From pre-models to models

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1 From pre-models to models normalization by Heyting algebras Olivier HERMANT 18 Mars 2008

2 Deduction System : natural deduction (NJ) first-order logic: function and predicate symbols, logical connectors:,,,, and quantifiers,. Γ, A A axiom Γ A Γ B Γ A B -i Γ A B -e1 Γ A Γ A B Γ B -e2 Γ, A B Γ A B -i Γ A B Γ A -e Γ B Γ xa[x] -e, any t Γ A[t] Γ A[x] -i, x free Γ xa[x]

3 Deduction modulo: allowed rewriting General form (free variables are possible): l r

4 Deduction modulo: allowed rewriting General form (free variables are possible): l r use: We replace t = σl by σr (unification). Rewriting could be deep in the term.

5 Deduction modulo: allowed rewriting General form (free variables are possible): l r use: We replace t = σl by σr (unification). Rewriting could be deep in the term. rewriting on terms: x + S(y) S(x + y)

6 Deduction modulo: allowed rewriting General form (free variables are possible): l r use: We replace t = σl by σr (unification). Rewriting could be deep in the term. rewriting on terms: x + S(y) S(x + y) and on propositions (predicate symbols): advantage: expressiveness x y = 0 x = 0 y = 0

7 Deduction modulo: allowed rewriting General form (free variables are possible): l r use: We replace t = σl by σr (unification). Rewriting could be deep in the term. rewriting on terms: x + S(y) S(x + y) and on propositions (predicate symbols): advantage: expressiveness x y = 0 x = 0 y = 0 we obtain a congruence modulo R (chosen set of rules):

8 Natural deduction modulo - first presentation Γ, A A axiom Γ A Γ B Γ A B -i Γ A B -e1 Γ A Γ A B Γ B -e2 -i Γ, A B Γ A B Γ A B Γ A -e Γ B Γ xa[x] -e, any t Γ A[t] Γ A[x] -i, x free Γ xa[x]

9 Natural deduction modulo - first presentation Γ, A A axiom Γ A Γ B Γ A B -i Γ A B -e1 Γ A Γ A B Γ B -e2 -i Γ, A B Γ A B Γ A B Γ A -e Γ B Γ xa[x] -e, any t Γ A[t] Γ A[x] -i, x free Γ xa[x] Add the following conversion rule Γ A Γ B A B

10 Natural deduction modulo, second version Γ, A B axiom, A B Γ A Γ B Γ C -i, C A B Γ C Γ A -e1,c A B Γ C Γ B -e2,c A B -i, C A B Γ, A B Γ C Γ C Γ A -e, C A B Γ B Γ A[x] Γ B -i, x free,b xa[x] Γ B -e, any t, B xa[x] Γ A[t]

11 Example: 3 consider the rewriting system R: P(0) A P(1) B xp(x) A B

12 Example: 3 consider the rewriting system R: P(0) A P(1) B xp(x) A xp(x) B -i xp(x) A B

13 Example: 3 consider the rewriting system R: P(0) A P(1) B -e xp(x) xp(x) xp(x) xp(x) -e xp(x) A xp(x) B -r xp(x) A B

14 Example: 3 consider the rewriting system R: P(0) A P(1) B xp(x) xp(x) xp(x) xp(x) -e -e xp(x) P(0) xp(x) P(1) conv conv xp(x) A xp(x) B -r xp(x) A B

15 Example: 3 consider the rewriting system R: P(0) A P(1) B -e xp(x) xp(x) xp(x) xp(x) -e xp(x) A xp(x) B -r xp(x) A B

16 Example: 3 consider the rewriting system R: P(0) A P(1) B axiom axiom xp(x) xp(x) xp(x) xp(x) -e -e xp(x) A xp(x) B -r xp(x) A B

17 A Cut: a detour Γ A Γ, A B -i Γ A B -e Γ B show Γ A and Γ, A B then, you have showed Γ B it is the application of a lemma.

18 A Cut: a detour π 1 π 2 Γ A Γ B -i Γ A B -e Γ A General pattern of a cut: an introduction rule, followed by an elimination on the same symbol. This is unnecessary, consider only π 1. π 1 Γ A

19 A Cut: a detour In deduction modulo: θ Γ A Γ B π Γ, A B -i, C A B Γ C -e, C A B need for cut elimination: the heart of logic.

20 A Cut: a detour In deduction modulo: θ Γ A Γ B π Γ, A B -i, C A B Γ C -e, C A B need for cut elimination: the heart of logic. two main methods: semantic: cut admissibility. syntactic: proof normalization.

21 A Cut: a detour In deduction modulo: θ Γ A Γ B π Γ, A B -i, C A B Γ C -e, C A B need for cut elimination: the heart of logic. two main methods: semantic: cut admissibility. syntactic: proof normalization. indecidable, need for conditions on R.

22 II The semantic method

23 The semantical method Γ A Gentzen Tait-Girard Dowek-Werner... Γ cf A soundness completeness Γ = A

24 The semantical method Γ A Gentzen Tait-Girard Dowek-Werner... Γ cf A soundness strong completeness Γ = A

25 Heyting algebras a universe Ω an order

26 Heyting algebras a universe Ω an order operations on it: lowest upper bound (join: ), greatest lower bound (meet: ), arrow (more that lattice). a b a a b b c a and c b implies c a b a a b b a b a c and b c implies a b c a b c iff a b c like Boolean algebras, with weaker complement

27 an example R and open sets (infinite g.l.b. is not infinite intersection)

28 an example R and open sets (infinite g.l.b. is not infinite intersection) complement is weaker: A 0 [ ] A

29 A model a domain D to interpret the first-order terms. a Heyting algebra Ω a interpretation function for each symbol: ˆf : D n D ˆP : D m Ω that we extend readily to all terms and all formulae and terms: (x) φ := φ(x) (f(t 1,, t n )) φ := ˆf(((t1 ) φ,, (t n) φ )) (P(t 1,, t n )) φ := ˆP(((t1 ) φ,, (t n) φ )) (A B) φ := (A) φ (B) φ

30 A model a domain D to interpret the first-order terms. a Heyting algebra Ω a interpretation function for each symbol: ˆf : D n D ˆP : D m Ω that we extend readily to all terms and all formulae and terms: (x) φ := φ(x) (f(t 1,, t n )) φ := ˆf(((t1 ) φ,, (t n) φ )) (P(t 1,, t n )) φ := ˆP(((t1 ) φ,, (t n) φ )) (A B) φ := (A) φ (B) φ degree of freedom: how to choose ˆf and ˆP. in deduction modulo, additional condition: A R B implies A = B

31 Cannonical model: Lindenbaum algebra defined for provability elements of Ω: the equivalence class of formulae [A]. [A] := {B A B} order: [A] [B] iff A B is provable meet: [A] [B] iff [A B]

32 Cannonical model: Lindenbaum algebra defined for provability elements of Ω: the equivalence class of formulae [A]. [A] := {B A B} order: [A] [B] iff A B is provable meet: [A] [B] iff [A B] and so on... (domain D: open terms). with this model, one proves completeness

33 Cannonical model: Lindenbaum algebra defined for provability with cuts elements of Ω: the equivalence class of formulae [A]. [A] := {B A B} intersection : [A] [B] iff [A B] order : [A] [B] iff A B and so on... (domain D: open terms) with this model, one proves completeness: cuts are needed for transitivity of the order.

34 Cut-free cannonical model defined for provability without cuts elements of Ω: the contexts proving A cut-free. [A] := {Γ Γ A} the [A] generate Ω with their (arbitrary) intersection and pseudo-union (l.u.b.): a b = {[A] a [A] and b [A]} order: a b iff a b and so on... with this model, one proves cut-free completeness.

35 Deduction modulo what about the domain? what about the validity of the rewrite rules? A R B implies A = B

36 Deduction modulo what about the domain: it depends on R (not always open term). what about the validity of the rewrite rules: choose carefully the interpretation of predicates and function symbols, depends on R.

37 An example: Simple Theory of Types aka higher-order (intuitionistic) logic. basic types o, ι, and arrow: o o, o ι,... constants of each type application (t u) and λ-abstraction or combinators: S, K logical connectors: constants : o o o,... e.g. we can form the formula: P.P same deduction rules as NJ plus lambda-conversion.

38 Cut admissibility in STT problem number one, circularity:..p(p P) (P P)

39 Cut admissibility in STT problem number one, circularity:..p(p P) ( P.(P P) P.(P P)) no more induction on the size of the formulae.

40 Cut admissibility in STT problem number one, circularity:..p(p P) ( P.(P P) P.(P P)) no more induction on the size of the formulae. solution, same as Girard: Define R A : quantify over all R B : Circular Avoid circularity: define C a priori, quantify over C instead, Prove a posteriori that R B C.

41 Cut admissibility in STT problem number one, circularity:..p(p P) ( P.(P P) P.(P P)) no more induction on the size of the formulae. solution, same as Girard: Define R A : quantify over all R B : Circular Avoid circularity: define C a priori, quantify over C instead, Prove a posteriori that R B C. define semantic candidates [Okada] for (A) without induction: {α Ω A α [A]}

42 Cut admissibility in STT problem number one, circularity:..p(p P) ( P.(P P) P.(P P)) no more induction on the size of the formulae. solution, same as Girard: Define R A : quantify over all R B : Circular Avoid circularity: define C a priori, quantify over C instead, Prove a posteriori that R B C. define semantic candidates [Okada] for (A) without induction: {α Ω A α [A]} then quantify over all truth-values candidates. Identifies which of the α is (A).

43 Cut admissibility in STT Problem 2: logical intensionality. In STT, as in λprolog: No logical extensionality rule: P(A A) P(A) P(A) A B P(B)

44 Cut admissibility in STT Problem 2: logical intensionality. In STT, as in λprolog: No logical extensionality rule: P(A A) P(A) P(A) A B P(B) implicates: although semantic truth value of A is in Ω, its domain of interpretation should not be Ω.

45 Cut admissibility in STT Problem 2: logical intensionality. In STT, as in λprolog: P(A A) P(A) No logical extensionality rule: P(A) A B P(B) implicates: although semantic truth value of A is in Ω, its domain of interpretation should not be Ω. usual trick: {α Ω A α [A]}

46 Cut admissibility in STT Problem 2: logical intensionality. In STT, as in λprolog: P(A A) P(A) No logical extensionality rule: P(A) A B P(B) implicates: although semantic truth value of A is in Ω, its domain of interpretation should not be Ω. usual trick: pairing (V-complexes). D o = { A, α A α [A]}

47 Cut admissibility in STT Problem 2: logical intensionality. In STT, as in λprolog: P(A A) P(A) No logical extensionality rule: P(A) A B P(B) implicates: although semantic truth value of A is in Ω, its domain of interpretation should not be Ω. usual trick: pairing (V-complexes). D o = { A, α A α [A]} interpret everything within those domains, e.g.: ˆ :=, λ B, b. B, λ C, c. B C, b c

48 Cut admissibility in STT Problem 2: logical intensionality. In STT, as in λprolog: P(A A) P(A) No logical extensionality rule: P(A) A B P(B) implicates: although semantic truth value of A is in Ω, its domain of interpretation should not be Ω. usual trick: pairing (V-complexes). D o = { A, α A α [A]} interpret everything within those domains, e.g.: ˆ :=, λ B, b. B, λ C, c. B C, b c then, extract the truth value: ω(a ) = π 2 (A )

49 STT in deduction modulo same types, same symbols,, application: K x y x S x y z (xz)(yz) how to express P.P in a first-order setting?

50 STT in deduction modulo same types, same symbols,, application: K x y x S x y z (xz)(yz) how to express P.P in a first-order setting? solution: embed P into ε(p), and define: ε( A B) ε(a) ε(b) ε( A) x.ε(ax)

51 STT in deduction modulo same types, same symbols,, application: K x y x S x y z (xz)(yz) how to express P.P in a first-order setting? solution: embed P into ε(p), and define: ε( A B) ε(a) ε(b) ε( A) x.ε(ax) duplication of connectors : (of the type hierarchy) connecting terms and, connecting propositions. two formulae : P, a term, and ε(p), at the logical level. ε is the only predicate symbol.

52 STT in deduction modulo same types, same symbols,, application: K x y x S x y z (xz)(yz) how to express P.P in a first-order setting? solution: embed P into ε(p), and define: ε( A B) ε(a) ε(b) ε( A) x.ε(ax) duplication of connectors : (of the type hierarchy) connecting terms and, connecting propositions. two formulae : P, a term, and ε(p), at the logical level. ε is the only predicate symbol. ε embeds in the syntax the ω is in the semantics: separates truth value and propositional content.

53 III - Normalization

54 Curry-Howard correspondence Notation for proofs. Give a name to each of the hypothesis: Γ = x 1 : A 1,..., x n : A n Γ, x : A x : A Axiom Γ π : A B Γ fst(π) : A -e1 Γ π 1 : A Γ π 2 : B -i Γ π 1, π 2 : A B Γ π : A B Γ snd(π) : A -e2 Γ, x : A π : B Γ λx.π : A B -i Γ π : A Γ π : A B Γ (ππ ) : B

55 Curry-Howard correspondence Notation for proofs. Give a name to each of the hypothesis: Γ = x 1 : A 1,..., x n : A n Γ, x : A x : A Axiom Γ π : A B Γ fst(π) : A -e1 Γ π 1 : A Γ π 2 : B -i Γ π 1, π 2 : A B Γ π : A B Γ snd(π) : A -e2 Γ, x : A π : B Γ λx.π : A B -i Γ π : A Γ π : A B Γ (ππ ) : B very similar to a type system

56 Curry-Howard correspondence Notation for proofs. Give a name to each of the hypothesis: Γ = x 1 : A 1,..., x n : A n Γ, x : A x : A Axiom Γ π : A B Γ fst(π) : A -e1 Γ π 1 : A Γ π 2 : B -i Γ π 1, π 2 : A B Γ π : A B Γ snd(π) : A -e2 Γ, x : A π : B Γ λx.π : A B -i Γ π : A Γ π : A B Γ (ππ ) : B very similar to a type system in deduction modulo, rewrite rules are silent: Γ π : A A B Γ π : B

57 Cut elimination with proof terms Cut elimination is a process, similar to function execution. Γ π 1 : A Γ π 2 : B -i Γ π 1, π 2 : A B -e Γ fst( π 1, π 2 ) : A Γ π 1 : A Γ, x : A π : B Γ θ : A Γ λx.π : A B Γ (λx.π)θ : B -i -e Γ {θ/x}π : B

58 Cut elimination with proof terms Cut elimination is a process, similar to function execution. Γ π 1 : A Γ π 2 : B -i Γ π 1, π 2 : A B -e Γ fst( π 1, π 2 ) : A Γ π 1 : A Γ, x : A π : B Γ θ : A Γ λx.π : A B Γ (λx.π)θ : B -i -e Γ {θ/x}π : B showing that every proof normalizes: the cut elimination process terminates.

59 Normalization [Dowek,Werner] deduction modulo is high-level: circularity hence reducibility candidates.

60 Normalization [Dowek,Werner] deduction modulo is high-level: circularity hence reducibility candidates. A reducibility candidate: a set of normalizing proof terms (and other closure properties).

61 Normalization [Dowek,Werner] deduction modulo is high-level: circularity hence reducibility candidates. A reducibility candidate: a set of normalizing proof terms (and other closure properties). to each formula A, associates a candidate A : this is a C-valued model (pre-model).

62 Normalization [Dowek,Werner] deduction modulo is high-level: circularity hence reducibility candidates. A reducibility candidate: a set of normalizing proof terms (and other closure properties). to each formula A, associates a candidate A : this is a C-valued model (pre-model). in deduction modulo, if A B, additional constraint: A = B

63 Normalization [Dowek,Werner] deduction modulo is high-level: circularity hence reducibility candidates. A reducibility candidate: a set of normalizing proof terms (and other closure properties). to each formula A, associates a candidate A : this is a C-valued model (pre-model). in deduction modulo, if A B, additional constraint: then prove the main theorem: A = B Theorem: if Γ π : A then for any ψ substitution, φ model assignment, θ environment (mapping α : B Γ to A φ ), we have θψπ A φ

64 IV From Normalization to usual semantics

65 such methods are defined in deduction modulo (Heyting arithmetic, higher-order logic, Zermelo s set theory,...)

66 such methods are defined in deduction modulo (Heyting arithmetic, higher-order logic, Zermelo s set theory,...) the pre-model have a structure: pseudo Heyting algebras, or truth value algebras (TVA) [Dowek].

67 Heyting algebras a universe Ω an order operations on it: lowest upper bound (join: ), greatest lower bound (meet: intersection). a b a a b b c a and c b implies c a b a a b b a b a c and b c implies a b c like Boolean algebras, with weaker complement

68 pseudo-heyting algebras, aka Truth Values Algebras a universe Ω a pre-order: a b and b a with a b possible. operations on it: lowest upper bound (join: pseudo union), greatest lower bound (meet: intersection). a b a a b b c a and c b implies c a b a a b b a b a c and b c implies a b c

69 Candidates form a pseudo-heyting algebra = = SN A B = A B and so on. pre-order: trivial one. But A A A only. of course: A B implies A = B

70 Super consistency the pre-model construction (domain,...) does not depends on the properties of C. consistency: there exists a model. condition in DM: A B implies A = B

71 Super consistency the pre-model construction (domain,...) does not depends on the properties of C. consistency: there exists a model. super-consistency: for every TVA, there exists a model (interpretation): construction has to be uniform. condition in DM: A B implies A = B

72 Super consistency the pre-model construction (domain,...) does not depends on the properties of C. consistency: there exists a model. super-consistency: for every TVA, there exists a model (interpretation): construction has to be uniform. condition in DM: A B implies A = B Super consistency implies cut elimination.

73 Super consistency e.g. higher-order logic is super-consistent: M ι = ι (dummy) M o = C M t u = M M t u

74 Super consistency e.g. higher-order logic is super-consistent: M ι = ι (dummy) M o = C M t u = M M t u hence, it has a model in the pseudo-heying Algebra of candidates Γ π : A implies π A. the system enjoys proof normalization.

75 Towards usual semantics Γ A Gentzen Tait-Girard Dowek-Werner... Γ cf A soundness strong completeness Γ = A

76 Super consistency e.g. higher-order logic is super-consistent: M ι = ι (dummy) M o = C M t u = M M t u

77 Super consistency e.g. higher-order logic is super-consistent: M ι = ι (dummy) M o = C M t u = M M t u hence, it has a model in the pseudo-heying Algebra of reducibility candidates but, A = {π such that...}

78 Towards usual semantics How to transform a TVA into a Heyting algebra. assume we have a model M, in the previous pseudo-heyting algebra of sequents. first idea: pseudo-heyting to Heyting by quotienting.

79 Towards usual semantics How to transform a TVA into a Heyting algebra. assume we have a model M, in the previous pseudo-heyting algebra of sequents. first idea: pseudo-heyting to Heyting by quotienting. trivial pseudo order implies =.

80 The link: extract contexts Assumption: we have a pre-model ( A φ, model M defined). Set: [A] σ φ = {Γ Γ π : σa, and for any environment θ, assignment ψ, θψπ A φ }

81 The link: extract contexts Assumption: we have a pre-model ( A φ, model M defined). Set: [A] σ φ = {Γ Γ π : σa, and for any environment θ, assignment ψ, θψπ A φ } A φ contains proof terms associated to π : B. Extract the contexts corresponding to A. this forms a Heyting algebra ([A]: basis)

82 The link: extract contexts Assumption: we have a pre-model ( A φ, model M defined). Set: [A] σ φ = {Γ Γ π : σa, and for any environment θ, assignment ψ, θψπ A φ } A φ contains proof terms associated to π : B. Extract the contexts corresponding to A. this forms a Heyting algebra ([A]: basis) interpretation of formulas in it: A = [A] σ φ

83 Wait a minute! interpretation? [A] σ φ.

84 Wait a minute! interpretation? [A] σ φ. Need for one single substitution. hybridization: σ φ. D = T M

85 Wait a minute! interpretation? [A] σ φ. Need for one single substitution. hybridization: σ φ. D = T M interpretation for symbols were, in C: ˆf M (d 1,..., d n ) M ˆPM (d 1,..., d n ) C

86 Wait a minute! interpretation? [A] σ φ. Need for one single substitution. hybridization: σ φ. D = T M interpretation for symbols were, in C: ˆf M (d 1,..., d n ) M ˆPM (d 1,..., d n ) C Now they are: ˆf D ( t 1, d 1,..., t n, d n ) = f(t 1,..., t n ),ˆf M (d 1,..., d n ) ˆP D ( t 1, d 1,..., t n, d n ) = [P] (t 1/x 1,...,t n /x n ) (d 1 /x 1,...,d n /x n ) = {Γ (Γ π : P( t )) P ( d / x ) }

87 Wait a minute! interpretation? [A] σ φ. Need for one single substitution. hybridization: σ φ. D = T M interpretation for symbols were, in C: ˆf M (d 1,..., d n ) M ˆPM (d 1,..., d n ) C Now they are: ˆf D ( t 1, d 1,..., t n, d n ) = f(t 1,..., t n ),ˆf M (d 1,..., d n ) ˆP D ( t 1, d 1,..., t n, d n ) = [P] (t 1/x 1,...,t n /x n ) (d 1 /x 1,...,d n /x n ) = {Γ (Γ π : P( t )) P ( d / x ) } Holds for any theory in DM. extends the V-complexes. pointwise application t, v t, v = (tt ), (vv ) instead of t, v t, v = (tt ), (v( t, v ))

88 Need to prove [A B] = [A] [B] to have a model interpretation. Usually (semantic cut elim), only: A B [A] [B] [A B]

89 Need to prove [A B] = [A] [B] to have a model interpretation. Usually (semantic cut elim), only: A B [A] [B] [A B] proof resembles the proof for normalization.

90 Cut admissibility Assume Γ A has a proof (with cuts) [Γ] [A] in D by (usual) soundness Γ [Γ] Γ [A] implies Γ cf A Q.E.D.

91 Cut admissibility Assume Γ A has a proof (with cuts) [Γ] [A] in D by (usual) soundness Γ [Γ] Γ [A] implies Γ cf A Q.E.D. compared to the former main lemma: Γ π : A implies π A, and hence π is SN.

92 Γ A Gentzen Tait-Girard Dowek-Werner... Γ cf A soundness strong completeness Γ = A This diagram does not commute in deduction modulo.

93 Further work what is the computational content of this algorithm? there is normalization by evaluation work, but in a Kripke style: links? do the proof terms (candidates) always have a pseudo- structure? realizing rewrite rule not with λx.x (not silently), could recover (some) normalization and make the previous diagram commute again. Γ π : A Γ π : B A B

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