Linear Logic and Constructive Mathematics, after Shulman
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This document discusses linear logic and its relationship to constructive mathematics. It begins by providing background on intuitionistic logic, classical logic, and the Brouwer-Heyting-Kolmogorov interpretation of intuitionistic connectives. It then introduces linear logic and discusses how linear logic can provide a better account of negation and proofs by contradiction for constructive mathematics. The document explores models of linear logic using Heyting algebras and the dialectica construction. It considers how linear logic may help explain traditionally non-constructive ideas in mathematics.
Linear Logic and Constructive Mathematics, after Shulman
- 1. 1/35 Introduction BHK Algebra Constructivism Linear Logic and Constructive Mathematics (Algebraic Dialectica for Logicians) Valeria de Paiva Topos Institute 21 de abril de 2021 Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 2. 2/35 Introduction BHK Algebra Constructivism Thanks Luiz Carlos! Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 3. 3/35 Introduction BHK Algebra Constructivism The courage of our convictions This discussion is mostly based on reading Mike Shulman’s ‘Linear Logic for Constructive Mathematics’. I am grateful to Mike for ideas and even slides. (the mistakes are my own, of course) I want to talk about an algebraic version of the dialectica construction, but to do that first classical vs. intuitionistic logic intuitionistic vs. linear logic linear mathematics? Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 4. 3/35 Introduction BHK Algebra Constructivism The courage of our convictions This discussion is mostly based on reading Mike Shulman’s ‘Linear Logic for Constructive Mathematics’. I am grateful to Mike for ideas and even slides. (the mistakes are my own, of course) I want to talk about an algebraic version of the dialectica construction, but to do that first classical vs. intuitionistic logic intuitionistic vs. linear logic linear mathematics? Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 5. 3/35 Introduction BHK Algebra Constructivism The courage of our convictions This discussion is mostly based on reading Mike Shulman’s ‘Linear Logic for Constructive Mathematics’. I am grateful to Mike for ideas and even slides. (the mistakes are my own, of course) I want to talk about an algebraic version of the dialectica construction, but to do that first classical vs. intuitionistic logic intuitionistic vs. linear logic linear mathematics? Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 6. 4/35 Introduction BHK Algebra Constructivism A Hundred Years Ago Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 7. 5/35 Introduction BHK Algebra Constructivism A Hundred Years Ago Hilbert (1927) ”To prohibit existence statements and the principle of excluded middle is tantamount to relinquishing the science of mathematics altogether.”Brouwer-Hilbert controversy (from wikipedia) Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 8. 6/35 Introduction BHK Algebra Constructivism Almost Fifty Years Ago Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 9. 7/35 Introduction BHK Algebra Constructivism More than Thirty Years Ago Girard shook the basis of logic several times “Broccoli logic”is still one of enduring jokes in the internet Linear Logic has been very influential Out of fashion now? Linear thinking and variations permeated logic and theoretical computing Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 10. 8/35 Introduction BHK Algebra Constructivism Categorical Proof Theory Types are formulae/objects in appropriate category, Terms/programs are proofs/morphisms in the category, Logical constructors are appropriate categorical constructions. Most important: Reduction is proof normalization (Tait) Outcome: Transfer results/tools from logic to CT to CSci How far can we push it? Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 11. 9/35 Introduction BHK Algebra Constructivism Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 12. 10/35 Introduction BHK Algebra Constructivism papers: Term calculus for intuitionistic linear logic (BBdePH1993), Term assignment for ILL (TR1992) and Linear λ-calculus and categorical models revisited (CSL1992) Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 13. 11/35 Introduction BHK Algebra Constructivism Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 14. 12/35 Introduction BHK Algebra Constructivism Intuitionistic Logic Brouwer wanted to eliminate non-constructive proofs. Heyting formulated intuitionistic logic where all valid proofs are necessarily constructive. Kolmogorov, Glivenko, Weyl, Bishop, and many others developed constructive maths http://dx.doi.org/10.1090/bull/1556 FIVE STAGES OF ACCEPTING CONSTRUCTIVE MATHEMATICS Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 15. 13/35 Introduction BHK Algebra Constructivism Intuitionistic Logic Proof by contradiction is not allowed a statement can be ’not false’ without being true: ¬¬P does not imply P De Morgan’s laws hold* except ¬(P ∧ Q) → (¬P ∨ ¬Q) Similarly,¬∀x.P(x) does not imply ∃x.¬P(x) The law of excluded middle P ∨ ¬P does not hold in general The three connectives ∧, ∨, → are independent: neither can be defined in terms of the others Negation is a defined connective ¬A := A → ⊥ Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 16. 14/35 Introduction BHK Algebra Constructivism Brouwer-Heyting-Kolmogorov (BHK) interpretation This an informal description of the meanings of intuitionistic connectives in terms of what counts as a proof of them A proof of P ∧ Q is a proof of P and a proof of Q A proof of P ∨ Q is a proof of P or a proof of Q (plus a marker of which one it is) A proof of P → Q is a construction transforming any proof of P into a proof of Q Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 17. 15/35 Introduction BHK Algebra Constructivism BHK interpretation of Negation Intuitionism defines ¬P to be P → ⊥, A proof of ¬P is a construction transforming any proof of P into a proof of a contradiction. This explains the properties of negation in intuitionistic logic: If it would be contradictory to have a construction transforming any proof of P into a contradiction, it does not follow that we have a proof of P. Hence ¬¬P does not imply P For an arbitrary P, we can not claim to have either a proof of P or a construction transforming any proof of P into a contradiction. (E.g. P might be the Riemann hypothesis.) So P ∨ ¬P does not hold. Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 18. 16/35 Introduction BHK Algebra Constructivism Is there a better Negation? Girard’s idea: formal de Morgan dual, the negation in LL ¬(P ∨ Q) =def ¬P ∧ ¬Q ¬(P ∧ Q) =def ¬P ∨ ¬Q ¬∃x.P(x) =def ∀x.¬P(x) ¬∀x.P(x) =def ∃x.¬P(x) A constructive proof of ∃x.P(x) must provide an example A constructive disproofof ∀x.P(x) should provide a counterexample Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 19. 17/35 Introduction BHK Algebra Constructivism Is there a better Negation? Shulman’s bold idea: LL’s involutive negation solves many of the issues of intuitionistic negation in mathematics To prove ∃x.P(x) by contradiction, we assume its negation ∀x.¬P(x). But in order to use this hypothesis at all, we have to apply it to some x! we are necessarily constructing something. Hence an involutive negation makes proofs by contradiction less objectionable Moreover, he produces examples showing that traditional uses of non-constructivity are disallowed and that convoluted ideas like ’apartness’ can be better explained in linear terms Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 20. 18/35 Introduction BHK Algebra Constructivism Linear Logic for Constructive Mathematics We divide the hypotheses into linear and nonlinear ones. The linear ones can only be used once in the course of a proof. All ‘hypotheses for contradiction’ in a proof by contradiction are linear hypotheses. Similarly, P −◦ Q is a linear implication that uses P only once. It is contraposable, ¬(P −◦ Q) = (¬Q −◦ ¬P) (here we’re talking about bi-implications) Linearity is the default status of assertions. We mark the nonlinear hypotheses with a modality, !P. Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 21. 19/35 Introduction BHK Algebra Constructivism Not so classical disjunctions In classical logic, (P ∨ Q) = (¬P → Q) = (¬Q → P). This is no longer true in intuitionistic logic. (connectives are independent) It also fails in linear logic for the ‘constructive’ disjunction ∨. classical Linear Logic does have (¬P −◦ Q) = (¬Q −◦ P), defining another kind of disjunction that is weaker than ∨. in classical linear logic P ` Q = (¬P −◦ Q) = (¬Q −◦ P). in CLL ∨-excluded middle P ∨ ¬P fails. But par-excluded middle (P ` ¬P) = (¬P −◦ ¬P) is a tautology. ∨ supports proof by cases; ` supports the disjunctive syllogism in FILL no excluded middle, 5 independent connectives Still the case that ”For an arbitrary P, we can not claim to have either a proof of P or a construction transforming any proof of P into a contradiction.” Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 22. 20/35 Introduction BHK Algebra Constructivism BHK for Linear Logic The BHK interpretation privileges proofs over refutations. A proof of P ∧ Q is a proof of P and a proof of Q. A refutation of P ∧ Q is a refutation of P or a refutation of Q A proof of P ∨ Q is a proof of P or a proof of Q. A refutation of P ∨ Q is a refutation of P and a refutation of Q. A proof of P ` Q is a construction transforming any refutation of P into a proof of Q, and any refutation of Q into a proof of P. A refutation of P ` Q is a refutation of P and a refutation of Q. A proof of P −◦ Q is a construction transforming any proof of P into a proof of Q, and a construction transforming any refutation of Q into a refutation of P. A refutation of P −◦ Q is a proof of P and a refutation of Q. Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 23. 21/35 Introduction BHK Algebra Constructivism Making this BHK more formal? Shulman’s sections: intuitionistic logic, linear logic, the standard interpretation, the hidden linear nature of constructive mathematics I cannot judge how good the linear logic modifications are for constructive mathematics, but I do have issues with what he calls the ’standard interpretation’. The Dialecica version over a Heyting algebra H and 0 is as good as the Chu construction Mike says ‘If constructive logic is the logic of affirmative propositions, then affine logic is the logic of propositions that are subject to both affirmation and refutation, and the Chu construction is the canonical embedding of the former in the latter. Why canonical? ‘in the Dialectica interpretation the forwards and backwards information is explicitly carried by functions, rather than proofs as in the Chu construction.’ hmm? Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 24. 22/35 Introduction BHK Algebra Constructivism Heyting algebras Definition A Heyting algebra is a cartesian closed lattice, i.e. a poset (H, ≤) with A top element > and bottom element ⊥, Meets P ∧ Q and joins P ∨ Q, An ’implication’ with (P ∧ Q) ≤ R iff P ≤ (Q → R) Heyting algebras are the algebraic semantics of intuitionistic propositional logic, just like Boolean algebras are for classical logic. For algebraic semantics of CLL/ILL/FILL a bit more complicated. I talked about lineales, which are simply posetal symmetric monoidal closed categories. Shulman wants to force units of tensor and product to be the same Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 25. 23/35 Introduction BHK Algebra Constructivism Dejà vu? Theorem (de Paiva CTCS1989) For any Heyting algebra H, consider the algebra Dial⊥(H): Elements are pairs P = (P+, P−) where P+, P− ∈ H and P+ ∧ P− = ⊥. (Think P+ = proofs, P− = refutations) Define P ≤ Q to mean (P+ ≤ Q+ and Q− ≤ P−) P ∧ Q = (P+ ∧ Q+, P− ∨ Q−) and P ∨ Q = (P+ ∨ Q+, P− ∧ Q−) and > = (>, ⊥) and ⊥ = (⊥, >) P ⊗ Q = (P+ ∧ Q+, (P+ → Q−) ∧ (Q+ → P−)) P ` Q = ((P+ → Q+) ∧ (Q+ → P+), P− ∧ Q−) P −◦ Q = ((P+ → Q+) ∧ (Q− → P−), P+ ∧ Q−) Then Dial⊥(H) is a model of Linear Logic (without exponentials). Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 26. 24/35 Introduction BHK Algebra Constructivism Bang Modality Digression: The other theorem of CTCS1989... For any Heyting algebra H which has free co-commutative monoids we can define a !-comonad that makes Dial⊥(H) is a model of IL. Too complicated? Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 27. 25/35 Introduction BHK Algebra Constructivism Back to constructivism: Mike says Girard was interested in Proof nets, Geometry of Interactions, Games, Ludics, etc Linear logicians were interested in having both LL and IL, constructivists use DTT Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 28. 26/35 Introduction BHK Algebra Constructivism Back to constructivism 6 and 7 get my money! Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 29. 27/35 Introduction BHK Algebra Constructivism Back to constructivism Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 30. 28/35 Introduction BHK Algebra Constructivism Back to constructivism Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 31. 29/35 Introduction BHK Algebra Constructivism Back to constructivism Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 32. 30/35 Introduction BHK Algebra Constructivism Back to constructivism Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 33. 31/35 Introduction BHK Algebra Constructivism Back to constructivism Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 34. 32/35 Introduction BHK Algebra Constructivism Intuition for Dialectica objects? Blass makes the case for thinking of problems in computational complexity. Intuitively an object of the dialectica construction A = (U, X, α) can be seen as representing a problem. The elements of U are instances of the problem, while the elements of X are possible answers to the problem instances. The relation α says whether the answer is correct for that instance of the problem or not. LL4CM only considers objects of the form (P+, P−) of proofs and refutations, the relation is always contradiction ⊥. Presumably sometimes one wants to have different relations... Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 35. 33/35 Introduction BHK Algebra Constructivism Examples of objects in Dialectica 1. The object (N, N, =) where n is related to m iff n = m. 2. The object (NN, N, α) where f is α-related to n iff f (n) = n. 3. The object (R, R, ≤) where r1 and r2 are related iff r1 ≤ r2 4. The objects (2, 2, =) and (2, 2, 6=) with usual equality and inequality. Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 36. 34/35 Introduction BHK Algebra Constructivism Conclusions Introduced you to Shulman’s bold idea of doing constructive mathematics with linear logic. Don’t see the canonicity of Chu’s construction. Believe FILL and Dial⊥(H) work just as well and have an associated linear λ-calculus Hinted at its importance for interdisciplinarity: Category Theory, Proofs and Programs Much more work needed for applications, LinearLean anyone? In particular work needed on connecting LL+IL with classical logic. Ecumenical logic ftw! Thank you! Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics
- 37. 35/35 Introduction BHK Algebra Constructivism Some References N.Benton, A mixed linear and non-linear logic: Proofs, terms and models. Computer Science Logic, CSL, (1994). A.Blass, Questions and Answers: A Category Arising in Linear Logic, Complexity Theory, and Set Theory, Advances in Linear Logic, London Math. Soc. Lecture Notes 222 (1995). de Paiva, The Dialectica Categories, Technical Report, Computer Lab, University of Cambridge, number 213, (1991). de Paiva, A dialectica-like model of linear logic, Category Theory and Computer Science, Springer, (1989) 341–356. de Paiva, The Dialectica Categories, In Proc of Categories in Computer Science and Logic, Boulder, CO, 1987. Contemporary Mathematics, vol 92, American Mathematical Society, 1989 (eds. J. Gray and A. Scedrov) Valeria de Paiva Topos Institute Linear Logic and Constructive Mathematics