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Semantically-Aligned Equation Generation
for Solving and Reasoning Math Word Problems
Ting-Rui Chiang and Yun-Nung (Vivian) Chen
https://github.com/MiuLab/E2EMathSolver
N T U M I U L A B Each notebook takes $0.5 and each pen takes $1. Tom has
$10. How many notebooks can he buy after buying 5 pens?
Math Word Problem
x = 10 − 1 × 5 ÷ 0.5
Reasoning & Solving
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Prior Work
Non-neural approaches
• Template-based
(Kushman et al., Upadhyay and Chang)
Rely on hand-crafted features!
Deep learning
• Seq2Seq
(Wang et al., Ling et al.)
Does not use the structure of math expression.
x = (? + ?) × ? - ?
x = (1+ 2) × 3 - 4
fill
Problem
x = (1+ 2) × 3 - 4
generate
Our model is end-to-end and structural!
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Decoder
Encoder
Overview of the Proposed Model
stack action
stack action
stack action
stack action
Each notebook takes $0.5 and each pen takes $1.
Tom has $10. How many notebooks can he buy after buying 5 pens?
x = 10 − 1 × 5 ÷ 0.5
N T U M I U L A B Each notebook takes $0.5 and each pen takes $1. Tom has
$10. How many notebooks can he buy after buying 5 pens?
Look Again at the Problem
$1
$10
$0.5
?
N T U M I U L A B Each notebook takes $0.5 and each pen takes $1. Tom has
$10. How many notebooks can he buy after buying 5 pens?
Semantic Meaning of the Operands
x = ( 10 − 1 × 5 ) ÷ 0.5
The amount of money Tom has
Price of a pen
Price of a notebook
Number of pens bought
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Idea: Bridging Symbolic and Semantic Worlds
Symbolic World Semantic World
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Preprocess
0.5 1 10
5
Preprocess
Symbolic Part
Each notebook takes $0.5 and each pen takes $1. Tom has
$10. How many notebooks can he buy after buying 5 pens?
N T U M I U L A B
Symbol Encoding
0.5 1 10
5
Preprocess
Symbolic Part
Each notebook takes $0.5 and each pen takes $1. Tom has
$10. How many notebooks can he buy after buying 5 pens?
Encode
Semantic Part
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Inside Encoder
Each notebook takes $ 0.5 and ...
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Semantic Generation for Unknown x
Each notebook takes $ 0.5 and ...
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Operands & Their Semantics
0.5 1 10
5
Symbolic Part x Semantic Part
Each notebook takes $0.5 and each pen takes $1. Tom has
$10. How many notebooks can he buy after buying 5 pens?
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Intuition of Using Semantics
x = ( 10 − 1 ? 5 )
Price of a pen.
Number of pens bought.
Each notebook takes $0.5 and each pen takes $1. Tom has
$10. How many notebooks can he buy after buying 5 pens?
N T U M I U L A B
x 10 1 5 × − 0.5 ÷ =
Equation Generation in Postfix
Each notebook takes $0.5 and each pen takes $1. Tom has
$10. How many notebooks can he buy after buying 5 pens?
N T U M I U L A B
• Stack is used
• The decoder generates stack actions.
• An equation is generated with actions on stack.
Equation Generation by Stack Actions
x = 10 − 1 × 5 ÷ 0.5
Decoder
stack action
stack action
stack action
stack
action
N T U M I U L A B Encoder
Action Selection in Each Step
Decoder
classifier
stack action
{+, -, × , ÷ , =, Push}
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Target Equation: x = 10 − 1 × 5 ÷ 0.5 Generated Actions:
Equation Generation by Stack Actions
0.5 1 10
5 x
Action: push
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x 0.5
1 10
5 x
10
Action: push
1
Target Equation: x = 10 − 1 × 5 ÷ 0.5 Generated Actions:
x
Equation Generation by Stack Actions
5
Target Equation: x = 10 − 1 × 5 ÷ 0.5 Generated Actions:
x 10
Target Equation: x = 10 − 1 × 5 ÷ 0.5 Generated Actions:
x 10
Target Equation: x = 10 − 1 × 5 ÷ 0.5 Generated Actions:
x 10 1
Target Equation: x = 10 − 1 × 5 ÷ 0.5 Generated Actions:
x 10 1 5
N T U M I U L A B
x 0.5
1 10
5 x
10 1
Target Equation: x = 10 − 1 × 5 ÷ 0.5 Generated Actions:
x
Equation Generation by Stack Actions
5
Target Equation: x = 10 − 1 × 5 ÷ 0.5 Generated Actions:
x 10
Target Equation: x = 10 − 1 × 5 ÷ 0.5 Generated Actions:
x 10
Target Equation: x = 10 − 1 × 5 ÷ 0.5 Generated Actions:
x 10 1
Target Equation: x = 10 − 1 × 5 ÷ 0.5 Generated Actions:
x 10 1 5
Action: ×
1 × 5
N T U M I U L A B
x=(10−1 × 5) ÷ 0.5 0.5
1 10
5 x
Target Equation: x = 10 − 1 × 5 ÷ 0.5 Generated Actions:
x
Equation Generation by Stack Actions
Target Equation: x = 10 − 1 × 5 ÷ 0.5 Generated Actions:
x 10
Target Equation: x = 10 − 1 × 5 ÷ 0.5 Generated Actions:
x 10
Target Equation: x = 10 − 1 × 5 ÷ 0.5 Generated Actions:
x 10 1
Target Equation: x = 10 − 1 × 5 ÷ 0.5 Generated Actions:
x 10 1 5 × 0.5 ÷ =
After many steps…
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• Target equation is given.
• Trained as Seq2Seq.
Training Process
Decoder Encoder
Each notebook takes $0.5 and each pen takes $1. Tom has $10. How many notebooks can he buy after buying 5 pens?
x 10 1 5 …
x 10 1 5 …
<bos>
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• Dataset: Math23k
• In Chinese
• 23000 math word problems.
• Operators: +, -, ×, ÷
Experiments
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Results
45 50 55 60 65 70
Retrieval BLSTM Self-Attention Seq2Seq w/SNI Proposed Hybrid
Acc.
Retrieval Template Generation Ensemble
≈ 8%
> 1%
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59 60 61 62 63 64 65 66
Char-Based Word-Based Word-Based -Semantic
Word-Based -Gate
Word-Based -Gate -Attention
Word-Based -Gate -Attention
-Stack
