Galois

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Theoretical background Fork algebras Fork algebras Limitation of relation algebras Relations algebras can express first-order predicates with at most three variables Fork algebras Extend relation algebras with a pairing operator Equivalent in expressive and deductive power to first-order logic Paulo Silva (UMinho) <strong>Galois</strong> PLMMS’09 18 / 28
Theoretical background Point-free transform summary Point-free transform Pointwise Pointfree ¬(bRa) b(¬R)a bRa ∧ bSa b(R ∩ S)a bSa ∨ bSa b(R ∪ S)a True b ⊤ a False b ⊥ a b = a b id a aRb bR ◦ a 〈∃ c :: bRc ∧ cSa〉 b(R ◦ S)a 〈∀ x :: xRb ⇒ xSa〉 b(R \ S)a 〈∀ x :: aRx ⇒ bSx〉 b(S/R)a bRa ∧ cSa (b, c)(R ∇ S)a bRa ∧ dSc (b, d)(R × S)(a, c) 〈∀ a, b :: bRa ⇒ bSa〉 R ⊆ S 〈∀ a, b :: bRa ⇔ bSa〉 R = S Paulo Silva (UMinho) <strong>Galois</strong> PLMMS’09 19 / 28
Page 1 and 2: Galois: A Language for Proofs Using
Page 3 and 4: Outline Introduction 1 Introduction
Page 5 and 6: Introduction Whole division specifi
Page 7 and 8: Introduction Whole division specifi
Page 9 and 10: Introduction Motivation Proof. n (
Page 11 and 12: Introduction Motivation Proof. n (
Page 13 and 14: Introduction Objectives Objectives
Page 15 and 16: Theoretical background Indirect equ
Page 23 and 24: Properties Theoretical background G
Page 25 and 26: Theoretical background Galois conne
Page 27 and 28: Theoretical background Fork algebra
Page 29: Theoretical background Fork algebra
Page 33 and 34: Outline Galois and Galculator 1 Int
Page 35 and 36: Galois and Galculator Architecture
Page 37 and 38: Summary Summary Summary Fork algebr
Page 39 and 40: Summary Future work Future work Mec

Galois

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