Advice about Debugger Construction Arthur Nunes-Harwitt Nassau - PDF document

Advice about Debugger Construction Arthur Nunes-Harwitt Nassau Community College One Education Drive Garden City, NY 11530 nunesa@ncc.edu http://www.matcmp.ncc.edu/ ~ nunesa/ 1

Background (define (interp exp env k) (cond ((constant? exp) (k exp)) ((variable? exp) (k (lookup exp env))) ((quote? exp) (k (unquote exp))) ...)) 2

Goals: Implement debug commands step and next such that... • they are efficient • they are correct 3

⊲ (fact 5) position: debug command> step ⊲ fact position: debug command> step fact ⊳ position: debug command> step ⊲ 5 position: debug command> step 5 ⊳ position: debug command> step ⊲ (if (= n 0) 1 (* n (fact (- n 1)))) position: debug command> step ⊲ (= n 0) position: debug command> next (= n 0) ⊳ position: debug command> continue value: 120 4

λ -calculus c op ∈ Operators c ∈ Constants ⊃ Operators x ∈ Variables V ∈ Values ::= c | x | ( λx.M ) M, N ∈ Λ ::= V | ( M M ) E ::= [] | E [( V [])] | E [([] M )] Example: (( λx.x ) ( succ ( succ 0))) E = (( λx.x ) ( succ [])) E [( succ 0)] − → E [1] 5

Specification step E [ · ( M N )] − → E [( · M N )] [step 1] step E [ · V ] − → E [ V · ] [step 2] step E [( V · M )] − → E [( V · M )] [step 3] δ ( c op c ) − → V [step 4] step E [( c op c · )] − → E [ V · ] β v (( λx.M ) V ) − → N [step 5] step E [(( λx.M ) V · )] − → E [ · N ] * − → V M [next 1] E [ · M ] next − → E [ V · ] * ( V 1 V 2 ) − → V [next 2] E [( V 1 V 2 · )] next − → E [ V · ] E [( V · M )] next − → E [( V · M )] [next 3] 6

Theoretical Implementation (part 1) CEK-machine E ∈ Env ::= [] | ( x, ( V, E )) :: E K ∈ K ::= k ∅ | fun ( V, E, K ) | arg ( M, E, K ) � ( V, E ) if x = y lookup ( x, ( y, ( V, E )) :: E ′ ) = lookup ( x, E ′ ) otherwise C ∈ C ::= � M, E, K � � ( M N ) , E, K � �− → � M, E, arg ( N, E, K ) � � V, E, arg ( N, E ′ , K ) � �− → � N, E ′ , fun ( V, E, K ) � if V / ∈ Variables � c, E, fun ( c op , E ′ , K ) � �− → � δ ( c op , c ) , [] , K � � V, E, fun (( λx.M ) , E ′ , K ) � �− → � M, ( x, ( V, E )) :: E ′ , K � if V / ∈ Variables � x, E, K � �− → � lookup ( x, E ) 1 , lookup ( x, E ) 2 , K � 7

Theoretical Implementation (part 2) � V, E, K d � �− → � V · , E, K � if V / ∈ Variables step �· ( M N ) , E, K � �− → �· M, E, arg ( N, E, K ) � step �· V, E, K � �− → � V, E, K d � step � V · , E, arg ( N, E ′ , K ) � �− → �· N, E ′ , fun ( V, E, K ) � step � c · , E, fun ( c op , E ′ , K ) � �− → � δ ( c op , c ) · , [] , K � step � V · , E, fun (( λx.M ) , E ′ , K ) � �− → �· M, ( x, ( V, E )) :: E ′ , K � next �· M, E, K � �− → � M, E, K d � next � V · , E, arg ( N, E ′ , K ) � �− → �· N, E ′ , fun ( V, E, K ) � next � c · , E, fun ( c op , E ′ , K ) � �− → � δ ( c op , c ) · , [] , K � next � V · , E, fun (( λx.M ) , E ′ , K ) � �− → � M, ( x, ( V, E )) :: E ′ , K d � 8

Correctness (of step) step E [ · ( M N )] − → E [( · M N )] step �· ( M N ) , E, K � �− → �· M, E, arg ( N, E, K ) � step E [ · V ] − → E [ V · ] step �· V, E, K � �− → � V, E, K d � step E [( V · M )] − → E [( V · M )] step � V · , E, arg ( M, E ′ , K ) � �− → �· M, E ′ , fun ( V, E, K ) � δ ( c op c ) − → V step E [( c op c · )] − → E [ V · ] step � c · , E, fun ( c op , E ′ , K ) � �− → � δ ( c op , c ) · , [] , K � β v (( λx.M ) V ) − → N step E [(( λx.M ) V · )] − → E [ · N ] step � V · , E, fun (( λx.M ) , E ′ , K ) � �− → �· M, ( x, ( V, E )) :: E ′ , K � 9

Correctness (of next) * M − → V E [ · M ] next − → E [ V · ] next �· M, E, K � �− → � M, E, K d � * ( V 1 V 2 ) − → V E [( V 1 V 2 · )] next − → E [ V · ] next � c · , E, fun ( c op , E ′ , K ) � �− → � δ ( c op , c ) · , [] , K � next � V · , E, fun (( λx.M ) , E ′ , K ) � �− → � M, ( x, ( V, E )) :: E ′ , K d � E [( V · M )] next − → E [( V · M )] next � V · , E, arg ( M, E ′ , K ) � �− → �· M, E ′ , fun ( V, E, K ) � 10

Practical Implementation (part 1) RK-machine � [] , eval ( c, E, K ) � �− → � c :: [] , K � � [] , eval ( x, E, K ) � �− → � lookup ( x, E ) :: [] , K � � [] , eval (( λ� x.M ) , E, K ) � �− → � closure ( � x, M, E ) :: [] , K � � [] , eval (( if M 0 M 1 M 2 ) , E, K ) � �− → � [] , eval ( M 0 , E, if ( M 1 , M 2 , E, K )) � � [] , eval (( M 0 M 1 . . . M n ) , E, K ) � �− → � [] , evalapp ([] , [ M 0 , M 1 , . . . , M n ] , E, K ) � � true :: [] , if ( M 1 , M 2 , E, K ) � �− → � [] , eval ( M 1 , E, K ) � � false :: [] , if ( M 1 , M 2 , E, K ) � �− → � [] , eval ( M 2 , E, K ) � � [] , evalapp ( c op :: � R, [] , E, K ) � �− → � δ ( c op , � R ) :: [] , K � x, M, E ) :: � R, [] , E ′ , K ) � � [] , evalapp ( closure ( � �− → x, � � [] , eval ( M, ( � R ) :: E, K ) � 11

Practical Implementation (part 2) � V, E, K d � �− → � V · , E, K � if V / ∈ Variables � R :: [] , K d � �− → � R · :: [] , K � step �· V, E, K � �− → � V, E, K d � step � [] , eval ( · V, E, K ) � �− → � [] , eval ( V, E, K d ) � step � V · , E, fun (( λx.M ) , E ′ , K ) � �− → �· M, ( x, ( V, E )) :: E ′ , K � x, M, E ) :: � R, [] , E ′ , K ) � � R n · :: [] , mevalapp ( closure ( � step x, � �− → � [] , eval ( · M, ( � R @[ R n ]) :: E, K ) � 12

Practical Implementation (part 3) (define (interp exp env k) (cond ((constant? exp) (apply-k k exp)) ((quote? exp) (apply-k k (unquote exp))) ((variable? exp) (apply-k k (lookup exp env))) ((abstraction? exp) (apply-k k (make-closure env exp))) ((if? exp) (interp (if-test exp) env (make-if-cont exp env k))) ((application? exp) (interp (rator exp) env (make-app-cont exp env k))) (else (error "unknown expression")))) 13

Practical Implementation (part 4) I [ [ c ] ] ρκ = κc I [ [ x ] ] ρκ = κρ ( x ) κ ( λι. ( λκ ′ . ( λv.ι [ ] ρ [ x �→ v ] κ ′ ))) I [ [( λx.M )] ] ρκ = [ M ] I [ [( M N )] ] ρκ = I [ [ M ] ] ρ ( λm. I [ [ N ] ] ρ ( λn. ((( m I ) κ ) n ))) I [ [ · M ] ] ρκ = B [ [ M ] ] ρκ step B [ [ c ] ] ρκ = A cκ step B [ [ x ] ] ρκ = A ρ ( x ) κ step B [ [( λx.M )] ] ρκ = A ( λι. ( λκ ′ . ( λv.ι [ ] ρ [ x �→ v ] κ ′ ))) κ [ M ] step B [ [( M N )] ] ρκ = B [ [ M ] ] ρ ( λm. I [ [ N ] ] ρ ( λn. ((( m I ) κ ) n ))) next B [ [ M ] ] ρκ = I [ [ M ] ] ρ ( λv. A vκ ) 14

step A v ( λn. ((( m I ) κ ) n )) = (( λn. ((( m B ) κ ) n )) v ) step A v ( λm. I [ ] ρ ( λn. ((( m I ) κ ) n ))) [ M ] = (( λm. B [ ] ρ ( λn. ((( m I ) κ ) n ))) v ) [ M ] next A v ( λn. ((( m I ) κ ) n )) = (( λn. ((( m I ) ( λv ′ . A v ′ κ )) n )) v ) next A v ( λm. I [ [ M ] ] ρ ( λn. ((( m I ) κ ) n ))) = (( λm. B [ [ M ] ] ρ ( λn. ((( m I ) κ ) n ))) v ) 15

Advice about Debugger Construction Arthur Nunes-Harwitt Nassau Community College One Education Drive Garden City, NY 11530 nunesa@ncc.edu http://www.matcmp.ncc.edu/ ~ nunesa/ 16