Automatic Mathematical Information Retrieval to Perform Translations - PowerPoint PPT Presentation

Automatic Mathematical Information Retrieval to Perform Translations up to Computer Algebra Systems André Greiner-Petter August 13, 2018 University of Konstanz Germany @GreinerPetter 1/9

Motivation & Problems

Motivation - Formulae Presentations DLMF 18.3 A Jacobi polynomial in different systems. Rendered Version: P ( α,β ) (cos( a Θ)) n Generic L A T EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Semantic L A T EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple : JacobiP(n,alpha,beta,cos(a*Theta)) CAS Mathematica : JacobiP[n,\[Alpha],\[Beta],Cos[a \[CapitalTheta]]] 2/9

Motivation - Formulae Presentations DLMF 18.3 A Jacobi polynomial in different systems. Rendered Version: P ( α,β ) (cos( a Θ)) n Generic L A T EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Semantic L A T EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple : JacobiP(n,alpha,beta,cos(a*Theta)) CAS Mathematica : JacobiP[n,\[Alpha],\[Beta],Cos[a \[CapitalTheta]]] 2/9

Motivation - Formulae Presentations DLMF 18.3 A Jacobi polynomial in different systems. Rendered Version: P ( α,β ) (cos( a Θ)) n Generic L A T EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Semantic L A T EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple : JacobiP(n,alpha,beta,cos(a*Theta)) CAS Mathematica : JacobiP[n,\[Alpha],\[Beta],Cos[a \[CapitalTheta]]] 2/9

Motivation - Formulae Presentations DLMF 18.3 A Jacobi polynomial in different systems. Rendered Version: P ( α,β ) (cos( a Θ)) n Generic L A T EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Semantic L A T EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple : JacobiP(n,alpha,beta,cos(a*Theta)) CAS Mathematica : JacobiP[n,\[Alpha],\[Beta],Cos[a \[CapitalTheta]]] 2/9

Presentation To Computation with semantic information

Problems of Translations DLMF 18.3 Semantic L A T EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple : JacobiP(n, alpha, beta, cos(a*Theta)) Potential Problems: • Differences in syntax • Function is not implemented in one system, • Function has multiple representations in one system, • Differences in definitions. 3/9

Problems of Translations DLMF 18.3 Semantic L A T EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple : JacobiP(n, alpha, beta, cos(a*Theta)) Potential Problems: • Differences in syntax • Function is not implemented in one system, • Function has multiple representations in one system, • Differences in definitions. 3/9

Problems of Translations DLMF 18.3 Semantic L A T EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple : JacobiP($2, $0, $1, $3) Potential Problems: • Differences in syntax ← solved by translation patterns • Function is not implemented in one system, • Function has multiple representations in one system, • Differences in definitions. 3/9

Problems of Translations DLMF 18.3 Semantic L A T EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple : JacobiP(n, alpha, beta, cos(a*Theta)) Potential Problems: • Differences in syntax ← solved by translation patterns • Function is not implemented in one system, • Function has multiple representations in one system, • Differences in definitions. 3/9

Problems of Translations DLMF 18.3 Semantic L A T EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} n � ℓ ( n + α + β + 1) ℓ ( α + ℓ + 1) n − ℓ � x − 1 CAS Maple : � ℓ ! ( n − ℓ )! 2 DLMF 18.5.7 ℓ =0 Potential Problems: • Differences in syntax ← solved by translation patterns • Function is not implemented in one system, translate equivalent presentations � • Function has multiple representations in one system, • Differences in definitions. 3/9

Problems of Translations DLMF 18.3 Semantic L A T EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple : JacobiP or Jacobi or JacobiPoly Potential Problems: • Differences in syntax ← solved by translation patterns • Function is not implemented in one system, translate equivalent presentations � • Function has multiple representations in one system, • Differences in definitions. 3/9

Problems of Translations DLMF 18.3 Semantic L A T EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple : JacobiP or Jacobi or JacobiPoly Potential Problems: • Differences in syntax ← solved by translation patterns • Function is not implemented in one system, translate equivalent presentations � • Function has multiple representations in one system, just pick a valid translation � • Differences in definitions. 3/9

Problems of Translations DLMF 18.3 Semantic L A T EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple : JacobiP(n, alpha, beta, cos(a*Theta)) Potential Problems: • Differences in syntax ← solved by translation patterns • Function is not implemented in one system, translate equivalent presentations � • Function has multiple representations in one system, just pick a valid translation � • Differences in definitions. 3/9

Problems of Translations DLMF 18.3 Semantic L A T EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple : JacobiP(n, alpha, beta, cos(a*Theta)) Potential Problems: • Differences in syntax ← solved by translation patterns • Function is not implemented in one system, translate equivalent presentations � • Function has multiple representations in one system, just pick a valid translation � • Differences in definitions. ← wait... What? 3/9

Problems of Translations DLMF 4.23.9 Maple Inv. Trig. Functions Rendered Version Semantic L A T EX CAS Maple arccot( z ) \acot@{z} arccot(z) 4/9

Problems of Translations DLMF 4.23.9 Maple Inv. Trig. Functions Rendered Version Semantic L A T EX CAS Maple arccot( z ) \acot@{z} arccot(z) Maple DLMF & Mathematica Figure 2: ℜ (arccot( z )) with Figure 1: ℜ (arccot( z )) with branch cut at [ − i, i ] . branch cut at [ −∞ i, − i ] , [ i, ∞ i ] . 4/9

Problems of Translations DLMF 4.23.9 Maple Inv. Trig. Functions Rendered Version Semantic L A T EX CAS Maple arccot( z ) \acot@{z} arccot(z) Maple DLMF & Mathematica Figure 2: ℜ (arccot( z )) with Figure 1: ℜ (arccot( z )) with branch cut at [ − i, i ] . branch cut at [ −∞ i, − i ] , [ i, ∞ i ] . 4/9

Problems of Translations DLMF 4.23.9 Maple Inv. Trig. Functions Rendered Version Semantic L A T EX CAS Maple arccot( z ) \acot@{z} arctan(1/z) Maple DLMF & Mathematica Figure 2: ℜ (arccot( z )) with Figure 1: ℜ (arccot( z )) with branch cut at [ − i, i ] . branch cut at [ −∞ i, − i ] , [ i, ∞ i ] . 4/9

Presentation To Computation (P2C) without semantic information

Problems of Generic L A T EX DLMF 18.3 Generic L A T EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Semantics: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} Potential Problems: • Is P a function, variable, constant? • Is cos( a Θ) an argument of P or part of a multiplication? • What are α , β , n , a , and Θ ? 5/9

Problems of Generic L A T EX DLMF 18.3 Generic L A T EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Semantics: Jacobi polynomial or Legendre function or Ferrers function or ... Potential Problems: • Is P a function, variable, constant? • Is cos( a Θ) an argument of P or part of a multiplication? • What are α , β , n , a , and Θ ? 5/9

Problems of Generic L A T EX DLMF 18.3 Generic L A T EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Semantics: P (cos( a Θ)) vs P · (cos( a Θ)) Potential Problems: • Is P a function, variable, constant? • Is cos( a Θ) an argument of P or part of a multiplication? • What are α , β , n , a , and Θ ? 5/9

Problems of Generic L A T EX DLMF 18.3 Generic L A T EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Semantics: Variable or 2 nd Feigenbaum constant or ... Potential Problems: • Is P a function, variable, constant? • Is cos( a Θ) an argument of P or part of a multiplication? • What are α , β , n , a , and Θ ? 5/9

Human Approach Rendered L A T EX: P ( α,β ) (cos( a Θ)) n 6/9

Human Approach Rendered L A T EX: P ( α,β ) (cos( a Θ)) n The Naive Approach How does a reader understands the mathematical formula? • he knows the symbols and structure, knowledge-based pattern recognition � • it was previously introduced in the paper (e.g. in definitions, the text or in other referenced publications), analyse the context from near to far � • he searching the formula in books or online dictionary-based pattern recognition � 6/9

Human Approach Rendered L A T EX: P ( α,β ) (cos( a Θ)) n The Naive Approach How does a reader understands the mathematical formula? • he knows the symbols and structure, knowledge-based pattern recognition � • it was previously introduced in the paper (e.g. in definitions, the text or in other referenced publications), analyse the context from near to far � • he searching the formula in books or online dictionary-based pattern recognition � 6/9

Human Approach Rendered L A T EX: P ( α,β ) (cos( a Θ)) n The Naive Approach How does a reader understands the mathematical formula? • he knows the symbols and structure, knowledge-based pattern recognition � • it was previously introduced in the paper (e.g. in definitions, the text or in other referenced publications), analyse the context from near to far � • he searching the formula in books or online dictionary-based pattern recognition � 6/9