BACAT2: Valeria suggested that we discussed some old work on linear type theory to get up to date with newer work. Here are some slides for a background talk on linear type theory, based on Term Assignment for Intuitionistic Linear Logic. (Benton, Bierman, de Paiva and Hyland). Technical Report 262, University of Cambridge Computer Laboratory 1992.

1. Linear Type Theory Revisited
Valeria de Paiva
Nuance Comms
February 21, 2014

2. Introduction

3. Goals
Discuss very old work (early 90’s) on linear type theories

4. Goals
Discuss very old work (early 90’s) on linear type theories
based on Term Assignment for Intuitionistic Linear Logic.
(Benton, Bierman, de Paiva and Hyland). Technical Report
262, University of Cambridge Computer Laboratory 1992.

5. Goals
Discuss very old work (early 90’s) on linear type theories
based on Term Assignment for Intuitionistic Linear Logic.
(Benton, Bierman, de Paiva and Hyland). Technical Report
262, University of Cambridge Computer Laboratory 1992.
available from
http://www.cs.bham.ac.uk/ vdp/publications/papers.html.

6. Goals
Discuss very old work (early 90’s) on linear type theories
based on Term Assignment for Intuitionistic Linear Logic.
(Benton, Bierman, de Paiva and Hyland). Technical Report
262, University of Cambridge Computer Laboratory 1992.
available from
http://www.cs.bham.ac.uk/ vdp/publications/papers.html.
then get to the state-of-the art...

7. Summary from Abstract
Investigate the problem of deriving a term assignment system
for Girard’s Intuitionistic Linear Logic

8. Summary from Abstract
Investigate the problem of deriving a term assignment system
for Girard’s Intuitionistic Linear Logic
both sequent calculus and natural deduction proof systems

9. Summary from Abstract
Investigate the problem of deriving a term assignment system
for Girard’s Intuitionistic Linear Logic
both sequent calculus and natural deduction proof systems
satisfying two important properties:
the substitution property (the set of valid deductions is closed
under substitution) and
subject reduction (reduction on terms is well-typed)

10. Summary from Abstract
Investigate the problem of deriving a term assignment system
for Girard’s Intuitionistic Linear Logic
both sequent calculus and natural deduction proof systems
satisfying two important properties:
the substitution property (the set of valid deductions is closed
under substitution) and
subject reduction (reduction on terms is well-typed)
a categorical model for Intuitionistic Linear Logic and how to
use it to derive the term assignment

11. Summary from Abstract
Investigate the problem of deriving a term assignment system
for Girard’s Intuitionistic Linear Logic
both sequent calculus and natural deduction proof systems
satisfying two important properties:
the substitution property (the set of valid deductions is closed
under substitution) and
subject reduction (reduction on terms is well-typed)
a categorical model for Intuitionistic Linear Logic and how to
use it to derive the term assignment
long (57 pages) but slow and easy...

12. Introduction
the problem of deriving a term assignment system for Girard’s
Intuitionistic Linear Logic

13. Introduction
the problem of deriving a term assignment system for Girard’s
Intuitionistic Linear Logic
Previous approaches have simply annotated the sequent
calculus with terms and have given little or no justification for
their choice

14. Introduction
the problem of deriving a term assignment system for Girard’s
Intuitionistic Linear Logic
Previous approaches have simply annotated the sequent
calculus with terms and have given little or no justification for
their choice
Phil Wadler: There’s no substitute for LL

15. Introduction
the problem of deriving a term assignment system for Girard’s
Intuitionistic Linear Logic
Previous approaches have simply annotated the sequent
calculus with terms and have given little or no justification for
their choice
Phil Wadler: There’s no substitute for LL
substitution lemma does not hold for the term assignment
system in Abramsky’s ‘Computational Interpretations of Linear
Logic’ (Cited by 546)

16. Digression: Other old work...
Linear types can change the world
Is there a use for linear logic?
A taste of linear logic
Operational interpretations of linear logic
Reference counting as a computational interpretation of linear
logic (Chirimar)

17. Introduction 2
solving the problem of deriving a term assignment system for
Girard’s Intuitionistic Linear Logic:

18. Introduction 2
solving the problem of deriving a term assignment system for
Girard’s Intuitionistic Linear Logic:
Two ways
By considering the sequent calculus formulation of the logic
and using the underlying categorical constructions to suggest a
term assignment system
By considering a linear natural deduction system

19. Introduction 2
solving the problem of deriving a term assignment system for
Girard’s Intuitionistic Linear Logic:
Two ways
By considering the sequent calculus formulation of the logic
and using the underlying categorical constructions to suggest a
term assignment system
By considering a linear natural deduction system
two approaches produce equivalent term assignment systems

20. Introduction 2
solving the problem of deriving a term assignment system for
Girard’s Intuitionistic Linear Logic:
Two ways
By considering the sequent calculus formulation of the logic
and using the underlying categorical constructions to suggest a
term assignment system
By considering a linear natural deduction system
two approaches produce equivalent term assignment systems
BUT for equality (via reduction of terms) matters are more
subtle: natural equalities for category theory are stronger than
those suggested by computational considerations

21. Outline of TR
Girard’s Intuitionistic Linear Logic:

22. Outline of TR
Girard’s Intuitionistic Linear Logic:
a simple categorical model of Intuitionistic Linear Logic
(sequent calculus)

23. Outline of TR
Girard’s Intuitionistic Linear Logic:
a simple categorical model of Intuitionistic Linear Logic
(sequent calculus)
a linear system of natural deduction

24. Outline of TR
Girard’s Intuitionistic Linear Logic:
a simple categorical model of Intuitionistic Linear Logic
(sequent calculus)
a linear system of natural deduction
how our two systems of Intuitionistic Linear Logic are related

25. Outline of TR
Girard’s Intuitionistic Linear Logic:
a simple categorical model of Intuitionistic Linear Logic
(sequent calculus)
a linear system of natural deduction
how our two systems of Intuitionistic Linear Logic are related
the process of proof normalisation within the linear natural
deduction system

26. Outline of TR
Girard’s Intuitionistic Linear Logic:
a simple categorical model of Intuitionistic Linear Logic
(sequent calculus)
a linear system of natural deduction
how our two systems of Intuitionistic Linear Logic are related
the process of proof normalisation within the linear natural
deduction system
model for Intuitionistic Linear Logic

27. Outline of TR
Girard’s Intuitionistic Linear Logic:
a simple categorical model of Intuitionistic Linear Logic
(sequent calculus)
a linear system of natural deduction
how our two systems of Intuitionistic Linear Logic are related
the process of proof normalisation within the linear natural
deduction system
model for Intuitionistic Linear Logic
cut-elimination in the sequent calculus

28. Outline of TR
Girard’s Intuitionistic Linear Logic:
a simple categorical model of Intuitionistic Linear Logic
(sequent calculus)
a linear system of natural deduction
how our two systems of Intuitionistic Linear Logic are related
the process of proof normalisation within the linear natural
deduction system
model for Intuitionistic Linear Logic
cut-elimination in the sequent calculus
conclusions

29. Intuitionistic Linear Logic
Recommended if sequent calculus/ND are not part of your vocab:
Jean Gallier: Constructive Logics. Part I: A Tutorial on Proof
Systems and Typed lambda-Calculi. Theoretical Computer
Science, 110(2), 249-239 (1993). from
http://www.cis.upenn.edu/ jean/gbooks/logic.html
multiplicative fragment of Intuitionistic Linear Logic (ILL)

30. Intuitionistic Linear Logic
Recommended if sequent calculus/ND are not part of your vocab:
Jean Gallier: Constructive Logics. Part I: A Tutorial on Proof
Systems and Typed lambda-Calculi. Theoretical Computer
Science, 110(2), 249-239 (1993). from
http://www.cis.upenn.edu/ jean/gbooks/logic.html
multiplicative fragment of Intuitionistic Linear Logic (ILL)
a refinement of intuitionistic logic (IL) where formulae must
be used exactly once

31. Intuitionistic Linear Logic
Recommended if sequent calculus/ND are not part of your vocab:
Jean Gallier: Constructive Logics. Part I: A Tutorial on Proof
Systems and Typed lambda-Calculi. Theoretical Computer
Science, 110(2), 249-239 (1993). from
http://www.cis.upenn.edu/ jean/gbooks/logic.html
multiplicative fragment of Intuitionistic Linear Logic (ILL)
a refinement of intuitionistic logic (IL) where formulae must
be used exactly once
Weakening and Contraction rules are removed

32. Intuitionistic Linear Logic
Recommended if sequent calculus/ND are not part of your vocab:
Jean Gallier: Constructive Logics. Part I: A Tutorial on Proof
Systems and Typed lambda-Calculi. Theoretical Computer
Science, 110(2), 249-239 (1993). from
http://www.cis.upenn.edu/ jean/gbooks/logic.html
multiplicative fragment of Intuitionistic Linear Logic (ILL)
a refinement of intuitionistic logic (IL) where formulae must
be used exactly once
Weakening and Contraction rules are removed
To regain the expressive power, rules returned in a controlled
manner using operator “!”

33. Intuitionistic Linear Logic
Recommended if sequent calculus/ND are not part of your vocab:
Jean Gallier: Constructive Logics. Part I: A Tutorial on Proof
Systems and Typed lambda-Calculi. Theoretical Computer
Science, 110(2), 249-239 (1993). from
http://www.cis.upenn.edu/ jean/gbooks/logic.html
multiplicative fragment of Intuitionistic Linear Logic (ILL)
a refinement of intuitionistic logic (IL) where formulae must
be used exactly once
Weakening and Contraction rules are removed
To regain the expressive power, rules returned in a controlled
manner using operator “!”
“!” similar to the modal necessity operator

34. Intuitionistic Linear Logic Rules

35. Generic Categorical considerations
sequent calculus: providing not proofs themselves but a
meta-theory concerning proofs
fundamental idea of categorical treatment of proof theory
propositions interpreted as objects of a category (or
multicategory or polycategory)

36. Generic Categorical considerations
sequent calculus: providing not proofs themselves but a
meta-theory concerning proofs
fundamental idea of categorical treatment of proof theory
propositions interpreted as objects of a category (or
multicategory or polycategory)
proofs interpreted as maps of the category

37. Generic Categorical considerations
sequent calculus: providing not proofs themselves but a
meta-theory concerning proofs
fundamental idea of categorical treatment of proof theory
propositions interpreted as objects of a category (or
multicategory or polycategory)
proofs interpreted as maps of the category
operations transforming proofs into proofs then correspond (if
possible) to natural transformations between appropriate
hom-functors

38. Categorical considerations
dealing with sequents Γ
multicategories
A in principle we should deal with

39. Categorical considerations
dealing with sequents Γ A in principle we should deal with
multicategories
simplifying assumption: multicategorical structure represented
by a tensor product .

40. Categorical considerations
dealing with sequents Γ A in principle we should deal with
multicategories
simplifying assumption: multicategorical structure represented
by a tensor product .
a sequent of form
C1 , C2 , . . . , Cn
A
will be represented by
C1 ⊗ C2 ⊗ . . . ⊗ Cn → A
sometimes written as Γ → A

41. Categorical considerations
dealing with sequents Γ A in principle we should deal with
multicategories
simplifying assumption: multicategorical structure represented
by a tensor product .
a sequent of form
C1 , C2 , . . . , Cn
A
will be represented by
C1 ⊗ C2 ⊗ . . . ⊗ Cn → A
sometimes written as Γ → A
(a coherence result is assumed)

42. Categorical considerations
dealing with sequents Γ A in principle we should deal with
multicategories
simplifying assumption: multicategorical structure represented
by a tensor product .
a sequent of form
C1 , C2 , . . . , Cn
A
will be represented by
C1 ⊗ C2 ⊗ . . . ⊗ Cn → A
sometimes written as Γ → A
(a coherence result is assumed)
We seek to enrich the sequent judgement to a term
assignment judgement of the form
x1 : C1 , x2 : C2 , . . . , xn : Cn
e: A
where the xi are distinct variables and e is a term.

43. Identity and Cut
The sequent representing the Identity rule is interpreted as the
canonical identity arrow idA : A → A

44. Identity and Cut
The sequent representing the Identity rule is interpreted as the
canonical identity arrow idA : A → A
The corresponding rule of term formation is x : A
x :A

45. Identity and Cut
The sequent representing the Identity rule is interpreted as the
canonical identity arrow idA : A → A
The corresponding rule of term formation is x : A
x :A
The rule of Exchange we interpret by assuming that we have a
symmetry for the tensor product

46. Identity and Cut
The sequent representing the Identity rule is interpreted as the
canonical identity arrow idA : A → A
The corresponding rule of term formation is x : A
x :A
The rule of Exchange we interpret by assuming that we have a
symmetry for the tensor product
We suppress Exchange and the corresponding symmetry,
considering multisets of formulae, so no term forming
operations result from this rule (others do diff...)

47. Identity and Cut
The sequent representing the Identity rule is interpreted as the
canonical identity arrow idA : A → A
The corresponding rule of term formation is x : A
x :A
The rule of Exchange we interpret by assuming that we have a
symmetry for the tensor product
We suppress Exchange and the corresponding symmetry,
considering multisets of formulae, so no term forming
operations result from this rule (others do diff...)
The cut rule is interpreted as a generalized form of
composition. if the maps f : Γ → A and g : A, ∆ → B are the
interpretations of hypotheses of the rule, then the composite
Γ ⊗ ∆ →f ⊗1∆ A ⊗ ∆ →g B is the interpretation of the
conclusion

48. Substitution in Terms...
Assumption: any logical rule corresponds to an operation on
maps of the category which is natural in the interpretations of
the components of the sequents which remain unchanged
during the application of a rule

49. Substitution in Terms...
Assumption: any logical rule corresponds to an operation on
maps of the category which is natural in the interpretations of
the components of the sequents which remain unchanged
during the application of a rule
Composition corresponds to Cut: our operations commute
where appropriate with Cut.

50. Substitution in Terms...
Assumption: any logical rule corresponds to an operation on
maps of the category which is natural in the interpretations of
the components of the sequents which remain unchanged
during the application of a rule
Composition corresponds to Cut: our operations commute
where appropriate with Cut.
Composition is interpreted by textual substitution thus the
free variables have to reflect the possibility for substitution.

51. Substitution in Terms...
Assumption: any logical rule corresponds to an operation on
maps of the category which is natural in the interpretations of
the components of the sequents which remain unchanged
during the application of a rule
Composition corresponds to Cut: our operations commute
where appropriate with Cut.
Composition is interpreted by textual substitution thus the
free variables have to reflect the possibility for substitution.
The rules for the constant I are parallel to the rules for the
tensor product ⊗, which we describe next.

52. Substitution in Terms...
Assumption: any logical rule corresponds to an operation on
maps of the category which is natural in the interpretations of
the components of the sequents which remain unchanged
during the application of a rule
Composition corresponds to Cut: our operations commute
where appropriate with Cut.
Composition is interpreted by textual substitution thus the
free variables have to reflect the possibility for substitution.
The rules for the constant I are parallel to the rules for the
tensor product ⊗, which we describe next.
The rules for (linear) implication are usual.

53. Substitution in Terms...
Assumption: any logical rule corresponds to an operation on
maps of the category which is natural in the interpretations of
the components of the sequents which remain unchanged
during the application of a rule
Composition corresponds to Cut: our operations commute
where appropriate with Cut.
Composition is interpreted by textual substitution thus the
free variables have to reflect the possibility for substitution.
The rules for the constant I are parallel to the rules for the
tensor product ⊗, which we describe next.
The rules for (linear) implication are usual.
The rules for the modality ! are the hard ones. these slides will
get us to page 12 of the report...

54. Text substitution Terms
use this and BB to get equation (2)

55. Tensor Terms
Abramsky introduced a let constructor for tensor products, a
reasonable syntax to bind variables x and y in the term f .

56. Tensor Terms
Abramsky introduced a let constructor for tensor products, a
reasonable syntax to bind variables x and y in the term f .
Composition is interpreted by textual substitution thus the
free variables have to reflect the possibility for substitution.

57. Tensor Terms
Abramsky introduced a let constructor for tensor products, a
reasonable syntax to bind variables x and y in the term f .
Composition is interpreted by textual substitution thus the
free variables have to reflect the possibility for substitution.
Naturality in C gives rise to the equation equation explaining
how lets interact with composition.

58. Tensor Terms
Abramsky introduced a let constructor for tensor products, a
reasonable syntax to bind variables x and y in the term f .
Composition is interpreted by textual substitution thus the
free variables have to reflect the possibility for substitution.
Naturality in C gives rise to the equation equation explaining
how lets interact with composition.
For the tensoring on the right, we simply multiply (or tensor)
the terms.

59. Tensor Terms
Abramsky introduced a let constructor for tensor products, a
reasonable syntax to bind variables x and y in the term f .
Composition is interpreted by textual substitution thus the
free variables have to reflect the possibility for substitution.
Naturality in C gives rise to the equation equation explaining
how lets interact with composition.
For the tensoring on the right, we simply multiply (or tensor)
the terms.
since the constant I is the unity for the tensor operation, its
rules are similar.

60. Implication Terms
Are just like usual implication (lambda) terms

61. Implication Terms
Are just like usual implication (lambda) terms
the difference is that the variable that your lambda binds has
to be present in the context and only once

62. Implication Terms
Are just like usual implication (lambda) terms
the difference is that the variable that your lambda binds has
to be present in the context and only once
There is an explanation for why the Yoneda lemma is
responsible for the simplification that one can do in the syntax
(and why Peter Schroeder-Heister is really right about his
formulation of Natural Deduction) but we don’t need to go
there.

63. Modality ! Term Assignment
The left rules are ok. The dereliction rules creates a binder, a
bit like the tensor and the I rules.

64. Modality ! Term Assignment
The left rules are ok. The dereliction rules creates a binder, a
bit like the tensor and the I rules.
The weakening and contraction rules discard and copy
variables, and have naturality conditions as the tensor rule.

65. Modality ! Term Assignment
The left rules are ok. The dereliction rules creates a binder, a
bit like the tensor and the I rules.
The weakening and contraction rules discard and copy
variables, and have naturality conditions as the tensor rule.
The problematic rule, ”promotion”

66. Multiplicative Term Assignment

67. Conclusions so Far...
The categorical model can give you guidance for the shape of the
type theory rules.

68. Conclusions so Far...
The categorical model can give you guidance for the shape of the
type theory rules.
Categorical rules are about full beta and eta equivalence plus some
naturality conditions (not very liked by FPers)

69. Conclusions so Far...
The categorical model can give you guidance for the shape of the
type theory rules.
Categorical rules are about full beta and eta equivalence plus some
naturality conditions (not very liked by FPers)

70. Conclusions so Far...
Newer models plus new term calculi via DILL (Dual Intuitionistic
Linear Logic) and monoidal adjunction models (Benton, Barber,
Mellies, who else?...)

71. Conclusions so Far...
Newer models plus new term calculi via DILL (Dual Intuitionistic
Linear Logic) and monoidal adjunction models (Benton, Barber,
Mellies, who else?...)
Which one to read about?

72. Conclusions so Far...
Newer models plus new term calculi via DILL (Dual Intuitionistic
Linear Logic) and monoidal adjunction models (Benton, Barber,
Mellies, who else?...)
Which one to read about?
Rewriting is a new ball game, which I would like to investigate too
cf. Barney Hilken paper “Towards a proof theory of rewriting: the
simply-typed 2-[lambda] calculus” Theor. Comp. Sci. Vol. 170,
1-2, pp 407-444. 1996.