Algebraic Dynamic Programming

Transcription

1 Algebraic Dynamic Programming Unit 2.b: Introduction to Bellman s GAP Robert Giegerich 1 (Lecture) Benedikt Löwes (Exercises) Faculty of Technology Bielefeld University Summer robert@techfak.uni-bielefeld.de Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

2 ADP recall Algebraic Dynamic Programming Separation of concerns Grammar Annotation Table Design Candidate Evaluation Algebras Grammar Search Space Desc. Candidate Description Signature Ban on indices ( No subscripts no errors! ) Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

3 ADP recall ADP implementation First implementation of ADP: as embedded domain specific language in Haskell executable via Haskell interpreter or compiler a compiler for this edsl (translation to C) Second generation ADP system: Bellman s GAP by Georg Sauthoff new Language for ADP programs C/Java-like Syntax new features new compiler with numerous optimizations released 2011 Others: ADPfusion, Scala-based implementation Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

4 Example Example: Optimal Matrix chain multiplication A closer look at the matrix chain problem (cf. Unit 1.b): Input: A R , B R 100 5, C R 5,50, X = A B C Goal: Compute X with the least number of elementary multiplications. Search space T 1 = (A B) C, T 2 = A (B C) Why is this an optimization problem at all? Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

5 Example Optimal Matrix chain multiplication (ctd.) Effort of a single multiplication: A a b B b c takes a b c scalar multiplications and gives an intermediate (a c)-matrix. Matrix multiplication is associative: (A B) C = A B C = A (B C) We can optimize the grouping of multiplications. For A , B 100 5, C 5 50 : (A B) C = = 7500 A (B C) = = ,. and things get more dramatic for longer matrix chain products (as they frequently arise in physics) Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

6 Example Example: Optimal Matrix chain multiplication (ctd.) Input: A R , B R 100 5, C R 5,50, X = A B C Goal: Compute X with the least number of elementary multiplications. Search space T 1 = (A B) C, T 2 = A (B C) C A A B B C Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

7 Example Example: Optimal matrix chain multiplication Tree grammar mult matrices matrices single matrices matrices rowcolumn rowcolumn Int, Int Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

8 Example Example: Optimal matrix chain multiplication Evaluation algebra For each intermediate chain, compute a triple (optimal number of mults, rows, columns) needed to compute and store the intermediate matrix. 1 s i n g l e (m) = t u p l e ( 0, rows (m), c o l s (m) ) 2 mult ( a, b ) = t u p l e ( ops ( a)+ops ( b)+ 3 rows ( a ) c o l s ( a ) c o l s ( b ), 4 rows ( a ), 5 c o l s ( b ) ) 6 h ( l ) = [ minimum ( l ) ] Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

9 Bellman s GAP The Bellman s GAP language Bellman s GAP by provides the language GAP-L and the compiler GAP-C Bellman s GAP is derived from Bellman s Principle, Grammars, Algebras and Products Signature 1 s i g n a t u r e S i g ( alphabet, answer ) 2 { 3 answer s i n g l e ( i n t, char, i n t ) ; 4 answer mult ( answer, char, answer ) ; 5 choice [ answer ] h ( [ answer ] ) ; 6 } Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

10 Bellman s GAP Example 1 a l g e b r a m i n m u l t i implements S i g ( alphabet = char, 2 answer = t u p l e ) 3 { 4 t u p l e s i n g l e ( i n t r, char a, i n t c ) 5 { 6 t u p l e x ; 7 x. ops = 0 ; 8 x. rows = r ; 9 x. c o l s = c ; 0 r e t u r n x ; 1 } 2 t u p l e mult ( t u p l e a, char x, t u p l e b ) {... } 3 choice [ t u p l e ] h ( [ t u p l e ] l ) 4 { 5 r e t u r n l i s t ( minimum ( l ) ) ; 6 } 7 } Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

11 Bellman s GAP Example Automatic algebras 1 a l g e b r a co auto count ; 2 a l g e b r a en auto enum ; Algebra extension 1 a l g e b r a maxmulti extends m i n m u l t i 2 { 3 choice [ t u p l e ] h ( [ t u p l e ] l ) 4 { 5 r e t u r n l i s t (maximum( l ) ) ; 6 } 7 } Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

12 Bellman s GAP Example Grammar 1 grammar mopt uses S i g ( axiom = m a t r i x ) { 2 3 m a t r i x = s i n g l e ( INT, CHAR(, ), INT ) 4 mult ( matrix, CHAR(, ), m a t r i x ) # h ; 5 6 } Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

13 Bellman s GAP Bellman s GAP Compiler BGAP Code BGAP Compiler C++ Code Makefile Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

14 Bellman s GAP Bellman s GAP Compiler ADP-aware type checking Semantic Analyses/Optimizations Runtime Table Design Yield Size Index elimination List elimination... Codegeneration Automatic parallelization for OpenMP Top-Down vs. Bottom-Up evaluators Different Backtracing schemes Sampling Window-Mode... Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

15 Bellman s GAP Multitrack Multitrack 1 skipr = s r ( < EMPTY, CHAR >, skipr ) 2 s k i p L # h ; 3 4 s k i p L = s l ( < CHAR, EMPTY >, s k i p L ) 5 a l i g n m e n t # h ; 6 7 a l i g n m e n t = n i l ( < SEQ, SEQ> ) 8 d e l ( < CHAR, EMPTY >, xdel ) 9 i n s ( < EMPTY, CHAR >, x I n s ) 0 match ( < CHAR, CHAR >, a l i g n m e n t ) # h ; 2 3 xdel = a l i g n m e n t 4 d e l x ( < CHAR, EMPTY >, xdel ) # h ; 5 6 x I n s = a l i g n m e n t 7 i n s x ( < EMPTY, CHAR >, x I n s ) # h ; Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

16 Bellman s GAP Grammar Filtering Terminal symbols and parsers Yield Size Parser min max CHAR, CHAR(arg) 1 1 STRING 1 n BASE 1 1 REGION 1 n REGION0 0 n FLOAT 1 n INT 1 n SEQ 1 n EMPTY 0 0 LOC Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

17 Bellman s GAP Grammar Filtering Grammar filtering 1 c l o s e d = { s t a c k h a i r p i n l e f t B r i g h t B 2 i l o o p m u l t i l o o p } 3 with s t a c k p a i r i n g # h ; 4 5 i l o o p = i l ( BASE, BASE, REGION with maxsize ( 3 0 ), 6 c l o s e d, 7 REGION with maxsize ( 3 0 ), BASE, BASE) 8 # h ; syntactic and semantic filters with overlay, e.g. f(a, B) with overlay samesize access on argument-subwords of a function symbol suchthat, suchthat overlay filter evaluated candidates Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

18 Conclusion Context Bellman s GAP is used for hand-written grammars with up to non-terminals there are grammar generators that generate Bellman s GAP code resulting grammars with non-terminals are not uncommon in this use case Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

19 Conclusion Conclusion Bellman s GAP is a domain specific language for specification of DP algorithms on sequences that simplifies the development of large DP algorithms Bellman s GAP compiler contains multiple semantic analyses and codegeneration options the generated C++-Code is competitive with hand-optimized code reduces development time Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 20

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