Compatible rewriting systems and completion for proving operator - PowerPoint PPT Presentation

Compatible rewriting systems and completion for proving operator identities (j.w. Clemens Hofstadler, Clemens G. Raab, and Georg Regensburger) Cyrille Chenavier Johannes Kepler University, Institute for Algebra Journées Nationales du Calcul Formel 2020 March 2, 2020 C. Chenavier JKU, Institute for Algebra March 2, 2020 1 / 14

Formal proofs of operator identities Motivations Operator identities and membership problems Objective: formally prove operator identities ( e.g. , for computing solutions/integrability conditions of a functional system of equations, proving analytic formulas, . . . ) � operators are terms constructed from basic symbols � "forgetting" the analytic meaning of operators Prove new identities � check membership in suitable algebraic structures, e.g. , • linear P.D.E.’s with constant/polynomial coeff. � polynomial/Weyl algebras • integro-diff. systems with smooth unknown functions � tensor algebras • other systems with mixed operations � Ore algbras/extensions, tensor rings Check membership � use rewriting theory • e.g. , (adaptations of) Gröbner/Janet bases, tensor reduction systems, . . . • simplify a syntactic expression into an equivalent one, e.g. , ∂ ◦ � A ◦ ∂ ◦ � = Id : ◦ B A ◦ B C. Chenavier JKU, Institute for Algebra March 2, 2020 2 / 14

Formal proofs of operator identities Motivations Operator identities and membership problems Objective: formally prove operator identities ( e.g. , for computing solutions/integrability conditions of a functional system of equations, proving analytic formulas, . . . ) � operators are terms constructed from basic symbols � "forgetting" the analytic meaning of operators Prove new identities � check membership in suitable algebraic structures, e.g. , • linear P.D.E.’s with constant/polynomial coeff. � polynomial/Weyl algebras • integro-diff. systems with smooth unknown functions � tensor algebras • other systems with mixed operations � Ore algbras/extensions, tensor rings Check membership � use rewriting theory • e.g. , (adaptations of) Gröbner/Janet bases, tensor reduction systems, . . . • simplify a syntactic expression into an equivalent one, e.g. , ∂ ◦ � A ◦ ∂ ◦ � = Id : ◦ B A ◦ B C. Chenavier JKU, Institute for Algebra March 2, 2020 2 / 14

Formal proofs of operator identities Motivations Operator identities and membership problems Objective: formally prove operator identities ( e.g. , for computing solutions/integrability conditions of a functional system of equations, proving analytic formulas, . . . ) � operators are terms constructed from basic symbols � "forgetting" the analytic meaning of operators Prove new identities � check membership in suitable algebraic structures , e.g. , • linear P.D.E. ’s with constant/polynomial coeff. � polynomial/Weyl algebras • integro-diff. systems with smooth unknown functions � tensor algebras • other systems with mixed operations � Ore algbras/extensions , tensor rings Check membership � use rewriting theory • e.g. , (adaptations of) Gröbner/Janet bases, tensor reduction systems, . . . • simplify a syntactic expression into an equivalent one, e.g. , ∂ ◦ � A ◦ ∂ ◦ � = Id : ◦ B A ◦ B C. Chenavier JKU, Institute for Algebra March 2, 2020 2 / 14

Formal proofs of operator identities Motivations Operator identities and membership problems Objective: formally prove operator identities ( e.g. , for computing solutions/integrability conditions of a functional system of equations, proving analytic formulas, . . . ) � operators are terms constructed from basic symbols � "forgetting" the analytic meaning of operators Prove new identities � check membership in suitable algebraic structures, e.g. , • linear P.D.E.’s with constant/polynomial coeff. � polynomial/Weyl algebras • integro-diff. systems with smooth unknown functions � tensor algebras • other systems with mixed operations � Ore algbras/extensions, tensor rings Check membership � use rewriting theory • e.g. , (adaptations of) Gröbner/Janet bases, tensor reduction systems, . . . • simplify a syntactic expression into an equivalent one, e.g. , ∂ ◦ � A ◦ ∂ ◦ � = Id : ◦ B A ◦ B C. Chenavier JKU, Institute for Algebra March 2, 2020 2 / 14

Formal proofs of operator identities Motivations Operator identities and membership problems Objective: formally prove operator identities ( e.g. , for computing solutions/integrability conditions of a functional system of equations, proving analytic formulas, . . . ) � operators are terms constructed from basic symbols � "forgetting" the analytic meaning of operators Prove new identities � check membership in suitable algebraic structures, e.g. , • linear P.D.E.’s with constant/polynomial coeff. � polynomial/Weyl algebras • integro-diff. systems with smooth unknown functions � tensor algebras • other systems with mixed operations � Ore algbras/extensions, tensor rings Check membership � use rewriting theory • e.g. , (adaptations of) Gröbner/Janet bases, tensor reduction systems, . . . • simplify a syntactic expression into an equivalent one, e.g. , ∂ ◦ � A ◦ ∂ ◦ � = Id : ◦ B A ◦ B C. Chenavier JKU, Institute for Algebra March 2, 2020 2 / 14

Formal proofs of operator identities Motivations Additionnal task Symbolic computations may not take into account compatibility conditions ⊲ multiplication of coefficients is not defined everywhere, e.g. , matrices ⊲ operators may have domains and codomains � composition is not defined everywhere � ∂ : C k +1 ( I ) → C k ( I ) , : C k ( I ) → C k +1 ( I ) e . g ., Solutions In this talk, we present two solutions to take compatibility conditions into account 1st solution: compute and then check compatibility • check membership using rew. • check "validity" at the end of the calculus 2nd solution: restrict to compatible rew. steps • check membership using only "valid" rew. steps C. Chenavier JKU, Institute for Algebra March 2, 2020 3 / 14

Formal proofs of operator identities Motivations Additionnal task Symbolic computations may not take into account compatibility conditions ⊲ multiplication of coefficients is not defined everywhere, e.g. , matrices ⊲ operators may have domains and codomains � composition is not defined everywhere � ∂ : C k +1 ( I ) → C k ( I ) , : C k ( I ) → C k +1 ( I ) e . g ., Solutions In this talk, we present two solutions to take compatibility conditions into account 1st solution: compute and then check compatibility • check membership using rew. • check "validity" at the end of the calculus 2nd solution: restrict to compatible rew. steps • check membership using only "valid" rew. steps C. Chenavier JKU, Institute for Algebra March 2, 2020 3 / 14

Formal proofs of operator identities Motivations Additionnal task Symbolic computations may not take into account compatibility conditions ⊲ multiplication of coefficients is not defined everywhere, e.g. , matrices ⊲ operators may have domains and codomains � composition is not defined everywhere � ∂ : C k + 1 ( I ) → C k ( I ) , : C k ( I ) → C k + 1 ( I ) e . g ., Solutions In this talk, we present two solutions to take compatibility conditions into account 1st solution: compute and then check compatibility • check membership using rew. • check "validity" at the end of the calculus 2nd solution: restrict to compatible rew. steps • check membership using only "valid" rew. steps C. Chenavier JKU, Institute for Algebra March 2, 2020 3 / 14

Formal proofs of operator identities Motivations Additionnal task Symbolic computations may not take into account compatibility conditions ⊲ multiplication of coefficients is not defined everywhere, e.g. , matrices ⊲ operators may have domains and codomains � composition is not defined everywhere � ∂ : C k +1 ( I ) → C k ( I ) , : C k ( I ) → C k +1 ( I ) e . g ., Solutions In this talk , we present two solutions to take compatibility conditions into account 1st solution: compute and then check compatibility • check membership using rew. • check "validity" at the end of the calculus 2nd solution: restrict to compatible rew. steps • check membership using only "valid" rew. steps C. Chenavier JKU, Institute for Algebra March 2, 2020 3 / 14

Formal proofs of operator identities Motivations Additionnal task Symbolic computations may not take into account compatibility conditions ⊲ multiplication of coefficients is not defined everywhere, e.g. , matrices ⊲ operators may have domains and codomains � composition is not defined everywhere � ∂ : C k +1 ( I ) → C k ( I ) , : C k ( I ) → C k +1 ( I ) e . g ., Solutions In this talk , we present two solutions to take compatibility conditions into account 1st solution: compute and then check compatibility • check membership using rew. • check "validity" at the end of the calculus 2nd solution: restrict to compatible rew. steps • check membership using only "valid" rew. steps C. Chenavier JKU, Institute for Algebra March 2, 2020 3 / 14