Mathematical Structures in Programming Introduction
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1 Mathematical Structures in Programming to Calculational Programming SOKENDAI/National Institute of Informatics URL: hu Sokendai, 2015 Mathematical Structures in Programming
2 Self Mathematical Structures in Programming
3 Self 1988: BS, Dept. of Computer Science, Shanghai Jiaotong Univ. Mathematical Structures in Programming
4 Self 1988: BS, Dept. of Computer Science, Shanghai Jiaotong Univ. 1991: MS, Dept. of Computer Science, Shanghai Jiaotong Univ. Mathematical Structures in Programming
5 Self 1988: BS, Dept. of Computer Science, Shanghai Jiaotong Univ. 1991: MS, Dept. of Computer Science, Shanghai Jiaotong Univ. 1996: PhD, Information Engineering, University of Tokyo Mathematical Structures in Programming
6 Self 1988: BS, Dept. of Computer Science, Shanghai Jiaotong Univ. 1991: MS, Dept. of Computer Science, Shanghai Jiaotong Univ. 1996: PhD, Information Engineering, University of Tokyo 1996: Research Associate (Assistant Professor), Univ. of Tokyo Mathematical Structures in Programming
7 Self 1988: BS, Dept. of Computer Science, Shanghai Jiaotong Univ. 1991: MS, Dept. of Computer Science, Shanghai Jiaotong Univ. 1996: PhD, Information Engineering, University of Tokyo 1996: Research Associate (Assistant Professor), Univ. of Tokyo 1997: Lecturer, Univ. of Tokyo Mathematical Structures in Programming
8 Self 1988: BS, Dept. of Computer Science, Shanghai Jiaotong Univ. 1991: MS, Dept. of Computer Science, Shanghai Jiaotong Univ. 1996: PhD, Information Engineering, University of Tokyo 1996: Research Associate (Assistant Professor), Univ. of Tokyo 1997: Lecturer, Univ. of Tokyo 2000: Associate Professor, Univ. of Tokyo Mathematical Structures in Programming
9 Self 1988: BS, Dept. of Computer Science, Shanghai Jiaotong Univ. 1991: MS, Dept. of Computer Science, Shanghai Jiaotong Univ. 1996: PhD, Information Engineering, University of Tokyo 1996: Research Associate (Assistant Professor), Univ. of Tokyo 1997: Lecturer, Univ. of Tokyo 2000: Associate Professor, Univ. of Tokyo 2008: Full Professor, National Institute of Informatics (NII) Mathematical Structures in Programming
10 Self 1988: BS, Dept. of Computer Science, Shanghai Jiaotong Univ. 1991: MS, Dept. of Computer Science, Shanghai Jiaotong Univ. 1996: PhD, Information Engineering, University of Tokyo 1996: Research Associate (Assistant Professor), Univ. of Tokyo 1997: Lecturer, Univ. of Tokyo 2000: Associate Professor, Univ. of Tokyo 2008: Full Professor, National Institute of Informatics (NII) Mathematical Structures in Programming
11 Research Interests Functional Programming (FP) Learned FP from Prof. Yonggiang Sun in SJTU since 1986 Serving on steering committees of ICFP, Haskell, APLAS Mathematical Structures in Programming
12 Research Interests Functional Programming (FP) Learned FP from Prof. Yonggiang Sun in SJTU since 1986 Serving on steering committees of ICFP, Haskell, APLAS Algorithmic Languages and Calculi PhD thesis: Calculational Approach to Program Optimization Serving as an IFIP WG 2.1 member (TC 2 Japan Representative) Mathematical Structures in Programming
13 Research Interests Functional Programming (FP) Learned FP from Prof. Yonggiang Sun in SJTU since 1986 Serving on steering committees of ICFP, Haskell, APLAS Algorithmic Languages and Calculi PhD thesis: Calculational Approach to Program Optimization Serving as an IFIP WG 2.1 member (TC 2 Japan Representative) Bidirectional Programming & Software Engineering Started in 2003 Serving on BX steering committee, IEEE TSE Associate Editor Mathematical Structures in Programming
14 Related Lectures Mathematical Structures in Programs Preliminary topics given to undergraduate students in Department of Mathematical Informatics, Univ. of Tokyo, Advanced topics given to graduate school of Information Science and Engineering, Univ. of Tokyo, , and Sokendai, Intensive course for graduate students in Kyoto Univ., 2009, and in Peking Univ., Mathematical Structures in Programming
15 Related Lectures Mathematical Structures in Programs Preliminary topics given to undergraduate students in Department of Mathematical Informatics, Univ. of Tokyo, Advanced topics given to graduate school of Information Science and Engineering, Univ. of Tokyo, , and Sokendai, Intensive course for graduate students in Kyoto Univ., 2009, and in Peking Univ., to Program Calculation/Transformation Summer School on Generative and Transformational Techniques in Software Engineering (GTTSE 2005, Braga, Portugal, 4-8 July, 2005) BASICS Summer School on Types and Programming Languages (BASICS 2010, Hangzhou, China, October 12-16, 2010) Mathematical Structures in Programming
16 Tsuru-Kame-Zan The Tsuru-Kame Problem Some cranes (tsuru) and tortoises (kame) are mixed in a cage. Known is that there are 6 heads and 20 legs. Find out the numbers of cranes and tortoises. Mathematical Structures in Programming
17 A Kindergarten Approach Mathematical Structures in Programming
18 A Kindergarten Approach A simple enumeration Mathematical Structures in Programming
19 A Kindergarten Approach A simple enumeration Mathematical Structures in Programming
20 Primary School Mathematical Structures in Programming
21 Primary School Reasoning Mathematical Structures in Programming
22 Primary School Reasoning if all 6 animals were cranes, there ought to be 6 2 = 12 legs. Mathematical Structures in Programming
23 Primary School Reasoning if all 6 animals were cranes, there ought to be 6 2 = 12 legs. However, there are in fact 20 legs, the extra = 8 legs must belong to some tortoises. Mathematical Structures in Programming
24 Primary School Reasoning if all 6 animals were cranes, there ought to be 6 2 = 12 legs. However, there are in fact 20 legs, the extra = 8 legs must belong to some tortoises. Since one tortoise can add 2 legs, we have 8/2 = 4 tortoises. Mathematical Structures in Programming
25 Primary School Reasoning if all 6 animals were cranes, there ought to be 6 2 = 12 legs. However, there are in fact 20 legs, the extra = 8 legs must belong to some tortoises. Since one tortoise can add 2 legs, we have 8/2 = 4 tortoises. So there must be 6 4 = 2 cranes. Mathematical Structures in Programming
26 Middle School Mathematical Structures in Programming
27 Middle School Algebra (Equation Theory) Mathematical Structures in Programming
28 Middle School Algebra (Equation Theory) x + y = 6 2x + 4y = 20 Mathematical Structures in Programming
29 Middle School Algebra (Equation Theory) x + y = 6 2x + 4y = 20 which gives x = 2 y = 4 Mathematical Structures in Programming
30 The same problem may have different difficulties depending on what weapons we have in hand. Many arithmetic problems can be easily solved if we use the equation theory. Mathematical Structures in Programming
31 The same problem may have different difficulties depending on what weapons we have in hand. Many arithmetic problems can be easily solved if we use the equation theory. What are weapons for solving programming problems? Do we have an equation theory for constructing correct and efficient programs? Mathematical Structures in Programming
32 The same problem may have different difficulties depending on what weapons we have in hand. Many arithmetic problems can be easily solved if we use the equation theory. What are weapons for solving programming problems? Do we have an equation theory for constructing correct and efficient programs? Calculational Programming Mathematical Structures in Programming
33 A Programming Problem Can you develop a correct linear-time program for solving the following problem? Maximum Segment Sum Problem Given a list of numbers, find the maximum of sums of all consecutive sublists. [ 1, 3, 3, 4, 1, 4, 2, 1] = 7 [ 1, 3, 1, 4, 1, 4, 2, 1] = 6 [ 1, 3, 1, 4, 1, 1, 2, 1] = 4 Mathematical Structures in Programming
34 A Simple Solution 1 Enumerating all segments (segs); Mathematical Structures in Programming
35 A Simple Solution 1 Enumerating all segments (segs); 2 Computing sum for each segment(sums); Mathematical Structures in Programming
36 A Simple Solution 1 Enumerating all segments (segs); 2 Computing sum for each segment(sums); 3 Calculating the maximum of all the sums (max). Mathematical Structures in Programming
37 A Simple Solution 1 Enumerating all segments (segs); 2 Computing sum for each segment(sums); 3 Calculating the maximum of all the sums (max). Mathematical Structures in Programming
38 A Simple Solution 1 Enumerating all segments (segs); 2 Computing sum for each segment(sums); 3 Calculating the maximum of all the sums (max). Exercise How many segments does a list of length n have? Mathematical Structures in Programming
39 A Simple Solution 1 Enumerating all segments (segs); 2 Computing sum for each segment(sums); 3 Calculating the maximum of all the sums (max). Exercise How many segments does a list of length n have? Exercise What is the time complexity of this simple solution? Mathematical Structures in Programming
40 There indeed exists a clever solution! mss=0; s=0; for(i=0;i<n;i++){ s += x[i]; if(s<0) s=0; if(mss<s) mss= s; } x[i] s mss Mathematical Structures in Programming
41 There is a big gap between the simple and clever solutions! Mathematical Structures in Programming
42 There is a big gap between the simple and clever solutions! Can we calculate the clever solution from the simple solution? Mathematical Structures in Programming
43 There is a big gap between the simple and clever solutions! Can we calculate the clever solution from the simple solution? What rules and theorems are necessary to do so? Mathematical Structures in Programming
44 There is a big gap between the simple and clever solutions! Can we calculate the clever solution from the simple solution? What rules and theorems are necessary to do so? How to apply the rules and theorems to do so? Mathematical Structures in Programming
45 There is a big gap between the simple and clever solutions! Can we calculate the clever solution from the simple solution? What rules and theorems are necessary to do so? How to apply the rules and theorems to do so? Can we reuse the derivation procedure to solve similar problems, say maximum increasing segment sum problme? Mathematical Structures in Programming
46 Transformational Programming One starts by writing clean and correct programs, and then use program transformation techniques to transform them step-by-step to more efficient equivalents. Specification: Clean and Correct programs Folding/Unfolding Program Transformation Efficient Programs Mathematical Structures in Programming
47 Program Calculation Program calculation is a kind of program transformation based on Constructive Algorithmics, a framework for developing laws/rules/theories for manipulating programs. Mathematical Structures in Programming
48 Program Calculation Program calculation is a kind of program transformation based on Constructive Algorithmics, a framework for developing laws/rules/theories for manipulating programs. Specification: Clean and Correct programs Folding-free Program Transformation Efficient Programs Mathematical Structures in Programming
49 Work on Program Calculation Algorithm Derivation Fold/Unfold-based Transformational Programming (Darlington&Burstall:77) Bird-Meertens Formalism (BMF) (Bird:87) Algebra of Programming (Bird&de Moor:96) Mathematical Structures in Programming
50 Work on Program Calculation Algorithm Derivation Fold/Unfold-based Transformational Programming (Darlington&Burstall:77) Bird-Meertens Formalism (BMF) (Bird:87) Algebra of Programming (Bird&de Moor:96) Our Work on Program Transformation in Calculation Form Fusion (ICFP 96) Tupling (ICFP 97) Accumulation (NGC 99) Inversion/Bidirectionalization (MPC 04, PEPM 07, ICFP 07, MPC 10, ICFP 10) Dynamic Programming (ICFP 00, ICFP 03, ICFP 08) Parallelization (POPL 98, ESOP 02, PLDI 07, POPL 09, ESOP 12) Mathematical Structures in Programming
51 What I will talk in this course? Algorithm Derivation Fold/Unfold-based Transformational Programming (Darlington&Burstall:77) Bird-Meertens Formalism (BMF) (Bird:87) Algebra of Programming (Bird&de Moor:96) Our Work on Program Transformation in Calculation Form Fusion (ICFP 96) Tupling (ICFP 97) Accumulation (NGC 99) Inversion/Bidirectionalization (MPC 04, PEPM 07, ICFP 07, MPC 10, ICFP 10) Dynamic Programming (ICFP 00, ICFP 03, ICFP 08) Parallelization (POPL 98, ESOP 02, PLDI 07, POPL 09, ESOP 12) Mathematical Structures in Programming
52 What I will talk in this course? Algorithm Derivation Fold/Unfold-based Transformational Programming (Darlington&Burstall:77) Bird-Meertens Formalism (BMF) (Bird:87) Algebra of Programming (Bird&de Moor:96) Our Work on Program Transformation in Calculation Form Fusion (ICFP 96) Tupling (ICFP 97) Accumulation (NGC 99) Inversion/Bidirectionalization (MPC 04, PEPM 07, ICFP 07, MPC 10, ICFP 10) Dynamic Programming (ICFP 00, ICFP 03, ICFP 08) Parallelization (POPL 98, ESOP 02, PLDI 07, POPL 09, ESOP 12) Functional Programming (basic concepts of algorithmic languages, program specification and reasoning) Mathematical Structures in Programming
53 Course Plan 1 to Functional Programming (about 4 lectures) Learn basic concepts of programming languages (FP as a meta language) Learn how to build programs constructively and how to reason about programs (FP as a programming language) Mathematical Structures in Programming
54 Course Plan 1 to Functional Programming (about 4 lectures) Learn basic concepts of programming languages (FP as a meta language) Learn how to build programs constructively and how to reason about programs (FP as a programming language) 2 Program Calculus: BMF (about 4 lectures) Learn basic programming theory for calculating programs from problem specifications Learn basic techniques for calculating programs Mathematical Structures in Programming
55 Course Plan 1 to Functional Programming (about 4 lectures) Learn basic concepts of programming languages (FP as a meta language) Learn how to build programs constructively and how to reason about programs (FP as a programming language) 2 Program Calculus: BMF (about 4 lectures) Learn basic programming theory for calculating programs from problem specifications Learn basic techniques for calculating programs 3 Applications of Calculational Programming (about 4 lectures) Learn how to solve a wide class of optimization problems Learn how to automatic parallelize sequential programs Mathematical Structures in Programming
56 Course Plan 1 to Functional Programming (about 4 lectures) Learn basic concepts of programming languages (FP as a meta language) Learn how to build programs constructively and how to reason about programs (FP as a programming language) 2 Program Calculus: BMF (about 4 lectures) Learn basic programming theory for calculating programs from problem specifications Learn basic techniques for calculating programs 3 Applications of Calculational Programming (about 4 lectures) Learn how to solve a wide class of optimization problems Learn how to automatic parallelize sequential programs 4 In-class Discussion (about 2 lectures) Discuss solutions to programming exercises in the class Mathematical Structures in Programming
57 Evaluation No Final Written Examination :-) Mathematical Structures in Programming
58 Evaluation No Final Written Examination :-) In-class activity: 50% Report: 20% Final Presentation: 30% Mathematical Structures in Programming
59 Evaluation No Final Written Examination :-) In-class activity: 50% Report: 20% Final Presentation: 30% You can get 100% unconditionally, if you can solve the open problem which will be discussed in the class Mathematical Structures in Programming
60 References Graham Hutton, Programming in Haskell, Cambridge University Press, Richard Bird, Lecture Notes on Constructive Functional Programming, Technical Monograph PRG-69, Oxford University, Anne Kaldewaij, Programming: The Derivation of Algorithms, Prentice Hall, Richard Bird and Oege de Moor, The Algebra of Programming, Prentice-Hall, Roland Backhouse, Program Construction: Calculating Implementation from Specification, Wiley, Slides will be available at the following website after the class. Mathematical Structures in Programming
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