Banana Algebra:

1
Banana Algebra
Syntactic Language Extension via an Algebra of
Languages and Transformations
Jacob Andersen jacand_at_cs.au.dk Aarhus
University
Claus Brabrand brabrand_at_itu.dk IT University
of Copenhagen
2
Abstract
We propose an algebra of languages and
transformations as a means for extending
languages syntactically. The algebra provides a
layer of high-level abstractions built on top of
languages (captured by CFGs) and transformations
(captured by constructive catamorphisms).
The algebra is self-contained in that any term
of the algebra specifying a transformation can be
reduced to a constant catamorphism, before the
transformation is run. Thus, the algebra comes
"for free" without sacrificing the strong safety
and efficiency properties of constructive
catamorphisms. The entire algebra as
presented in the paper is implemented as the
Banana Algebra Tool which may be used to
syntactically extend languages in an incremental
and modular fashion via algebraic composition of
previously defined languages and transformations.
We demonstrate and evaluate the tool via several
kinds of extensions.
3
The Banana Algebra Café

• The "Banana Algebra Café"
• Located in Costa Rica
• City of Cahuita (pop. 3,000)

Banana Breakfast!
Cahuita!
4
Outline

• Introduction "What is a Banana?"
• Bananas for Language Transformation
• Language Extension Pattern
• Banana Algebra
• Examples
• Implementation
• Related Work
• Conclusion

5
What is a 'Banana' ?

• Datatype "list"
• Banana ("sum-of-list")
• Implicit recursion on input structure
• Separation of recursion and evaluation
• bottom-up re-combination of intermediate results

list Num int Cons int list
list ? int
Num n n Cons n l n l
(aka. "Catamorphism" )
( ?n.n , ?(n,l ).nl )
Another Ex. "length-of-list"
6
Language Transformation

• Bananas (statically typed)
• Source language 'LS'
• Target language 'LT'
• Nonterminal-typing '? '
• Reconstructors 'c '

?
( LS -gt LT ? c )
LS -gt LT
list Num int Cons int list
tree Nil Leaf int Node tree tree
list -gt tree
Num n Leaf n Cons n l Node (Leaf
n) l
Type-check'able!
7
Banana Properties

• Banana properties
• Simple (corresponds to simple recursion)
• Safe (syntactically safe always
  terminate)
• Efficient (linear time in size of input
  output)
• (Expressive) (enough for interesting
  extensions)
• Banana Algebra for free (16 banana ops)
• Modular
• Incremental
• Simple
• Safe
• Efficient
• (Expressive)

Been around for many years
We now propose
?
8
Outline

• Introduction "What is a Banana?"
• Bananas for Language Transformation
• Language Extension Pattern
• Banana Algebra
• Examples
• Implementation
• Related Work
• Conclusion

9
Language Extension Pattern
Numeral extension
Lambda-Calculus
'LS'
'LT'
Exp var Id lam Id Exp app Exp
Exp zero succ Exp pred Exp
Exp var Id lam Id Exp app Exp
Exp
'? '
Nonterminal typing
Exp -gt Exp
Reconstructors
'c '
var V var V lam V E lam V
E app E1 E2 app E1 E2
zero lam z (var z) succ E lam s
E pred E app E (lam z (var z))
Catamorphism
( LS -gt LT ? c )
Using very simple numeral encoding
10
Algebraic Solution
lnl ? l

ln ? l
( ln -gt l Exp -gt Exp zero lam z (var
z) succ E lam s E pred E app E
... )
l ? l
idx
ln
l
Exp var Id lam Id Exp app Exp
Exp
Exp zero succ Exp pred Exp
11
Extending Java
javarepeat ? java

repeat ? java
java ? java
( repeat -gt java Stm -gt Stm,
Exp -gt Exp repeat S E do-while
S (not E) )
idx
repeat
java
Java grammar( 575 lines !!! ) ...
Stm "repeat" Stm "until" "(" Exp ")" ""
12
Banana Algebra

• Languages (L)
• l
• v
• L \ L
• L L
• src ( X )
• tgt ( X )
• let v L in L
• letx w X in L

• Transformations (X)
• x
• w
• X \ L
• X X
• X X
• idx ( L )
• let v L in X
• letx w X in X

( L -gt L ? c )
?CFG?
13
Algebraic Laws

• Idempotency of ''
• Commutativity of ''
• Associativity of ''
• Source-identity

L ? L L
L1 L2 ? L2 L1
L1 (L2 L3) ? (L1 L2) L3
Target-identity
L ? tgt(idx(L))
L ? src(idx(L))
14
Outline

• Introduction "What is a Banana?"
• Bananas for Language Transformation
• Language Extension Pattern
• Banana Algebra
• Examples
• Implementation
• Related Work
• Conclusion

15
Example Revisited
--- "ln2l.x" ---
let l "l.l" in let ln "ln.l" in idx(l)
( ln -gt l Exp -gt Exp Exp.zero
'\z.z' Exp.succ '\s.1'
Exp.pred '(1 \z.z)' )
--- "l.l" ---
--- "ln.l" ---
Id a-z a-z0-9 Exp.var Id
Exp.lam "\\" Id "." Exp
Exp.app "(" Exp Exp ")"
Exp.zero "zero" Exp.succ
"succ" "(" Exp ")" Exp.pred "pred" "("
Exp ")"
16
Numerals Booleans
llnlb ? l
with Nums Bools?

with Nums
with Bools
lbl ? l
lnl ? l


l ? l
lb ? l
ln? l
idx
idx
l
lb
l
ln
17
Java Repeat
--- "java.l" ---
575 lines
Java ... "try" Stm "catch" ...
Name.id Id
--- "repeat.l" ---
Stm.repeat "repeat" Stm "until" "(" Exp
")" ""
--- "repeat2java.x" ---
let java "java.l" in let repeat
"repeat.l" in idx(java) ( repeat -gt
java Exp -gt Exp, Stm -gt Stm
Stm.repeat 'do 1 while (!(2))' )
7 lines !
18
Concrete vs. Abstract Syntax
Concrete syntax
Stm.repeat 'do 1 while (!(2))'
Exp (with explicit assoc./prec.)
Abstract syntax
Exp.or Exp1 "" Exp .exp1 Exp1
Exp1.and Exp2 "" Exp1 .exp2
Exp2 Exp2.add Exp3 "" Exp2
.exp3 Exp3 ? Exp7.neg
"!" Exp8 .exp8 Exp8
Exp8.par "(" Exp ")" .var Id
.num IntConst
Stm.repeat Stm.do(lt1gt, Exp.exp1(
Exp1.exp2( Exp2.exp3(
Exp3.exp4( Exp4.exp5(
Exp5.exp6( Exp6.exp7(
Exp7.neg(
Exp8.par(lt2gt) )))))))))
(unambiguous concrete ? abstract)
NB Tool supports BOTH !
19
"FUN" Example
The "FUN" Language used for Teaching
Functional Programming
(at Aarhus University)
Fun
Basically The Lambda Calculus with numerals,
booleans, arithmetic, boolean logic, local
definitions, pairs, literals, lists, signs,
comparisons, dynamic types, fixed-point
combinators,
Fun grammar transform
Literals
Literals?Nums
Unsigned arithmetic booleans
definitions pairs
Nums??
Bools??
Defs??
Pairs??



Lambda Calculus
20
"FUN" Example
Component re-use
Fun
Fun
FunSigned

Fun grammar transform
Fun grammar transform
FunSigned GT

Literals
Literals?Nums
Literals?Nums
Signed arith?Nums
Unsigned arithmetic booleans
definitions pairs
Nums??
Bools??
Defs??
Pairs??



Lambda Calculus
21
"FUN" Example
Fun
FunSigned

FunCompare
FunTypesafe


Fun GT
FunSigned GT

FunCompare GT

FunTypesafe GT

245x Banana Algebra ops ? 4 MB Banana !
Unsigned arithmetic booleans
definitions pairs
Nums??
Bools??
Defs??
Pairs??



Lambda Calculus
22
"FUN" Usage Statistics

• Usage statistics (245x operators) in "FUN"
• 58x cfg Constant languages
• 51x "file.l" Language inclusions
• 28x L L Language additions
• 23x v Language variables
• 17x (L ? L ? c) Constant
  transformations
• 17x X X Transformation additions
• 14x "file.x" Transformation inclusions
• 10x let-in Local definitions
• 9x idx(L) Identity transformations
• 8x X X Compositions
• 4x L \ L Language restriction
• 4x w Transformation variables
• 2x src(X) Source extractions

23
Incremental Development
EXERCISE
--- "l.l" ---
--- "li.l" ---
Id a-z a-z0-9 Exp.var Id
Exp.lam "\\" Id "." Exp
Exp.app "(" Exp Exp ")"
Exp.id "id"
--- "li2l.x" ---
let l "l.l" in idx(l) ( "li.l" -gt l
Exp -gt Exp Exp.id '\z.z' )
--- "ln.l" ---
Exp.zero "zero" Exp.succ
"succ" Exp Exp.pred "pred" Exp
--- "ln2li.x" ---
--- "ln2l.x" ---
let l "l.l" in idx(l) ( "ln.l" -gt l
Exp -gt Exp Exp.zero '\z.z'
Exp.succ '\x.1' Exp.pred
'(1 \z.z)' )
let l "l.l" in idx(l) ( ln -gt l"li.l"
Exp -gt Exp Exp.zero 'id'
Exp.succ '\x.1' Exp.pred '(1
id)' )
--- "ln2l.x" ---
"li2l.x" o "ln2li.x"
24
Example cont'd

• Both statically reduce to same catamorphism

( Exp.app Exp.app(1, 2)
Exp.lam Exp.lam(1, 2) Exp.pred
Exp.app(1, Exp.lam(Id("z"), Exp.var(Id("z"))))
Exp.succ Exp.lam(Id("x"), 1)
Exp.var Exp.var(1) Exp.zero
Exp.lam(Id("z"), Exp.var(Id("z"))) )
Id a-z 0-9a-z Exp.app "(" Exp
Exp ")" Exp.lam "\" Id "." Exp
Exp.pred "pred" Exp Exp.succ "succ" Exp
Exp.var Id Exp.zero "zero"
Id a-z 0-9a-z Exp.app "(" Exp
Exp ")" Exp.lam "\" Id "." Exp Exp.var
Id
-gt
Exp -gt Exp, Id-gtId
25
Other Examples

• Self-Application (The tool on itself!)
• SQL embedding (in ltbigwiggt)
• My-Java (endless variations)

L1 ltlt L2 '(L1 \ L2) L2' X1 ltlt X2
'(X1 \ src(X2)) X2'
Stm.select 'factor (lt2gt) if (lt3gt)
return ( \ (lt1gt) ) '
java
( sql)
( \ loops)
o syntaxe_francais
26
Implementation
The 'Banana Algebra' Tool (3,600 lines of O'Caml)
http//www.itu.dk/people/brabrand/banana-algebra
/
Uses (underlying technologies)
'dk.brics.grammar' for parsing, unparsing, and
ambiguity analysis ! 'XSugar' for
transformation "concrete syntax ? abstract XML
syntax" 'XSLT' for transformation "XML ? XML"
27
Outline

• Introduction "What is a Banana?"
• Bananas for Language Transformation
• Language Extension Pattern
• Banana Algebra
• Examples
• Implementation
• Related Work
• Conclusion

28
Related Work (I/III)

• Macro Systems

• "Growing Languages with Metamorphic Syntax
  Macros" Claus Brabrand Michael
  Schwartzbach ( PEPM 2002 )

• "The metafront System Safe and Extensible
  Parsing and Transformation" Claus Brabrand
  Michael Schwartzbach ( LDTA 2003 , SCP J.
  2007 )

29
Related Work (II/III)

• Attribute Grammars
• Language transformation (and extension)
• via computation on AST's
• (using "inherited" or "synthesized" or
  attributes)
• E.g., Eli, JastAdd, Silver,
• Rewrite Systems
• Language transformation (and extension)
• via syntactic rewriting, using encodings
• gradually rewrite "S-syntax" to
  "T-syntax"
• gradually rewrite "S-syntax" to
  "T-syntax"
• E.g., Elan, TXL, ASFSDF, Stratego/XT,

S ? T
S ? T
30
Related Work (III/III)

• Functional Programming
• Catas mimicked by "disciplined style" of fun.
  programming
• aided by
• Traversal functions (auto-synthesized from
  datatypes)
• Combinator libraries
• "Shortcut fusion" (to eliminate ' ' at
  compile-time)
• Category Theory
• A lot of this work can be viewed as Category
  Theory

Basically ye olde issue GPL vs. DSL
31
Conclusion

• IF bananas are sufficiently
• (Expressive)
• THEN you get
• Banana Algebra for free (16 banana ops)
• Incremental
• Modular
• Simple
• Safe
• Efficient

"Niche"
?
32
BONUS SLIDES

• - Reduction Semantics -

If you want all the details

• "Syntactic Language Extension via an Algebra of
  Languages and Transformations" Jacob
  Andersen Claus Brabrand ( ITU Technical
  Report, Dec. 2008 )

33
Reduction Semantics

• Environments
• Reduction relations
• Abbreviations
• ...as a short-hand for
• ...as a short-hand for

ENVL VARL ? EXPL
environment of languages
ENVX VARX ? EXPX
environment of transformations
ENVL ? ENVX ? EXPL ? EXPL ? '?L'
ENVL ? ENVX ? EXPX ? EXPX ? '?X'
?,? - L ?L l
(?,?,L,l) ? '?L'
(?,?,X,x) ? '?X'
?,? - X ?X x
34
Semantics (L)
CONL
VARL
l
?,? l ?L l
wfl
?,? v ?L ?(v)
?,? L ?L l
?,? L' ?L l'
RESL
?,? L \ L' ?L l l'
l
?,? L ?L l
?,? L' ?L l'
l l'
ADDL
l
?,? L L' ?L l l'
l
35
Semantics (L)
?,? X ?X ( lS -gt lT ? c )
SRCL
?,? src (X) ?L lS
?,? X ?X ( lS -gt lT ? c )
TGTL
?,? tgt (X) ?L lT
?vl,? L' ?L l'
?,? L ?L l
LETL
?,? let vL in L' ?L l'
36
Semantics (X)
( lS -gt lT ? c )
?,? LT ?L lT
?,? LS ?L lS
wfx
CONX
?,? ( LS -gt LT ? c ) ?X ( lS -gt lT ? c
)
?,? X ?X x
?,? L ?L l
VARX
RESX
?,? w ?X ?(w)
?,? X \ L ?X x l
x
?,? X ?X x
?,? X' ?X x'
x x'
ADDX
x
?,? X X' ?X x x'
x
37
Semantics (X)
?,? X ?X ( lS -gt lT ? c )
?,? X' ?X ( lS' -gt lT' ?' c' )
lT lS'
COMPX
l
?,? X' X ?X ( lS -gt lT' ?' ? c' c )
?,? L ?L l
IDXX
?,? idx (L) ?X ( l -gt l id?(l) idc(l) )
?,?wx X' ?X x'
?,? X ?X x
LETL
?,? letx wX in X' ?X x'
38
BONUS SLIDES

• - More Examples -

39
Numeral Boolean Extension

• Numeral Extension (catamorphism)
• Boolean Extension (catamorphism)

var V var V lam V E lam
V E app E1 E2 app E1 E2
zero lam z (var z) succ E lam s
E pred E app E (lam z (var z))
Exp var Id lam Id Exp app Exp
Exp zero succ Exp pred Exp
Exp var Id lam Id Exp app Exp
Exp
var V var V lam V E lam V
E app E1 E2 app E1 E2
true lam a (lam b (var a)) false
lam a (lam b (var b)) if E1 E2 E3 app (app
E1 E2) E3
Exp var Id lam Id Exp app Exp
Exp true false if Exp Exp Exp
Exp var Id lam Id Exp app Exp
Exp
40
Lambda with Booleans
lbl ? l

lb ? l
( lb -gt l Exp -gt Exp true '\a.\b .
a' false '\a.\b . b' if E1 E2 E3
'((E1 E2) E3)' )
l ? l
idx
lb
l
Exp true false if Exp Exp Exp
Exp var Id lam Id Exp app Exp
Exp
41
Incremental Development
--- "l.l" ---
--- "li.l" ---
Id a-z a-z0-9 Exp.var Id
Exp.lam "\\" Id "." Exp
Exp.app "(" Exp Exp ")"
Exp.id "id"
--- "li2l.x" ---
let l "l.l" in idx(l) ( "li.l" -gt l
Exp -gt Exp Exp.id '\z.z' )
--- "ln.l" ---
Exp.zero "zero" Exp.succ
"succ" Exp Exp.pred "pred" Exp
--- "ln2li.x" ---
--- "ln2l.x" ---
let l "l.l" in idx(l) ( "ln.l" -gt l
Exp -gt Exp Exp.zero '\z.z'
Exp.succ '\x.1' Exp.pred
'(1 \z.z)' )
let l "l.l" in idx(l) ( ln -gt l"li.l"
Exp -gt Exp Exp.zero 'id'
Exp.succ '\x.1' Exp.pred '(1
id)' )
--- "ln2l.x" ---
"li2l.x" o "ln2li.x"
42
Java Repeat
javarepeat ? java

repeat ? java
java ? java
( repeat -gt java Exp -gt Exp,
Stm -gt Stm repeat S until (E) )
idx
java
Java ... ... Exp Exp "" Exp ... Stm
Exp "" "if" "(" Exp ")" Stm "while"
"(" Exp ")" Stm ...
repeat
575 lines
Stm "repeat" Stm "until" "(" Exp ")" ""
Entire extension 7 lines !
43
Usage Scenarios

• Programmers
• May extend existing languages ( syntax macros)
• Developers
• May embed DSLs into host languages (SQL in Java)
• Developers (and teachers)
• May incrementally specify multi-layered languages
• Compiler writers
• May rely on tool and implement only a small core
• (and then specify the rest externally as
  extensions)

44
BONUS SLIDES

• - Parsing Error Reporting -

45
Parsing

• Parsing (XSugar)
• Variant of Earley's algorithm O ( ?3 )
• Can parse any context-free grammar
• Closed under union of languages
• Support for production priority
• Tool easily adapts to other parsing algorithms

46
Ambiguity parsing?unparsing
.
.
ASTL / L
L
.
.

• Unparsing
• Canonical whitespace

• Parsing
• Grammar ambiguity

?
?
47
Ambiguity Analysis

• Ambiguity Analysis
• Using implementation ( )
  on
• Source language
• Target language and/or
• all intermediate languages (somewhat expensive)
• (Note Ambiguity analysis comes with XSugar tool)

• "Analyzing Ambiguity of Context-Free Grammars"
  Claus Brabrand Robert Giegerich Anders
  Møller ( CIAA 2007 )

"dk.brics.grammar" by Anders Møller
48
Error Reporting

• Error reporting
• Static parse-error (O'Caml-lex)
• Static transformation error (XSugar)
• (is actually a parse-error in a cata
  reconstructor)
• Dynamic parse-error (XSugar)
• Dynamic transformation error
• impossible -)

Prototype
In ln2l.x (4,4)-(4,7) Parse error at
"Exp"
Parse error at character 6 (line 1, column 7)
in /tmp/shape84e645.txt
Could be improved
Parse error at character 23 (line 1, column
24) in /dev/stdin