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Lambda Calculus, Linear Logic and Symbolic Computation Habilitation diriger des recherches Giulio Manzonetto giulio.manzonetto@lipn.univ-paris13.fr LIPN IRIF Universit Paris Nord Universit Paris Diderot March the 7th, 2017 Giulio

  1. λ -Calculus — Its Syntax and Semantics Syntactic and semantic tools Observational Theories Denotational Tree-like structures Semantics Scott’s continuous semantics: Böhm Trees Data ∈ D poset + extensional equalities Program = continuous function Full abstraction for H ∗ [Hyland/Wadsworth’76] ⇒ H ∗ ⊢ M = N D ∞ | = M = N ⇐ Full abstraction for H + [Coppo-Dezani-Zacchi’87] ⇒ H + ⊢ M = N D cdz | = M = N ⇐ Giulio Manzonetto HdR Defense 07/03/17 9 / 38

  2. λ -Calculus — Its Syntax and Semantics The Relational Semantics The cartesian closed category MRel : objects: sets A , B , . . . morphisms: A → B are relations from M f ( A ) to B Relational Graph Models D = ( D , i ) : injection: i : M f ( D ) × D → D Approximation Theorem [ManzonettoR, MFPS 2014] In every rgm D , � M � D = { � t � D : t is a finite approximant of BT ( M ) } Characterization of Full Abstraction for H + [BreuvartMPR, FSCD 2016] Th ( D ) = H + ⇐ ⇒ D is extensional and λ -König. Characterization of Full Abstraction for H ∗ [BreuvartMR 2017] Th ( D ) = H ∗ ⇐ ⇒ D is extensional and hyperimmune. Giulio Manzonetto HdR Defense 07/03/17 10 / 38

  3. λ -Calculus — Its Syntax and Semantics The Relational Semantics The cartesian closed category MRel : objects: sets A , B , . . . morphisms: A → B are relations from M f ( A ) to B Relational Graph Models D = ( D , i ) : injection: i : M f ( D ) × D → D Approximation Theorem [ManzonettoR, MFPS 2014] In every rgm D , � M � D = { � t � D : t is a finite approximant of BT ( M ) } Characterization of Full Abstraction for H + [BreuvartMPR, FSCD 2016] Th ( D ) = H + ⇐ ⇒ D is extensional and λ -König. Characterization of Full Abstraction for H ∗ [BreuvartMR 2017] Th ( D ) = H ∗ ⇐ ⇒ D is extensional and hyperimmune. Giulio Manzonetto HdR Defense 07/03/17 10 / 38

  4. λ -Calculus — Its Syntax and Semantics The Relational Semantics The cartesian closed category MRel : objects: sets A , B , . . . morphisms: A → B are relations from M f ( A ) to B Relational Graph Models D = ( D , i ) : injection: i : M f ( D ) × D → D Approximation Theorem [ManzonettoR, MFPS 2014] In every rgm D , � M � D = { � t � D : t is a finite approximant of BT ( M ) } Characterization of Full Abstraction for H + [BreuvartMPR, FSCD 2016] Th ( D ) = H + ⇐ ⇒ D is extensional and λ -König. Characterization of Full Abstraction for H ∗ [BreuvartMR 2017] Th ( D ) = H ∗ ⇐ ⇒ D is extensional and hyperimmune. Giulio Manzonetto HdR Defense 07/03/17 10 / 38

  5. λ -Calculus — Its Syntax and Semantics The Relational Semantics The cartesian closed category MRel : objects: sets A , B , . . . morphisms: A → B are relations from M f ( A ) to B Are all theories T in the interval [ H + , H ∗ ] Relational Graph Models D = ( D , i ) : representable by relational graph models? injection: i : M f ( D ) × D → D Approximation Theorem [ManzonettoR, MFPS 2014] In every rgm D , � M � D = { � t � D : t is a finite approximant of BT ( M ) } Characterization of Full Abstraction for H + [BreuvartMPR, FSCD 2016] Th ( D ) = H + ⇐ ⇒ D is extensional and λ -König. Characterization of Full Abstraction for H ∗ [BreuvartMR 2017] Th ( D ) = H ∗ ⇐ ⇒ D is extensional and hyperimmune. Giulio Manzonetto HdR Defense 07/03/17 10 / 38

  6. λ -Calculus — Its Syntax and Semantics The Relational Semantics The cartesian closed category MRel : objects: sets A , B , . . . morphisms: A → B are relations from M f ( A ) to B Are all theories T in the interval [ H + , H ∗ ] Relational Graph Models D = ( D , i ) : representable by relational graph models? injection: i : M f ( D ) × D → D Approximation Theorem [ManzonettoR, MFPS 2014] In every rgm D , � M � D = { � t � D : t is a finite approximant of BT ( M ) } Otherwise, is it possible to characterize all representable theories? Characterization of Full Abstraction for H + [BreuvartMPR, FSCD 2016] Th ( D ) = H + ⇐ ⇒ D is extensional and λ -König. Characterization of Full Abstraction for H ∗ [BreuvartMR 2017] Th ( D ) = H ∗ ⇐ ⇒ D is extensional and hyperimmune. Giulio Manzonetto HdR Defense 07/03/17 10 / 38

  7. λ -Calculus — Its Syntax and Semantics The Relational Semantics The cartesian closed category MRel : objects: sets A , B , . . . morphisms: A → B are relations from M f ( A ) to B Are all theories T in the interval [ H + , H ∗ ] Relational Graph Models D = ( D , i ) : representable by relational graph models? injection: i : M f ( D ) × D → D Approximation Theorem [ManzonettoR, MFPS 2014] In every rgm D , � M � D = { � t � D : t is a finite approximant of BT ( M ) } Otherwise, is it possible to characterize all representable theories? Characterization of Full Abstraction for H + [BreuvartMPR, FSCD 2016] Th ( D ) = H + ⇐ ⇒ D is extensional and λ -König. What can we say about non-extensional Characterization of Full Abstraction for H ∗ [BreuvartMR 2017] representable theories? Th ( D ) = H ∗ ⇐ ⇒ D is extensional and hyperimmune. Giulio Manzonetto HdR Defense 07/03/17 10 / 38

  8. Quantitative Semantics 2. Quantitative Properties in Nondeterministic Settings Giulio Manzonetto HdR Defense 07/03/17 11 / 38

  9. Quantitative Semantics Semantics of Non-Determinism Nondeterministic Functional Programming Languages Denotational characterize Operational properties Semantics Semantics Execution properties I/O flow Termination Contextual equivalence Giulio Manzonetto HdR Defense 07/03/17 12 / 38

  10. Quantitative Semantics Semantics of Non-Determinism Nondeterministic Functional Programming Languages Denotational characterize Operational properties Semantics Semantics Properties of the model Execution properties Soundness I/O flow Adequacy Termination Full Abstraction Contextual equivalence Giulio Manzonetto HdR Defense 07/03/17 12 / 38

  11. Quantitative Semantics Semantics of Non-Determinism Nondeterministic Functional Programming Languages Denotational characterize Operational properties Semantics Semantics Properties of the model Qualitative properties Soundness I/O flow Adequacy Termination Full Abstraction Contextual equivalence Giulio Manzonetto HdR Defense 07/03/17 12 / 38

  12. Quantitative Semantics Quantitative properties input output P → · → · → · · · → · → · → i o “black box” Intensional view on Programs We need to “open the box”. . . Giulio Manzonetto HdR Defense 07/03/17 13 / 38

  13. Quantitative Semantics Quantitative properties input output → · → · → · · · → · → · → i o Intensional view on Programs We need to “open the box”. . . Quantitative Properties Number of steps to termination, Number of calls to the argument at runtime, Amount of resources used during the computation, Non-deterministic setting: Number of “ways” to get the output. Giulio Manzonetto HdR Defense 07/03/17 13 / 38

  14. Quantitative Semantics Quantitative properties input output · · · · · · · i o · · · · Intensional view on Programs We need to “open the box”. . . Quantitative Properties Number of steps to termination, Number of calls to the argument at runtime, Amount of resources used during the computation, Non-deterministic setting: Number of “ways” to get the output. Giulio Manzonetto HdR Defense 07/03/17 13 / 38

  15. Quantitative Semantics Semantics of Non-Determinism Nondeterministic Functional Programming Languages Denotational characterize Operational properties Semantics Semantics Properties of the model Qualitative properties Soundness I/O flow Adequacy Termination Full Abstraction Contextual equivalence Quantitative properties # steps to termination ? # calls to argument # resources needed # ways to get the output Giulio Manzonetto HdR Defense 07/03/17 14 / 38

  16. Quantitative Semantics Semantics of Non-Determinism Nondeterministic Functional Programming Languages Denotational characterize Operational properties Semantics Semantics Properties of the model Qualitative properties Soundness I/O flow Adequacy Termination Full Abstraction Contextual equivalence Quantitative properties # steps to termination We need quantitative models! # calls to argument # resources needed # ways to get the output Giulio Manzonetto HdR Defense 07/03/17 14 / 38

  17. Quantitative Semantics The Relational Semantics is Quantitative The category MRel Data/Objects: sets Program/Morphism A → B : relation from M f ( A ) and B . # calls to the argument output � P � ⊆ M f ( A ) × B Example A program like M = λ n . n ∗ n : Nat → Nat is interpreted by the relation � M � = { ([ n 1 , n 2 ] , n 1 ∗ n 2 ) | n 1 , n 2 ∈ N } Multiset of cardinality two ⇒ M uses its argument twice. Giulio Manzonetto HdR Defense 07/03/17 15 / 38

  18. Quantitative Semantics The Relational Semantics is Quantitative The category MRel Data/Objects: sets Program/Morphism A → B : relation from M f ( A ) and B . # calls to the argument output � P � ⊆ M f ( A ) × B Example A program like M = λ n . n ∗ n : Nat → Nat is interpreted by the relation � M � = { ([ n 1 , n 2 ] , n 1 ∗ n 2 ) | n 1 , n 2 ∈ N } Multiset of cardinality two ⇒ M uses its argument twice. Giulio Manzonetto HdR Defense 07/03/17 15 / 38

  19. Quantitative Semantics The Relational Semantics is Quantitative The category MRel Data/Objects: sets Program/Morphism A → B : relation from M f ( A ) and B . # calls to the argument output � P � ⊆ M f ( A ) × B Example A program like M = λ n . n ∗ n : Nat → Nat is interpreted by the relation � M � = { ([ n 1 , n 2 ] , n 1 ∗ n 2 ) | n 1 , n 2 ∈ N } NB: Non-deterministic setting! Giulio Manzonetto HdR Defense 07/03/17 15 / 38

  20. Quantitative Semantics Replace qualitative by quantitative methods Qualitative methods Quantitative methods Continuous semantics Relational Semantics � M � = cont. function A → B � M � = relation ⊆ M f ( A ) × B � M � = � A ∈ App ( M ) � A � Appr. Thm. � M � = � t ∈T ( M ) � t � Appr. Thm. Böhm’s approximants Resource approximants λ� x | λ x . s | s · [ t 1 , . . . , t n ] (linear term) x . yA 1 · · · A n (finite tree) Taylor Expansion Böhm trees ∞ 1 � BT ( M ) = � A T ( MN ) = n ! M [ N 1 , . . . , N n ] A ∈ App ( BT ( M )) n = 0 Giulio Manzonetto HdR Defense 07/03/17 16 / 38

  21. Quantitative Semantics Replace qualitative by quantitative methods Qualitative methods Quantitative methods For the resource Continuous semantics Relational Semantics calculus, look in the HdR thesis! � M � = cont. function A → B � M � = relation ⊆ M f ( A ) × B � M � = � A ∈ App ( M ) � A � Appr. Thm. � M � = � t ∈T ( M ) � t � Appr. Thm. Böhm’s approximants Resource approximants λ� x | λ x . s | s · [ t 1 , . . . , t n ] (linear term) x . yA 1 · · · A n (finite tree) Taylor Expansion Böhm trees ∞ 1 � BT ( M ) = � A T ( MN ) = n ! M [ N 1 , . . . , N n ] A ∈ App ( BT ( M )) n = 0 Manzonetto et Al. [2010-2013] Syntactic properties + model theory for Ehrhard’s resource calculus. Giulio Manzonetto HdR Defense 07/03/17 16 / 38

  22. Quantitative Semantics Some Quantitative Properties Bound on the length of the (linear) head reduction: σ ∈ � M � ⇒ M normalizes in # σ linear-steps Theorems [DeCarvalho 2008] Call-By-Name, [Ehrhard 2012] Call-By-Value, [Díaz-CaroMP 2013] Call-By-Value with may/must non-determinism. [BreuvartMR 2017] Combinatorial proof of Approximation Theorem Giulio Manzonetto HdR Defense 07/03/17 17 / 38

  23. Quantitative Semantics Can we do better? Giulio Manzonetto HdR Defense 07/03/17 18 / 38

  24. Quantitative Semantics The Weighted Relational Model Relations as matrices A relation from M f ( A ) to B is a matrix in Bool M f ( A ) × B Generalization: Continuous semi-rings From Bool to arbitrary commutative semi-rings R = ( R , · , 1 , � ) , such that: R is a positively ordered cpo with a bottom element 0 the operators · and � are continuous. The interpretation of ⊢ M : A → B becomes a matrix � M � : M f ( A ) × B → R Giulio Manzonetto HdR Defense 07/03/17 19 / 38

  25. Quantitative Semantics The program interpretation Program ⊢ P : Nat → Nat . � P � 0 1 2 3 4 5 · · · [] p 1 , 1 p 1 , 2 p 1 , 3 p 1 , 4 p 1 , 5 p 1 , 6 · · · [ 0 ] p 2 , 1 p 2 , 2 p 2 , 3 p 2 , 4 p 2 , 5 p 2 , 6 · · · [ 0 , 1 ] p 3 , 1 p 3 , 2 p 3 , 3 p 3 , 4 p 3 , 5 p 2 , 6 · · · [ 0 , 0 ] p 4 , 1 p 4 , 2 p 4 , 3 p 4 , 4 p 4 , 5 p 3 , 6 · · · [ 1 , 1 ] p 5 , 1 p 5 , 2 p 5 , 3 p 5 , 4 p 5 , 5 p 4 , 6 · · · [ 0 , 0 , 1 ] p 6 , 1 p 6 , 2 p 6 , 3 p 6 , 4 p 6 , 5 p 6 , 6 · · · . . . . . . . ... . . . . . . . . . . . . . . Programs become matrices. . . what is the meaning of the scalars? Giulio Manzonetto HdR Defense 07/03/17 20 / 38

  26. Quantitative Semantics The program interpretation Program ⊢ P : Nat → Nat . � P � 0 1 2 3 4 5 · · · [] p 1 , 1 p 1 , 2 p 1 , 3 p 1 , 4 p 1 , 5 p 1 , 6 · · · [ 0 ] p 2 , 1 p 2 , 2 p 2 , 3 p 2 , 4 p 2 , 5 p 2 , 6 · · · [ 0 , 1 ] p 3 , 1 p 3 , 2 p 3 , 3 p 3 , 4 p 3 , 5 p 2 , 6 · · · [ 0 , 0 ] p 4 , 1 p 4 , 2 p 4 , 3 p 4 , 4 p 4 , 5 p 3 , 6 · · · [ 1 , 1 ] p 5 , 1 p 5 , 2 p 5 , 3 p 5 , 4 p 5 , 5 p 4 , 6 · · · [ 0 , 0 , 1 ] p 6 , 1 p 6 , 2 p 6 , 3 p 6 , 4 p 6 , 5 p 6 , 6 · · · . . . . . . . ... . . . . . . . . . . . . . . Programs become matrices. . . what is the meaning of the scalars? Giulio Manzonetto HdR Defense 07/03/17 20 / 38

  27. Quantitative Semantics Capturing the operational semantics Consider PCF or , with weighted reductions: M Giulio Manzonetto HdR Defense 07/03/17 21 / 38

  28. Quantitative Semantics Capturing the operational semantics Consider PCF or , with weighted reductions: p 1 M 1 M p 2 M 2 Giulio Manzonetto HdR Defense 07/03/17 21 / 38

  29. Quantitative Semantics Capturing the operational semantics Consider PCF or , with weighted reductions: M 1 , 1 p 1 , 1 p 1 M 1 , 2 M 1 p 1 , 2 M p 2 , 1 p 2 M 2 , 1 M 2 p 2 , 2 M 2 , 2 Giulio Manzonetto HdR Defense 07/03/17 21 / 38

  30. Quantitative Semantics Capturing the operational semantics Consider PCF or , with weighted reductions: n M 1 , 1 k p 1 , 1 k ′ p 1 M 1 , 2 n M 1 p 1 , 2 m M p 2 , 1 p 2 n M 2 M 2 , 1 n p 2 , 2 m M 2 , 2 Giulio Manzonetto HdR Defense 07/03/17 21 / 38

  31. Quantitative Semantics Capturing the operational semantics Consider PCF or , with weighted reductions: n M 1 , 1 k p 1 , 1 k ′ p 1 M 1 , 2 n M 1 p 1 , 2 m M p 2 , 1 p 2 n M 2 M 2 , 1 n p 2 , 2 m M 2 , 2 Parametric Adequacy [Laird, Manzonetto, McCusker, Pagani LICS’13] Given M : Nat we have: � M � ( n ) = Sum of the product of the scalars along the paths M ։ n Giulio Manzonetto HdR Defense 07/03/17 21 / 38

  32. Quantitative Semantics Parametrized Quantitative Results for PCF or What are we actually counting? This depends on the chosen semi-ring. . . x | λ x . M | MN | Y M | zero | succ M | pred M | ifz ( M , N , P ) | M or N Giulio Manzonetto HdR Defense 07/03/17 22 / 38

  33. Quantitative Semantics Parametrized Quantitative Results for PCF or What are we actually counting? This depends on the chosen semi-ring. . . x | λ x . M | MN | Y M | zero | succ M | pred M | ifz ( M , N , P ) | M or N � M � R Π R ! = Red ( M , n ) n Giulio Manzonetto HdR Defense 07/03/17 22 / 38

  34. Quantitative Semantics Parametrized Quantitative Results for PCF or What are we actually counting? This depends on the chosen semi-ring. . . x | λ x . M | MN | Y M | zero | succ M | pred M | ifz ( M , N , P ) | M or N � M � R Π R ! = Red ( M , n ) n ( N , · , 1 , � , ≤ ) number of paths from M to n Giulio Manzonetto HdR Defense 07/03/17 22 / 38

  35. Quantitative Semantics Parametrized Quantitative Results for PCF or What are we actually counting? This depends on the chosen semi-ring. . . x | λ x . M | MN | Y M | zero | succ M | pred M | ifz ( M , N , P ) | 1 2 M or 1 2 N � M � R Π R ! = Red ( M , n ) n ( N , · , 1 , � , ≤ ) number of paths from M to n ( R + , · , 1 , � , ≤ ) probability that M reduces to n [Danos-Ehrhard’11] Giulio Manzonetto HdR Defense 07/03/17 22 / 38

  36. Quantitative Semantics Parametrized Quantitative Results for PCF or What are we actually counting? This depends on the chosen semi-ring. . . x | λ x . 1 M | MN | Y ( 1 M ) | zero | succ M | pred M | ifz ( M , N , P ) | M or N � M � R Π R ! = Red ( M , n ) n ( N , · , 1 , � , ≤ ) number of paths from M to n ( R + , · , 1 , � , ≤ ) probability that M reduces to n [Danos-Ehrhard’11] ( N , + , 0 , min , ≥ ) minimum number of β, Y steps to n Giulio Manzonetto HdR Defense 07/03/17 22 / 38

  37. Quantitative Semantics Parametrized Quantitative Results for PCF or What are we actually counting? This depends on the chosen semi-ring. . . x | λ x . 1 M | MN | Y ( 1 M ) | zero | succ M | pred M | ifz ( M , N , P ) | M or N � M � R Π R ! = Red ( M , n ) n ( N , · , 1 , � , ≤ ) number of paths from M to n ( R + , · , 1 , � , ≤ ) probability that M reduces to n [Danos-Ehrhard’11] ( N , + , 0 , min , ≥ ) minimum number of β, Y steps to n ( N ⊥ , + , 0 , max , ≤ ) maximum number of β, Y steps to n Giulio Manzonetto HdR Defense 07/03/17 22 / 38

  38. Quantitative Semantics Parametrized Quantitative Results for PCF or What are we actually counting? This depends on the chosen semi-ring. . . x | λ x . 1 M | MN | Y ( 1 M ) | zero | succ M | pred M | ifz ( M , N , P ) | M or N � M � R Π R ! = Red ( M , n ) n ( N , · , 1 , � , ≤ ) number of paths from M to n ( R + , · , 1 , � , ≤ ) probability that M reduces to n [Danos-Ehrhard’11] ( N , + , 0 , min , ≥ ) minimum number of β, Y steps to n ( N ⊥ , + , 0 , max , ≤ ) maximum number of β, Y steps to n And more. . . See [Laird-Manzonetto-McCusker-Pagani LICS’2013] Giulio Manzonetto HdR Defense 07/03/17 22 / 38

  39. Quantitative Semantics Parametrized Quantitative Results for PCF or What are we actually counting? This depends on the chosen semi-ring. . . Are there other quantitative properties that can be captured by suitable semirings? x | λ x . 1 M | MN | Y ( 1 M ) | zero | succ M | pred M | ifz ( M , N , P ) | M or N � M � R Π R ! = Red ( M , n ) n ( N , · , 1 , � , ≤ ) number of paths from M to n ( R + , · , 1 , � , ≤ ) probability that M reduces to n [Danos-Ehrhard’11] ( N , + , 0 , min , ≥ ) minimum number of β, Y steps to n ( N ⊥ , + , 0 , max , ≤ ) maximum number of β, Y steps to n And more. . . See [Laird-Manzonetto-McCusker-Pagani LICS’2013] Giulio Manzonetto HdR Defense 07/03/17 22 / 38

  40. Quantitative Semantics Parametrized Quantitative Results for PCF or What are we actually counting? This depends on the chosen semi-ring. . . Are there other quantitative properties that can be captured by suitable semirings? x | λ x . 1 M | MN | Y ( 1 M ) | zero | succ M | pred M | ifz ( M , N , P ) | M or N � M � R Π R ! = Red ( M , n ) n ( N , · , 1 , � , ≤ ) number of paths from M to n Is it possible to get rid of commutativity? ( R + , · , 1 , � , ≤ ) probability that M reduces to n [Danos-Ehrhard’11] ( N , + , 0 , min , ≥ ) minimum number of β, Y steps to n ( N ⊥ , + , 0 , max , ≤ ) maximum number of β, Y steps to n And more. . . See [Laird-Manzonetto-McCusker-Pagani LICS’2013] Giulio Manzonetto HdR Defense 07/03/17 22 / 38

  41. Quantitative Semantics Parametrized Quantitative Results for PCF or What are we actually counting? This depends on the chosen semi-ring. . . Are there other quantitative properties that can be captured by suitable semirings? x | λ x . 1 M | MN | Y ( 1 M ) | zero | succ M | pred M | ifz ( M , N , P ) | M or N � M � R Π R ! = Red ( M , n ) n ( N , · , 1 , � , ≤ ) number of paths from M to n Is it possible to get rid of commutativity? ( R + , · , 1 , � , ≤ ) probability that M reduces to n [Danos-Ehrhard’11] ( N , + , 0 , min , ≥ ) minimum number of β, Y steps to n ( N ⊥ , + , 0 , max , ≤ ) maximum number of β, Y steps to n What about the profunctorial semantics? And more. . . See [Laird-Manzonetto-McCusker-Pagani LICS’2013] Giulio Manzonetto HdR Defense 07/03/17 22 / 38

  42. Factor Algebras at Work 3. Universal Algebra and Symbolic Computation Giulio Manzonetto HdR Defense 07/03/17 23 / 38

  43. Factor Algebras at Work Classical Logic — The Usual Algebraic Approach Propositional variables �→ algebraic variables Connectives �→ Boolean operations Propositional formula �→ Boolean algebraic term Tautology φ �→ equation of Boolean algebra φ = t Peirce Law: (( P → Q ) → P ) → P P Q P → Q ( P → Q ) → P (( P → Q ) → P ) → P t t t t t t f f t t f t t f t f f t f t Logical gates P P P ∧ Q P ∨ Q P ¬ P AND OR NOT Q Q Giulio Manzonetto HdR Defense 07/03/17 24 / 38

  44. Factor Algebras at Work Classical Logic — The Usual Algebraic Approach Propositional variables �→ algebraic variables Connectives �→ Boolean operations Propositional formula �→ Boolean algebraic term Tautology φ �→ equation of Boolean algebra φ = t Peirce Law: (( P → Q ) → P ) → P P Q P → Q ( P → Q ) → P (( P → Q ) → P ) → P t t t t t t f f t t f t t f t f f t f t Logical gates P P P ∧ Q P ∨ Q P ¬ P AND OR NOT Q Q Giulio Manzonetto HdR Defense 07/03/17 24 / 38

  45. Factor Algebras at Work Classical Logic — The Usual Algebraic Approach Propositional variables �→ algebraic variables Connectives �→ Boolean operations Propositional formula �→ Boolean algebraic term Tautology φ �→ equation of Boolean algebra φ = t Peirce Law: (( P → Q ) → P ) → P P Q P → Q ( P → Q ) → P (( P → Q ) → P ) → P t t t t t t f f t t f t t f t f f t f t tautology Logical gates P P P ∧ Q P ∨ Q P ¬ P AND OR NOT Q Q Giulio Manzonetto HdR Defense 07/03/17 24 / 38

  46. Factor Algebras at Work Classical Logic — The Usual Algebraic Approach Propositional variables �→ algebraic variables Connectives �→ Boolean operations Propositional formula �→ Boolean algebraic term Tautology φ �→ equation of Boolean algebra φ = t Peirce Law: (( P → Q ) → P ) → P P Q P → Q ( P → Q ) → P (( P → Q ) → P ) → P t t t t t t f f t t f t t f t f f t f t tautology Logical gates P P P ∧ Q P ∨ Q P ¬ P AND OR NOT Q Q Giulio Manzonetto HdR Defense 07/03/17 24 / 38

  47. Factor Algebras at Work Multi-Valued Matrix Logics: The Usual Approach Propositional variables �→ algebraic variables Connectives �→ operations of some algebra �→ Propositional formula algebraic term Tautology φ �→ equation φ = t n -valued Logics Algebras Łukasiewicz Logic Heyting algebras Gödel Logic MV-algebras Post Logic Post algebras . . etc. . Giulio Manzonetto HdR Defense 07/03/17 25 / 38

  48. Factor Algebras at Work In Practice: Binary Decision Diagrams P 1 P 2 P 3 P 1 f f f f f f t t f t f f P 2 P 2 f t t t t f f f P 3 P 3 P 3 P 3 t f t t t t f t f t f t f t t t t t t t ROBDDs Reduced: maximal sharing, Ordered: to ensure canonicity. Giulio Manzonetto HdR Defense 07/03/17 26 / 38

  49. Factor Algebras at Work In Practice: Binary Decision Diagrams P 1 P 2 P 3 P 3 f f f f f f t t f t f f P 2 f t t t t f f f P 1 t f t t t t f t t f t t t t ROBDDs Reduced: maximal sharing, Ordered: to ensure canonicity. Giulio Manzonetto HdR Defense 07/03/17 26 / 38

  50. Factor Algebras at Work Approach based on Decomposition Operators A τ -algebra A is decomposed into (simpler) factors B , C when: A ∼ = B × C A decomposition operator f : A × A → A is a map satisfying: ( F 1 ) f ( x , x ) = x . ( F 2 ) f ( f ( x , y ) , f ( w , z )) = f ( x , z ) . ( F 3 ) f is an algebra homomorphism. Each f induces a pair of complementary factor congruences ϕ, ¯ ϕ : x ϕ y ⇐ ⇒ f ( x , y ) = x ; x ¯ ϕ y ⇐ ⇒ f ( x , y ) = y . Giulio Manzonetto HdR Defense 07/03/17 27 / 38

  51. Factor Algebras at Work Approach based on Decomposition Operators A τ -algebra A is decomposed into (simpler) factors B , C when: A ∼ = B × C A decomposition operator f : A × A → A is a map satisfying: ( F 1 ) f ( x , x ) = x . ( F 2 ) f ( f ( x , y ) , f ( w , z )) = f ( x , z ) . ( F 3 ) f is an algebra homomorphism. Each f induces a pair of complementary factor congruences ϕ, ¯ ϕ : x ϕ y ⇐ ⇒ f ( x , y ) = x ; x ¯ ϕ y ⇐ ⇒ f ( x , y ) = y . Giulio Manzonetto HdR Defense 07/03/17 27 / 38

  52. Factor Algebras at Work Approach based on Decomposition Operators A τ -algebra A is decomposed into (simpler) factors B ∼ = A /ϕ, C ∼ = A / ¯ ϕ when: A ∼ = A /ϕ × A / ¯ ϕ A decomposition operator f : A × A → A is a map satisfying: ( F 1 ) f ( x , x ) = x . ( F 2 ) f ( f ( x , y ) , f ( w , z )) = f ( x , z ) . ( F 3 ) f is an algebra homomorphism. Each f induces a pair of complementary factor congruences ϕ, ¯ ϕ : x ϕ y ⇐ ⇒ f ( x , y ) = x ; x ¯ ϕ y ⇐ ⇒ f ( x , y ) = y . Giulio Manzonetto HdR Defense 07/03/17 27 / 38

  53. Factor Algebras at Work Classical Logic — Our Approach Truth values t , f �→ algebraic variables ξ f , ξ t ; Propositional variables P �→ decomposition operators f P ( ξ f , ξ t ); �→ Connectives disappear: implemented via substitutions and Boolean operations on indices The Translation t ∗ = ξ t f ∗ = ξ f P ∗ = P ( ξ f , ξ t ) ( ¬ φ ) ∗ φ ∗ ( ξ ¬ f , ξ ¬ t ); = ( φ ∧ ψ ) ∗ ψ ∗ ( φ ∗ ( ξ f ∧ f , ξ f ∧ t ) , φ ∗ ( ξ t ∧ f , ξ t ∧ t )); = ( φ ∨ ψ ) ∗ ψ ∗ ( φ ∗ ( ξ f ∨ f , ξ f ∨ t ) , φ ∗ ( ξ t ∨ f , ξ t ∨ t )); = ( φ → ψ ) ∗ ( ¬ φ ∨ ψ ) ∗ . = Giulio Manzonetto HdR Defense 07/03/17 28 / 38

  54. Factor Algebras at Work Classical Logic — Our Approach Truth values t , f �→ algebraic variables ξ f , ξ t ; Propositional variables P �→ decomposition operators f P ( ξ f , ξ t ); �→ Connectives disappear: implemented via substitutions and Boolean operations on indices The Translation t ∗ = ξ t f ∗ = ξ f P ∗ = P ( ξ f , ξ t ) ( ¬ φ ) ∗ φ ∗ ( ξ t , ξ f ); = ( φ ∧ ψ ) ∗ ψ ∗ ( φ ∗ ( ξ f , ξ f ) , φ ∗ ( ξ f , ξ t )); = ( φ ∨ ψ ) ∗ ψ ∗ ( φ ∗ ( ξ f , ξ t ) , φ ∗ ( ξ t , ξ t )); = ( φ → ψ ) ∗ ψ ∗ ( φ ∗ ( ξ t , ξ f ) , φ ∗ ( ξ t , ξ t )) . = Peirce Law: ((( P → Q ) → P ) → P ) ∗ = P ( P ( Q ( P ( t , f ) , P ( t , t )) , Q ( P ( f , f ) , P ( f , f ))) , P ( Q ( P ( t , t ) , P ( t , t )) , Q ( P ( t , t ) , P ( t , t )))) Giulio Manzonetto HdR Defense 07/03/17 28 / 38

  55. Factor Algebras at Work Classical Logic — Our Approach Truth values t , f �→ algebraic variables ξ f , ξ t ; Propositional variables P �→ decomposition operators f P ( ξ f , ξ t ); �→ Connectives disappear: implemented via substitutions and Boolean operations on indices The Translation t ∗ = ξ t f ∗ = ξ f P ∗ = P ( ξ f , ξ t ) ( ¬ φ ) ∗ φ ∗ ( ξ t , ξ f ); = ( φ ∧ ψ ) ∗ ψ ∗ ( φ ∗ ( ξ f , ξ f ) , φ ∗ ( ξ f , ξ t )); = ( φ ∨ ψ ) ∗ ψ ∗ ( φ ∗ ( ξ f , ξ t ) , φ ∗ ( ξ t , ξ t )); = ( φ → ψ ) ∗ ψ ∗ ( φ ∗ ( ξ t , ξ f ) , φ ∗ ( ξ t , ξ t )) . = The Axioms (F1) P ( x , x ) = x ; (F2) P ( P ( x , y ) , P ( w , z )) = P ( x , z ) ; (F3) P ( Q ( x , y ) , Q ( w , z )) = Q ( P ( x , w ) , P ( y , z )) . Giulio Manzonetto HdR Defense 07/03/17 28 / 38

  56. Factor Algebras at Work Classical Logic — Our Approach These are ROBDDs! Truth values t , f �→ algebraic variables ξ f , ξ t ; Propositional variables P �→ decomposition operators f P ( ξ f , ξ t ); �→ Connectives disappear: implemented via substitutions and Boolean operations on indices The Translation t ∗ = ξ t f ∗ = ξ f P ∗ = P ( ξ f , ξ t ) ( ¬ φ ) ∗ φ ∗ ( ξ t , ξ f ); = ( φ ∧ ψ ) ∗ ψ ∗ ( φ ∗ ( ξ f , ξ f ) , φ ∗ ( ξ f , ξ t )); = ( φ ∨ ψ ) ∗ ψ ∗ ( φ ∗ ( ξ f , ξ t ) , φ ∗ ( ξ t , ξ t )); = ( φ → ψ ) ∗ ψ ∗ ( φ ∗ ( ξ t , ξ f ) , φ ∗ ( ξ t , ξ t )) . = The Axioms (F1) P ( x , x ) = x ; (F2) P ( P ( x , y ) , P ( w , z )) = P ( x , z ) ; (F3) P ( Q ( x , y ) , Q ( w , z )) = Q ( P ( x , w ) , P ( y , z )) . Giulio Manzonetto HdR Defense 07/03/17 28 / 38

  57. Factor Algebras at Work Factor Algebras and Factor Varieties An evaluation of the propositional variable P becomes: P is true ⇐ ⇒ f P ( ξ f , ξ t ) = ξ t ; P is false ⇐ ⇒ f P ( ξ f , ξ t ) = ξ f . Factor Algebras Algebras having decomposition operators f P ( − , − ) which are projections. Factor Variety We study classical propositional logic through the variety of algebras generated by these factor algebras. Giulio Manzonetto HdR Defense 07/03/17 29 / 38

  58. Factor Algebras at Work A Uniform Approach for MV-Matrix Logics Truth values V = { v 1 , . . . , v p } �→ algebraic variables ξ 1 , . . . , ξ p ; �→ n -ary decomposition operators f P ( ξ 1 , . . . , ξ p ); Propositional variables P �→ Connectives disappear: implemented via substitutions and logical operations on indices The translations generalizes without problems. . . Giulio Manzonetto HdR Defense 07/03/17 30 / 38

  59. Factor Algebras at Work A Uniform Approach for MV-Matrix Logics Truth values V = { v 1 , . . . , v p } �→ algebraic variables ξ 1 , . . . , ξ p ; Propositional variables P �→ n -ary decomposition operators f P ( ξ 1 , . . . , ξ p ); �→ Connectives disappear: implemented via substitutions and logical operations on indices The translations generalizes without problems. . . Main Theorem (Propositional Case) [SalibraMF, LICS16] A propositional formula φ is a tautology in a MV-matrix logic L if and only if ξ.φ ∗ = ξ t in the variety generated by all factor algebras. = ∀ � V | Giulio Manzonetto HdR Defense 07/03/17 30 / 38

  60. Factor Circuits Decomposition Gate Factor Circuits New notion of logic circuit Only one kind of gate: P s i 1 o i p - p input ports i 1 , . . . , i p (one for each truth value); - a switch s , called the selector switch ; - an output port o . D -gates represent decomposition operators of an algebra A ∈ Factor Variety. When A is a factor algebra, the gate behaves as a multiplexer. Giulio Manzonetto HdR Defense 07/03/17 31 / 38

  61. Factor Circuits Decomposition Gate Factor Circuits New notion of logic circuit Only one kind of gate: i 1 o P i p - p input ports i 1 , . . . , i p (one for each truth value); - a switch s , called the selector switch ; - an output port o . D -gates represent decomposition operators of an algebra A ∈ Factor Variety. When A is a factor algebra, the gate behaves as a multiplexer. Giulio Manzonetto HdR Defense 07/03/17 31 / 38

  62. Factor Circuits Comparison factor circuit gate AND D-gate ξ f ξ f Q P P ∧ Q P P ξ t Q ξ t logic gate AND D-gate operation connective ∧ decomposition operator f P meaning static ( AND ) dynamic (depends on P ) arity of ∧ # V no. of inputs input values prop. variables P , Q algebraic variables ξ f , ξ t signals carried truth values elements of by the wires the algebra A P ∧ Q f P ( ξ f , ξ t ) output Classical simplification processes for logical circuits are based on Boolean identities, Karnaugh maps, Quine-McClusky methods, . . . Giulio Manzonetto HdR Defense 07/03/17 32 / 38

  63. Factor Circuits Simplification by Term Rewriting F 1 x P i x ֌ x F r x F ℓ x P i P i 2 2 y P i y ֌ ֋ z P i P i z z x x P i F ℓ P j 3 P j y y ֌ P P i i z z x P i F r x P i y 3 P j y ֌ P j P i z z Theorem A propositional formula φ is a tautology if and only if nf ( φ ∗ ) = ξ t Giulio Manzonetto HdR Defense 07/03/17 33 / 38

  64. Factor Circuits Simplification by Term Rewriting F 1 x P i x ֌ x F r x F ℓ x P i P i 2 2 y P i y ֌ ֋ z P i P i z z x x P i F ℓ P j 3 P j y y ֌ P i > j P i i z z x P i F r x P i y 3 P j y ֌ P j i > j P i z z Theorem A propositional formula φ is a tautology if and only if nf ( φ ∗ ) = ξ t Giulio Manzonetto HdR Defense 07/03/17 33 / 38

  65. Factor Circuits Simplification by Term Rewriting F 1 x P i x ֌ x F r x F ℓ x P i P i 2 2 y P i y ֌ ֋ z P i P i z z x x P i F ℓ P j 3 P j y y ֌ P i > j P i i z z x P i F r x P i y 3 P j y ֌ P j i > j P i z z Theorem A propositional formula φ is a tautology if and only if nf ( φ ∗ ) = ξ t Giulio Manzonetto HdR Defense 07/03/17 33 / 38

  66. Factor Circuits Generalization I: Quantified Matrix Logics p -valued Quantified Matrix Logics Φ ::= R ( t 1 , . . . , t m ) | o (Φ 1 , . . . , Φ n ) | ∀ x . Φ | ∃ x . Φ R ( a 1 , . . . , a m ) �→ f R ( a 1 , . . . , a m , ξ 1 , . . . , ξ p ) decomp. operator of arity m + p bijection Factor Algebras ← → Structures Theorem [SalibraMF, LICS16] = ∀ ξ 1 . . . ξ p . Φ ∗ = ξ t holds A universal sentence Φ is a logical truth ⇐ ⇒ V ˆ ν | For classical logic we can do more. . . Giulio Manzonetto HdR Defense 07/03/17 34 / 38

  67. Factor Circuits Generalization I: Quantified Matrix Logics p -valued Quantified Matrix Logics Φ ::= R ( t 1 , . . . , t m ) | o (Φ 1 , . . . , Φ n ) | ∀ x . Φ | ∃ x . Φ R ( a 1 , . . . , a m ) �→ f R ( a 1 , . . . , a m , ξ 1 , . . . , ξ p ) decomp. operator of arity m + p bijection (proper) Factor Algebras ← → (proper) Structures Theorem [SalibraMF, LICS16] = ∀ ξ 1 . . . ξ p . Φ ∗ = ξ t holds A universal sentence Φ is a logical truth ⇐ ⇒ V ˆ ν | and the propositional translation of Φ is a tautology (in presence of equality). For classical logic we can do more. . . Giulio Manzonetto HdR Defense 07/03/17 34 / 38

  68. Factor Circuits Generalization II: First-order Logic First-order Classical Logic Ψ 1 , . . . , Ψ n | = Φ is reduced to an equational problem. Reduction procedure exploiting prenex normal form, Skolemization, etc. Theorem (Completeness) [SalibraMF, LICS16] Ψ 1 , . . . , Ψ n | = Φ if and only if Ax ( V Σ ) ⊢ eq ∀ x ∀ y ( x = y ) and the propositional translation of Ψ 1 ∧ · · · ∧ Ψ n → Φ is a tautology. Giulio Manzonetto HdR Defense 07/03/17 35 / 38

  69. Factor Circuits Generalization II: First-order Logic First-order Classical Logic Ψ 1 , . . . , Ψ n | = Φ is reduced to an equational problem. Reduction procedure exploiting prenex normal form, Skolemization, etc. Is the method generalizable to infinite logics? Theorem (Completeness) [SalibraMF, LICS16] Ψ 1 , . . . , Ψ n | = Φ if and only if Ax ( V Σ ) ⊢ eq ∀ x ∀ y ( x = y ) and the propositional translation of Ψ 1 ∧ · · · ∧ Ψ n → Φ is a tautology. Giulio Manzonetto HdR Defense 07/03/17 35 / 38

  70. Factor Circuits Generalization II: First-order Logic First-order Classical Logic Ψ 1 , . . . , Ψ n | = Φ is reduced to an equational problem. Reduction procedure exploiting prenex normal form, Skolemization, etc. Is the method generalizable to infinite logics? Theorem (Completeness) [SalibraMF, LICS16] Are these techniques useful to theorem provers? Ψ 1 , . . . , Ψ n | = Φ if and only if Ax ( V Σ ) ⊢ eq ∀ x ∀ y ( x = y ) and the propositional translation of Ψ 1 ∧ · · · ∧ Ψ n → Φ is a tautology. Giulio Manzonetto HdR Defense 07/03/17 35 / 38

  71. Conclusions The Big Picture Giulio Manzonetto HdR Defense 07/03/17 36 / 38

  72. Conclusions Algebraic Logic Quantitative Properties Nondeterminism PhD ( λ ) 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 Λ → ⇐ ⇒ Λ ∧ Differential λ -calculus Morris’s Theory SN ML F Giulio Manzonetto HdR Defense 07/03/17 37 / 38

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