Linear Type Theory Revisited
Valeria de PaivaNuance Comms
February 21, 2014
Introduction
Goals
I Discuss very old work (early 90’s) on linear type theories
I based on Term Assignment for Intuitionistic Linear Logic.(Benton, Bierman, de Paiva and Hyland). Technical Report262, University of Cambridge Computer Laboratory 1992.
I available fromhttp://www.cs.bham.ac.uk/ vdp/publications/papers.html.
I then get to the state-of-the art...
Goals
I Discuss very old work (early 90’s) on linear type theories
I based on Term Assignment for Intuitionistic Linear Logic.(Benton, Bierman, de Paiva and Hyland). Technical Report262, University of Cambridge Computer Laboratory 1992.
I available fromhttp://www.cs.bham.ac.uk/ vdp/publications/papers.html.
I then get to the state-of-the art...
Goals
I Discuss very old work (early 90’s) on linear type theories
I based on Term Assignment for Intuitionistic Linear Logic.(Benton, Bierman, de Paiva and Hyland). Technical Report262, University of Cambridge Computer Laboratory 1992.
I available fromhttp://www.cs.bham.ac.uk/ vdp/publications/papers.html.
I then get to the state-of-the art...
Goals
I Discuss very old work (early 90’s) on linear type theories
I based on Term Assignment for Intuitionistic Linear Logic.(Benton, Bierman, de Paiva and Hyland). Technical Report262, University of Cambridge Computer Laboratory 1992.
I available fromhttp://www.cs.bham.ac.uk/ vdp/publications/papers.html.
I then get to the state-of-the art...
Summary from Abstract
I Investigate the problem of deriving a term assignment systemfor Girard’s Intuitionistic Linear Logic
I both sequent calculus and natural deduction proof systemsI satisfying two important properties:
I the substitution property (the set of valid deductions is closedunder substitution) and
I subject reduction (reduction on terms is well-typed)
I a categorical model for Intuitionistic Linear Logic and how touse it to derive the term assignment
I long (57 pages) but slow and easy...
Summary from Abstract
I Investigate the problem of deriving a term assignment systemfor Girard’s Intuitionistic Linear Logic
I both sequent calculus and natural deduction proof systems
I satisfying two important properties:I the substitution property (the set of valid deductions is closed
under substitution) andI subject reduction (reduction on terms is well-typed)
I a categorical model for Intuitionistic Linear Logic and how touse it to derive the term assignment
I long (57 pages) but slow and easy...
Summary from Abstract
I Investigate the problem of deriving a term assignment systemfor Girard’s Intuitionistic Linear Logic
I both sequent calculus and natural deduction proof systemsI satisfying two important properties:
I the substitution property (the set of valid deductions is closedunder substitution) and
I subject reduction (reduction on terms is well-typed)
I a categorical model for Intuitionistic Linear Logic and how touse it to derive the term assignment
I long (57 pages) but slow and easy...
Summary from Abstract
I Investigate the problem of deriving a term assignment systemfor Girard’s Intuitionistic Linear Logic
I both sequent calculus and natural deduction proof systemsI satisfying two important properties:
I the substitution property (the set of valid deductions is closedunder substitution) and
I subject reduction (reduction on terms is well-typed)
I a categorical model for Intuitionistic Linear Logic and how touse it to derive the term assignment
I long (57 pages) but slow and easy...
Summary from Abstract
I Investigate the problem of deriving a term assignment systemfor Girard’s Intuitionistic Linear Logic
I both sequent calculus and natural deduction proof systemsI satisfying two important properties:
I the substitution property (the set of valid deductions is closedunder substitution) and
I subject reduction (reduction on terms is well-typed)
I a categorical model for Intuitionistic Linear Logic and how touse it to derive the term assignment
I long (57 pages) but slow and easy...
Introduction
I the problem of deriving a term assignment system for Girard’sIntuitionistic Linear Logic
I Previous approaches have simply annotated the sequentcalculus with terms and have given little or no justification fortheir choice
I Phil Wadler: There’s no substitute for LL
I substitution lemma does not hold for the term assignmentsystem in Abramsky’s ‘Computational Interpretations of LinearLogic’ (Cited by 546)
Introduction
I the problem of deriving a term assignment system for Girard’sIntuitionistic Linear Logic
I Previous approaches have simply annotated the sequentcalculus with terms and have given little or no justification fortheir choice
I Phil Wadler: There’s no substitute for LL
I substitution lemma does not hold for the term assignmentsystem in Abramsky’s ‘Computational Interpretations of LinearLogic’ (Cited by 546)
Introduction
I the problem of deriving a term assignment system for Girard’sIntuitionistic Linear Logic
I Previous approaches have simply annotated the sequentcalculus with terms and have given little or no justification fortheir choice
I Phil Wadler: There’s no substitute for LL
I substitution lemma does not hold for the term assignmentsystem in Abramsky’s ‘Computational Interpretations of LinearLogic’ (Cited by 546)
Introduction
I the problem of deriving a term assignment system for Girard’sIntuitionistic Linear Logic
I Previous approaches have simply annotated the sequentcalculus with terms and have given little or no justification fortheir choice
I Phil Wadler: There’s no substitute for LL
I substitution lemma does not hold for the term assignmentsystem in Abramsky’s ‘Computational Interpretations of LinearLogic’ (Cited by 546)
Digression: Other old work...
I Linear types can change the world
I Is there a use for linear logic?
I A taste of linear logic
I Operational interpretations of linear logic
I Reference counting as a computational interpretation of linearlogic (Chirimar)
Introduction 2
I solving the problem of deriving a term assignment system forGirard’s Intuitionistic Linear Logic:
I Two waysI By considering the sequent calculus formulation of the logic
and using the underlying categorical constructions to suggest aterm assignment system
I By considering a linear natural deduction system
I two approaches produce equivalent term assignment systems
I BUT for equality (via reduction of terms) matters are moresubtle: natural equalities for category theory are stronger thanthose suggested by computational considerations
Introduction 2
I solving the problem of deriving a term assignment system forGirard’s Intuitionistic Linear Logic:
I Two waysI By considering the sequent calculus formulation of the logic
and using the underlying categorical constructions to suggest aterm assignment system
I By considering a linear natural deduction system
I two approaches produce equivalent term assignment systems
I BUT for equality (via reduction of terms) matters are moresubtle: natural equalities for category theory are stronger thanthose suggested by computational considerations
Introduction 2
I solving the problem of deriving a term assignment system forGirard’s Intuitionistic Linear Logic:
I Two waysI By considering the sequent calculus formulation of the logic
and using the underlying categorical constructions to suggest aterm assignment system
I By considering a linear natural deduction system
I two approaches produce equivalent term assignment systems
I BUT for equality (via reduction of terms) matters are moresubtle: natural equalities for category theory are stronger thanthose suggested by computational considerations
Introduction 2
I solving the problem of deriving a term assignment system forGirard’s Intuitionistic Linear Logic:
I Two waysI By considering the sequent calculus formulation of the logic
and using the underlying categorical constructions to suggest aterm assignment system
I By considering a linear natural deduction system
I two approaches produce equivalent term assignment systems
I BUT for equality (via reduction of terms) matters are moresubtle: natural equalities for category theory are stronger thanthose suggested by computational considerations
Outline of TR
I Girard’s Intuitionistic Linear Logic:
I a simple categorical model of Intuitionistic Linear Logic(sequent calculus)
I a linear system of natural deduction
I how our two systems of Intuitionistic Linear Logic are related
I the process of proof normalisation within the linear naturaldeduction system
I model for Intuitionistic Linear Logic
I cut-elimination in the sequent calculus
I conclusions
Outline of TR
I Girard’s Intuitionistic Linear Logic:
I a simple categorical model of Intuitionistic Linear Logic(sequent calculus)
I a linear system of natural deduction
I how our two systems of Intuitionistic Linear Logic are related
I the process of proof normalisation within the linear naturaldeduction system
I model for Intuitionistic Linear Logic
I cut-elimination in the sequent calculus
I conclusions
Outline of TR
I Girard’s Intuitionistic Linear Logic:
I a simple categorical model of Intuitionistic Linear Logic(sequent calculus)
I a linear system of natural deduction
I how our two systems of Intuitionistic Linear Logic are related
I the process of proof normalisation within the linear naturaldeduction system
I model for Intuitionistic Linear Logic
I cut-elimination in the sequent calculus
I conclusions
Outline of TR
I Girard’s Intuitionistic Linear Logic:
I a simple categorical model of Intuitionistic Linear Logic(sequent calculus)
I a linear system of natural deduction
I how our two systems of Intuitionistic Linear Logic are related
I the process of proof normalisation within the linear naturaldeduction system
I model for Intuitionistic Linear Logic
I cut-elimination in the sequent calculus
I conclusions
Outline of TR
I Girard’s Intuitionistic Linear Logic:
I a simple categorical model of Intuitionistic Linear Logic(sequent calculus)
I a linear system of natural deduction
I how our two systems of Intuitionistic Linear Logic are related
I the process of proof normalisation within the linear naturaldeduction system
I model for Intuitionistic Linear Logic
I cut-elimination in the sequent calculus
I conclusions
Outline of TR
I Girard’s Intuitionistic Linear Logic:
I a simple categorical model of Intuitionistic Linear Logic(sequent calculus)
I a linear system of natural deduction
I how our two systems of Intuitionistic Linear Logic are related
I the process of proof normalisation within the linear naturaldeduction system
I model for Intuitionistic Linear Logic
I cut-elimination in the sequent calculus
I conclusions
Outline of TR
I Girard’s Intuitionistic Linear Logic:
I a simple categorical model of Intuitionistic Linear Logic(sequent calculus)
I a linear system of natural deduction
I how our two systems of Intuitionistic Linear Logic are related
I the process of proof normalisation within the linear naturaldeduction system
I model for Intuitionistic Linear Logic
I cut-elimination in the sequent calculus
I conclusions
Outline of TR
I Girard’s Intuitionistic Linear Logic:
I a simple categorical model of Intuitionistic Linear Logic(sequent calculus)
I a linear system of natural deduction
I how our two systems of Intuitionistic Linear Logic are related
I the process of proof normalisation within the linear naturaldeduction system
I model for Intuitionistic Linear Logic
I cut-elimination in the sequent calculus
I conclusions
Intuitionistic Linear Logic
Recommended if sequent calculus/ND are not part of your vocab:Jean Gallier: Constructive Logics. Part I: A Tutorial on ProofSystems and Typed lambda-Calculi. Theoretical ComputerScience, 110(2), 249-239 (1993). fromhttp://www.cis.upenn.edu/ jean/gbooks/logic.html
I multiplicative fragment of Intuitionistic Linear Logic (ILL)
I a refinement of intuitionistic logic (IL) where formulae mustbe used exactly once
I Weakening and Contraction rules are removed
I To regain the expressive power, rules returned in a controlledmanner using operator “!”
I “!” similar to the modal necessity operator �
Intuitionistic Linear Logic
Recommended if sequent calculus/ND are not part of your vocab:Jean Gallier: Constructive Logics. Part I: A Tutorial on ProofSystems and Typed lambda-Calculi. Theoretical ComputerScience, 110(2), 249-239 (1993). fromhttp://www.cis.upenn.edu/ jean/gbooks/logic.html
I multiplicative fragment of Intuitionistic Linear Logic (ILL)
I a refinement of intuitionistic logic (IL) where formulae mustbe used exactly once
I Weakening and Contraction rules are removed
I To regain the expressive power, rules returned in a controlledmanner using operator “!”
I “!” similar to the modal necessity operator �
Intuitionistic Linear Logic
Recommended if sequent calculus/ND are not part of your vocab:Jean Gallier: Constructive Logics. Part I: A Tutorial on ProofSystems and Typed lambda-Calculi. Theoretical ComputerScience, 110(2), 249-239 (1993). fromhttp://www.cis.upenn.edu/ jean/gbooks/logic.html
I multiplicative fragment of Intuitionistic Linear Logic (ILL)
I a refinement of intuitionistic logic (IL) where formulae mustbe used exactly once
I Weakening and Contraction rules are removed
I To regain the expressive power, rules returned in a controlledmanner using operator “!”
I “!” similar to the modal necessity operator �
Intuitionistic Linear Logic
Recommended if sequent calculus/ND are not part of your vocab:Jean Gallier: Constructive Logics. Part I: A Tutorial on ProofSystems and Typed lambda-Calculi. Theoretical ComputerScience, 110(2), 249-239 (1993). fromhttp://www.cis.upenn.edu/ jean/gbooks/logic.html
I multiplicative fragment of Intuitionistic Linear Logic (ILL)
I a refinement of intuitionistic logic (IL) where formulae mustbe used exactly once
I Weakening and Contraction rules are removed
I To regain the expressive power, rules returned in a controlledmanner using operator “!”
I “!” similar to the modal necessity operator �
Intuitionistic Linear Logic
Recommended if sequent calculus/ND are not part of your vocab:Jean Gallier: Constructive Logics. Part I: A Tutorial on ProofSystems and Typed lambda-Calculi. Theoretical ComputerScience, 110(2), 249-239 (1993). fromhttp://www.cis.upenn.edu/ jean/gbooks/logic.html
I multiplicative fragment of Intuitionistic Linear Logic (ILL)
I a refinement of intuitionistic logic (IL) where formulae mustbe used exactly once
I Weakening and Contraction rules are removed
I To regain the expressive power, rules returned in a controlledmanner using operator “!”
I “!” similar to the modal necessity operator �
Intuitionistic Linear Logic Rules
Generic Categorical considerations
I sequent calculus: providing not proofs themselves but ameta-theory concerning proofs
I fundamental idea of categorical treatment of proof theoryI propositions interpreted as objects of a category (or
multicategory or polycategory)
I proofs interpreted as maps of the categoryI operations transforming proofs into proofs then correspond (if
possible) to natural transformations between appropriatehom-functors
Generic Categorical considerations
I sequent calculus: providing not proofs themselves but ameta-theory concerning proofs
I fundamental idea of categorical treatment of proof theoryI propositions interpreted as objects of a category (or
multicategory or polycategory)I proofs interpreted as maps of the category
I operations transforming proofs into proofs then correspond (ifpossible) to natural transformations between appropriatehom-functors
Generic Categorical considerations
I sequent calculus: providing not proofs themselves but ameta-theory concerning proofs
I fundamental idea of categorical treatment of proof theoryI propositions interpreted as objects of a category (or
multicategory or polycategory)I proofs interpreted as maps of the categoryI operations transforming proofs into proofs then correspond (if
possible) to natural transformations between appropriatehom-functors
Categorical considerationsI dealing with sequents Γ ` A in principle we should deal with
multicategories
I simplifying assumption: multicategorical structure representedby a tensor product .
I a sequent of form
C1,C2, . . . ,Cn ` A
will be represented by
C1 ⊗ C2 ⊗ . . .⊗ Cn → A
sometimes written as Γ→ AI (a coherence result is assumed)I 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.
Categorical considerationsI dealing with sequents Γ ` A in principle we should deal with
multicategoriesI simplifying assumption: multicategorical structure represented
by a tensor product .
I a sequent of form
C1,C2, . . . ,Cn ` A
will be represented by
C1 ⊗ C2 ⊗ . . .⊗ Cn → A
sometimes written as Γ→ AI (a coherence result is assumed)I 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.
Categorical considerationsI dealing with sequents Γ ` A in principle we should deal with
multicategoriesI simplifying assumption: multicategorical structure represented
by a tensor product .I a sequent of form
C1,C2, . . . ,Cn ` A
will be represented by
C1 ⊗ C2 ⊗ . . .⊗ Cn → A
sometimes written as Γ→ A
I (a coherence result is assumed)I 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.
Categorical considerationsI dealing with sequents Γ ` A in principle we should deal with
multicategoriesI simplifying assumption: multicategorical structure represented
by a tensor product .I a sequent of form
C1,C2, . . . ,Cn ` A
will be represented by
C1 ⊗ C2 ⊗ . . .⊗ Cn → A
sometimes written as Γ→ AI (a coherence result is assumed)
I We seek to enrich the sequent judgement to a termassignment judgement of the form
x1 : C1, x2 : C2, . . . , xn : Cn ` e : A
where the xi are distinct variables and e is a term.
Categorical considerationsI dealing with sequents Γ ` A in principle we should deal with
multicategoriesI simplifying assumption: multicategorical structure represented
by a tensor product .I a sequent of form
C1,C2, . . . ,Cn ` A
will be represented by
C1 ⊗ C2 ⊗ . . .⊗ Cn → A
sometimes written as Γ→ AI (a coherence result is assumed)I 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.
Identity and Cut
I The sequent representing the Identity rule is interpreted as thecanonical identity arrow idA : A→ A
I The corresponding rule of term formation is x : A ` x : A
I The rule of Exchange we interpret by assuming that we have asymmetry for the tensor product
I We suppress Exchange and the corresponding symmetry,considering multisets of formulae, so no term formingoperations result from this rule (others do diff...)
I The cut rule is interpreted as a generalized form ofcomposition. if the maps f : Γ→ A and g : A,∆→ B are theinterpretations of hypotheses of the rule, then the compositeΓ⊗∆→f⊗1∆ A⊗∆→g B is the interpretation of theconclusion
Identity and Cut
I The sequent representing the Identity rule is interpreted as thecanonical identity arrow idA : A→ A
I The corresponding rule of term formation is x : A ` x : A
I The rule of Exchange we interpret by assuming that we have asymmetry for the tensor product
I We suppress Exchange and the corresponding symmetry,considering multisets of formulae, so no term formingoperations result from this rule (others do diff...)
I The cut rule is interpreted as a generalized form ofcomposition. if the maps f : Γ→ A and g : A,∆→ B are theinterpretations of hypotheses of the rule, then the compositeΓ⊗∆→f⊗1∆ A⊗∆→g B is the interpretation of theconclusion
Identity and Cut
I The sequent representing the Identity rule is interpreted as thecanonical identity arrow idA : A→ A
I The corresponding rule of term formation is x : A ` x : A
I The rule of Exchange we interpret by assuming that we have asymmetry for the tensor product
I We suppress Exchange and the corresponding symmetry,considering multisets of formulae, so no term formingoperations result from this rule (others do diff...)
I The cut rule is interpreted as a generalized form ofcomposition. if the maps f : Γ→ A and g : A,∆→ B are theinterpretations of hypotheses of the rule, then the compositeΓ⊗∆→f⊗1∆ A⊗∆→g B is the interpretation of theconclusion
Identity and Cut
I The sequent representing the Identity rule is interpreted as thecanonical identity arrow idA : A→ A
I The corresponding rule of term formation is x : A ` x : A
I The rule of Exchange we interpret by assuming that we have asymmetry for the tensor product
I We suppress Exchange and the corresponding symmetry,considering multisets of formulae, so no term formingoperations result from this rule (others do diff...)
I The cut rule is interpreted as a generalized form ofcomposition. if the maps f : Γ→ A and g : A,∆→ B are theinterpretations of hypotheses of the rule, then the compositeΓ⊗∆→f⊗1∆ A⊗∆→g B is the interpretation of theconclusion
Identity and Cut
I The sequent representing the Identity rule is interpreted as thecanonical identity arrow idA : A→ A
I The corresponding rule of term formation is x : A ` x : A
I The rule of Exchange we interpret by assuming that we have asymmetry for the tensor product
I We suppress Exchange and the corresponding symmetry,considering multisets of formulae, so no term formingoperations result from this rule (others do diff...)
I The cut rule is interpreted as a generalized form ofcomposition. if the maps f : Γ→ A and g : A,∆→ B are theinterpretations of hypotheses of the rule, then the compositeΓ⊗∆→f⊗1∆ A⊗∆→g B is the interpretation of theconclusion
Substitution in Terms...
I Assumption: any logical rule corresponds to an operation onmaps of the category which is natural in the interpretations ofthe components of the sequents which remain unchangedduring the application of a rule
I Composition corresponds to Cut: our operations commutewhere appropriate with Cut.
I Composition is interpreted by textual substitution thus thefree variables have to reflect the possibility for substitution.
I The rules for the constant I are parallel to the rules for thetensor product ⊗, which we describe next.
I The rules for (linear) implication are usual.
I The rules for the modality ! are the hard ones. these slides will
get us to page 12 of the report...
Substitution in Terms...
I Assumption: any logical rule corresponds to an operation onmaps of the category which is natural in the interpretations ofthe components of the sequents which remain unchangedduring the application of a rule
I Composition corresponds to Cut: our operations commutewhere appropriate with Cut.
I Composition is interpreted by textual substitution thus thefree variables have to reflect the possibility for substitution.
I The rules for the constant I are parallel to the rules for thetensor product ⊗, which we describe next.
I The rules for (linear) implication are usual.
I The rules for the modality ! are the hard ones. these slides will
get us to page 12 of the report...
Substitution in Terms...
I Assumption: any logical rule corresponds to an operation onmaps of the category which is natural in the interpretations ofthe components of the sequents which remain unchangedduring the application of a rule
I Composition corresponds to Cut: our operations commutewhere appropriate with Cut.
I Composition is interpreted by textual substitution thus thefree variables have to reflect the possibility for substitution.
I The rules for the constant I are parallel to the rules for thetensor product ⊗, which we describe next.
I The rules for (linear) implication are usual.
I The rules for the modality ! are the hard ones. these slides will
get us to page 12 of the report...
Substitution in Terms...
I Assumption: any logical rule corresponds to an operation onmaps of the category which is natural in the interpretations ofthe components of the sequents which remain unchangedduring the application of a rule
I Composition corresponds to Cut: our operations commutewhere appropriate with Cut.
I Composition is interpreted by textual substitution thus thefree variables have to reflect the possibility for substitution.
I The rules for the constant I are parallel to the rules for thetensor product ⊗, which we describe next.
I The rules for (linear) implication are usual.
I The rules for the modality ! are the hard ones. these slides will
get us to page 12 of the report...
Substitution in Terms...
I Assumption: any logical rule corresponds to an operation onmaps of the category which is natural in the interpretations ofthe components of the sequents which remain unchangedduring the application of a rule
I Composition corresponds to Cut: our operations commutewhere appropriate with Cut.
I Composition is interpreted by textual substitution thus thefree variables have to reflect the possibility for substitution.
I The rules for the constant I are parallel to the rules for thetensor product ⊗, which we describe next.
I The rules for (linear) implication are usual.
I The rules for the modality ! are the hard ones. these slides will
get us to page 12 of the report...
Substitution in Terms...
I Assumption: any logical rule corresponds to an operation onmaps of the category which is natural in the interpretations ofthe components of the sequents which remain unchangedduring the application of a rule
I Composition corresponds to Cut: our operations commutewhere appropriate with Cut.
I Composition is interpreted by textual substitution thus thefree variables have to reflect the possibility for substitution.
I The rules for the constant I are parallel to the rules for thetensor product ⊗, which we describe next.
I The rules for (linear) implication are usual.
I The rules for the modality ! are the hard ones. these slides will
get us to page 12 of the report...
Text substitution Terms
use this and BB to get equation (2)
Tensor Terms
I Abramsky introduced a let constructor for tensor products, areasonable syntax to bind variables x and y in the term f .
I Composition is interpreted by textual substitution thus thefree variables have to reflect the possibility for substitution.
I Naturality in C gives rise to the equation equation explaininghow lets interact with composition.
I For the tensoring on the right, we simply multiply (or tensor)the terms.
I since the constant I is the unity for the tensor operation, itsrules are similar.
Tensor Terms
I Abramsky introduced a let constructor for tensor products, areasonable syntax to bind variables x and y in the term f .
I Composition is interpreted by textual substitution thus thefree variables have to reflect the possibility for substitution.
I Naturality in C gives rise to the equation equation explaininghow lets interact with composition.
I For the tensoring on the right, we simply multiply (or tensor)the terms.
I since the constant I is the unity for the tensor operation, itsrules are similar.
Tensor Terms
I Abramsky introduced a let constructor for tensor products, areasonable syntax to bind variables x and y in the term f .
I Composition is interpreted by textual substitution thus thefree variables have to reflect the possibility for substitution.
I Naturality in C gives rise to the equation equation explaininghow lets interact with composition.
I For the tensoring on the right, we simply multiply (or tensor)the terms.
I since the constant I is the unity for the tensor operation, itsrules are similar.
Tensor Terms
I Abramsky introduced a let constructor for tensor products, areasonable syntax to bind variables x and y in the term f .
I Composition is interpreted by textual substitution thus thefree variables have to reflect the possibility for substitution.
I Naturality in C gives rise to the equation equation explaininghow lets interact with composition.
I For the tensoring on the right, we simply multiply (or tensor)the terms.
I since the constant I is the unity for the tensor operation, itsrules are similar.
Tensor Terms
I Abramsky introduced a let constructor for tensor products, areasonable syntax to bind variables x and y in the term f .
I Composition is interpreted by textual substitution thus thefree variables have to reflect the possibility for substitution.
I Naturality in C gives rise to the equation equation explaininghow lets interact with composition.
I For the tensoring on the right, we simply multiply (or tensor)the terms.
I since the constant I is the unity for the tensor operation, itsrules are similar.
Implication Terms
I Are just like usual implication (lambda) terms
I the difference is that the variable that your lambda binds hasto be present in the context and only once
I There is an explanation for why the Yoneda lemma isresponsible for the simplification that one can do in the syntax(and why Peter Schroeder-Heister is really right about hisformulation of Natural Deduction) but we don’t need to gothere.
Implication Terms
I Are just like usual implication (lambda) terms
I the difference is that the variable that your lambda binds hasto be present in the context and only once
I There is an explanation for why the Yoneda lemma isresponsible for the simplification that one can do in the syntax(and why Peter Schroeder-Heister is really right about hisformulation of Natural Deduction) but we don’t need to gothere.
Implication Terms
I Are just like usual implication (lambda) terms
I the difference is that the variable that your lambda binds hasto be present in the context and only once
I There is an explanation for why the Yoneda lemma isresponsible for the simplification that one can do in the syntax(and why Peter Schroeder-Heister is really right about hisformulation of Natural Deduction) but we don’t need to gothere.
Modality ! Term Assignment
I The left rules are ok. The dereliction rules creates a binder, abit like the tensor and the I rules.
I The weakening and contraction rules discard and copyvariables, and have naturality conditions as the tensor rule.
I The problematic rule, ”promotion”
Modality ! Term Assignment
I The left rules are ok. The dereliction rules creates a binder, abit like the tensor and the I rules.
I The weakening and contraction rules discard and copyvariables, and have naturality conditions as the tensor rule.
I The problematic rule, ”promotion”
Modality ! Term Assignment
I The left rules are ok. The dereliction rules creates a binder, abit like the tensor and the I rules.
I The weakening and contraction rules discard and copyvariables, and have naturality conditions as the tensor rule.
I The problematic rule, ”promotion”
Multiplicative Term Assignment
Conclusions so Far...
I The categorical model can give you guidance for the shape of thetype theory rules.
I Categorical rules are about full beta and eta equivalence plus somenaturality conditions (not very liked by FPers)
I
Conclusions so Far...
I The categorical model can give you guidance for the shape of thetype theory rules.
I Categorical rules are about full beta and eta equivalence plus somenaturality conditions (not very liked by FPers)
I
Conclusions so Far...
I The categorical model can give you guidance for the shape of thetype theory rules.
I Categorical rules are about full beta and eta equivalence plus somenaturality conditions (not very liked by FPers)
I
Conclusions so Far...
I Newer models plus new term calculi via DILL (Dual IntuitionisticLinear Logic) and monoidal adjunction models (Benton, Barber,Mellies, who else?...)
I Which one to read about?
I Rewriting is a new ball game, which I would like to investigate toocf. Barney Hilken paper “Towards a proof theory of rewriting: thesimply-typed 2-[lambda] calculus” Theor. Comp. Sci. Vol. 170,1-2, pp 407-444. 1996.
Conclusions so Far...
I Newer models plus new term calculi via DILL (Dual IntuitionisticLinear Logic) and monoidal adjunction models (Benton, Barber,Mellies, who else?...)
I Which one to read about?
I Rewriting is a new ball game, which I would like to investigate toocf. Barney Hilken paper “Towards a proof theory of rewriting: thesimply-typed 2-[lambda] calculus” Theor. Comp. Sci. Vol. 170,1-2, pp 407-444. 1996.
Conclusions so Far...
I Newer models plus new term calculi via DILL (Dual IntuitionisticLinear Logic) and monoidal adjunction models (Benton, Barber,Mellies, who else?...)
I Which one to read about?
I Rewriting is a new ball game, which I would like to investigate toocf. Barney Hilken paper “Towards a proof theory of rewriting: thesimply-typed 2-[lambda] calculus” Theor. Comp. Sci. Vol. 170,1-2, pp 407-444. 1996.