Lambda Calculus, Linear Logic and Symbolic Computation Habilitation - - PowerPoint PPT Presentation
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
Habilitation à diriger des recherches Giulio Manzonetto
giulio.manzonetto@lipn.univ-paris13.fr LIPN IRIF Université Paris Nord Université Paris Diderot
March the 7th, 2017
Giulio Manzonetto HdR Defense 07/03/17 1 / 38
Introduction Giulio Manzonetto HdR Defense 07/03/17 2 / 38
Introduction
Giulio Manzonetto HdR Defense 07/03/17 2 / 38
Introduction
Giulio Manzonetto HdR Defense 07/03/17 2 / 38
Introduction
Giulio Manzonetto HdR Defense 07/03/17 2 / 38
λ-Calculus — Its Syntax and Semantics
Giulio Manzonetto HdR Defense 07/03/17 3 / 38
λ-Calculus — Its Syntax and Semantics
Coder
Giulio Manzonetto HdR Defense 07/03/17 4 / 38
λ-Calculus — Its Syntax and Semantics
Coder Semanticist
Giulio Manzonetto HdR Defense 07/03/17 4 / 38
λ-Calculus — Its Syntax and Semantics
Coder P Semanticist
Giulio Manzonetto HdR Defense 07/03/17 4 / 38
λ-Calculus — Its Syntax and Semantics
Coder P Semanticist M | = P = ⊥
Giulio Manzonetto HdR Defense 07/03/17 4 / 38
λ-Calculus — Its Syntax and Semantics
Coder P Semanticist M | = P = ⊥ When are two programs P and P′ equivalent?
Giulio Manzonetto HdR Defense 07/03/17 4 / 38
λ-Calculus — Its Syntax and Semantics
P P’ When are two λ-terms equivalent? P ∼ =O P′ ⇐ ⇒ ∀C . C(P) ։ o1 ∈ O ⇐ ⇒ C(P′) ։ o2 ∈ O
Giulio Manzonetto HdR Defense 07/03/17 5 / 38
λ-Calculus — Its Syntax and Semantics
C P
i
C P’
i
When are two λ-terms equivalent? P ∼ =O P′ ⇐ ⇒ ∀C . C(P) ։ o1 ∈ O ⇐ ⇒ C(P′) ։ o2 ∈ O
Giulio Manzonetto HdR Defense 07/03/17 5 / 38
λ-Calculus — Its Syntax and Semantics
C P
i
C P’
i
When are two λ-terms equivalent? (It depends on the Observables) P ∼ =O P′ ⇐ ⇒ ∀C . C(P) ։ o1 ∈ O ⇐ ⇒ C(P′) ։ o2 ∈ O
Giulio Manzonetto HdR Defense 07/03/17 5 / 38
λ-Calculus — Its Syntax and Semantics
C P
i
C P’
i
When are two λ-terms equivalent? (It depends on the Observables) P ∼ =O P′ ⇐ ⇒ ∀C . C(P) ։ o1 ∈ O ⇐ ⇒ C(P′) ։ o2 ∈ O Hyland/Wadsworth equivalence: O = head-normal forms (H∗)
Giulio Manzonetto HdR Defense 07/03/17 5 / 38
λ-Calculus — Its Syntax and Semantics
C P
i
C P’
i
When are two λ-terms equivalent? (It depends on the Observables) P ∼ =O P′ ⇐ ⇒ ∀C . C(P) ։ o1 ∈ O ⇐ ⇒ C(P′) ։ o2 ∈ O Hyland/Wadsworth equivalence: O = head-normal forms (H∗) Morris equivalence: O = β-normal forms (H+)
Giulio Manzonetto HdR Defense 07/03/17 5 / 38
λ-Calculus — Its Syntax and Semantics
C P
i
C P’
i
When are two λ-terms equivalent? (It depends on the Observables) P ∼ =O P′ ⇐ ⇒ ∀C . C(P) ։ o1 ∈ O ⇐ ⇒ C(P′) ։ o2 ∈ O Hyland/Wadsworth equivalence: O = head-normal forms (H∗) Morris equivalence: O = β-normal forms (H+) Difficult to work with this definition! Because of the quantification over all possible contexts.
Giulio Manzonetto HdR Defense 07/03/17 5 / 38
λ-Calculus — Its Syntax and Semantics
Observational Theories
Tree-like structures
characterization
Giulio Manzonetto HdR Defense 07/03/17 6 / 38
λ-Calculus — Its Syntax and Semantics
BT([I]n∈N)
x λy.y x λy.y x λy.y BT([1]n∈N)
λz.x λy.y z λz.x λy.y z λz.x λy.y z BT([1n]n∈N)
η1(x) λy.y η2(x) λy.y η3(x) λy.y BT([J]n∈N)
λz0.x λy.y λz1.z0 η∞(x) λy.y λz2.z1 η∞(x) λy.y
Barendregt Adding the η-rule to B is not enough! Hyland/Lévy 1976 H+ ⊢ M = N ⇐ ⇒ BT(M) = BT(N) “up to” finitary η-expansions Hyland/Wadsworth 1975 H∗ ⊢ M = N ⇐ ⇒ BT(M) = BT(N) “up to” possibly infinite η-expansions
Giulio Manzonetto HdR Defense 07/03/17 7 / 38
λ-Calculus — Its Syntax and Semantics
BT([I]n∈N) = Bη
x λy.y x λy.y x λy.y BT([1]n∈N)
λz.x λy.y z λz.x λy.y z λz.x λy.y z BT([1n]n∈N)
η1(x) λy.y η2(x) λy.y η3(x) λy.y BT([J]n∈N)
λz0.x λy.y λz1.z0 η∞(x) λy.y λz2.z1 η∞(x) λy.y
Barendregt Adding the η-rule to B is not enough! Hyland/Lévy 1976 H+ ⊢ M = N ⇐ ⇒ BT(M) = BT(N) “up to” finitary η-expansions Hyland/Wadsworth 1975 H∗ ⊢ M = N ⇐ ⇒ BT(M) = BT(N) “up to” possibly infinite η-expansions
Giulio Manzonetto HdR Defense 07/03/17 7 / 38
λ-Calculus — Its Syntax and Semantics
BT([I]n∈N) = Bη
x λy.y x λy.y x λy.y BT([1]n∈N) = H+
λz.x λy.y z λz.x λy.y z λz.x λy.y z BT([1n]n∈N)
η1(x) λy.y η2(x) λy.y η3(x) λy.y BT([J]n∈N)
λz0.x λy.y λz1.z0 η∞(x) λy.y λz2.z1 η∞(x) λy.y
Barendregt Adding the η-rule to B is not enough! Hyland/Lévy 1976 H+ ⊢ M = N ⇐ ⇒ BT(M) = BT(N) “up to” finitary η-expansions Hyland/Wadsworth 1975 H∗ ⊢ M = N ⇐ ⇒ BT(M) = BT(N) “up to” possibly infinite η-expansions
Giulio Manzonetto HdR Defense 07/03/17 7 / 38
λ-Calculus — Its Syntax and Semantics
BT([I]n∈N) = Bη
x λy.y x λy.y x λy.y BT([1]n∈N) = H+
λz.x λy.y z λz.x λy.y z λz.x λy.y z BT([1n]n∈N) = H∗
η1(x) λy.y η2(x) λy.y η3(x) λy.y BT([J]n∈N)
λz0.x λy.y λz1.z0 η∞(x) λy.y λz2.z1 η∞(x) λy.y
Barendregt Adding the η-rule to B is not enough! Hyland/Lévy 1976 H+ ⊢ M = N ⇐ ⇒ BT(M) = BT(N) “up to” finitary η-expansions Hyland/Wadsworth 1975 H∗ ⊢ M = N ⇐ ⇒ BT(M) = BT(N) “up to” possibly infinite η-expansions
Giulio Manzonetto HdR Defense 07/03/17 7 / 38
λ-Calculus — Its Syntax and Semantics
The ω-Rule Strong form of extensionality: ∀P ∈ Λo.MP =T NP entails M =T N Does T ⊢ ω ? λβ λβη H Hη Hω Bη λβω B Bω H∗
Giulio Manzonetto HdR Defense 07/03/17 8 / 38
λ-Calculus — Its Syntax and Semantics
The ω-Rule Strong form of extensionality: ∀P ∈ Λo.MP =T NP entails M =T N Does T ⊢ ω ? Does H+ satisfies ω? λβ λβη H Hη Hω Bη λβω B Bω H∗
Giulio Manzonetto HdR Defense 07/03/17 8 / 38
λ-Calculus — Its Syntax and Semantics
The ω-Rule Strong form of extensionality: ∀P ∈ Λo.MP =T NP entails M =T N Does T ⊢ ω ? Does H+ satisfies ω? [BreuvartMPR 2016] By Morris’s Weak Separability Lemma: YES! Sallé’s conjecture [Barendregt 1984] Bω H+ λβ λβη H Hη Hω Bη λβω B Bω ? • H+ H∗
Giulio Manzonetto HdR Defense 07/03/17 8 / 38
λ-Calculus — Its Syntax and Semantics
The ω-Rule Strong form of extensionality: ∀P ∈ Λo.MP =T NP entails M =T N Does T ⊢ ω ? Does H+ satisfies ω? [BreuvartMPR 2016] By Morris’s Weak Separability Lemma: YES! Sallé’s conjecture [Barendregt 1984] Bω H+ FALSE! [IntrigilaMP 2017] λβ λβη H Hη Hω Bη λβω B Bω = H+ H∗
Giulio Manzonetto HdR Defense 07/03/17 8 / 38
λ-Calculus — Its Syntax and Semantics
Observational Theories
Tree-like structures
Böhm Trees + extensional equalities
Giulio Manzonetto HdR Defense 07/03/17 9 / 38
λ-Calculus — Its Syntax and Semantics
Observational Theories
Tree-like structures
Böhm Trees + extensional equalities
Denotational Semantics
Scott’s continuous semantics: Data ∈ D poset Program = continuous function
Giulio Manzonetto HdR Defense 07/03/17 9 / 38
λ-Calculus — Its Syntax and Semantics
Observational Theories
Tree-like structures
Böhm Trees + extensional equalities
Denotational Semantics
Scott’s continuous semantics: Data ∈ D poset Program = continuous function Full abstraction for H∗ [Hyland/Wadsworth’76] D∞ | = M = N ⇐ ⇒ H∗ ⊢ M = N Full abstraction for H+ [Coppo-Dezani-Zacchi’87] Dcdz | = M = N ⇐ ⇒ H+ ⊢ M = N
Giulio Manzonetto HdR Defense 07/03/17 9 / 38
λ-Calculus — Its Syntax and Semantics
The cartesian closed category MRel:
sets A, B, . . . morphisms: A → B are relations from Mf(A) to B Relational Graph Models D = (D, i): injection: i : Mf(D) × D → D Approximation Theorem [ManzonettoR, MFPS 2014] In every rgm D, MD = {tD : 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
λ-Calculus — Its Syntax and Semantics
The cartesian closed category MRel:
sets A, B, . . . morphisms: A → B are relations from Mf(A) to B Relational Graph Models D = (D, i): injection: i : Mf(D) × D → D Approximation Theorem [ManzonettoR, MFPS 2014] In every rgm D, MD = {tD : 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
λ-Calculus — Its Syntax and Semantics
The cartesian closed category MRel:
sets A, B, . . . morphisms: A → B are relations from Mf(A) to B Relational Graph Models D = (D, i): injection: i : Mf(D) × D → D Approximation Theorem [ManzonettoR, MFPS 2014] In every rgm D, MD = {tD : 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
λ-Calculus — Its Syntax and Semantics
The cartesian closed category MRel:
sets A, B, . . . morphisms: A → B are relations from Mf(A) to B Relational Graph Models D = (D, i): injection: i : Mf(D) × D → D Approximation Theorem [ManzonettoR, MFPS 2014] In every rgm D, MD = {tD : 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. Are all theories T in the interval [H+, H∗] representable by relational graph models?
Giulio Manzonetto HdR Defense 07/03/17 10 / 38
λ-Calculus — Its Syntax and Semantics
The cartesian closed category MRel:
sets A, B, . . . morphisms: A → B are relations from Mf(A) to B Relational Graph Models D = (D, i): injection: i : Mf(D) × D → D Approximation Theorem [ManzonettoR, MFPS 2014] In every rgm D, MD = {tD : 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. Are all theories T in the interval [H+, H∗] representable by relational graph models? Otherwise, is it possible to characterize all representable theories?
Giulio Manzonetto HdR Defense 07/03/17 10 / 38
λ-Calculus — Its Syntax and Semantics
The cartesian closed category MRel:
sets A, B, . . . morphisms: A → B are relations from Mf(A) to B Relational Graph Models D = (D, i): injection: i : Mf(D) × D → D Approximation Theorem [ManzonettoR, MFPS 2014] In every rgm D, MD = {tD : 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. Are all theories T in the interval [H+, H∗] representable by relational graph models? Otherwise, is it possible to characterize all representable theories? What can we say about non-extensional representable theories?
Giulio Manzonetto HdR Defense 07/03/17 10 / 38
Quantitative Semantics
Giulio Manzonetto HdR Defense 07/03/17 11 / 38
Quantitative Semantics
Nondeterministic Functional Programming Languages
Operational Semantics Denotational Semantics
characterize properties
Execution properties I/O flow Termination Contextual equivalence
Giulio Manzonetto HdR Defense 07/03/17 12 / 38
Quantitative Semantics
Nondeterministic Functional Programming Languages
Operational Semantics Denotational Semantics
characterize properties
Execution properties I/O flow Termination Contextual equivalence Properties of the model Soundness Adequacy Full Abstraction
Giulio Manzonetto HdR Defense 07/03/17 12 / 38
Quantitative Semantics
Nondeterministic Functional Programming Languages
Operational Semantics Denotational Semantics
characterize properties
Qualitative properties I/O flow Termination Contextual equivalence Properties of the model Soundness Adequacy Full Abstraction
Giulio Manzonetto HdR Defense 07/03/17 12 / 38
Quantitative Semantics
P
“black box” input i
Intensional view on Programs We need to “open the box”. . .
Giulio Manzonetto HdR Defense 07/03/17 13 / 38
Quantitative Semantics
input i
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
Quantitative Semantics
input i
· · · · · · · · · · 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
Quantitative Semantics
Nondeterministic Functional Programming Languages
Operational Semantics Denotational Semantics
characterize properties
Qualitative properties I/O flow Termination Contextual equivalence Properties of the model Soundness Adequacy Full Abstraction Quantitative properties # steps to termination # calls to argument # resources needed # ways to get the output
Giulio Manzonetto HdR Defense 07/03/17 14 / 38
Quantitative Semantics
Nondeterministic Functional Programming Languages
Operational Semantics Denotational Semantics
characterize properties
Qualitative properties I/O flow Termination Contextual equivalence Properties of the model Soundness Adequacy Full Abstraction Quantitative properties # steps to termination # calls to argument # resources needed # ways to get the output
We need quantitative models!
Giulio Manzonetto HdR Defense 07/03/17 14 / 38
Quantitative Semantics
The category MRel Data/Objects: sets Program/Morphism A → B: relation from Mf(A) and B.
# calls to the argument
P ⊆ Mf(A) × B Example A program like M = λn.n ∗ n : Nat → Nat is interpreted by the relation M = {([n1, n2], n1 ∗ n2) | n1, n2 ∈ N} Multiset of cardinality two ⇒ M uses its argument twice.
Giulio Manzonetto HdR Defense 07/03/17 15 / 38
Quantitative Semantics
The category MRel Data/Objects: sets Program/Morphism A → B: relation from Mf(A) and B.
# calls to the argument
P ⊆ Mf(A) × B Example A program like M = λn.n ∗ n : Nat → Nat is interpreted by the relation M = {([n1, n2], n1 ∗ n2) | n1, n2 ∈ N} Multiset of cardinality two ⇒ M uses its argument twice.
Giulio Manzonetto HdR Defense 07/03/17 15 / 38
Quantitative Semantics
The category MRel Data/Objects: sets Program/Morphism A → B: relation from Mf(A) and B.
# calls to the argument
P ⊆ Mf(A) × B Example A program like M = λn.n ∗ n : Nat → Nat is interpreted by the relation M = {([n1, n2], n1 ∗ n2) | n1, n2 ∈ N} NB: Non-deterministic setting!
Giulio Manzonetto HdR Defense 07/03/17 15 / 38
Quantitative Semantics
Qualitative methods Continuous semantics M = cont. function A → B M =
A∈App(M)A Appr. Thm.
Böhm’s approximants λ x.yA1 · · · An (finite tree) Böhm trees BT(M) =
A Quantitative methods Relational Semantics M = relation ⊆ Mf(A) × B M =
t∈T (M)t Appr. Thm.
Resource approximants x | λx.s | s · [t1, . . . , tn] (linear term) Taylor Expansion T (MN) =
∞
1 n!M[N1, . . . , Nn]
Giulio Manzonetto HdR Defense 07/03/17 16 / 38
Quantitative Semantics
Qualitative methods Continuous semantics M = cont. function A → B M =
A∈App(M)A Appr. Thm.
Böhm’s approximants λ x.yA1 · · · An (finite tree) Böhm trees BT(M) =
A Quantitative methods Relational Semantics M = relation ⊆ Mf(A) × B M =
t∈T (M)t Appr. Thm.
Resource approximants x | λx.s | s · [t1, . . . , tn] (linear term) Taylor Expansion T (MN) =
∞
1 n!M[N1, . . . , Nn] Manzonetto et Al. [2010-2013] Syntactic properties + model theory for Ehrhard’s resource calculus. For the resource calculus, look in the HdR thesis!
Giulio Manzonetto HdR Defense 07/03/17 16 / 38
Quantitative Semantics
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
Quantitative Semantics
Giulio Manzonetto HdR Defense 07/03/17 18 / 38
Quantitative Semantics
Relations as matrices A relation from Mf(A) to B is a matrix in BoolMf (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 : Mf(A) × B → R
Giulio Manzonetto HdR Defense 07/03/17 19 / 38
Quantitative Semantics
Program ⊢ P : Nat → Nat. P 1 2 3 4 5 · · · [] p1,1 p1,2 p1,3 p1,4 p1,5 p1,6 · · · [0] p2,1 p2,2 p2,3 p2,4 p2,5 p2,6 · · · [0, 1] p3,1 p3,2 p3,3 p3,4 p3,5 p2,6 · · · [0, 0] p4,1 p4,2 p4,3 p4,4 p4,5 p3,6 · · · [1, 1] p5,1 p5,2 p5,3 p5,4 p5,5 p4,6 · · · [0, 0, 1] p6,1 p6,2 p6,3 p6,4 p6,5 p6,6 · · · . . . . . . . . . . . . . . . . . . . . . ...
Programs become matrices. . . what is the meaning of the scalars?
Giulio Manzonetto HdR Defense 07/03/17 20 / 38
Quantitative Semantics
Program ⊢ P : Nat → Nat. P 1 2 3 4 5 · · · [] p1,1 p1,2 p1,3 p1,4 p1,5 p1,6 · · · [0] p2,1 p2,2 p2,3 p2,4 p2,5 p2,6 · · · [0, 1] p3,1 p3,2 p3,3 p3,4 p3,5 p2,6 · · · [0, 0] p4,1 p4,2 p4,3 p4,4 p4,5 p3,6 · · · [1, 1] p5,1 p5,2 p5,3 p5,4 p5,5 p4,6 · · · [0, 0, 1] p6,1 p6,2 p6,3 p6,4 p6,5 p6,6 · · · . . . . . . . . . . . . . . . . . . . . . ...
Programs become matrices. . . what is the meaning of the scalars?
Giulio Manzonetto HdR Defense 07/03/17 20 / 38
Quantitative Semantics
Consider PCFor, with weighted reductions: M
Giulio Manzonetto HdR Defense 07/03/17 21 / 38
Quantitative Semantics
Consider PCFor, with weighted reductions: M M1 M2 p1 p2
Giulio Manzonetto HdR Defense 07/03/17 21 / 38
Quantitative Semantics
Consider PCFor, with weighted reductions: M M1 M2 p1 p2 M1,1 M1,2 p1,1 p1,2 M2,1 M2,2 p2,1 p2,2
Giulio Manzonetto HdR Defense 07/03/17 21 / 38
Quantitative Semantics
Consider PCFor, with weighted reductions: M M1 M2 p1 p2 M1,1 M1,2 p1,1 p1,2 M2,1 M2,2 p2,1 p2,2 n k k′ n m n n m
Giulio Manzonetto HdR Defense 07/03/17 21 / 38
Quantitative Semantics
Consider PCFor, with weighted reductions: M M1 M2 p1 p2 M1,1 M1,2 p1,1 p1,2 M2,1 M2,2 p2,1 p2,2 n k k′ n m n n m 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
Quantitative Semantics
What are we actually counting? This depends on the chosen semi-ring. . . x | λx.M | MN | YM | zero | succ M | pred M | ifz(M, N, P) | M or N
Giulio Manzonetto HdR Defense 07/03/17 22 / 38
Quantitative Semantics
What are we actually counting? This depends on the chosen semi-ring. . . x | λx.M | MN | YM | zero | succ M | pred M | ifz(M, N, P) | M or N R MRΠ
!
n
= Red(M, n)
Giulio Manzonetto HdR Defense 07/03/17 22 / 38
Quantitative Semantics
What are we actually counting? This depends on the chosen semi-ring. . . x | λx.M | MN | YM | zero | succ M | pred M | ifz(M, N, P) | M or N R MRΠ
!
n
= Red(M, n) (N, ·, 1, , ≤) number of paths from M to n
Giulio Manzonetto HdR Defense 07/03/17 22 / 38
Quantitative Semantics
What are we actually counting? This depends on the chosen semi-ring. . . x | λx.M | MN | YM | zero | succ M | pred M | ifz(M, N, P) | 1 2M or 1 2N R MRΠ
!
n
= Red(M, 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
Quantitative Semantics
What are we actually counting? This depends on the chosen semi-ring. . . x | λx.1M | MN | Y(1M) | zero | succ M | pred M | ifz(M, N, P) | M or N R MRΠ
!
n
= Red(M, 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
Quantitative Semantics
What are we actually counting? This depends on the chosen semi-ring. . . x | λx.1M | MN | Y(1M) | zero | succ M | pred M | ifz(M, N, P) | M or N R MRΠ
!
n
= Red(M, 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
Quantitative Semantics
What are we actually counting? This depends on the chosen semi-ring. . . x | λx.1M | MN | Y(1M) | zero | succ M | pred M | ifz(M, N, P) | M or N R MRΠ
!
n
= Red(M, 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
Quantitative Semantics
What are we actually counting? This depends on the chosen semi-ring. . . x | λx.1M | MN | Y(1M) | zero | succ M | pred M | ifz(M, N, P) | M or N R MRΠ
!
n
= Red(M, 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] Are there other quantitative properties that can be captured by suitable semirings?
Giulio Manzonetto HdR Defense 07/03/17 22 / 38
Quantitative Semantics
What are we actually counting? This depends on the chosen semi-ring. . . x | λx.1M | MN | Y(1M) | zero | succ M | pred M | ifz(M, N, P) | M or N R MRΠ
!
n
= Red(M, 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] Are there other quantitative properties that can be captured by suitable semirings? Is it possible to get rid of commutativity?
Giulio Manzonetto HdR Defense 07/03/17 22 / 38
Quantitative Semantics
What are we actually counting? This depends on the chosen semi-ring. . . x | λx.1M | MN | Y(1M) | zero | succ M | pred M | ifz(M, N, P) | M or N R MRΠ
!
n
= Red(M, 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] Are there other quantitative properties that can be captured by suitable semirings? Is it possible to get rid of commutativity? What about the profunctorial semantics?
Giulio Manzonetto HdR Defense 07/03/17 22 / 38
Factor Algebras at Work
Giulio Manzonetto HdR Defense 07/03/17 23 / 38
Factor Algebras at Work
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
AND
P Q P ∧ Q
OR
P Q P ∨ Q
NOT
P ¬P
Giulio Manzonetto HdR Defense 07/03/17 24 / 38
Factor Algebras at Work
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
AND
P Q P ∧ Q
OR
P Q P ∨ Q
NOT
P ¬P
Giulio Manzonetto HdR Defense 07/03/17 24 / 38
Factor Algebras at Work
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
AND
P Q P ∧ Q
OR
P Q P ∨ Q
NOT
P ¬P
Giulio Manzonetto HdR Defense 07/03/17 24 / 38
Factor Algebras at Work
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
AND
P Q P ∧ Q
OR
P Q P ∨ Q
NOT
P ¬P
Giulio Manzonetto HdR Defense 07/03/17 24 / 38
Factor Algebras at Work
Propositional variables → algebraic variables Connectives →
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
Factor Algebras at Work
P1 P2 P3 f t P3 f t P2 P3 f t P3 t t P1 P2 P3 f f f f f f t t f t f f f t t t t f f f t f t t t t f t t t t t ROBDDs Reduced: maximal sharing, Ordered: to ensure canonicity.
Giulio Manzonetto HdR Defense 07/03/17 26 / 38
Factor Algebras at Work
P3 P2 P1 t f P1 P2 P3 f f f f f f t t f t f f f t t t t f f f t f t t t t f t t t t t ROBDDs Reduced: maximal sharing, Ordered: to ensure canonicity.
Giulio Manzonetto HdR Defense 07/03/17 26 / 38
Factor Algebras at Work
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: (F1) f(x, x) = x. (F2) f(f(x, y), f(w, z)) = f(x, z). (F3) 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
Factor Algebras at Work
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: (F1) f(x, x) = x. (F2) f(f(x, y), f(w, z)) = f(x, z). (F3) 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
Factor Algebras at Work
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: (F1) f(x, x) = x. (F2) f(f(x, y), f(w, z)) = f(x, z). (F3) 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
Factor Algebras at Work
Truth values t, f → algebraic variables ξf, ξt; Propositional variables P → decomposition operators fP(ξf, ξt); Connectives → disappear: implemented via substitutions and Boolean operations on indices The Translation t∗ = ξt f∗ = ξf P∗ = P(ξf, ξt) (¬φ)∗ = φ∗(ξ¬f, ξ¬t); (φ ∧ ψ)∗ = ψ∗(φ∗(ξf∧f, ξf∧t), φ∗(ξt∧f, ξt∧t)); (φ ∨ ψ)∗ = ψ∗(φ∗(ξf∨f, ξf∨t), φ∗(ξt∨f, ξt∨t)); (φ → ψ)∗ = (¬φ ∨ ψ)∗.
Giulio Manzonetto HdR Defense 07/03/17 28 / 38
Factor Algebras at Work
Truth values t, f → algebraic variables ξf, ξt; Propositional variables P → decomposition operators fP(ξf, ξt); Connectives → disappear: implemented via substitutions and Boolean operations on indices The Translation t∗ = ξt f∗ = ξf P∗ = P(ξf, ξt) (¬φ)∗ = φ∗(ξt, ξf); (φ ∧ ψ)∗ = ψ∗(φ∗(ξf, ξf), φ∗(ξf, ξt)); (φ ∨ ψ)∗ = ψ∗(φ∗(ξf, ξt), φ∗(ξt, ξt)); (φ → ψ)∗ = ψ∗(φ∗(ξt, ξf), φ∗(ξt, ξt)). 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
Factor Algebras at Work
Truth values t, f → algebraic variables ξf, ξt; Propositional variables P → decomposition operators fP(ξf, ξt); Connectives → disappear: implemented via substitutions and Boolean operations on indices The Translation t∗ = ξt f∗ = ξf P∗ = P(ξf, ξt) (¬φ)∗ = φ∗(ξt, ξf); (φ ∧ ψ)∗ = ψ∗(φ∗(ξf, ξf), φ∗(ξf, ξt)); (φ ∨ ψ)∗ = ψ∗(φ∗(ξf, ξt), φ∗(ξt, ξt)); (φ → ψ)∗ = ψ∗(φ∗(ξt, ξf), φ∗(ξt, ξt)). 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
Factor Algebras at Work
Truth values t, f → algebraic variables ξf, ξt; Propositional variables P → decomposition operators fP(ξf, ξt); Connectives → disappear: implemented via substitutions and Boolean operations on indices The Translation t∗ = ξt f∗ = ξf P∗ = P(ξf, ξt) (¬φ)∗ = φ∗(ξt, ξf); (φ ∧ ψ)∗ = ψ∗(φ∗(ξf, ξf), φ∗(ξf, ξt)); (φ ∨ ψ)∗ = ψ∗(φ∗(ξf, ξt), φ∗(ξt, ξt)); (φ → ψ)∗ = ψ∗(φ∗(ξt, ξf), φ∗(ξt, ξt)). 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)). These are ROBDDs!
Giulio Manzonetto HdR Defense 07/03/17 28 / 38
Factor Algebras at Work
An evaluation of the propositional variable P becomes: P is true ⇐ ⇒ fP(ξf, ξt) = ξt; P is false ⇐ ⇒ fP(ξf, ξt) = ξf. Factor Algebras Algebras having decomposition operators fP(−, −) 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
Factor Algebras at Work
Truth values V = {v1, . . . , vp} → algebraic variables ξ1, . . . , ξp; Propositional variables P → n-ary decomposition operators fP(ξ1, . . . , ξp); Connectives → disappear: implemented via substitutions and logical operations on indices The translations generalizes without problems. . .
Giulio Manzonetto HdR Defense 07/03/17 30 / 38
Factor Algebras at Work
Truth values V = {v1, . . . , vp} → algebraic variables ξ1, . . . , ξp; Propositional variables P → n-ary decomposition operators fP(ξ1, . . . , ξp); 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 V | = ∀ ξ.φ∗ = ξt in the variety generated by all factor algebras.
Giulio Manzonetto HdR Defense 07/03/17 30 / 38
Factor Circuits
New notion of logic circuit Only one kind of gate: P s i1 ip
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
Factor Circuits
New notion of logic circuit Only one kind of gate: P i1 ip
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
Factor Circuits
gate AND P Q P ∧ Q P D-gate ξf ξt P Q factor circuit ξf ξt logic gate AND D-gate
connective ∧ decomposition operator fP meaning static (AND) dynamic (depends on P)
arity of ∧ # V input values
algebraic variables ξf, ξt signals carried truth values elements of by the wires the algebra A
P ∧ Q fP(ξf, ξt) 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
Factor Circuits
F1
Pi x x
F ℓ
2
Pi x y Pi z Pi x z
F r
2
Pi Pi x y z
F ℓ
3
P
j
x y P
i
z Pi Pi Pj x y z Pj Pi x y z
F r
3
Pi Pi Pj x y z Theorem A propositional formula φ is a tautology if and only if nf(φ∗) = ξt
Giulio Manzonetto HdR Defense 07/03/17 33 / 38
Factor Circuits
F1
Pi x x
F ℓ
2
Pi x y Pi z Pi x z
F r
2
Pi Pi x y z
F ℓ
3
i>j
P
j
x y P
i
z Pi Pi Pj x y z Pj Pi x y z
F r
3
i>j
Pi Pi Pj x y z Theorem A propositional formula φ is a tautology if and only if nf(φ∗) = ξt
Giulio Manzonetto HdR Defense 07/03/17 33 / 38
Factor Circuits
F1
Pi x x
F ℓ
2
Pi x y Pi z Pi x z
F r
2
Pi Pi x y z
F ℓ
3
i>j
P
j
x y P
i
z Pi Pi Pj x y z Pj Pi x y z
F r
3
i>j
Pi Pi Pj x y z Theorem A propositional formula φ is a tautology if and only if nf(φ∗) = ξt
Giulio Manzonetto HdR Defense 07/03/17 33 / 38
Factor Circuits
p-valued Quantified Matrix Logics
Φ ::= R(t1, . . . , tm) | o(Φ1, . . . , Φn) | ∀x.Φ | ∃x.Φ R(a1, . . . , am) → fR(a1, . . . , am, ξ1, . . . , ξp) decomp. operator of arity m + p Factor Algebras
bijection
← → Structures Theorem [SalibraMF, LICS16] A universal sentence Φ is a logical truth ⇐ ⇒ Vˆ
ν |
= ∀ξ1 . . . ξp.Φ∗ = ξt holds For classical logic we can do more. . .
Giulio Manzonetto HdR Defense 07/03/17 34 / 38
Factor Circuits
p-valued Quantified Matrix Logics
Φ ::= R(t1, . . . , tm) | o(Φ1, . . . , Φn) | ∀x.Φ | ∃x.Φ R(a1, . . . , am) → fR(a1, . . . , am, ξ1, . . . , ξp) decomp. operator of arity m + p (proper) Factor Algebras
bijection
← → (proper) Structures Theorem [SalibraMF, LICS16] A universal sentence Φ is a logical truth ⇐ ⇒ Vˆ
ν |
= ∀ξ1 . . . ξp.Φ∗ = ξt holds 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
Factor Circuits
First-order Classical Logic
Ψ1, . . . , Ψn | = Φ is reduced to an equational problem. Reduction procedure exploiting prenex normal form, Skolemization, etc. Theorem (Completeness) [SalibraMF, LICS16] Ψ1, . . . , Ψn | = Φ if and only if Ax(VΣ) ⊢eq ∀x∀y(x = y) and the propositional translation of Ψ1 ∧ · · · ∧ Ψn → Φ is a tautology.
Giulio Manzonetto HdR Defense 07/03/17 35 / 38
Factor Circuits
First-order Classical Logic
Ψ1, . . . , Ψn | = Φ is reduced to an equational problem. Reduction procedure exploiting prenex normal form, Skolemization, etc. Theorem (Completeness) [SalibraMF, LICS16] Ψ1, . . . , Ψn | = Φ if and only if Ax(VΣ) ⊢eq ∀x∀y(x = y) and the propositional translation of Ψ1 ∧ · · · ∧ Ψn → Φ is a tautology. Is the method generalizable to infinite logics?
Giulio Manzonetto HdR Defense 07/03/17 35 / 38
Factor Circuits
First-order Classical Logic
Ψ1, . . . , Ψn | = Φ is reduced to an equational problem. Reduction procedure exploiting prenex normal form, Skolemization, etc. Theorem (Completeness) [SalibraMF, LICS16] Ψ1, . . . , Ψn | = Φ if and only if Ax(VΣ) ⊢eq ∀x∀y(x = y) and the propositional translation of Ψ1 ∧ · · · ∧ Ψn → Φ is a tautology. Is the method generalizable to infinite logics? Are these techniques useful to theorem provers?
Giulio Manzonetto HdR Defense 07/03/17 35 / 38
Conclusions
Giulio Manzonetto HdR Defense 07/03/17 36 / 38
Conclusions
2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 PhD (λ) Nondeterminism Quantitative Properties SN MLF Λ→⇐ ⇒ Λ∧ Differential λ-calculus Morris’s Theory Algebraic Logic
Giulio Manzonetto HdR Defense 07/03/17 37 / 38
Conclusions
2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 PhD (λ) Nondeterminism Quantitative Properties SN MLF Λ→⇐ ⇒ Λ∧ Differential λ-calculus Morris’s Theory Algebraic Logic LICS06 2×CSL07 MFCS08 LFCS09 MSCS MFCS09 MFPS10 JLC CSL11 TLCA11 ICALP11 ICALP12 LMCS MSCS APAL TCS MFCS10 LFCS13 LICS13 InfComp MFPS14 LICS16 FSCD16
Giulio Manzonetto HdR Defense 07/03/17 37 / 38
Conclusions
2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 PhD (λ) Nondeterminism Quantitative Properties SN MLF Λ→⇐ ⇒ Λ∧ Differential λ-calculus Morris’s Theory Algebraic Logic
Giulio Manzonetto HdR Defense 07/03/17 37 / 38
Conclusions
2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 PhD (λ) Nondeterminism Quantitative Properties SN MLF Λ→⇐ ⇒ Λ∧ Differential λ-calculus Morris’s Theory Algebraic Logic
Giulio Manzonetto HdR Defense 07/03/17 37 / 38
Conclusions Giulio Manzonetto HdR Defense 07/03/17 38 / 38