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Free-cut elimination in linear logic and an application to a feasible arithmetic

Anupam Das Patrick Baillot

LIP, Universit´ e de Lyon, CNRS, ENS de Lyon, INRIA, Universit´ e Claude-Bernard Lyon 1, Milyon

6th October, 2016 Bologne ELICA meeting

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Outline

Introduction Normal forms in first-order linear logic An arithmetic in linear logic Bellantoni-Cook programs and the WFM for IΣN+

1

Conclusions

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Implicit computational complexity (ICC)

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Implicit computational complexity (ICC)

In a nutshell: ICC studies correspondences between features of logic and complexity classes

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Implicit computational complexity (ICC)

In a nutshell: ICC studies correspondences between features of logic and complexity classes Proof-theoretic approach For a logic or theory, ‘representable’ functions = given complexity class where representability can mean definability, typability etc.

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Implicit computational complexity (ICC)

In a nutshell: ICC studies correspondences between features of logic and complexity classes Proof-theoretic approach For a logic or theory, ‘representable’ functions = given complexity class where representability can mean definability, typability etc. We distinguish the following two methodologies:

1 Theories whose definable functions = given complexity class. 2 Logics that type terms with normalisation complexity of a given class.

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Implicit computational complexity (ICC)

In a nutshell: ICC studies correspondences between features of logic and complexity classes Proof-theoretic approach For a logic or theory, ‘representable’ functions = given complexity class where representability can mean definability, typability etc. We distinguish the following two methodologies:

1 Theories whose definable functions = given complexity class. 2 Logics that type terms with normalisation complexity of a given class.

This work is about the first methodology.

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Provably convergent functions

Correspondence between a theory T and a class C: T ⊢ ∀x.∃y.A(x, y) ⇔ N | = ∀x.A(x, f(x)) for some f ∈ C

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Provably convergent functions

Correspondence between a theory T and a class C: T ⊢ ∀x.∃y.A(x, y) ⇔ N | = ∀x.A(x, f(x)) for some f ∈ C For example:

Theorem (Parsons ’68, Mints ’73, Buss ’95)

IΣ1 proves the totality of precisely the primitive recursive functions.

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Provably convergent functions

Correspondence between a theory T and a class C: T ⊢ ∀x.∃y.A(x, y) ⇔ N | = ∀x.A(x, f(x)) for some f ∈ C For example:

Theorem (Parsons ’68, Mints ’73, Buss ’95)

IΣ1 proves the totality of precisely the primitive recursive functions.

Parsons’ proof.

• Via a Dialectica-style functional interpretation.
• Extracted programs: higher-order variant of primitive recursive functions.

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Provably convergent functions

Correspondence between a theory T and a class C: T ⊢ ∀x.∃y.A(x, y) ⇔ N | = ∀x.A(x, f(x)) for some f ∈ C For example:

Theorem (Parsons ’68, Mints ’73, Buss ’95)

IΣ1 proves the totality of precisely the primitive recursive functions.

Parsons’ proof.

• Via a Dialectica-style functional interpretation.
• Extracted programs: higher-order variant of primitive recursive functions.

Buss’ and Mints’ proof.

• Via the witness function method.
• Extracted programs: regular primitive recursive functions of ground type.

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The witness function method (WFM)

The idea

• A formal witness predicate over N for each ‘tame’ formula.
• Arithmetic proofs functions from witnesses to witnesses:

π

Γ ⊢ ∆

• f π :

witnesses

• f Γ
• →

witnesses

• f ∆
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The witness function method (WFM)

The idea

• A formal witness predicate over N for each ‘tame’ formula.
• Arithmetic proofs functions from witnesses to witnesses:

π

Γ ⊢ ∆

• f π :

witnesses

• f Γ
• →

witnesses

• f ∆
• Crucial points
• π free-cut free: tames the complexity of formulae; no bad ∀.

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The witness function method (WFM)

The idea

• A formal witness predicate over N for each ‘tame’ formula.
• Arithmetic proofs functions from witnesses to witnesses:

π

Γ ⊢ ∆

• f π :

witnesses

• f Γ
• →

witnesses

• f ∆
• Crucial points
• π free-cut free: tames the complexity of formulae; no bad ∀.
• De Morgan normal form: only functions at ground type, i.e. Nk → N.

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The witness function method (WFM)

The idea

• A formal witness predicate over N for each ‘tame’ formula.
• Arithmetic proofs functions from witnesses to witnesses:

π

Γ ⊢ ∆

• f π :

witnesses

• f Γ
• →

witnesses

• f ∆
• Crucial points
• π free-cut free: tames the complexity of formulae; no bad ∀.
• De Morgan normal form: only functions at ground type, i.e. Nk → N.
• Right-contraction: tests the witness predicate (should be decidable).

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Context and motivation

Free-cut elimination

• Used in various forms by Gentzen, Parikh, Paris & Wilkie, Cook,

Kraj´ ıcek,...

• First presented for general fragments of PA by Takeuti.
• Further generalised by Buss and others.

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Context and motivation

Free-cut elimination

• Used in various forms by Gentzen, Parikh, Paris & Wilkie, Cook,

Kraj´ ıcek,...

• First presented for general fragments of PA by Takeuti.
• Further generalised by Buss and others.

Witness function method

• Due to Buss and Mints.
• bounded arithmetic. Theories for NCi, ACi, P, PH,...
• The best method available to delineate hierarchies of classical theories.

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Context and motivation

Free-cut elimination

• Used in various forms by Gentzen, Parikh, Paris & Wilkie, Cook,

Kraj´ ıcek,...

• First presented for general fragments of PA by Takeuti.
• Further generalised by Buss and others.

Witness function method

• Due to Buss and Mints.
• bounded arithmetic. Theories for NCi, ACi, P, PH,...
• The best method available to delineate hierarchies of classical theories.

Question

Can WFM be useful for characterising complexity classes via linear logic?

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Outline

Introduction Normal forms in first-order linear logic An arithmetic in linear logic Bellantoni-Cook programs and the WFM for IΣN+

1

Conclusions

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Linear logic (LL)

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Linear logic (LL)

• LL is a substructural logic:

A ` A A A A ` B

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Linear logic (LL)

• LL is a substructural logic:

A ` A A A A ` B

• It distinguishes multiplicative and additive rules by separate connectives:

Γ, A ∆, B Γ, ∆, A ⊗ B Γ, A Γ, B Γ, A&B

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Linear logic (LL)

• LL is a substructural logic:

A ` A A A A ` B

• It distinguishes multiplicative and additive rules by separate connectives:

Γ, A ∆, B Γ, ∆, A ⊗ B Γ, A Γ, B Γ, A&B

• Controlled access to structural rules via modalities:

!A ⊢ !A⊗!A

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Linear logic (LL)

• LL is a substructural logic:

A ` A A A A ` B

• It distinguishes multiplicative and additive rules by separate connectives:

Γ, A ∆, B Γ, ∆, A ⊗ B Γ, A Γ, B Γ, A&B

• Controlled access to structural rules via modalities:

!A ⊢ !A⊗!A (otherwise ! behaves just like in S4)

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Linear logic (LL)

• LL is a substructural logic:

A ` A A A A ` B

• It distinguishes multiplicative and additive rules by separate connectives:

Γ, A ∆, B Γ, ∆, A ⊗ B Γ, A Γ, B Γ, A&B

• Controlled access to structural rules via modalities:

!A ⊢ !A⊗!A (otherwise ! behaves just like in S4)

• De Morgan duality is everywhere!

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Free-cut elimination in linear logic

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Free-cut elimination in linear logic

A nonlogical rule has the following format: {!Γ, Σi ⊢ ∆i, ?Π}i∈I !Γ, Σ ⊢ ∆, ?Π The formulae in Σ and ∆ are considered principal.

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Free-cut elimination in linear logic

A nonlogical rule has the following format: {!Γ, Σi ⊢ ∆i, ?Π}i∈I !Γ, Σ ⊢ ∆, ?Π The formulae in Σ and ∆ are considered principal. A cut step is anchored if:

• its cut-formulae are (almost) principal on both sides.
• on at least one side it is (almost) principal for a nonlogical step.

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Free-cut elimination in linear logic

A nonlogical rule has the following format: {!Γ, Σi ⊢ ∆i, ?Π}i∈I !Γ, Σ ⊢ ∆, ?Π The formulae in Σ and ∆ are considered principal. A cut step is anchored if:

• its cut-formulae are (almost) principal on both sides.
• on at least one side it is (almost) principal for a nonlogical step.

Theorem

Any linear logic proof can be transformed into one where all cuts are anchored.

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Free-cut elimination in linear logic

A nonlogical rule has the following format: {!Γ, Σi ⊢ ∆i, ?Π}i∈I !Γ, Σ ⊢ ∆, ?Π The formulae in Σ and ∆ are considered principal. A cut step is anchored if:

• its cut-formulae are (almost) principal on both sides.
• on at least one side it is (almost) principal for a nonlogical step.

Theorem

Any linear logic proof can be transformed into one where all cuts are anchored.

• Proof similar to usual cut-elimination arguments.
• Special cases due to Lincoln et al., Baelde & Miller,...

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Free-cut elimination in linear logic

A nonlogical rule has the following format: {!Γ, Σi ⊢ ∆i, ?Π}i∈I !Γ, Σ ⊢ ∆, ?Π The formulae in Σ and ∆ are considered principal. A cut step is anchored if:

• its cut-formulae are (almost) principal on both sides.
• on at least one side it is (almost) principal for a nonlogical step.

Theorem

Any linear logic proof can be transformed into one where all cuts are anchored.

• Proof similar to usual cut-elimination arguments.
• Special cases due to Lincoln et al., Baelde & Miller,...

Corollary

Every theorem has a proof where all formulae are subformulae of the conclusion

• r a principal formula of a nonlogical step.

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Outline

Introduction Normal forms in first-order linear logic An arithmetic in linear logic Bellantoni-Cook programs and the WFM for IΣN+

1

Conclusions

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An arithmetic in linear logic

We consider an axiomatisation inspired by Bellantoni & Hofmann: Ncntr ∀xN.(N(x) ⊗ N(x)) Nε N(ε) N0 ∀xN.N(s0x) N1 ∀xN.N(s1x) ε ∀xN.(ε = s0x ⊗ ε = s1x) inj0 ∀xN, y N.(s0x = s0y ⊸ x = y) inj1 ∀xN, y N.(s1x = s1y ⊸ x = y) tree ∀xN.s0x = s1x surj ∀xN.(x = ε ⊕ ∃y N.x = s0y ⊕ ∃y N.x = s1y) PIND A(ε) ⊸ !(∀x!N.(A(x) ⊸ A(s0x))) ⊸ !(∀x!N.(A(x) ⊸ A(s1x))) ⊸ ∀x!N.A(x) Peano’s N predicate: N(t) := “t is a natural number”.

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Convergence

Functions are specified by equational programs. E.g.: Φ            add(0, x) = x add(su, x) = s(add(u, y)) mult(0, x) = mult(su, x) = add(x, mult(u, x))

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Convergence

Functions are specified by equational programs. E.g.: Φ            add(0, x) = x add(su, x) = s(add(u, y)) mult(0, x) = mult(su, x) = add(x, mult(u, x)) Convergence statement: ∀x!N.Φ(x) ⊸ ∀xN, y N.N(mult(x, y))

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Convergence

Functions are specified by equational programs. E.g.: Φ            add(0, x) = x add(su, x) = s(add(u, y)) mult(0, x) = mult(su, x) = add(x, mult(u, x)) Convergence statement: ∀x!N.Φ(x) ⊸ ∀xN, y N.N(mult(x, y)) We will consider the theory IΣN+

1 , admitting PIND only over:

E ::= N(t) | s = t | s = t | E ` E | E ⊗ E | ∃x.E

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Outline

Introduction Normal forms in first-order linear logic An arithmetic in linear logic Bellantoni-Cook programs and the WFM for IΣN+

1

Conclusions

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Bellantoni-Cook characterisation of polytime functions

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Bellantoni-Cook characterisation of polytime functions

Arguments of a function are separated into normal and safe inputs: f (u; x) Normal: left of ; so u above. Safe: right of ; so x above.

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Bellantoni-Cook characterisation of polytime functions

Arguments of a function are separated into normal and safe inputs: f (u; x) Normal: left of ; so u above. Safe: right of ; so x above. Predicative recursion on notation If g, h0, h1 are BC then so is f defined by: f (ε, v; x) = g(v; x) f (s0u, v; x) = h0(u, v; x, f (u, v; x)) f (s1u, v; x) = h1(u, v; x, f (u, v; x))

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Bellantoni-Cook characterisation of polytime functions

Arguments of a function are separated into normal and safe inputs: f (u; x) Normal: left of ; so u above. Safe: right of ; so x above. Predicative recursion on notation If g, h0, h1 are BC then so is f defined by: f (ε, v; x) = g(v; x) f (s0u, v; x) = h0(u, v; x, f (u, v; x)) f (s1u, v; x) = h1(u, v; x, f (u, v; x)) Safe composition Can compose functions as long as safe inputs are hereditarily safe.

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Bellantoni-Cook characterisation of polytime functions

Arguments of a function are separated into normal and safe inputs: f (u; x) Normal: left of ; so u above. Safe: right of ; so x above. Predicative recursion on notation If g, h0, h1 are BC then so is f defined by: f (ε, v; x) = g(v; x) f (s0u, v; x) = h0(u, v; x, f (u, v; x)) f (s1u, v; x) = h1(u, v; x, f (u, v; x)) Safe composition Can compose functions as long as safe inputs are hereditarily safe. (Also an adequate stock of initial functions.)

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Bellantoni-Cook characterisation of polytime functions

Arguments of a function are separated into normal and safe inputs: f (u; x) Normal: left of ; so u above. Safe: right of ; so x above. Predicative recursion on notation If g, h0, h1 are BC then so is f defined by: f (ε, v; x) = g(v; x) f (s0u, v; x) = h0(u, v; x, f (u, v; x)) f (s1u, v; x) = h1(u, v; x, f (u, v; x)) Safe composition Can compose functions as long as safe inputs are hereditarily safe. (Also an adequate stock of initial functions.)

Theorem (Bellantoni & Cook ’92)

BC programs compute just the polynomial-time functions.

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Multiplication again. . .

add(0; x) = x add(su; x) = s(add(u; y)) mult(0, x; ) = mult(su, x; ) = add(x; mult(u, x))

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Main results

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Main results

Theorem

Every BC program is provably convergent in IΣN+

1 .

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Main results

Theorem

Every BC program is provably convergent in IΣN+

1 .

Proof.

• Straightforward. Similar to previous arguments by Leivant, Bellantoni &

Hofmann, Cantini, taking care to respect linear considerations.

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Main results

Theorem

Every BC program is provably convergent in IΣN+

1 .

Proof.

• Straightforward. Similar to previous arguments by Leivant, Bellantoni &

Hofmann, Cantini, taking care to respect linear considerations.

Theorem

Every function provably convergent in IΣN+

1

is polynomial-time computable.

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Main results

Theorem

Every BC program is provably convergent in IΣN+

1 .

Proof.

• Straightforward. Similar to previous arguments by Leivant, Bellantoni &

Hofmann, Cantini, taking care to respect linear considerations.

Theorem

Every function provably convergent in IΣN+

1

is polynomial-time computable.

Main proof intuitions.

Via the WFM.

• Free-cut elimination:

No ∀-formulae witness predicates of ground type, no ∀-right. No ?-formulae no contraction-right.

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Main results

Theorem

Every BC program is provably convergent in IΣN+

1 .

Proof.

• Straightforward. Similar to previous arguments by Leivant, Bellantoni &

Hofmann, Cantini, taking care to respect linear considerations.

Theorem

Every function provably convergent in IΣN+

1

is polynomial-time computable.

Main proof intuitions.

Via the WFM.

• Free-cut elimination:

No ∀-formulae witness predicates of ground type, no ∀-right. No ?-formulae no contraction-right.

• !-formulae: normal inputs for the witness functions.

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Main results

Theorem

Every BC program is provably convergent in IΣN+

1 .

Proof.

• Straightforward. Similar to previous arguments by Leivant, Bellantoni &

Hofmann, Cantini, taking care to respect linear considerations.

Theorem

Every function provably convergent in IΣN+

1

is polynomial-time computable.

Main proof intuitions.

Via the WFM.

• Free-cut elimination:

No ∀-formulae witness predicates of ground type, no ∀-right. No ?-formulae no contraction-right.

• !-formulae: normal inputs for the witness functions.
• !, ?-free PIND: predicative recursion.

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Main results

Theorem

Every BC program is provably convergent in IΣN+

1 .

Proof.

• Straightforward. Similar to previous arguments by Leivant, Bellantoni &

Hofmann, Cantini, taking care to respect linear considerations.

Theorem

Every function provably convergent in IΣN+

1

is polynomial-time computable.

Main proof intuitions.

Via the WFM.

• Free-cut elimination:

No ∀-formulae witness predicates of ground type, no ∀-right. No ?-formulae no contraction-right.

• !-formulae: normal inputs for the witness functions.
• !, ?-free PIND: predicative recursion.
• Anchored cuts: safe composition of functions.

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Example case: induction

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Example case: induction

• By deduction and invertibility, we can assume PIND occurs as:

!Γ ⊢ A(ε), ?∆ !N(a), !Γ, A(a) ⊢ A(s0a), ?∆ !N(a), !Γ, A(a) ⊢ A(s0a), ?∆ !N(t), !Γ ⊢ A(t), ?∆

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Example case: induction

• By deduction and invertibility, we can assume PIND occurs as:

!Γ ⊢ A(ε), ?∆ !N(a), !Γ, A(a) ⊢ A(s0a), ?∆ !N(a), !Γ, A(a) ⊢ A(s0a), ?∆ !N(t), !Γ ⊢ A(t), ?∆

• By free-cut elimination we can assume ∆ is empty.

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Example case: induction

• By deduction and invertibility, we can assume PIND occurs as:

!Γ ⊢ A(ε), ?∆ !N(a), !Γ, A(a) ⊢ A(s0a), ?∆ !N(a), !Γ, A(a) ⊢ A(s0a), ?∆ !N(t), !Γ ⊢ A(t), ?∆

• By free-cut elimination we can assume ∆ is empty.
• By inductive hypothesis suppose we have functions:

g

• uΓ;
• hi
• uN(a), uΓ; xA(a)

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Example case: induction

• By deduction and invertibility, we can assume PIND occurs as:

!Γ ⊢ A(ε), ?∆ !N(a), !Γ, A(a) ⊢ A(s0a), ?∆ !N(a), !Γ, A(a) ⊢ A(s0a), ?∆ !N(t), !Γ ⊢ A(t), ?∆

• By free-cut elimination we can assume ∆ is empty.
• By inductive hypothesis suppose we have functions:

g

• uΓ;
• hi
• uN(a), uΓ; xA(a)
• Define f by PRN:

f

• 0, uΓ;
• :=

g

• uΓ;
• f
• siuN(t), uΓ;
• :=

hi

• uN(t), uΓ; f
• uN(t), uΓ;
• 17 / 20

Outline

Introduction Normal forms in first-order linear logic An arithmetic in linear logic Bellantoni-Cook programs and the WFM for IΣN+

1

Conclusions

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Conclusions

Summary

• General form of free-cut elimination for first-order linear logic.
• Induces useful normal forms for arithmetic proofs.
• Soundness and completeness of an arithmetic for BC-programs.

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Conclusions

Summary

• General form of free-cut elimination for first-order linear logic.
• Induces useful normal forms for arithmetic proofs.
• Soundness and completeness of an arithmetic for BC-programs.

Further work

• Bounded arithmetic style approach.

Finer use of the witness predicate: evaluation in polynomial-time. Relationships to BC-versions of equational theory PV ?

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Conclusions

Summary

• General form of free-cut elimination for first-order linear logic.
• Induces useful normal forms for arithmetic proofs.
• Soundness and completeness of an arithmetic for BC-programs.

Further work

• Bounded arithmetic style approach.

Finer use of the witness predicate: evaluation in polynomial-time. Relationships to BC-versions of equational theory PV ?

• Characterise polynomial hierarchy via minimisation principles.

Functions conditional on Σp

i tests.

Relies on evaluation of witness predicate in ∆p

i .

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Thank you.

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