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Transductions in affjne logic

Nguyễn Lê Thành Dũng (a.k.a. Tito) — nltd@nguyentito.eu LIPN, Université Paris 13 Trends in Linear Logic and Applications, Dortmund, June 30th, 2019

1/18

Transductions in typed λ-calculi

• 1. Motivation: the simply-typed λ-calculus

functions on Church encodings and automata theory

• Claim:

λ-defjnable string functions are related to transducers, i.e. automata with output (many possible variants).

• 2. Capturing a class of transductions with affjne typing,

via linearity in streaming string transducers Details on (2) in my 5-page abstract. I’ve come to think that (1) is more important. But (2) came fjrst, triggered by a question by M. Bojańczyk. Counterexample to “provinciality” of linear logic.

2/18

Transductions in typed λ-calculi

• 1. Motivation: the simply-typed λ-calculus

functions on Church encodings and automata theory

• Claim:

λ-defjnable string functions are related to transducers, i.e. automata with output (many possible variants).

• 2. Capturing a class of transductions with affjne typing,

via linearity in streaming string transducers Details on (2) in my 5-page abstract. I’ve come to think that (1) is more important. But (2) came fjrst, triggered by a question by M. Bojańczyk. − → Counterexample to “provinciality” of linear logic.

2/18

Simply typed functions on Church numerals (1)

Church encodings of (unary) natural numbers:

• Nat = o → (o → o) → o
• n ∈ N ⇝ n = λx. λf. f (. . . (f x) . . .) : Nat with n times f

For t : Nat → Nat, F(t)(n) = m ⇐ ⇒ t n =β m. Theorem (Schwichtenberg 1975) The functions defjnable by simply-typed λ-terms of type Nat → Nat are the extended polynomials (generated by 0,1,+,×,ifzero). Let’s add a bit of (meta-level) polymorphism: t Nat A Nat where Nat A Nat A o A A A A Open question t A simple type t Nat A Nat

3/18

Simply typed functions on Church numerals (1)

Church encodings of (unary) natural numbers:

• Nat = o → (o → o) → o
• n ∈ N ⇝ n = λx. λf. f (. . . (f x) . . .) : Nat with n times f

For t : Nat → Nat, F(t)(n) = m ⇐ ⇒ t n =β m. Theorem (Schwichtenberg 1975) The functions defjnable by simply-typed λ-terms of type Nat → Nat are the extended polynomials (generated by 0,1,+,×,ifzero). Let’s add a bit of (meta-level) polymorphism: t = Nat[A] → Nat where Nat[A] = Nat[A/o] = A → (A → A) → A Open question {F(t) | A simple type, t : Nat[A] → Nat} = ?

3/18

Simply typed functions on Church numerals (2)

Take mult = λn.λm.λf. n (m f) : Nat → Nat → Nat. mult 2 : Nat → Nat can be iterated by a Nat[Nat]… − → exp2 = λn. n (mult 2) 1 : Nat[Nat] → Nat which cannot be iterated! Smaller types, still heterogenous: exp2 n n 2 Nat o

• Nat

Towers of exponentials of any fjxed height Nat T A Nat A . This is the fastest possible growth for simply typed

• terms.

4/18

Simply typed functions on Church numerals (2)

Take mult = λn.λm.λf. n (m f) : Nat → Nat → Nat. mult 2 : Nat → Nat can be iterated by a Nat[Nat]… − → exp2 = λn. n (mult 2) 1 : Nat[Nat] → Nat which cannot be iterated! Smaller types, still heterogenous: exp2 = λn. n 2 : Nat[o → o] → Nat Towers of exponentials of any fjxed height Nat T A Nat A . This is the fastest possible growth for simply typed

• terms.

4/18

Simply typed functions on Church numerals (2)

Take mult = λn.λm.λf. n (m f) : Nat → Nat → Nat. mult 2 : Nat → Nat can be iterated by a Nat[Nat]… − → exp2 = λn. n (mult 2) 1 : Nat[Nat] → Nat which cannot be iterated! Smaller types, still heterogenous: exp2 = λn. n 2 : Nat[o → o] → Nat Towers of exponentials of any fjxed height Nat[T[A]] → Nat[A]. This is the fastest possible growth for simply typed λ-terms.

4/18

Simply typed functions on Church numerals (3)

Towers of exponentials of any fjxed height Nat[T[A]] → Nat[A]. On the other hand: Theorem (Statman 198?) Subtraction cannot be defjned as a simply typed λ-term of type Nat[A] → Nat[B] → Nat. More weirdness: some “easy” 1-variable functions, e.g. n → ⌊√n⌋, are also undefjnable. Does this really correspond to any interesting class, or is

• ur open problem a “typical misguided natural question”?

(cf. Gromov, Spaces and questions)

Might explain lack of progress on this question…

5/18

Simply typed functions on Church numerals (3)

Towers of exponentials of any fjxed height Nat[T[A]] → Nat[A]. On the other hand: Theorem (Statman 198?) Subtraction cannot be defjned as a simply typed λ-term of type Nat[A] → Nat[B] → Nat. More weirdness: some “easy” 1-variable functions, e.g. n → ⌊√n⌋, are also undefjnable. − → Does this really correspond to any interesting class, or is

• ur open problem a “typical misguided natural question”?

(cf. Gromov, Spaces and questions)

Might explain lack of progress on this question…

5/18

A fjrst interesting result: predicates on Church integers

My POV: the question is actually interesting. Illustration on restricted case: predicates (Bool = o → o → o). Theorem (Joly 2001) A subset of N is decidable by some t : Nat[A] → Bool if and only if it is ultimately periodic. Corollary For all simple types A and all t Nat A Nat, X ultimately periodic t

1 X ultimately periodic.

A not quite trivial necessary condition!

6/18

A fjrst interesting result: predicates on Church integers

My POV: the question is actually interesting. Illustration on restricted case: predicates (Bool = o → o → o). Theorem (Joly 2001) A subset of N is decidable by some t : Nat[A] → Bool if and only if it is ultimately periodic. Corollary For all simple types A and all t : Nat[A] → Nat, X ⊆ N ultimately periodic = ⇒ F(t)−1(X) ultimately periodic. A not quite trivial necessary condition!

6/18

Simply typed functions on Church-encoded strings

To gain more insight, let’s generalize! Nat = Str{1} Church encodings of strings over alphabet Σ = {a1, . . . , a|Σ|}:

• StrΣ = o → (o → o) → . . . |Σ| times . . . → (o → o) → o
• w ∈ Σ∗ ⇝ w = λx. λf1. . . . λf|Σ|. fi1 (. . . (fin x) . . .) : StrΣ

where w = ai1 . . . ain Characterizations of {F(t) | t : StrΓ → StrΣ} (no subst.) exist1. Open question {F(t) | A simple type, t : StrΓ[A] → StrΣ} = ?

1e.g. Zaionc 1987, Word operation defjnable in the typed λ-calculus

7/18

Simply typed predicates on Church-encoded strings

Theorem (Hillebrand & Kanellakis 1995) A subset of Σ∗ is decidable by some t : StrΣ[A] → Bool ifg it is a regular language. (Note: X ⊆ {1}∗ regular ⇐ ⇒ X ⊆ N ultimately periodic) Corollary L ⊆ Σ∗ regular = ⇒ F(t)−1(L) regular for t : StrΓ[A] → StrΣ. − → look for an automata-theoretic function class with regularity preservation + hyperexponential growth

8/18

Register transducers

For input alphabet Γ, output alphabet Σ:

• Finite set of Σ∗-valued registers e.g. R = {X, Y}
• Initial values R → Σ∗ e.g. Xinit = Yinit = ε
• Register update function u : Γ → (R → (Σ ∪ R)∗) e.g.

a →    X → Xa Y → aY b →    X → Xb Y → bY

• “output function” ∈ (Σ ∪ R)∗ e.g. out = XY

Execution over : start with X Y

9/18

Register transducers

For input alphabet Γ, output alphabet Σ:

• Finite set of Σ∗-valued registers e.g. R = {X, Y}
• Initial values R → Σ∗ e.g. Xinit = Yinit = ε
• Register update function u : Γ → (R → (Σ ∪ R)∗) e.g.

a →    X := Xa Y := aY b →    X := Xb Y := bY

• “output function” ∈ (Σ ∪ R)∗ e.g. out = XY

Execution over : start with X Y

9/18

Register transducers

For input alphabet Γ, output alphabet Σ:

• Finite set of Σ∗-valued registers e.g. R = {X, Y}
• Initial values R → Σ∗ e.g. Xinit = Yinit = ε
• Register update function u : Γ → (R → (Σ ∪ R)∗) e.g.

a →    X := Xa Y := aY b →    X := Xb Y := bY

• “output function” ∈ (Σ ∪ R)∗ e.g. out = XY

Execution over abaa: start with X = ε Y = ε

9/18

Register transducers

For input alphabet Γ, output alphabet Σ:

• Finite set of Σ∗-valued registers e.g. R = {X, Y}
• Initial values R → Σ∗ e.g. Xinit = Yinit = ε
• Register update function u : Γ → (R → (Σ ∪ R)∗) e.g.

a →    X := Xa Y := aY b →    X := Xb Y := bY

• “output function” ∈ (Σ ∪ R)∗ e.g. out = XY

Execution over abaa: X = a Y = a

9/18

Register transducers

For input alphabet Γ, output alphabet Σ:

• Finite set of Σ∗-valued registers e.g. R = {X, Y}
• Initial values R → Σ∗ e.g. Xinit = Yinit = ε
• Register update function u : Γ → (R → (Σ ∪ R)∗) e.g.

a →    X := Xa Y := aY b →    X := Xb Y := bY

• “output function” ∈ (Σ ∪ R)∗ e.g. out = XY

Execution over abaa: X = ab Y = ba

9/18

Register transducers

For input alphabet Γ, output alphabet Σ:

• Finite set of Σ∗-valued registers e.g. R = {X, Y}
• Initial values R → Σ∗ e.g. Xinit = Yinit = ε
• Register update function u : Γ → (R → (Σ ∪ R)∗) e.g.

a →    X := Xa Y := aY b →    X := Xb Y := bY

• “output function” ∈ (Σ ∪ R)∗ e.g. out = XY

Execution over abaa: X = aba Y = aba

9/18

Register transducers

For input alphabet Γ, output alphabet Σ:

• Finite set of Σ∗-valued registers e.g. R = {X, Y}
• Initial values R → Σ∗ e.g. Xinit = Yinit = ε
• Register update function u : Γ → (R → (Σ ∪ R)∗) e.g.

a →    X := Xa Y := aY b →    X := Xb Y := bY

• “output function” ∈ (Σ ∪ R)∗ e.g. out = XY

Execution over abaa: X = abaa Y = aaba

9/18

Register transducers

For input alphabet Γ, output alphabet Σ:

• Finite set of Σ∗-valued registers e.g. R = {X, Y}
• Initial values R → Σ∗ e.g. Xinit = Yinit = ε
• Register update function u : Γ → (R → (Σ ∪ R)∗) e.g.

a →    X := Xa Y := aY b →    X := Xb Y := bY

• “output function” ∈ (Σ ∪ R)∗ e.g. out = XY

Execution over abaa: f(abaa) = abaaaaba X = abaa Y = aaba

9/18

Register transducers

For input alphabet Γ, output alphabet Σ:

• Finite set of Σ∗-valued registers e.g. R = {X, Y}
• Initial values R → Σ∗ e.g. Xinit = Yinit = ε
• Register update function u : Γ → (R → (Σ ∪ R)∗) e.g.

a →    X := Xa Y := aY b →    X := Xb Y := bY

• “output function” ∈ (Σ ∪ R)∗ e.g. out = XY

Execution over abaa: f(abaa) = abaaaaba, f : w → w · reverse(w) X = abaa Y = aaba

9/18

Register transducers with states

• Finite set of Σ∗-valued registers R, initial values R → Σ∗
• fjnite state machine (i.e. DFA) (Q, qinit ∈ Q, δ : Γ × Q → Q)
• updates Γ × Q → (Σ ∪ R)∗, “output function” Q → (Σ ∪ R)∗

Theorem (Filiot & Reynier 2017) Let f . The following are equivalent:

• f can be computed by a register transducer
• f can be computed by a register transducer with states
• f can be specifjed by a HDT0L system

(variant of Lindenmayer systems from the 1960s) Furthermore, L regular f

1 L regular for such f. 10/18

Register transducers with states

• Finite set of Σ∗-valued registers R, initial values R → Σ∗
• fjnite state machine (i.e. DFA) (Q, qinit ∈ Q, δ : Γ × Q → Q)
• updates Γ × Q → (Σ ∪ R)∗, “output function” Q → (Σ ∪ R)∗

Theorem (Filiot & Reynier 2017) Let f : Γ∗ → Σ∗. The following are equivalent:

• f can be computed by a register transducer
• f can be computed by a register transducer with states
• f can be specifjed by a HDT0L system

(variant of Lindenmayer systems from the 1960s) Furthermore, L regular = ⇒ f−1(L) regular for such f.

10/18

Exponential growth of HDT0L transductions

Γ = Σ = {1} R = {X} Xinit = 1 X := XX = ⇒ f : Γ∗ → Σ∗ such that |f(w)| = 2|w| Proposition (Exponential upper bound) Conversely, for any HDT0L f , f w 2O w . Corollary HDT0L transductions are not closed under composition. Their closure under comp. has elementary growth. HDT0L+composition all transduction classes that I know

(to be fair, I don’t know many)

See also: Sénizergues 2007, Sequences of level 1,2,…,k,…

11/18

Exponential growth of HDT0L transductions

Γ = Σ = {1} R = {X} Xinit = 1 X := XX = ⇒ f : Γ∗ → Σ∗ such that |f(w)| = 2|w| Proposition (Exponential upper bound) Conversely, for any HDT0L f : Γ∗ → Σ∗, |f(w)| = 2O(|w|). Corollary HDT0L transductions are not closed under composition. Their closure under comp. has elementary growth. HDT0L+composition all transduction classes that I know

(to be fair, I don’t know many)

See also: Sénizergues 2007, Sequences of level 1,2,…,k,…

11/18

Exponential growth of HDT0L transductions

Γ = Σ = {1} R = {X} Xinit = 1 X := XX = ⇒ f : Γ∗ → Σ∗ such that |f(w)| = 2|w| Proposition (Exponential upper bound) Conversely, for any HDT0L f : Γ∗ → Σ∗, |f(w)| = 2O(|w|). Corollary HDT0L transductions are not closed under composition. Their closure under comp. has elementary growth. HDT0L+composition ⊃ all transduction classes that I know

(to be fair, I don’t know many)

See also: Sénizergues 2007, Sequences of level 1,2,…,k,…

11/18

Returning to λ-defjnable functions

Theorem The class of functions defjned by simply typed λ-terms of type StrΓ[A] → StrΣ contains HDT0L+composition. Proof. It is closed under composition and can encode register transducers (programming exercise). ST subsumes many classes of regularity-preserving functions! Open question (concrete proposal) What about the converse? That is, ST -defjnable = HDT0L+composition? Possible connections with iterated pushdown automata.

12/18

Returning to λ-defjnable functions

Theorem The class of functions defjned by simply typed λ-terms of type StrΓ[A] → StrΣ contains HDT0L+composition. Proof. It is closed under composition and can encode register transducers (programming exercise). STλ subsumes many classes of regularity-preserving functions! Open question (concrete proposal) What about the converse? That is, STλ-defjnable = HDT0L+composition? Possible connections with iterated pushdown automata.

12/18

Linearity in register transducers

X := XX is a non-linear register update − → as usual, duplication causes complexity (2O(n) growth) Whereas affjne updates lead to O(n) growth, e.g. |f(w)| = 2|w| for f : w → w · reverse(w)    X := Xa Y := aY ↭ x ⊗ y → xa ⊗ ay (In the automata theory literature: “copyless assignments”.) Defjnition A streaming string transducer (SST) is a register transducer with states and with affjne register updates.

13/18

Linearity in register transducers

X := XX is a non-linear register update − → as usual, duplication causes complexity (2O(n) growth) Whereas affjne updates lead to O(n) growth, e.g. |f(w)| = 2|w| for f : w → w · reverse(w)    X := Xa Y := aY ↭ x ⊗ y → xa ⊗ ay (In the automata theory literature: “copyless assignments”.) Defjnition A streaming string transducer (SST) is a register transducer with states and with affjne register updates.

13/18

Regular functions

Defjnition A streaming string transducer (SST) is a register transducer with states and with affjne register updates. Theorem (Alur & Černý 2010) The functions computed by SSTs are the regular functions. Closed under composition, linear growth, f−1(reg. lang) is reg. Many other characterizations:

• two-way fjnite state transducers without registers
• Monadic Second-Order Logic
• basic functions + combinators (at least 3 possible variants)
• etc.

14/18

Inspiration from implicit complexity in linear logic

Goal: regular functions in an affjne programming language. We use the elementary affjne λ-calculus (EAλ). Theorem (Baillot, De Benedetti & Ronchi Della Rocca 2014) The languages defjned by EAλ-terms with type fjxpoints of type !StrΣ ⊸ !!Bool are exactly those decidable in polynomial time. Where StrΣ = ∀α. (!(α ⊸ α))|Σ| ⊸ !(α ⊸ α) Theorem (my talk at TLLA 2018!) The languages defjned by EA -terms of type Str Bool are exactly the regular languages over . Good news: Str Str regularity-preserving functions.

15/18

Inspiration from implicit complexity in linear logic

Goal: regular functions in an affjne programming language. We use the elementary affjne λ-calculus (EAλ). Theorem (Baillot, De Benedetti & Ronchi Della Rocca 2014) The languages defjned by EAλ-terms with type fjxpoints of type !StrΣ ⊸ !!Bool are exactly those decidable in polynomial time. Where StrΣ = ∀α. (!(α ⊸ α))|Σ| ⊸ !(α ⊸ α) Theorem (my talk at TLLA 2018!) The languages defjned by EAλ-terms of type !StrΣ ⊸ !!Bool are exactly the regular languages over Σ. Good news: !StrΓ ⊸ !StrΣ ⇝ regularity-preserving functions.

15/18

A characterization of regular functions

Good news: !StrΓ ⊸ !StrΣ ⇝ regularity-preserving functions. Bad news: they have super-linear growth (Θ(nk) for any k). Idea: drop the exponentials on the Str. Theorem f regular f defjned by EA term t Str Str . Proof of . Encode the streaming string transducer in EA . Proof of (with Pistone, Seiller & Tortora de Falco). Much more diffjcult: we need to construct a kind of “fjnitely parameterized semantics” of 2nd-order affjne MLL (+ constants) via combinatorial study of proof nets

(looks like concurrent games?)

16/18

A characterization of regular functions

Good news: !StrΓ ⊸ !StrΣ ⇝ regularity-preserving functions. Bad news: they have super-linear growth (Θ(nk) for any k). Idea: drop the exponentials on the Str. Theorem f : Γ∗ → Σ∗ regular ⇐ ⇒ f defjned by EAλ term t : StrΓ ⊸ StrΣ. Proof of . Encode the streaming string transducer in EA . Proof of (with Pistone, Seiller & Tortora de Falco). Much more diffjcult: we need to construct a kind of “fjnitely parameterized semantics” of 2nd-order affjne MLL (+ constants) via combinatorial study of proof nets

(looks like concurrent games?)

16/18

A characterization of regular functions

Good news: !StrΓ ⊸ !StrΣ ⇝ regularity-preserving functions. Bad news: they have super-linear growth (Θ(nk) for any k). Idea: drop the exponentials on the Str. Theorem f : Γ∗ → Σ∗ regular ⇐ ⇒ f defjned by EAλ term t : StrΓ ⊸ StrΣ. Proof of ( = ⇒ ). Encode the streaming string transducer in EAλ. Proof of (⇐ =) (with Pistone, Seiller & Tortora de Falco). Much more diffjcult: we need to construct a kind of “fjnitely parameterized semantics” of 2nd-order affjne MLL (+ constants) via combinatorial study of proof nets

(looks like concurrent games?)

16/18

Relationship with implicit complexity in linear logic, again

Theorem (Simplifjcation of a result by Baillot et al. 2014) The functions defjned by EAλ-terms with type fjxpoints of type !StrΓ ⊸ !StrΣ are exactly those computable in polynomial time. So, what about !StrΓ ⊸ !StrΣ in EAλ w/o type fjxpoints? Some class of functions that must be regularity-preserving… regular this polyregular functions (Bojańczyk 2018)

(last inclusion probably strict; also, polyregular HDT0L+composition)

“Slogan: Polyregular functions are the polynomial growth fjnite state transducers.” Unpublished paper (arXiv:1810.08760).

17/18

Relationship with implicit complexity in linear logic, again

Theorem (Simplifjcation of a result by Baillot et al. 2014) The functions defjned by EAλ-terms with type fjxpoints of type !StrΓ ⊸ !StrΣ are exactly those computable in polynomial time. So, what about !StrΓ ⊸ !StrΣ in EAλ w/o type fjxpoints? Some class of functions that must be regularity-preserving… regular ⊊ this ⊆ polyregular functions (Bojańczyk 2018)

(last inclusion probably strict; also, polyregular ⊊ HDT0L+composition)

“Slogan: Polyregular functions are the polynomial growth fjnite state transducers.” Unpublished paper (arXiv:1810.08760).

17/18

Conclusion

This is only the start! Our concrete results for now:

• register transducers (HDT0L) + composition

can be encoded in simply typed λ-calculus (STλ)

• regular functions can be characterized

in the elementary affjne λ-calculus

• not mentioned here: tree transducers vs additive ‘&’

Why this might be interesting:

• reviving an old question on ST
• recent surge of interest in transductions with super-linear

growth (HDT0L systems, polyregular functions)

• ST

gives the largest (?) such “natural” class known

Paper: https://nguyentito.eu/tmp/transducer-p1.pdf

18/18

Conclusion

This is only the start! Our concrete results for now:

• register transducers (HDT0L) + composition

can be encoded in simply typed λ-calculus (STλ)

• regular functions can be characterized

in the elementary affjne λ-calculus

• not mentioned here: tree transducers vs additive ‘&’

Why this might be interesting:

• reviving an old question on STλ
• recent surge of interest in transductions with super-linear

growth (HDT0L systems, polyregular functions)

• STλ gives the largest (?) such “natural” class known

Paper: https://nguyentito.eu/tmp/transducer-p1.pdf

18/18

Conclusion

This is only the start! Our concrete results for now:

• register transducers (HDT0L) + composition

can be encoded in simply typed λ-calculus (STλ)

• regular functions can be characterized

in the elementary affjne λ-calculus

• not mentioned here: tree transducers vs additive ‘&’

Why this might be interesting:

• reviving an old question on STλ
• recent surge of interest in transductions with super-linear

growth (HDT0L systems, polyregular functions)

• STλ gives the largest (?) such “natural” class known

Paper: https://nguyentito.eu/tmp/transducer-p1.pdf

18/18