Pla an Traversal Fu unctions (1) ● Booleans ● Many functions have the e characteristic that they traverse the tree recursiv traverse the tree recursiv vely and only do vely and only do ● Steps towards a Pico e S d Pi nvironment i something interesting at t a few nodes – Step 1: define syntax ● Example: count the iden E l t th id ntifiers in a program tifi i – Step 2: define a type che ecker ● Using a recursive (induc ctive) definition: – Step 3: define an evaluat Step 3: define an evaluat tor tor – # of equations is equal to o number of syntax rules – Step 4: define a compile er – think about Cobol or Jav think about Cobol or Jav va with hundreds of rules va with hundreds of rules ● Traversal functions T l f ti ● Traversal functions auto omate recursion ● Methodology Introduction to ASF+SDF 1 Introduction to ASF+SDF 2 Traversal Fu unctions (2) Top-down vers sus Bottom-up There are two important as spects of traversal Top-down Bottom-up functions: functions: 1 7 – the kind of traversal 2 5 3 ● accumulate a value durin ng traversal 6 ● transform the tree during g traversal – the order of traversal h d f l 3 4 6 7 1 2 5 4 ● top-down versus bottom- -up ● left-to-right versus right- -to-left (we only have the first) ● break or continue after a visit Introduction to ASF+SDF 3 Introduction to ASF+SDF 4
Traversal Cube: v Traversal Cube: v visiting behaviour visiting behaviour Three kinds of traversals ● Accumulator: traversal (accu) – accumulate a value durin l l d i ng traversal l ● Transformer: traversal( (trafo) Top-down – perform local transforma ations mer: traversal(accu, trafo) mer: traversal(accu trafo) ● Accumulating transform ● Accumulating transform B tt Bottom-up Right-to-left – accumulate and transform m Left-to-right Break Break Conti Conti inue inue Introduction to ASF+SDF 5 Introduction to ASF+SDF 6 Simple e Trees Count node s (classical) Count the nodes in a tree module Tree-cnt module Tree-syntax Simple trees containing imports Tree-syntax numbers as leaves and imports Naturals exports p constructors f g or h f , g , or h exports t constructors context-free syntax sorts TREE These equations are needed cnt(TREE) -> NAT context-free syntax to visit all nodes in the tree equations equations NAT -> TREE E [1] cnt(N) = 1 f(TREE, TREE) -> TREE E A new equation has to be [2] cnt(f(T1,T2)) = 1+cnt(T1)+cnt(T2) ) g(TREE TREE) -> TREE g(TREE, TREE) > TREE E added for each new constructor added for each new constructor [3] cnt(g(T1,T2)) = 1+cnt(T1)+cnt(T2) ( ( )) ( ) ( ) ) ) h(TREE, TREE) -> TREE E [4] cnt(h(T1,T2)) = 1+cnt(T1)+cnt(T2 ) variables “N”[0-9]* -> NAT “N”[0 9]* NAT T Count this node Count this node Count nodes in both subtrees “T”[0-9]* -> TREE E cnt( f( g( f(1,2), 3 ), 11 g( g( , ), g( g(4,5), 6 )), )), ) Introduction to ASF+SDF 7 Introduction to ASF+SDF 8
Exam mple Using Acc cumulators cnt( f( g( 7, 8), 9) ) Le L f ft-most innermost reduction: i d i ● Goal: traverse term and accumulate a value [2] cnt(f(T1,T2)) = 1+cnt(T1)+cnt(T2) [ ● fun(Tree, Accu) -> Accu { f ( , ) { {traversal(accu, ...)} { ( , )} 1 + cnt(g( 7, 8)) + cnt(9) ● Tree: term to be traversed [3] cnt(g(T1,T2)) = 1+cnt(T1)+cnt(T2) [ (always the first argument) 1 1 1 + 1 +cnt(7) + cnt(8) + cnt(9) t(7) t(8) t(9) ● Accu : value to be accumul ated (always second Addition of integers argument) 2 +cnt(7) + cnt(8) + cnt(9) ( ) ( ) ( ) [1] cnt(N) = 1 ● Important: the sorts of seco ond argument and result are 2 + 1 + cnt(8) + cnt(9) always equal. y q ● Optional: extra arguments ... Similar reductions 5 5 ● fun(Tree, Accu, A1, ...) -> f (T A A1 ) A Accu {traversal(...)} {t l( )} Introduction to ASF+SDF 9 Introduction to ASF+SDF 10 Example: accu, Count nodes s (traversals) ,bottom-up,continue A bottom up accumulator that A bottom-up accumulator that cnt( f( g( 7, 8), 9), 0 ) module Tree-cnt 5 continues after each matching node imports Tree-syntax cnt( , 0) cnt( , 0) exports exports f f context-free syntax cnt(TREE, NAT) -> NAT {traversa al(accu,bottom-up,continue)} equations g 9 [1] cnt(T, N) = N + 1 Acc cumulated value Traversed tree (mat tches every node) 7 7 8 8 cnt( f( g( f(1,2), 3 ), g( g(4,5), 6 )), 11 [1] cnt(T,N) = N + 1 [1] cnt(T,N) N 1 0) 0) Introduction to ASF+SDF 11 Introduction to ASF+SDF 12
Using Tra ansformers Example: accu, bottom-up,continue cnt( f( g( 7, 8), 9), 0 ) 5 ● fun(Tree) -> Tree {trave ersal(trafo, ...)} cnt( , 0) ( , ) f ● Tree: term to be traversed cnt( , 4) (always the first argument) ● Important: the sorts of the f p first argument and result are g g 9 cnt( , 3) c always equal. cnt( , 2) ● Optional: extra arguments Optional: extra arguments ● fun(Tree, A1, A2, ...) -> Tree {traversal(...)} cnt( , 0) ( 7 , ) 8 cnt( , 1) cnt( , 1) [ ] [1] cnt(T, N) = N + 1 ( , N) N Introduction to ASF+SDF 13 Introduction to ASF+SDF 14 Incremen nt leaves Exam mple trafo,bottom-up,continue A bottom-up transformer that A bottom-up transformer that module Tree-inc d l T i inc( f( g( 7, 8), 9)) f( g( 8, 9), 10) continues after each matching node imports Tree-syntax exports p f inc( ) nc( ) f f inc( ) context-free syntax inc(TREE) -> TREE {traversa al(trafo,bottom-up,continue)} equations equations g g 9 10 [1] inc(N) = N + 1 i inc( ) inc( ) replaced by N+1 is leaf N inc( ) ( 7 ) 8 inc( ) inc( ) 8 9 i nc( f( g( f(1,2), 3 ), f( g( f(2,3), 4 ), g( g(4,5), 6 )) ) g( g(5,6), 7 )) [1] inc(T, N) = N + 1 Introduction to ASF+SDF 15 Introduction to ASF+SDF 16
Increment leaves w with explicit amount Exam mple trafo,bottom-up,continue module Tree-incp d l T i A bottom-up transformer that A bottom up transformer that inc( f( g( 7, 8), 9), 5) f( g( 12, 13), 14) continues after each matching node imports Tree-syntax exports p inc( , 5) nc( , 5) f f f inc( , 5) context-free syntax Amount inc(TREE, NAT) -> TREE {traver rsal(trafo,bottom-up,continue)} equations equations g 9 g [1] inc(N1, N2) = N1 + N2 i inc( , 5) 14 inc( , 5) lace N1 by N1+N2 repl amount N2 leaf N1 inc( , 5) ( 7 , ) 8 inc( , 5) inc( , 5) 12 12 13 13 i nc( f( g( f(1,2), 3 ), f( g( f( 8, 9), 10), g( g(4,5), 6 )), [1] inc(N1, N2) = N1 + N2 g( g(11,12), 13)) g( g( , ), )) 7 ) 7 ) Introduction to ASF+SDF 17 Introduction to ASF+SDF 18 Term Rep placement Deep rep lacement Auxiliary constructor i Auxiliary constructor i module Tree-drepl d l T d l imports Tree-syntax ● Deep replacement: repla ace only occurrences close exports A bottom-up transformer that to the leaves to the leaves context-free syntax stops after first matching node f fi hi d i(TREE, TREE) -> TREE ● Shallow replacement: re eplace only occurrences drepl(TREE) -> TREE {tra drepl(TREE) TREE {tra versal(trafo,bottom up,break)} versal(trafo bottom-up break)} close to the root l t th t equations [1] drepl(g(T1, T2)) = i(T1, T2) ● Full replacement: replac ce all occurrences nces of g Only the deepest occurren Shallow are replaced f( i( f(1,2), 3 ), Full drepl( f( g( f(1,2), 3 ), Deep g( i(4,5), 6 )) g( g(4,5), 6 )) ) Introduction to ASF+SDF 19 Introduction to ASF+SDF 20
Shallow re eplacement Exam mple trafo,bottom-up,break module Tree-srepl d l T l drepl( g( g( 7, 8), 9)) g( i( 7, 8), 9) imports Tree-syntax A top-down transformer that drepl( ) p ( ) exports g g g g stops after first matching node stops after first matching node context-free syntax i(TREE, TREE) -> TREE srepl(TREE) -> TREE {tra srepl(TREE) TREE {tra aversal(trafo, top down, break)} aversal(trafo top-down break)} g 9 equations i dr repl( ) drepl( ) 9 [1] srepl(g(T1, T2)) = i(T1, T2) ) ences of g Only the outermost occurre are replaced drepl( ) p ( 7 ) 8 drepl( ) drepl( ) 7 7 8 8 f( i( f(1,2), 3 ), srepl( f( g( f(1,2), 3 ), i( g(4,5), 6 )) g( g(4,5), 6 )) ) [1] drepl(g(T1, T2)) = i(T1, T2) Introduction to ASF+SDF 21 Introduction to ASF+SDF 22 Full repl lacement Exam mple trafo, top-down, break module Tree-frepl d l T f l srepl( g( g( 7, 8), 9)) i( g( 7, 8), 9) imports Tree-syntax A top-down transformer that srepl( ) p ( ) exports g g i i continues after each matching node continues after each matching node context-free syntax i(TREE, TREE) -> TREE frepl(TREE) TREE {tra frepl(TREE) -> TREE {tra aversal(trafo top-down continue)} aversal(trafo,top down,continue)} g 9 equations g 9 [1] frepl(g(T1, T2)) = i(T1, T2) ) top-down and bottom-up have bottom up have g are re here the same effect All occurrences of eplaced 7 8 7 7 8 8 f( i( f(1,2), 3 ), frepl( f( g( f(1,2), 3 ), i( i(4,5), 6 )) g( g(4,5), 6 )) ) [1] srepl(g(T1, T2)) = i(T1, T2) Introduction to ASF+SDF 23 Introduction to ASF+SDF 24
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