E[v E 1 ] is carried out safel. The intuition is that the argument E 1 is passed to the function (λv.e) b substituting E 1 for the formal parameter v. An epression of the form ((λv.e) E 1 ) is called a β-rede (reduction epression), i.e. an epression that can be β-reduced. [17] 520 [18] η-reductions η-reductions... η-reductions allow us to get rid of etra lambda abstractions. For eample, we can sa (λ.(square )) η square square is a function that squares its argument, while (λ.(square )) is a function of that squares. (λ.(square )) reminds us that square takes an argument, but square is a bit less mess. If E is a lambda epression that denotes a function, and v has no free occurences in E, then The following reduction (λv.(e v)) η E (λ.(add )) η (add ) is invalid since (add ) has a free occurence of.

6 Termination Reduction Strategies Question: Can ever lambda epression be reduced to a normal form? Consider the epression which reduces to itself: ((λ.( )) (λ.( ))) ((λ.( )) (λ.( ))) β ((λ.( )) (λ.( ))) β Answer: No. [21] 520 Lambda calculus contains non-terminating reductions. [22] Paths Paths... Question: Is there more than one wa to reduce a lambda epression? Consider the epression (((λ.(λ.(add ((λz.(mul z)) 3)))) 7) 5) Let s tr reducing it in two different was, to see if we ll arrive at different normal forms. A lambda epression is in a normal form if we can appl no more β-reductions or δ-reductions. (((λ.(λ.(add ((λz.(mul z)) 3)))) 7) 5) = (((λ.(λ.(add ((λz.(mul z)) 3)))) 7) 5) β ((λ.(add ((λz.(mul 7 z)) 3))) 5) β (add 5 ((λz.(mul 7 z)) 3)) β (add 5 (mul 7 3)) δ (add 5 21) δ 26

7 Paths... Paths (((λ.(λ.(add ((λz.(mul z)) 3)))) 7) 5) = (((λ.(λ.(add ((λz.(mul z)) 3)))) 7) 5) β (((λ.(λ.(add (mul 3)))) 7) 5) β Question: Is there more than one wa to reduce a lambda epression? Answer: Yes. ((λ.(add (mul 7 3))) 5) δ ((λ.(add 21)) 5) δ (add 5 21) δ 26 [25] 520 [26] Application Order Application Order... Consider the epression ((λ.5) ((λ.( )) (λ.( )))) We can either reduce the leftmost rede first: ((λ.5) ((λ.( )) (λ.( )))) β 5 or, we can evaluate the rightmost rede ever time: ((λ.5) ((λ.( )) (λ.( )))) β ((λ.5) ((λ.( )) (λ.( )))) β The leftmost rede is that rede whose λ is tetuall to the left of all other redees within the epression. An outermost rede is defined to be a rede which is not contained within an other rede. An innermost rede is defined to be a rede which contains no other rede. A normal order reduction alwas reduces the leftmost outermost β-rede (or δ-rede) first. A applicative order reduction alwas reduces the leftmost innermost β-rede (or δ-rede) first. ((λ.5) ((λ.( )) (λ.( )))) β

8 Finding Redees Finding Redees... Remembering that a rede is an epression of the form ((λ.e) ), find the redees in this epression: (((λ.(λ.(add ))) ((λz.(succ z)) 5)) ((λw.(sqr w)) 7)) = (((λ.(λ.(add ))) ((λz.(succ z)) 5) ) ((λw.(sqr w)) 7)) = }{{} (((λ.(λ.(add ))) ((λz.(succ z)) 5) ) ((λw.(sqr w)) 7) }{{}}{{} ) = ( ((λ.(λ.(add { ))) }} ((λz.(succ z)) 5) { ) ((λw.(sqr w)) 7) }{{}}{{} ) If we have problems finding the redees of an epression, we can first draw it as an abstract snta tree. This is a tree that shows the structure of an epression, ignoring sntactic details. lamb (λ.) ( ) z z lamb z lamb ( ( z)) ((z ) ) ( (λ.z)) (λ.( z)) z [29] 520 stands for an application, lamb for an abstraction. [30] Finding Redees... During evaluation we look for a rede, a tree with the following structure: lamb E ((λ.e) ) I.e., a rede is an abstraction joined with another epression to be passed to the function. Finding Redees... Again, consider the epression: ( { ((λ.(λ.(add ))) }} ((λz.(succ z)) 5) { ) ((λw.(sqr w)) 7) }{{}}{{} ) Its abstract snta tree is given on the net slide. The leftmost outermost rede is ( ((λ.(λ.(add ))) ((λz.(succ z)) 5)) }{{} The leftmost innermost rede is ((λw.(sqr w)) 7)) (((λ.(λ.(add ))) ((λz.(succ z)) 5) ) ((λw.(sqr w)) 7)) }{{}

9 Finding Redees... Church-Rosser Theorem I lamb lamb add z lamb succ Outermost z 5 Innermost w lamb sqr w 7 Outermost and Innermost [33] 520 Question: If there is more than one reduction strateg, does each one lead to the same normal form epression? Theorem: For an lambda epressions E, F and G, if E F and E G, there is a lambda epression Z such that F Z and G Z. diamond propert confluence propert [34] F E Z G Church-Rosser Theorem I... Corollar: For an lambda epressions E, M and N, if E M and E N, where M and N are in normal form, M and N are variants of each other (ecept for changes in variables, using α-reductions). Answer: Yes, if a lambda epression is in normal form, it is unique, ecept for changes in bound variables. Church-Rosser Theorem II Question: Is there a reduction strateg that will guarantee that a normal form epression will be produced, if one eists? Theorem: For an lambda epressions E and N, if E N where N is in normal form, there is a normal order reduction from E to N. Answer: Yes, normal order reduction will produce a normal form lambda epression, if on eists.

10 Normal Order Reduction Church s Theses A normal-order reduction can have these outcomes: 1. A unique normal form lambda epression is reached (up to α-reduction). 2. The reduction never terminates. The effectivel computable functions on the positive integers are precisel those functions definable in the pure lambda calculus (and computable b Turing Machines). [37] 520 It can be shown that ever Turing Machine can be simulated b a lambda epression. It can be shown that ever lambda epression can can be realized b a Turing Machine. Hence, Turing Machines and lambda calculus are equivalent. Since it s not possible to determine whether a Turing Machine will terminate, it s not possible to determine whether a normal order reduction will terminate. [38] Alonzo Church People From hbe/church.pdf Born on June 14 (Flag Da), 1903, in Washington, D.C. His great-grandfather (also named Alonzo Church) was a professor of mathematics and astronom. His grandfather Alonzo Webster Church was at one time Librarian of the U.S. Senate. An airgun incident in high school left Church blind in one ee. Church enrolled at Princeton, where his uncles had attended college. He worked part-time in the dining hall to help pa his wa.

11 Alonzo Church... Haskell Curr He was an eceptional student; his first published paper was written while he was an undergraduate. He continued graduate work at Princeton, completing a Ph.D. in three ears While a graduate student, he married Mar Julia Kuczinski, who was training to be a nurse. (This in spite of the fact that his senior class had voted him the most likel to remain a bachelor. In the summer of 1924, Church stepped o the curb and was hit b a trolle car coming from his blind side; Mar was a nurse-trainee at the hospital. Mar was an ecellent cook; over the ears man a mathematician enjoed dining at the Church home. He and Mar had three children. Alonzo Church, Jr., was born in 1929, Mar Ann in 1933, and Mildred in (Mar Ann later married the logician John Addison.) Alonzo Church had the polite manners of a gentleman who had grown up in Virginia. He was never known to be rude, even with people with whom he had strong disagreements. A deepl religious person, he was a lifelong member of the Presbterian church. From histor/mathematicians/curr.html Haskell did not show particular interest in mathematics when at high school and when he graduated in 1916 he full intended to stud medicine. He entered Harvard College. A major influence on the direction that his studies took was the entr of the United States into World War I in the spring of Curr wanted to serve his countr, and decided that he would be more likel to see action if he had a mathematics training rather than if he continued the pre-medical course he was on. The war, however, ended shortl after this (in November) and on 9 December 1918 Curr left the arm. [41] 520 [42] Haskell Curr... Readings and References Curr now decided that he would look for a career in electrical engineering and he took a job with the General Electric Compan which allowed him to stud electrical engineering part-time at the Massachusetts Institute of Technolog. However he soon discovered that he had a different attitude to the others taking the courses, for he wanted to know wh a result was correct when for everone else it onl mattered that it was correct. Read pp , in Snta and Semantics of Programming Languages, b Ken Slonneger and Barr Kurtz, slonnegr/plf/book. Read pp , in Scott. If one imagines that from 1924 when Curr embarked on his doctorate in mathematics at Harvard he at last found the topic for him, then one would be mistaken. He was given a topic in the theor of differential equations b George Birkhoff but he began reading books on logic which seemed to him far more interesting that his research topic.

12 Acknowledgments Much of the material in this lecture on Lambda Calculus is taken from the book Snta and Semantics of Programming Languages, b Ken Slonneger and Barr Kurtz, slonnegr/plf/book. [45]