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A Language for Mathematical Knowledge Management. Steve Kieffer Carnegie Mellon University. What is Mathematical Knowledge?. Theorems Definitions Proofs. How do we “manage” it?. Experts Books Wikipedia?. Idea for MKM: Encyclopedia Entries, Stored in Computers. - PowerPoint PPT Presentation
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A Language for Mathematical Knowledge Management
Steve Kieffer
Carnegie Mellon University
2
What is Mathematical Knowledge?
Theorems Definitions Proofs
3
How do we “manage” it?
Experts Books Wikipedia?
4
Idea for MKM: Encyclopedia Entries, Stored in Computers
5
How transparent is the entry, to the computer?
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What do we want to do with mathematical knowledge?
Learn it Add to it Study its history Formally verify it Study its logical structure ...
7
My Work
1. Parser 2. Database of definitions 3. Translator 4. Statistics on logical structure 5. GUI for concept exploration
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Choice of Language
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Choice of Language
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Mizar
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LPT
Designed by Friedman Adds nice features to language of set theory
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Terms in LPT
Ordered tuples:
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Terms in LPT
Function evaluation:
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Terms in LPT
Infix functions:
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Terms in LPT
Sets:
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Terms in LPT
Descriptions:
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Terms in LPT
Lambdaabstraction:
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Formulas in LPT
Predication:
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Formulas in LPT
Infix relations:
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Formulas in LPT
Quantifiers:
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R is a partial order on A
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R is a partial order on A
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What is parsing?
(a a) a
ET
F
a(
)
TFE
T E
aF
a
TF
[[[([[[a]F]T[[[a]F]T]E]E)]F[[a]F]T]T]E[[[([[[a]F]T[[[a]F]T]E]E)]F[[a]F]T]T]E[[[([[[a]F]T[[[a]F]T]E]E)]F[[a]F]T]T]E[[[([[[a]F]T[[[a]F]T]E]E)]F[[a]F]T]T]E[[[([[[a]F]T[[[a]F]T]E]E)]F[[a]F]T]T]E[[[([[[a]F]T[[[a]F]T]E]E)]F[[a]F]T]T]E[[[([[[a]F]T[[[a]F]T]E]E)]F[[a]F]T]T]E[[[([[[a]F]T[[[a]F]T]E]E)]F[[a]F]T]T]E[[[([[[a]F]T[[[a]F]T]E]E)]F[[a]F]T]T]E[[[([[[a]F]T[[[a]F]T]E]E)]F[[a]F]T]T]E[[[([[[a]F]T[[[a]F]T]E]E)]F[[a]F]T]T]E[[[([[[a]F]T[[[a]F]T]E]E)]F[[a]F]T]T]E
ET
F
a(
)
TFE
T E
aF
a
TF
[ ][ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]var var var var var var varrel’n ][
formula
[ ]tuple
[ ]formula
[ ]term
formula
FCN [ f ] f = { < x , y > : f ( x ) = y }
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Parsing method
Earley algorithm n3 runtime parses any context free grammar
Simple example
E T EE T T F T T F F ( E )F a
Grammar:
Input:
( a a ) a
E
T
F T
Fa
( )E
T E
T
Fa
Fa
Earley algorithm
E T EE T T F T T F F ( E )F a
Grammar:
Input:
( a a ) a
I0
[E T E, 0][E T, 0][T F T, 0][T F, 0][F (E), 0][F a, 0]
I1
[F ( E), 0][E T E, 1][E T, 1][T F T, 1][T F, 1][F (E), 1][F a, 1]
I2
[F a , 1][T F T, 1][T F , 1][E T E, 1][E T , 1][F (E ), 0]
I3
[E T E, 1][E T E, 3][E T, 3][T F T, 3][T F, 3][F (E), 3][F a, 3]
I4
[F a , 3][T F T, 3][T F , 3][E T E, 3][E T , 3][E T E , 1][F (E ), 0]
I5
[F (E) , 0][T F T, 0][T F , 0][E T E, 0][E T , 0]
I6
[T F T, 0][T F T, 6][T F, 6][F (E), 6][F a, 6]
I7
[F a , 6][T F T, 6][T F , 6][T F T , 0][E T E, 0][E T , 0]
(1)(2)(3)(4)(5)(6)
I7
[F a , 6][T F T, 6][T F , 6][T F T , 0][E T E, 0][E T , 0]
64642156432
I7
[F a , 6][T F T, 6][T F , 6][T F T , 0][E T E, 0][E T , 0]
64642156432
I7
[F a , 6][T F T, 6][T F , 6][T F T , 0][E T E, 0][E T , 0]
64642156432
I7
[F a , 6][T F T, 6][T F , 6][T F T , 0][E T E, 0][E T , 0]
64642156432
I5
[F (E) , 0][T F T, 0][T F , 0][E T E, 0][E T , 0]
64642156432
I4
[F a , 3][T F T, 3][T F , 3][E T E, 3][E T , 3][E T E , 1][F (E ), 0]
64642156432
I4
[F a , 3][T F T, 3][T F , 3][E T E, 3][E T , 3][E T E , 1][F (E ), 0]
6464215643264642156432
I4
[F a , 3][T F T, 3][T F , 3][E T E, 3][E T , 3][E T E , 1][F (E ), 0]
64642156432
I4
[F a , 3][T F T, 3][T F , 3][E T E, 3][E T , 3][E T E , 1][F (E ), 0]
64642156432
I2
[F a , 1][T F T, 1][T F , 1][E T E, 1][E T , 1][F (E ), 0]
64642156432
I2
[F a , 1][T F T, 1][T F , 1][E T E, 1][E T , 1][F (E ), 0]
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My Work
1. Parser 2. Database of definitions 3. Translator 4. Statistics on logical structure 5. GUI for concept exploration
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Translation
LPT as language for proof system? Set up translation to make database useable. Database has set-theoretic foundational
definitions (e.g. von Neumann ordinals). Translate into DZFC (“Definitional ZFC”), a
conservative extension of ZFC.
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Comparison: LPT vs. DZFC
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Comparison: LPT vs. DZFC
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Comparison: LPT vs. DZFC
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My Work
1. Parser 2. Database of definitions 3. Translator 4. Statistics on logical structure 5. GUI for concept exploration
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Directed Acyclic Graph (DAG) of Conceptual Dependencies
Depth: 4 Size: 5
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DAG data
Max MeanDepth 32 10.77Size 110 29.56
DAG depths
0
5
10
15
20
25
30
35
40
0 5 10 15 20 25 30 35
Depth
Fre
qu
en
cy
DAG depths
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Quantifier depth: alternating or non-alternating
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Expanding formulas
Definitional axiom for PORD in DZFC:
To expand, locate the definiens for each defined conceptappearing above.
:
:
:
Then plug in.
Definiens for PORD:
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Expanding formulas
The result:
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Three expansion levels
1. No expansion 2. Total expansion 3. Partial – lowest foundational concepts left
unexpanded
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Eight data points
LPT unexpanded DZFC fully expanded DZFC partially expanded DZFC alt. LPT alt. unexpanded DZFC alt. fully expanded DZFC alt. partially expanded DZFC
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Quantifier depth data
Max MeanLPT 4 0.66unexpanded DZFC 5 1.31fully expanded DZFC 1235 78.68partially expanded DZFC 552 38.54
alt. LPT 3 0.63alt. unexpanded DZFC 5 1.18alt. fully expanded DZFC 422 36.19alt.partially expanded DZFC 239 22.16
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Quantifier depth data
Depth LPT alt. LPT0 178 1781 118 1202 30 353 14 84 1 0
Occurrences
(out of 341 definitions)
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My Work
1. Parser 2. Database of definitions 3. Translator 4. Statistics on logical structure 5. GUI for concept exploration
46
Summary
1. Parser 2. Database of definitions 3. Translator 4. Statistics on logical structure 5. GUI for concept exploration
