denis kuperberg laureline pinault damien pousperso.ens-lyon.fr/laureline.pinault/slides.pdf ·...
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Cyclic Proofs and jumping automata
Denis Kuperberg Laureline Pinault Damien Pous
LIP, ENS Lyon
Journées 2019 du GT ScalpLyon
Friday 18th October 2019
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Cyclic proofs
Regular expressions
e, f := 1 | a ∈ A | e · f | e+ f | e∗
Context: Cyclic proofs for inclusion of expressions [Das, Pous ’17]
Infinite proof trees, with root of the form e ` f .
(Ax)1 ` 1
1 ` a∗(Ax)
a ` a a∗ ` a∗a, a∗ ` a∗
a∗ ` a∗
Validity condition on infinite branches
∃ proof of e ` f ⇔ L(e) ⊆ L(f).
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 2 / 12
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Cyclic proofs
Regular expressions
e, f := 1 | a ∈ A | e · f | e+ f | e∗
Context: Cyclic proofs for inclusion of expressions [Das, Pous ’17]
Infinite proof trees, with root of the form e ` f .
(Ax)1 ` 1
1 ` a∗(Ax)
a ` a a∗ ` a∗a, a∗ ` a∗
a∗ ` a∗
Validity condition on infinite branches
∃ proof of e ` f ⇔ L(e) ⊆ L(f).
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 2 / 12
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Cyclic proofs
Regular expressions
e, f := 1 | a ∈ A | e · f | e+ f | e∗
Context: Cyclic proofs for inclusion of expressions [Das, Pous ’17]
Infinite proof trees, with root of the form e ` f .
(Ax)1 ` 1
1 ` a∗(Ax)
a ` a a∗ ` a∗a, a∗ ` a∗
a∗ ` a∗
Validity condition on infinite branches
∃ proof of e ` f ⇔ L(e) ⊆ L(f).
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 2 / 12
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Computational interpretation
Proof of e ` f
Program with input from e and output in f .
Computational interpretation
Several proofs of the same statementm
Several programs of the same type
Example:a ` a+ ainl or inr
Curry-Howard isomorphism, typed programming,. . .Well-understood for finite proofs, active field for infinite proofs.
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 3 / 12
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Computational interpretation
Proof of e ` f
Program with input from e and output in f .
Computational interpretation
Several proofs of the same statementm
Several programs of the same type
Example:a ` a+ ainl or inr
Curry-Howard isomorphism, typed programming,. . .Well-understood for finite proofs, active field for infinite proofs.
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 3 / 12
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Computational interpretation
Proof of e ` f
Program with input from e and output in f .
Computational interpretation
Several proofs of the same statementm
Several programs of the same type
Example:a ` a+ ainl or inr
Curry-Howard isomorphism, typed programming,. . .Well-understood for finite proofs, active field for infinite proofs.
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 3 / 12
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This work
Boolean type 2 = 1 + 1
Add structural rules corresponding to simple natural programs
Study the expressive power of regular proofs (finite graphs)
Focus on proofs for languages:
Proof π of A∗ ` 2 Language L(π) ⊆ A∗
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 4 / 12
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Proof system
Expressions e := A | A∗
Sequents E,F = e1, e2, . . . , en
Proof system with extra rules for basic data manipulation:
(tt)` 2
(ff)` 2
E,F ` 2(wkn)
E, e, F ` 2
E, e, e, F ` 2(ctr)
E, e, F ` 2
(E,F ` 2)a∈A(A)
E,A, F ` 2
E,F ` 2 E,A,A∗, F ` 2(∗)
E,A∗, F ` 2
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 5 / 12
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Proofs as language acceptors
What are the languages computed by cyclic proofs ?
Example on alphabet {a, b}: b∗
(tt)` 2
(ff)` 2
(wkn)(A∗ ` 2)a (A∗ ` 2)b
(A)A,A∗ ` 2
(∗)A∗ ` 2
LemmaWithout contraction, the system captures exactly regular languages.
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 6 / 12
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Proofs as language acceptors
What are the languages computed by cyclic proofs ?
Example on alphabet {a, b}: b∗
(tt)` 2
(ff)` 2
(wkn)(A∗ ` 2)a (A∗ ` 2)b
(A)A,A∗ ` 2
(∗)A∗ ` 2
LemmaWithout contraction, the system captures exactly regular languages.
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 6 / 12
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With contractions: what class of language?
Example on alphabet {a, b}: anbn
(tt)` 2
(ff)` 2
(wkn)A,A∗ ` 2
(∗)A∗ ` 2
(A∗, A∗ ` 2)a
(ff)` 2
(wkn)A∗ ` 2 A,A∗, A∗ ` 2
(∗)(A∗, A∗ ` 2)b
(A)A∗, A,A∗ ` 2
(∗)A∗, A∗ ` 2
(ctr)A∗ ` 2
1st step : create a copy of the input and delete the first a’s.
Example on alphabet {a, b}: anbncn
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 7 / 12
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With contractions: what class of language?
Example on alphabet {a, b}: anbn
(tt)` 2
(ff)` 2
(wkn)A,A∗ ` 2
(∗)A∗ ` 2
(A∗, A∗ ` 2)a
(ff)` 2
(wkn)A∗ ` 2 A,A∗, A∗ ` 2
(∗)(A∗, A∗ ` 2)b
(A)A∗, A,A∗ ` 2
(∗)A∗, A∗ ` 2
(ctr)A∗ ` 2
1st step : create a copy of the input and delete the first a’s.
Example on alphabet {a, b}: anbncn
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 7 / 12
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With contractions: what class of language?
Example on alphabet {a, b}: anbn
(tt)` 2
(ff)` 2
(wkn)A,A∗ ` 2
(∗)A∗ ` 2
(A∗, A∗ ` 2)a
(ff)` 2
(wkn)A∗ ` 2 A,A∗, A∗ ` 2
(∗)(A∗, A∗ ` 2)b
(A)A∗, A,A∗ ` 2
(∗)A∗, A∗ ` 2
(ctr)A∗ ` 2
1st step : create a copy of the input and delete the first a’s.
Example on alphabet {a, b}: anbncn
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 7 / 12
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With contractions: what class of language?
Example on alphabet {a, b}: anbn
...
A∗ ` 2
(ff)` 2
(wkn)(A∗, A∗ ` 2)a (A∗, A∗ ` 2)b
(A)A∗, A,A∗ ` 2
(∗)(A∗, A∗ ` 2)a
(ff)` 2
(wkn)(A∗, A∗ ` 2)b
(A)A,A∗, A∗ ` 2
(∗)(A∗, A∗ ` 2)b
2nd step : check that for each b of the second copy we have a a in the firstone.
Example on alphabet {a, b}: anbncn
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 7 / 12
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With contractions: what class of language?
Example on alphabet {a, b}: anbn
...
A∗ ` 2
(ff)` 2
(wkn)(A∗, A∗ ` 2)a (A∗, A∗ ` 2)b
(A)A∗, A,A∗ ` 2
(∗)(A∗, A∗ ` 2)a
(ff)` 2
(wkn)(A∗, A∗ ` 2)b
(A)A,A∗, A∗ ` 2
(∗)(A∗, A∗ ` 2)b
2nd step : check that for each b of the second copy we have a a in the firstone.
Example on alphabet {a, b}: anbncn
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 7 / 12
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With contractions: what class of language?
Example on alphabet {a, b}: anbn
...
A∗ ` 2
(ff)` 2
(wkn)(A∗, A∗ ` 2)a (A∗, A∗ ` 2)b
(A)A∗, A,A∗ ` 2
(∗)(A∗, A∗ ` 2)a
(ff)` 2
(wkn)(A∗, A∗ ` 2)b
(A)A,A∗, A∗ ` 2
(∗)(A∗, A∗ ` 2)b
2nd step : check that for each b of the second copy we have a a in the firstone.
Example on alphabet {a, b}: anbncn
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 7 / 12
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With contractions: what class of language?
Example on alphabet {a, b}: anbn
...
A∗ ` 2
(ff)` 2
(wkn)(A∗, A∗ ` 2)a (A∗, A∗ ` 2)b
(A)A∗, A,A∗ ` 2
(∗)(A∗, A∗ ` 2)a
(ff)` 2
(wkn)(A∗, A∗ ` 2)b
(A)A,A∗, A∗ ` 2
(∗)(A∗, A∗ ` 2)b
2nd step : check that for each b of the second copy we have a a in the firstone.
(ff)` 2
(ff)` 2
(wkn)(A∗ ` 2)a
(tt)` 2
(wkn)(A∗ ` 2)b
(A)A,A∗ ` 2
(∗)A∗ ` 2
3rd step : checking that we have no more a’s
Example on alphabet {a, b}: anbncn
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With contractions: what class of language?
Example on alphabet {a, b}: anbn
(ff)` 2
(ff)` 2
(wkn)(A∗ ` 2)a
(tt)` 2
(wkn)(A∗ ` 2)b
(A)A,A∗ ` 2
(∗)A∗ ` 2
3rd step : checking that we have no more a’s
Example on alphabet {a, b}: anbncn
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 7 / 12
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With contractions: what class of language?
Example on alphabet {a, b}: anbn
(ff)` 2
(ff)` 2
(wkn)(A∗ ` 2)a
(tt)` 2
(wkn)(A∗ ` 2)b
(A)A,A∗ ` 2
(∗)A∗ ` 2
3rd step : checking that we have no more a’s
Example on alphabet {a, b}: anbncn
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 7 / 12
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With contractions: a new automaton model
Jumping Multihead Automata
A JMA is an automaton with k reading heads.
Transitions: Q×Ak → Q× {�,�, J1, . . . , Jk}k
a a b a a b b a
q
12 3
� : advance one step�: stay in placeJi: jump to the position of head i
+ Equivalent of the validity criterion
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 8 / 12
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With contractions: a new automaton model
Jumping Multihead Automata
A JMA is an automaton with k reading heads.
Transitions: Q×Ak → Q× {�,�, J1, . . . , Jk}k
a a b a a b b a
q
12 3
� : advance one step�: stay in placeJi: jump to the position of head i
+ Equivalent of the validity criterion
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 8 / 12
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Example of JMA
Example: {a2n | n ∈ N} is accepted by a 2-head JMA.
a a a a a a a a
ACCEPT
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 9 / 12
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Example of JMA
Example: {a2n | n ∈ N} is accepted by a 2-head JMA.
a a a a a a a a
ACCEPT
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 9 / 12
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Example of JMA
Example: {a2n | n ∈ N} is accepted by a 2-head JMA.
a a a a a a a a
ACCEPT
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 9 / 12
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Example of JMA
Example: {a2n | n ∈ N} is accepted by a 2-head JMA.
a a a a a a a a
ACCEPT
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 9 / 12
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Example of JMA
Example: {a2n | n ∈ N} is accepted by a 2-head JMA.
a a a a a a a a
ACCEPT
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 9 / 12
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Example of JMA
Example: {a2n | n ∈ N} is accepted by a 2-head JMA.
a a a a a a a a
ACCEPT
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 9 / 12
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Example of JMA
Example: {a2n | n ∈ N} is accepted by a 2-head JMA.
a a a a a a a a
ACCEPT
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 9 / 12
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Example of JMA
Example: {a2n | n ∈ N} is accepted by a 2-head JMA.
a a a a a a a a
ACCEPT
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 9 / 12
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Example of JMA
Example: {a2n | n ∈ N} is accepted by a 2-head JMA.
a a a a a a a a
ACCEPT
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 9 / 12
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Example of JMA
Example: {a2n | n ∈ N} is accepted by a 2-head JMA.
a a a a a a a a
ACCEPT
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 9 / 12
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Example of JMA
Example: {a2n | n ∈ N} is accepted by a 2-head JMA.
a a a a a a a a
ACCEPT
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 9 / 12
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Example of JMA
Example: {a2n | n ∈ N} is accepted by a 2-head JMA.
a a a a a a a a
ACCEPT
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Example of JMA
Example: {a2n | n ∈ N} is accepted by a 2-head JMA.
a a a a a a a a
ACCEPT
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Example of JMA
Example: {a2n | n ∈ N} is accepted by a 2-head JMA.
a a a a a a a a
ACCEPT
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 9 / 12
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Example of JMA
Example: {a2n | n ∈ N} is accepted by a 2-head JMA.
a a a a a a a a
ACCEPT
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 9 / 12
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Example of JMA
Example: {a2n | n ∈ N} is accepted by a 2-head JMA.
a a a a a a a a
ACCEPT
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 9 / 12
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Example of JMA
Example: {a2n | n ∈ N} is accepted by a 2-head JMA.
a a a a a a a a
ACCEPT
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 9 / 12
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Example of JMA
Example: {a2n | n ∈ N} is accepted by a 2-head JMA.
a a a a a a a a
ACCEPT
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 9 / 12
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Expressive power of JMA
Comparison with Multihead Automata in Litterature:[Holzer, Kutrib, Malcher 2008]
1-way Multihead
⊆ JMA ⊆
2-way Multihead
Emptiness Undecidable DLogSpace
∀k, JMA(2) 6⊆ 1DFA(k) ?=
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 10 / 12
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Expressive power of JMA
Comparison with Multihead Automata in Litterature:[Holzer, Kutrib, Malcher 2008]
1-way Multihead ⊆ JMA ⊆ 2-way Multihead
Emptiness Undecidable DLogSpace
∀k, JMA(2) 6⊆ 1DFA(k) ?=
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 10 / 12
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Expressive power of JMA
Comparison with Multihead Automata in Litterature:[Holzer, Kutrib, Malcher 2008]
1-way Multihead ⊆ JMA ⊆ 2-way Multihead
Emptiness Undecidable DLogSpace
∀k, JMA(2) 6⊆ 1DFA(k) ?=
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 10 / 12
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Expressive power of JMA
Comparison with Multihead Automata in Litterature:[Holzer, Kutrib, Malcher 2008]
1-way Multihead ⊆ JMA ⊆ 2-way Multihead
Emptiness Undecidable DLogSpace
∀k, JMA(2) 6⊆ 1DFA(k)
?=
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 10 / 12
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Expressive power of JMA
Comparison with Multihead Automata in Litterature:[Holzer, Kutrib, Malcher 2008]
1-way Multihead ⊆ JMA ⊆ 2-way Multihead
Emptiness Undecidable DLogSpace
∀k, JMA(2) 6⊆ 1DFA(k) ?=
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 10 / 12
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Main result
TheoremCyclic proofs and JMA recognize the same class of languages.
States of the automaton ∼ Positions in the proof tree
Accepting / Rejecting state ∼ True / False axiom
Multiple heads ∼ Multiple copies of A∗
Reading a letter ∼ Applying ∗ and (A) rules
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What next ?
Add the cut rule
Corresponds to composition of functions
Sequents (1∗)k ` 1∗: functions Nk → N
Work in progress:
No contraction = Primitive Recursive
With contraction = System T
Thank you for your attention !
[Denis Kuperberg, Laureline Pinault and Damien Pous, FSTTCS 19]
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What next ?
Add the cut rule
Corresponds to composition of functions
Sequents (1∗)k ` 1∗: functions Nk → N
Work in progress:
No contraction = Primitive Recursive
With contraction = System T
Thank you for your attention !
[Denis Kuperberg, Laureline Pinault and Damien Pous, FSTTCS 19]
L. Pinault Cyclic Proofs and jumping automata Friday 18th October 2019 12 / 12