Geometric constructions, first order logic Problematics An example - PowerPoint PPT Presentation

Geometric constructions Pascal Schreck Introduction Geometric constructions, first order logic Problematics An example and implementation First order logic Ruler and compass Formalization of geometry Signature and Expressiveness Pascal Schreck Axiomatic and inferences Implementation Universit´ e de Strasbourg - LSIIT, UMR CNRS 7005 Different kinds of inference Permutation, decomposition, exception 5th WS on Formal And Automated Theorem Proving Geometric proofs High level rules and Applications Conclusion February 2012

Geometric Some domains where geometric constructions constructions (could) appear Pascal Schreck Introduction ◮ Education: Statement → program of construction Problematics An example Let d 1 and d 2 be 2 parallel lines, A ∈ d 1 and B ∈ d 2 be two points, and O be any point, how First order logic Ruler and compass to construct a line ∆ passing through O and meeting d 1 in M and d 2 in N such as Formalization of geometry AM + BN = k, (k is a given constant). Signature and Expressiveness ◮ Technical drawing: sketch → precise drawing Axiomatic and inferences o Implementation Different kinds of inference n Permutation, p+q=l=60 j=57 decomposition, exception m Geometric proofs q High level rules p d1 d2 Conclusion a b i=40 k=55 ◮ Architecture, photogrammetry (projections → 3D-objects), curves et surfaces, molecule problem, robotic . . . This talk is focused on the first domain.

Geometric constructions Pascal Schreck Introduction Problematics An example First order logic Ruler and compass Formalization of geometry Signature and Expressiveness Back to school Axiomatic and inferences Implementation Different kinds of inference Permutation, decomposition, exception Geometric proofs High level rules Conclusion

Geometric Back to school constructions Pascal Schreck Introduction Problematics Exercice An example First order logic Let d 1 and d 2 be 2 parallel lines, A ∈ d 1 and B ∈ d 2 be two Ruler and compass points, and O be any point, how to construct a line ∆ Formalization of geometry Signature and passing through O and meeting d 1 in M and d 2 in N such as Expressiveness Axiomatic and AM + BN = k, (k is a given constant). inferences Implementation Different kinds of inference Permutation, decomposition, exception Geometric proofs High level rules Conclusion

Geometric Back to school constructions Pascal Schreck Introduction Problematics Exercice An example First order logic Let d 1 and d 2 be 2 parallel lines, A ∈ d 1 and B ∈ d 2 be two Ruler and compass points, and O be any point, how to construct a line ∆ Formalization of geometry Signature and passing through O and meeting d 1 in M and d 2 in N such as Expressiveness Axiomatic and AM + BN = k, (k is a given constant). inferences Implementation Different kinds of inference Let P be on d 1 at dis- Permutation, decomposition, exception tance k from A Geometric proofs High level rules AM+MP= k =AM+BN Conclusion it is easy to see that ( M , P , N , B ) is a paral- lelogram

Geometric Back to school constructions Pascal Schreck Introduction Problematics Exercice An example First order logic Let d 1 and d 2 be 2 parallel lines, A ∈ d 1 and B ∈ d 2 be two Ruler and compass points, and O be any point, how to construct a line ∆ Formalization of geometry Signature and passing through O and meeting d 1 in M and d 2 in N such as Expressiveness Axiomatic and AM + BN = k, (k is a given constant). inferences Implementation Different kinds of inference construction : Permutation, decomposition, exception Draw point P on d1 at Geometric proofs High level rules distance k from A Conclusion Construct point I as the midpoint of P and A Draw ∆ as line (OI)

Geometric Back to school constructions Pascal Schreck Introduction Problematics Exercice An example First order logic Let d 1 and d 2 be 2 parallel lines, A ∈ d 1 and B ∈ d 2 be two Ruler and compass points, and O be any point, how to construct a line ∆ Formalization of geometry Signature and passing through O and meeting d 1 in M and d 2 in N such as Expressiveness Axiomatic and AM + BN = k, (k is a given constant). inferences Implementation Different kinds of inference A , B , O , d 1 , k : free Permutation, decomposition, exception ( A is on d 1 ) Geometric proofs High level rules d 2 = lpd( B , dir( d 1 )) Conclusion P = interlc( d 1 , cir( A , k )) I = mid( P , B ) ∆ = lpp( O , I )

Geometric Testing the construction ... constructions Pascal Schreck Introduction Problematics An example First order logic Ruler and compass M2 M1 P Formalization of A geometry Signature and I Expressiveness Axiomatic and N2 N1 B inferences Implementation O Different kinds of inference Permutation, decomposition, exception Geometric proofs High level rules Conclusion

Geometric Testing the construction ... constructions Pascal Schreck Introduction Problematics An example First order logic Ruler and compass M2 P M1 Formalization of A geometry Signature and I Expressiveness Axiomatic and N2 N1 B inferences Implementation O Different kinds of inference Permutation, decomposition, exception Geometric proofs High level rules Conclusion

Geometric Testing the construction ... constructions Pascal Schreck Introduction Problematics An example First order logic Ruler and compass M2 P M1 Formalization of A geometry Signature and I Expressiveness Axiomatic and N2 N1 B inferences Implementation O Different kinds of inference Permutation, decomposition, exception Geometric proofs High level rules Conclusion

Geometric Testing the construction ... constructions Pascal Schreck Introduction Problematics An example First order logic Ruler and compass M2 P M1 Formalization of A geometry Signature and I Expressiveness Axiomatic and N2 N1 B inferences Implementation O Different kinds of inference Permutation, decomposition, exception Geometric proofs High level rules Conclusion Explanation Point O being in this position, ( M , P , N , B ) is no more a parallelogram, but ( M , P , B , N ) is. This leads to another construction where: ∆ = lpd( O ,dir(lpp( P , B ))).

Geometric Discussion (1) constructions Pascal Schreck Introduction Problematics An example First order logic Ruler and compass Formalization of geometry We have two cases to consider, but there are other flaws : Signature and Expressiveness Axiomatic and inferences Implementation P = interlc( d 1 , cir( A , k )) there is two such points Different kinds of inference I = mid( P , B ) ok Permutation, decomposition, exception ∆ = lpp( O , I ) not defined if O = I Geometric proofs High level rules Conclusion

Geometric Discussion (2) ... a lot of cases constructions Pascal Schreck Introduction Problematics An example First order logic Ruler and compass Formalization of geometry Signature and Expressiveness Axiomatic and inferences Implementation Different kinds of inference Permutation, decomposition, exception Geometric proofs High level rules Conclusion

Geometric A Program of Construction constructions Pascal Schreck Introduction A, B, O, k, di : free Problematics d1 = lpd(A, di) An example First order logic d2 = lpd(B, di) Ruler and compass Formalization of C = cir(A, k) geometry Signature and for P in interlc(d1, C) Expressiveness Axiomatic and for case inferences case pll(M,P,B,N): Implementation case pll(M,P,N,B): Different kinds of inference if P <> B then I = mid(P,B) Permutation, decomposition, d3 = lpp(P,B) if I <>O then exception Geometric proofs di3 = dir(d3) Delta = lpp(O,I) High level rules Delta = lpd(O, di3) Conclusion else else fail fail endif endif endcase endfor

Geometric constructions Pascal Schreck Introduction Problematics An example First order logic Ruler and compass Formalization of geometry Signature and Formalization and first order logic Expressiveness Axiomatic and inferences Implementation Different kinds of inference Permutation, decomposition, exception Geometric proofs High level rules Conclusion

Geometric Ruler and compass constructions constructions Pascal Schreck Introduction Problematics An example Definition First order logic Ruler and compass A point P is said RC-constructible from base points Formalization of geometry { B 0 , . . . , B k } if there is a finite sequence of points Signature and Expressiveness Axiomatic and { P 0 , . . . , P n } such that each point P i is either a base point, inferences or a the intersection of lines or circles built from Implementation Different kinds of { P 0 , . . . , P i − 1 } and P = P n inference Permutation, decomposition, exception Result Geometric proofs High level rules The problem of ruler and compass construction is not Conclusion expressible in first order logic because of the notion of finiteness.