On sequent calculi vs natural deductions in logic and computer - - PowerPoint PPT Presentation

▶
on sequent calculi vs natural deductions in logic and
SMART_READER_LITE
LIVE PREVIEW

On sequent calculi vs natural deductions in logic and computer - - PowerPoint PPT Presentation

Feb 07, 2023 301 likes •1.08k views

On sequent calculi vs natural deductions in logic and computer science L. Gordeev Uni-T ubingen, Uni-Ghent, PUC-Rio PUC-Rio, Rio de Janeiro, October 13, 2015 L. Gordeev On sequent calculi vs natural deductions in logic and computer science

slide-1
SLIDE 1

On sequent calculi vs natural deductions in logic and computer science

  • L. Gordeev

Uni-T¨ ubingen, Uni-Ghent, PUC-Rio

PUC-Rio, Rio de Janeiro, October 13, 2015

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-2
SLIDE 2

§1. Sequent calculus (SC): Basics -1-

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-3
SLIDE 3

§1. Sequent calculus (SC): Basics -1-

Gentzen invented sequent calculus in order to prove Hilbert’s consistency (more precisely, contradiction-free) assertion for pure logic and Peano Arithmetic. He succeeded in both cases, although the latter proof required consistency of Cantor’s basic system of ordinals below ε0 .

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-4
SLIDE 4

§1. Sequent calculus (SC): Basics -1-

Gentzen invented sequent calculus in order to prove Hilbert’s consistency (more precisely, contradiction-free) assertion for pure logic and Peano Arithmetic. He succeeded in both cases, although the latter proof required consistency of Cantor’s basic system of ordinals below ε0 . To this end he replaced a familiar Hilbert-style logic formalism based on the rule of detachment (aka modus ponens) α α → β β by a system R of direct inferences having subformula property: ‘premise formulas occur as (sub)formulas in the conclusion’.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-5
SLIDE 5

§1. Sequent calculus (SC): Basics -1-

Gentzen invented sequent calculus in order to prove Hilbert’s consistency (more precisely, contradiction-free) assertion for pure logic and Peano Arithmetic. He succeeded in both cases, although the latter proof required consistency of Cantor’s basic system of ordinals below ε0 . To this end he replaced a familiar Hilbert-style logic formalism based on the rule of detachment (aka modus ponens) α α → β β by a system R of direct inferences having subformula property: ‘premise formulas occur as (sub)formulas in the conclusion’. Such R (finitary, generally well-founded) is consistent, since ⊥ (or 0 = 1) has no proper subformula, and hence not derivable.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-6
SLIDE 6

§1. Sequent calculus: Basics -2-

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-7
SLIDE 7

§1. Sequent calculus: Basics -2-

To complete the consistency proof it remains to show that modus ponens is admissible in S.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-8
SLIDE 8

§1. Sequent calculus: Basics -2-

To complete the consistency proof it remains to show that modus ponens is admissible in S. In sequent form, modus ponens is called cut and looks like this Γ ⇒ α Γ, α ⇒ β Γ ⇒ β (int.) or Γ, α Γ, ¬α Γ (class.)

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-9
SLIDE 9

§1. Sequent calculus: Basics -2-

To complete the consistency proof it remains to show that modus ponens is admissible in S. In sequent form, modus ponens is called cut and looks like this Γ ⇒ α Γ, α ⇒ β Γ ⇒ β (int.) or Γ, α Γ, ¬α Γ (class.) So cut elimination theorem does the job.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-10
SLIDE 10

§1. Sequent calculus: Basics -2-

To complete the consistency proof it remains to show that modus ponens is admissible in S. In sequent form, modus ponens is called cut and looks like this Γ ⇒ α Γ, α ⇒ β Γ ⇒ β (int.) or Γ, α Γ, ¬α Γ (class.) So cut elimination theorem does the job. Theorem (cut elimination)

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-11
SLIDE 11

§1. Sequent calculus: Basics -2-

To complete the consistency proof it remains to show that modus ponens is admissible in S. In sequent form, modus ponens is called cut and looks like this Γ ⇒ α Γ, α ⇒ β Γ ⇒ β (int.) or Γ, α Γ, ¬α Γ (class.) So cut elimination theorem does the job. Theorem (cut elimination)

1 Logic: Every sequent derivable in R ∪ {cut} is derivable in R .

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-12
SLIDE 12

§1. Sequent calculus: Basics -2-

To complete the consistency proof it remains to show that modus ponens is admissible in S. In sequent form, modus ponens is called cut and looks like this Γ ⇒ α Γ, α ⇒ β Γ ⇒ β (int.) or Γ, α Γ, ¬α Γ (class.) So cut elimination theorem does the job. Theorem (cut elimination)

1 Logic: Every sequent derivable in R ∪ {cut} is derivable in R . 2 Peano Arithmetic: Every qf-sequent derivable in RPA ∪ {cut}

is derivable in RPA .

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-13
SLIDE 13

§1.1. Sequent calculus: Conservative extensions

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-14
SLIDE 14

§1.1. Sequent calculus: Conservative extensions

Due to Kreisel’s observation one can use cut elimination techniques to establish proof-theoretic conservations : ‘ every formula provable in T is provable in sub-theory S ’.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-15
SLIDE 15

§1.1. Sequent calculus: Conservative extensions

Due to Kreisel’s observation one can use cut elimination techniques to establish proof-theoretic conservations : ‘ every formula provable in T is provable in sub-theory S ’. The trick: express syntax a/o axioms of T \ S using appropriate cuts which can be eliminated from sequent calculus of T.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-16
SLIDE 16

§1.1. Sequent calculus: Conservative extensions

Due to Kreisel’s observation one can use cut elimination techniques to establish proof-theoretic conservations : ‘ every formula provable in T is provable in sub-theory S ’. The trick: express syntax a/o axioms of T \ S using appropriate cuts which can be eliminated from sequent calculus of T. Example (ACA0 is conservative extension of PA)

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-17
SLIDE 17

§1.1. Sequent calculus: Conservative extensions

Due to Kreisel’s observation one can use cut elimination techniques to establish proof-theoretic conservations : ‘ every formula provable in T is provable in sub-theory S ’. The trick: express syntax a/o axioms of T \ S using appropriate cuts which can be eliminated from sequent calculus of T. Example (ACA0 is conservative extension of PA) Every 1-order formula provable in ACA0 is provable in PA, where ACA0 extends PA by adding 2-order set-variables together with (corresponding logic and) axioms for 1-order comprehension and induction restricted to sets.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-18
SLIDE 18

§1.2. Sequent calculus: Ordinal analysis and beyond

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-19
SLIDE 19

§1.2. Sequent calculus: Ordinal analysis and beyond

Sch¨ utte (and followers) generalized Gentzen’s arithmetical consistency proof working with infinite well-founded tree-like derivations supplied with ordinal labels. This yields deeper insight into proof-theoretic ordinals.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-20
SLIDE 20

§1.2. Sequent calculus: Ordinal analysis and beyond

Sch¨ utte (and followers) generalized Gentzen’s arithmetical consistency proof working with infinite well-founded tree-like derivations supplied with ordinal labels. This yields deeper insight into proof-theoretic ordinals. Namely, for much stronger than PA theories T it’s possible to describe proof-theoretic ordinals αT >> ε0 which characterize theorems of T as follows:

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-21
SLIDE 21

§1.2. Sequent calculus: Ordinal analysis and beyond

Sch¨ utte (and followers) generalized Gentzen’s arithmetical consistency proof working with infinite well-founded tree-like derivations supplied with ordinal labels. This yields deeper insight into proof-theoretic ordinals. Namely, for much stronger than PA theories T it’s possible to describe proof-theoretic ordinals αT >> ε0 which characterize theorems of T as follows: ‘ every arithmetical theorem of T is provable in PA extended by transfinite induction below αT ’.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-22
SLIDE 22

§1.2. Sequent calculus: Ordinal analysis and beyond

Sch¨ utte (and followers) generalized Gentzen’s arithmetical consistency proof working with infinite well-founded tree-like derivations supplied with ordinal labels. This yields deeper insight into proof-theoretic ordinals. Namely, for much stronger than PA theories T it’s possible to describe proof-theoretic ordinals αT >> ε0 which characterize theorems of T as follows: ‘ every arithmetical theorem of T is provable in PA extended by transfinite induction below αT ’. More recent research (initiated by Harvey Friedman) enables us to replace ordinals αT (which are very involved for strong T) by more transparent quasi-ordinals characterized by extended Kruskal-style tree theorems.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-23
SLIDE 23

§1.2. Sequent calculus: Ordinal analysis and beyond

Sch¨ utte (and followers) generalized Gentzen’s arithmetical consistency proof working with infinite well-founded tree-like derivations supplied with ordinal labels. This yields deeper insight into proof-theoretic ordinals. Namely, for much stronger than PA theories T it’s possible to describe proof-theoretic ordinals αT >> ε0 which characterize theorems of T as follows: ‘ every arithmetical theorem of T is provable in PA extended by transfinite induction below αT ’. More recent research (initiated by Harvey Friedman) enables us to replace ordinals αT (which are very involved for strong T) by more transparent quasi-ordinals characterized by extended Kruskal-style tree theorems. This stuff is obviously related to (say, extended) Hilbert’s Program concerning logic foundations of mathematics.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-24
SLIDE 24

§1.3. Sequent calculus: Some consequences

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-25
SLIDE 25

§1.3. Sequent calculus: Some consequences

To put it in a nutshell, cut elimination provides an extremely powerful tool in Hilbert-style proof theory. Moreover it is constructive, and hence yields by now strongest conservation results for various intuitionistic theories [L.G]. Besides, it enables to work with cutfree systems of direct sequent rules. Also note:

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-26
SLIDE 26

§1.3. Sequent calculus: Some consequences

To put it in a nutshell, cut elimination provides an extremely powerful tool in Hilbert-style proof theory. Moreover it is constructive, and hence yields by now strongest conservation results for various intuitionistic theories [L.G]. Besides, it enables to work with cutfree systems of direct sequent rules. Also note: Cutfree sequent calculi have better proof search.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-27
SLIDE 27

§1.3. Sequent calculus: Some consequences

To put it in a nutshell, cut elimination provides an extremely powerful tool in Hilbert-style proof theory. Moreover it is constructive, and hence yields by now strongest conservation results for various intuitionistic theories [L.G]. Besides, it enables to work with cutfree systems of direct sequent rules. Also note: Cutfree sequent calculi have better proof search. However, there are complexity problems (re: speed-up).

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-28
SLIDE 28

§1.3. Sequent calculus: Some consequences

To put it in a nutshell, cut elimination provides an extremely powerful tool in Hilbert-style proof theory. Moreover it is constructive, and hence yields by now strongest conservation results for various intuitionistic theories [L.G]. Besides, it enables to work with cutfree systems of direct sequent rules. Also note: Cutfree sequent calculi have better proof search. However, there are complexity problems (re: speed-up). What to do? Dag-like cutfree derivations and substitution rule!

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-29
SLIDE 29

§1.3. Sequent calculus: Some consequences

To put it in a nutshell, cut elimination provides an extremely powerful tool in Hilbert-style proof theory. Moreover it is constructive, and hence yields by now strongest conservation results for various intuitionistic theories [L.G]. Besides, it enables to work with cutfree systems of direct sequent rules. Also note: Cutfree sequent calculi have better proof search. However, there are complexity problems (re: speed-up). What to do? Dag-like cutfree derivations and substitution rule! Full dag-like compression with substitution may provide a solution (at least in the propositional case).

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-30
SLIDE 30

§1.3. Sequent calculus: Some consequences

To put it in a nutshell, cut elimination provides an extremely powerful tool in Hilbert-style proof theory. Moreover it is constructive, and hence yields by now strongest conservation results for various intuitionistic theories [L.G]. Besides, it enables to work with cutfree systems of direct sequent rules. Also note: Cutfree sequent calculi have better proof search. However, there are complexity problems (re: speed-up). What to do? Dag-like cutfree derivations and substitution rule! Full dag-like compression with substitution may provide a solution (at least in the propositional case). But main complexity problem remains open (re: NP vs coNP).

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-31
SLIDE 31

§2. Natural deductions (ND): Basic comparison -1-

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-32
SLIDE 32

§2. Natural deductions (ND): Basic comparison -1-

Prawitz formalized (Gentzen’s notion of) natural deductions. Here’s a short overview of ND features (compared to SC).

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-33
SLIDE 33

§2. Natural deductions (ND): Basic comparison -1-

Prawitz formalized (Gentzen’s notion of) natural deductions. Here’s a short overview of ND features (compared to SC).

1 Lambda calculus connections. (Disadvantage: geometry fails.)

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-34
SLIDE 34

§2. Natural deductions (ND): Basic comparison -1-

Prawitz formalized (Gentzen’s notion of) natural deductions. Here’s a short overview of ND features (compared to SC).

1 Lambda calculus connections. (Disadvantage: geometry fails.) 2 Normalization (also constructive) instead of cut elimination.

Normal ND also have global subformula property (but, unlike cutfree SC, not the local one).

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-35
SLIDE 35

§2. Natural deductions (ND): Basic comparison -1-

Prawitz formalized (Gentzen’s notion of) natural deductions. Here’s a short overview of ND features (compared to SC).

1 Lambda calculus connections. (Disadvantage: geometry fails.) 2 Normalization (also constructive) instead of cut elimination.

Normal ND also have global subformula property (but, unlike cutfree SC, not the local one). Remember that normal ND are not like cutfree SC; actually they include modus ponens explicitly.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-36
SLIDE 36

§2. Natural deductions (ND): Basic comparison -1-

Prawitz formalized (Gentzen’s notion of) natural deductions. Here’s a short overview of ND features (compared to SC).

1 Lambda calculus connections. (Disadvantage: geometry fails.) 2 Normalization (also constructive) instead of cut elimination.

Normal ND also have global subformula property (but, unlike cutfree SC, not the local one). Remember that normal ND are not like cutfree SC; actually they include modus ponens explicitly.

3 ND normalization is weaker than SC cut elimination. And/but

it also features exponential speed-up.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-37
SLIDE 37

§2. Natural deductions (ND): Basic comparison -1-

Prawitz formalized (Gentzen’s notion of) natural deductions. Here’s a short overview of ND features (compared to SC).

1 Lambda calculus connections. (Disadvantage: geometry fails.) 2 Normalization (also constructive) instead of cut elimination.

Normal ND also have global subformula property (but, unlike cutfree SC, not the local one). Remember that normal ND are not like cutfree SC; actually they include modus ponens explicitly.

3 ND normalization is weaker than SC cut elimination. And/but

it also features exponential speed-up.

4 Ordinal analysis?

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-38
SLIDE 38

§2. Natural deductions (ND): Basic comparison -1-

Prawitz formalized (Gentzen’s notion of) natural deductions. Here’s a short overview of ND features (compared to SC).

1 Lambda calculus connections. (Disadvantage: geometry fails.) 2 Normalization (also constructive) instead of cut elimination.

Normal ND also have global subformula property (but, unlike cutfree SC, not the local one). Remember that normal ND are not like cutfree SC; actually they include modus ponens explicitly.

3 ND normalization is weaker than SC cut elimination. And/but

it also features exponential speed-up.

4 Ordinal analysis? 5 Stronger ties to complexity theory?

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-39
SLIDE 39

§2. Natural deductions: Basic comparison -2-

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-40
SLIDE 40

§2. Natural deductions: Basic comparison -2-

Due to lack of time let’s go straight to final clause 5.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-41
SLIDE 41

§2. Natural deductions: Basic comparison -2-

Due to lack of time let’s go straight to final clause 5. Main question: How to get a short propositional ND?

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-42
SLIDE 42

§2. Natural deductions: Basic comparison -2-

Due to lack of time let’s go straight to final clause 5. Main question: How to get a short propositional ND? Recall that sequent derivations (proofs) admit full dag-like

  • compressions. (In the sequel we consider propositional logic.)
  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-43
SLIDE 43

§2. Natural deductions: Basic comparison -2-

Due to lack of time let’s go straight to final clause 5. Main question: How to get a short propositional ND? Recall that sequent derivations (proofs) admit full dag-like

  • compressions. (In the sequel we consider propositional logic.)

Theorem (L.G.)

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-44
SLIDE 44

§2. Natural deductions: Basic comparison -2-

Due to lack of time let’s go straight to final clause 5. Main question: How to get a short propositional ND? Recall that sequent derivations (proofs) admit full dag-like

  • compressions. (In the sequel we consider propositional logic.)

Theorem (L.G.) Any given tree-like sequent proof T (whether cutfree or not) is constructively compressible to a dag-like sequent proof D of the same endsequent such that the total number of pairwise distinct nodes in D is less, or equal, than the total number of pairwise distinct sequents occurring in T. Loosely speaking this holds also in the presence of substitution rule(s).

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-45
SLIDE 45

§2. Natural deductions: Basic comparison -3-

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-46
SLIDE 46

§2. Natural deductions: Basic comparison -3-

Clearly this result yields a significant space reduction.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-47
SLIDE 47

§2. Natural deductions: Basic comparison -3-

Clearly this result yields a significant space reduction. But there is one catch: We can’t properly control the number

  • f pairwise distinct sequents (as being sets of formulas)
  • ccurring in T even if we know that all formulas in question

are subformulas of the conclusion.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-48
SLIDE 48

§2. Natural deductions: Basic comparison -3-

Clearly this result yields a significant space reduction. But there is one catch: We can’t properly control the number

  • f pairwise distinct sequents (as being sets of formulas)
  • ccurring in T even if we know that all formulas in question

are subformulas of the conclusion. It’s still exponential upper bound! What to do?

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-49
SLIDE 49

§2. Natural deductions: Basic comparison -3-

Clearly this result yields a significant space reduction. But there is one catch: We can’t properly control the number

  • f pairwise distinct sequents (as being sets of formulas)
  • ccurring in T even if we know that all formulas in question

are subformulas of the conclusion. It’s still exponential upper bound! What to do? Recall that, by contrast, ND’s consist of single formulas. Moreover, the normal ones share the same subformula

  • property. Is it possible to compress them analogously and
  • btain polynomial upper bounds on the total number of

nodes?

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-50
SLIDE 50

§2. Natural deductions: Basic comparison -4-

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-51
SLIDE 51

§2. Natural deductions: Basic comparison -4-

Warning: There is one obstacle when it comes to compression of natural

  • deductions. Namely, we should avoid vertical compressions

due to possible confusion caused by discharged assumptions. (This problem is irrelevant to proof compression theory in sequent calculi).

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-52
SLIDE 52

§2. Natural deductions: Basic comparison -4-

Warning: There is one obstacle when it comes to compression of natural

  • deductions. Namely, we should avoid vertical compressions

due to possible confusion caused by discharged assumptions. (This problem is irrelevant to proof compression theory in sequent calculi). What can be done is a sort of horizontal dag-like compression.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-53
SLIDE 53

§2. Natural deductions: Basic comparison -4-

Warning: There is one obstacle when it comes to compression of natural

  • deductions. Namely, we should avoid vertical compressions

due to possible confusion caused by discharged assumptions. (This problem is irrelevant to proof compression theory in sequent calculi). What can be done is a sort of horizontal dag-like compression. Underlying idea: If an input tree-like deduction has merely polynomial height and every horizontal section is fully dag-like compressible (i.e. reducible to pairwise distinct formulas), then the resulting dag-like deduction has polynomial size. Voil` a!

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-54
SLIDE 54

§2.1. Dag-like natural deductions -1-

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-55
SLIDE 55

§2.1. Dag-like natural deductions -1-

The assumptions on the tree-like complexity in question can be obtained by embedding into ND calculus Hudelmaier’s sequent calculus for minimal a/o intuitionistic logic (without disjunction).

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-56
SLIDE 56

§2.1. Dag-like natural deductions -1-

The assumptions on the tree-like complexity in question can be obtained by embedding into ND calculus Hudelmaier’s sequent calculus for minimal a/o intuitionistic logic (without disjunction). All in all such dag-like compression will infer NP = PSPACE.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-57
SLIDE 57

§2.1. Dag-like natural deductions -1-

The assumptions on the tree-like complexity in question can be obtained by embedding into ND calculus Hudelmaier’s sequent calculus for minimal a/o intuitionistic logic (without disjunction). All in all such dag-like compression will infer NP = PSPACE. But to this end we have to formalize dag-like deducibility in Prawitz’s world. Recall that ‘dag’ stands for directed acyclic graph (edges directed upwards).

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-58
SLIDE 58

§2.1. Dag-like natural deductions -1-

The assumptions on the tree-like complexity in question can be obtained by embedding into ND calculus Hudelmaier’s sequent calculus for minimal a/o intuitionistic logic (without disjunction). All in all such dag-like compression will infer NP = PSPACE. But to this end we have to formalize dag-like deducibility in Prawitz’s world. Recall that ‘dag’ stands for directed acyclic graph (edges directed upwards). The main difference between tree-like and dag-like natural deductions is caused by the art of discharging, as the following example shows.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-59
SLIDE 59

§2.1. Dag-like natural deductions -2-

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-60
SLIDE 60

§2.1. Dag-like natural deductions -2-

Example

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-61
SLIDE 61

§2.1. Dag-like natural deductions -2-

Example Consider a dag-like natural deduction ∂ = (→ E) (→ E) Γ

  • ∵

β → α [α]1 α → β β (→ E) α α → β [1] (→ I) β in which the right-hand side premise of second (→ E) coincides with (→ I) premise β. Note that the assumption α above β is discharged by this (→ I). However, we can only infer that ∂ deduces β from Γ ∪ {α, α → β}, instead of expected Γ ∪ {α → β}, which leaves the option Γ ∪ {α → β} β open, if α / ∈ Γ.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-62
SLIDE 62

§2.1. Dag-like natural deductions -3-

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-63
SLIDE 63

§2.1. Dag-like natural deductions -3-

Example (continued)

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-64
SLIDE 64

§2.1. Dag-like natural deductions -3-

Example (continued) This becomes obvious if we replace ∂ by its “unfolded” tree-like version ∂u = α α → β β Γ

  • ∵

β → α α [α]1 α → β β α → β [1] β Clearly ∂u deduces β from Γ∪{α, α → β}, instead of Γ∪{α → β}, which leaves the option Γ ∪ {α → β} β open, if α / ∈ Γ.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-65
SLIDE 65

§2.1. Dag-like natural deductions -4-

1unless ∂ is tree-like, in which case both propeties are in P.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-66
SLIDE 66

§2.1. Dag-like natural deductions -4-

Keeping this in mind we’ll say that in a dag-like natural deduction ∂, a given leaf u is an open (or undischarged) assumption-node iff there exists a thread θ connecting u with the root and such that no w ∈ θ is the (→ I) conclusion assigned with α → β, provided that α is assigned to u. Other leaves are called closed (or discharged) in ∂. Note that the corresponding condition ‘u is open (resp. closed) in ∂’ belongs merely to NP (resp. coNP) 1, and hence ad hoc is inappropriate for polynomial time proof verification.

1unless ∂ is tree-like, in which case both propeties are in P.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-67
SLIDE 67

§2.1. Dag-like natural deductions -4-

Keeping this in mind we’ll say that in a dag-like natural deduction ∂, a given leaf u is an open (or undischarged) assumption-node iff there exists a thread θ connecting u with the root and such that no w ∈ θ is the (→ I) conclusion assigned with α → β, provided that α is assigned to u. Other leaves are called closed (or discharged) in ∂. Note that the corresponding condition ‘u is open (resp. closed) in ∂’ belongs merely to NP (resp. coNP) 1, and hence ad hoc is inappropriate for polynomial time proof verification. We overcome this trouble by a suitable modification of the notion of local correctness that includes special vertex-labeling function ℓd : v(D) × f (D) → {0, 1}, where v(D) and f(D) are respectively the vertices (= nodes) and formulas of the underlying dag D.

1unless ∂ is tree-like, in which case both propeties are in P.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-68
SLIDE 68

§2.1. Dag-like natural deductions -5-

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-69
SLIDE 69

§2.1. Dag-like natural deductions -5-

Other two (standard) labeling functions ℓf : v(D) → F and ℓr : v(D) → R ∪ {∅} assign formulas and rule-names. Now for any given locally correct labeled dag D =

  • D, ℓf, ℓr, ℓd

we call ΓD :=

  • ℓf (u) : 0 = −

→ deg (u) = ℓd (u, ℓf (u))

  • u∈v(D) the

set of open (or undischarged) assumptions, in D. Moreover D =

  • D, ℓf, ℓr, ℓd

is called an encoded dag-like natural deduction of ℓf (root (D)) from ΓD. In particular, if ΓD = ∅, then D is called an encoded dag-like proof of ℓf (root (D)). Lemma There is a 1 − 1 correspondence between plain and encoded dag-like natural deductions (in particular, proofs).

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-70
SLIDE 70

§2.2. More on local correctness

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-71
SLIDE 71

§2.2. More on local correctness

Definition

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-72
SLIDE 72

§2.2. More on local correctness

Definition Local correctness conditions for ℓd are as follows, where ℓrf (u) = (→ I)α stands for ℓr (u) = (→ I) ∧ ℓf (u) = α → ℓf u(1) .

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-73
SLIDE 73

§2.2. More on local correctness

Definition Local correctness conditions for ℓd are as follows, where ℓrf (u) = (→ I)α stands for ℓr (u) = (→ I) ∧ ℓf (u) = α → ℓf u(1) .

1 If ←

− deg (u) = 0, i.e. u is the root, then ℓd (u, α) = 1 iff ℓrf (u) = (→ I)α .

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-74
SLIDE 74

§2.2. More on local correctness

Definition Local correctness conditions for ℓd are as follows, where ℓrf (u) = (→ I)α stands for ℓr (u) = (→ I) ∧ ℓf (u) = α → ℓf u(1) .

1 If ←

− deg (u) = 0, i.e. u is the root, then ℓd (u, α) = 1 iff ℓrf (u) = (→ I)α .

2 If ℓrf (u) = (→ I)α, then ℓd (u, α) = 1.

  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science

slide-75
SLIDE 75

§2.2. More on local correctness

Definition Local correctness conditions for ℓd are as follows, where ℓrf (u) = (→ I)α stands for ℓr (u) = (→ I) ∧ ℓf (u) = α → ℓf u(1) .

1 If ←

− deg (u) = 0, i.e. u is the root, then ℓd (u, α) = 1 iff ℓrf (u) = (→ I)α .

2 If ℓrf (u) = (→ I)α, then ℓd (u, α) = 1. 3 If ←

− deg (u) > 0 and ℓrf (u) = (→ I)α, then ℓd (u, α) =

← − deg(u)

  • i=1

ℓd u(i), α

  • .
  • L. Gordeev

On sequent calculi vs natural deductions in logic and computer science