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Edwardian Proofs as futuristic Programs for Personal Assistants

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Invited Talk at North American Association of Symbolic Logic Meeting, Boulder, CO, May 2014

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Edwardian Proofs as futuristic Programs for Personal Assistants

1. 1. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Edwardian Proofs as Futuristic Programs for Personal Assistants Valeria de Paiva Nuance Communications, CA May, 2014 Valeria de Paiva ASL 2014 – Boulder, CO
2. 2. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Thanks!... Valeria de Paiva ASL 2014 – Boulder, CO
3. 3. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Introduction I’m a logician, a proof-theorist and a category theorist. I work in industry, have done so for the last 15 years, applying the purest of pure mathematics, in surprising ways. Today I want to show you what I think is a most under-appreciated piece of mathematics on the 20th century. The Curry-Howard Correspondence Categorical Proof Theory (as much as time permits) my small part on that... Valeria de Paiva ASL 2014 – Boulder, CO
4. 4. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Introduction I’m a logician, a proof-theorist and a category theorist. I work in industry, have done so for the last 15 years, applying the purest of pure mathematics, in surprising ways. Today I want to show you what I think is a most under-appreciated piece of mathematics on the 20th century. The Curry-Howard Correspondence Categorical Proof Theory (as much as time permits) my small part on that... Valeria de Paiva ASL 2014 – Boulder, CO
5. 5. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Introduction I’m a logician, a proof-theorist and a category theorist. I work in industry, have done so for the last 15 years, applying the purest of pure mathematics, in surprising ways. Today I want to show you what I think is a most under-appreciated piece of mathematics on the 20th century. The Curry-Howard Correspondence Categorical Proof Theory (as much as time permits) my small part on that... Valeria de Paiva ASL 2014 – Boulder, CO
6. 6. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Mathematics is full of surprises... It often happens that there are similarities between the solutions to problems. Sometimes, these similarities point to more general phenomena that simultaneously explain several diﬀerent pieces of mathematics. These more general phenomena can be very diﬃcult to discover, but when they are discovered, they have a very important simplifying and organizing role, and can lead to the solutions of further problems, or raise new and fascinating questions. – T. Gowers, The Importance of Mathematics, 2000 Valeria de Paiva ASL 2014 – Boulder, CO
7. 7. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Proofs are Programs? The bulk of mathematics today got crystallized in the last years of the 19th century, ﬁrst years of the 20th century. The shock is still being felt. A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today Frank Quinn (Notices of the AMS, Jan 2012) Today: the relationship between Algebra, Proofs and Programs Valeria de Paiva ASL 2014 – Boulder, CO
8. 8. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Birth of Algebra [...] a fundamental shift occurred in mathematics from about 1880 to 1940–the consideration of a wide variety of mathematical ”structures,”deﬁned axiomatically and studied both individually and as the classes of structures satisfying those axioms. This approach is so common now that it is almost superﬂuous to mention it explicitly, but it represented a major conceptual shift in answering the question: What is mathematics? The axiomatization of Linear Algebra, Moore, Historia Mathematica, 1995. Valeria de Paiva ASL 2014 – Boulder, CO
9. 9. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Edwardian Algebra Bourbaki on Abstract Algebra The axiomatization of algebra was begun by Dedekind and Hilbert, and then vigorously pursued by Steinitz (1910). It was then completed in the years following 1920 by Artin, Noether and their colleagues at G¨ottingen (Hasse, Krull, Schreier, van der Waerden). It was presented to the world in complete form by van der Waerden’s book (1930). Valeria de Paiva ASL 2014 – Boulder, CO
10. 10. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Bourbaki didn’t say: Algebra became Category Theory... Category Theory: there’s an underlying unity of mathematical concepts/theories. More important than the mathematical concepts themselves is how they relate to each other. Topological spaces come with continuous maps, while vector spaces come with linear transformations. Morphisms: how structures transform into others in the (most reasonable) way to organize the mathematical ediﬁce. Abstract Nonsense... Valeria de Paiva ASL 2014 – Boulder, CO
11. 11. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Edwardian Proofs Frege: one of the founders of modern symbolic logic put forward the view that mathematics is reducible to logic. Begriﬀsschrift, 1879 Was the ﬁrst to write proofs using a collection of abstract symbols: instead of B → A and B hence A Valeria de Paiva ASL 2014 – Boulder, CO
12. 12. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Why Proofs? Mathematics in turmoil in the turn of the century because of paradoxes e.g. Russell’s Paradox Hilbert’s Program: Base all of mathematics in ﬁnitistic methods Proving the consistency of Arithmetic: the big quest Read the graphic novel!! Valeria de Paiva ASL 2014 – Boulder, CO
13. 13. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Edwardian Turmoil... Hilbert’s program:provide secure foundations for all mathematics. How? Formalization all mathematical statements should be written in a precise formal language, and manipulated according to well deﬁned rules. There is no ignorabimus in mathematics.. . Sounds good, doesn’t it? Valeria de Paiva ASL 2014 – Boulder, CO
14. 14. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Hilbert’s Program Consistent: no contradiction can be obtained in the formalism of mathematics. Complete: all true mathematical statements can be proven in the formalism. Consistency proof use only “ﬁnitistic”reasoning about ﬁnite mathematical objects. Conservative: any result about “real objects”obtained using reasoning about “ideal objects”(such as uncountable sets) can be proved without ideal objects. Decidable: an algorithm for deciding the truth or falsity of any mathematical statement. Valeria de Paiva ASL 2014 – Boulder, CO
15. 15. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic G¨odel’s Incompleteness Theorems (1931) Hilbert’s program impossible, if interpreted in the most obvious way. BUT: The development of proof theory itself is an outgrowth of Hilbert’s program. Gentzen’s development of natural deduction and the sequent calculus [too]. G¨odel obtained his incompleteness theorems while trying to prove the consistency of analysis. The tradition of reductive proof theory of the Gentzen-Sch¨utte school is itself a direct continuation of Hilbert’s program. Valeria de Paiva ASL 2014 – Boulder, CO
16. 16. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Proof theory: poor sister or cinderella? Logic traditionally divided into: Model Theory, Proof Theory, Set Theory and Recursion Theory. What about Complexity Theory? Valeria de Paiva ASL 2014 – Boulder, CO
17. 17. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic 20th Century Proofs To prove the consistency of Arithmetic Gentzen invented his systems of NATURAL DEDUCTION (how mathematicians think) SEQUENT CALCULUS (how he could formalize the thinking to obtain the main result he needed, his Hauptsatz. (1934)) Valeria de Paiva ASL 2014 – Boulder, CO
18. 18. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Church and lambda-calculus Alonzo Church: the lambda calculus (1932) Church realized that lambda terms could be used to express every function that could ever be computed by a machine. Instead of “the function f where f (x) = t”, he simply wrote λx.t. The lambda calculus is an universal programming language. The Curry-Howard correspondence: logicians and computer scientists developed a cornucopia of new logics/program constructs based on the correspondence between proofs and programs. Valeria de Paiva ASL 2014 – Boulder, CO
19. 19. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Curry-Howard for Implication Natural deduction rules for implication (without λ-terms) A → B A B [A] · · · · π B A → B Natural deduction rules for implication (with λ-terms) M : A → B N : A M(N): B [x : A] · · · · π M : B λx.M : A → B function application abstraction Valeria de Paiva ASL 2014 – Boulder, CO
20. 20. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Proofs are Programs! 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 Valeria de Paiva ASL 2014 – Boulder, CO
21. 21. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Valeria de Paiva ASL 2014 – Boulder, CO
22. 22. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Proof Theory using Categories... Category: a collection of objects and of morphisms, satisfying obvious laws Functors: the natural notion of morphism between categories Natural transformations: the natural notion of morphisms between functors Constructors: products, sums, limits, duals.... Adjunctions: an abstract version of equality How does this relate to logic? Where’s the theorem? A long time coming: Curry, Schoenﬁnkel, Howard (1969, published in 1980) Valeria de Paiva ASL 2014 – Boulder, CO
23. 23. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Categorical Proof Theory Model derivations/proofs, not whether theorems are true or not Proofs deﬁnitely ﬁrst-class citizens How? Uses extended Curry-Howard correspondence Why is it good? Modeling derivations useful in linguistics, functional programming, compilers.. Why is it important? Widespread use of logic/algebra in CS means new important problems to solve with our favorite tools. Why so little impact on logic itself? Valeria de Paiva ASL 2014 – Boulder, CO
24. 24. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic How many Curry-Howard Correspondences? Easier to count, if thinking about the logics: Intuitionistic Propositional Logic, System F, Dependent Type Theory (Martin-L¨of), Linear Logic, Constructive Modal Logics, various versions of Classical Logic since the early 90’s. The programs corresponding to these logical systems are futuristic programs. The logics inform the design of new type systems, that can be used in new applications. Valeria de Paiva ASL 2014 – Boulder, CO
25. 25. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Valeria de Paiva ASL 2014 – Boulder, CO
26. 26. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Dialectica Interpretation If we cannot do Hilbert’s program with ﬁnitistic means, can we do it some other way? Can we, at least, prove consistency of arithmetic? Try: liberalized version of Hilbert’s programme – justify classical systems in terms of notions as intuitively clear as possible. G¨odel’s approach: computable (or primitive recursive) functionals of ﬁnite type (System T), using the Dialectica Interpretation (named after the Swiss journal Dialectica, special volume dedicated to Paul Bernays 70th birthday) in 1958. Valeria de Paiva ASL 2014 – Boulder, CO
27. 27. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Dialectica Categories Hyland suggested that to provide a categorical model of the Dialectica Interpretation, one should look at the functionals corresponding to the interpretation of logical implication. The categories in my thesis proved to be a model of Linear Logic... Linear Logic introduced by Girard (1987) as a proof-theoretic tool: the symmetries of classical logic plus the constructive content of proofs of intuitionistic logic. Linear Logic: a tool for semantics of Computing. Valeria de Paiva ASL 2014 – Boulder, CO
28. 28. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Linear Logic A proof theoretic logic described by Jean-Yves Girard in 1986. Basic idea: assumptions cannot be discarded or duplicated. They must be used exactly once – just like dollar bills... Other approaches to accounting for logical resources before. Great win of Linear Logic: Account for resources when you want to, otherwise fall back on traditional logic, A → B iﬀ !A −◦ B Valeria de Paiva ASL 2014 – Boulder, CO
29. 29. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Dialectica Categories as Models of Linear Logic In Linear Logic formulas denote resources. Resources are premises, assumptions and conclusions, as they are used in logical proofs. For example: \$1 −◦ latte If I have a dollar, I can get a Latte \$1 −◦ cappuccino If I have a dollar, I can get a Cappuccino \$1 I have a dollar Can conclude either latte or cappuccino — But using my dollar and one of the premisses above, say \$1 −◦ latte gives me a latte but the dollar is gone — Usual logic doesn’t pay attention to uses of premisses, A implies B and A gives me B but I still have A... Valeria de Paiva ASL 2014 – Boulder, CO
30. 30. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Linear Implication and (Multiplicative) Conjunction Traditional implication: A, A → B B A, A → B A ∧ B Re-use A Linear implication: A, A −◦ B B A, A −◦ B A ⊗ B Cannot re-use A Traditional conjunction: A ∧ B A Discard B Linear conjunction: A ⊗ B A Cannot discard B Of course: !A A⊗!A Re-use !(A) ⊗ B B Discard Valeria de Paiva ASL 2014 – Boulder, CO
31. 31. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic The challenges of modeling Linear Logic Traditional categorical modeling of intuitionistic logic: formula A object A of appropriate category A ∧ B A × B (real product) A → B BA (set of functions from A to B) But these are real products, so we have projections (A × B → A) and diagonals (A → A × A) which correspond to deletion and duplication of resources. Not linear!!! Need to use tensor products and internal homs in Category Theory. Hard to decide how to deﬁne the “make-everything-as-usual”operator ”!”. Valeria de Paiva ASL 2014 – Boulder, CO
32. 32. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic My version of Curry-Howard: Dialectica Categories Based on G¨odel’s Dialectica Interpretation (1958): Result: an interpretation of intuitionistic arithmetic HA in a quantiﬁer-free theory of functionals of ﬁnite type T. Idea: translate every formula A of HA to AD = ∃u∀x.AD, where AD is quantiﬁer-free. Use: If HA proves A then T proves AD(t, y) where y is string of variables for functionals of ﬁnite type, t a suitable sequence of terms not containing y Goal: to be as constructive as possible while being able to interpret all of classical arithmetic Valeria de Paiva ASL 2014 – Boulder, CO
33. 33. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Motivations and interpretations. . . For G¨odel (in 1958) the Dialectica interpretation was a way of proving consistency of arithmetic. For me (in 1988) an internal way of modelling Dialectica turned out to produce models of Linear Logic instead of models of Intuitionistic Logic, which were expected... For Blass (in 1995) a way of connecting work of Votj´as in Set Theory with mine and also his own work on Linear Logic and cardinalities of the continuum. Valeria de Paiva ASL 2014 – Boulder, CO
34. 34. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Dialectica Categories Objects of the Dialectica category DDial2(Sets) are triples, a generic object is A = (U, X, R), where U and X are sets and R ⊆ U × X is an usual set-theoretic relation. A morphism from A to B = (V , Y , S) is a pair of functions f : U → V and F : U × Y → X such that uRF(u, y) → fuSy. (Note direction!) Theorem: You have to ﬁnd the right structure. . . (de Paiva 1987) The category DDial2(Sets) has a symmetric monoi- dal closed structure, which makes it a model of (exponential-free) intuitionistic multiplicative linear logic. Theorem(Hard part): You also want usual logic. . . There is a comonad ! which models exponentials/modalities and recovers Intuitionistic and Classical Logic. Valeria de Paiva ASL 2014 – Boulder, CO
35. 35. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Two Kinds of Dialectica Categories Girard’s sugestion in Boulder: Dialectica category Dial2(Sets) objects are triples, a generic object is A = (U, X, R), where U and X are sets and R ⊆ U × X is a set-theoretic relation. A morphism from A to B = (V , Y , S) is a pair of functions f : U → V and F : Y → X such that uRFy → fuSy. (Simpliﬁed maps!) Theorem: You just have to ﬁnd the right structure. . . (de Paiva 1989) The category Dial2(Sets) has a symmetric mo- noidal closed structure, and involution which makes it a model of (exponential-free) classical multiplicative linear logic. Theorem (Even Harder part): You still want usual logic. . . There is a comonad ! which models exponentials/modalities, hence recovers Intuitionistic and Classical Logic. Valeria de Paiva ASL 2014 – Boulder, CO
36. 36. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Can we give some intuition for these morphisms? Blass makes the case for thinking of problems in computational complexity. Intuitively an object of Dial2(Sets) (U, X, R) 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 R says whether the answer is correct for that instance of the problem or not. Valeria de Paiva ASL 2014 – Boulder, CO
37. 37. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Examples of objects in Dial2(Sets) 1. The object (N, N, =) where n is related to m iﬀ n = m. 2. The object (NN, N, R) where f is R-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, =) with usual equality/inequality. Valeria de Paiva ASL 2014 – Boulder, CO
38. 38. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic The Right Structure? To “internalize”the notion of map between problems, we need to consider the collection of all maps from U to V , V U, the collection of all maps from Y to X, XY and we need to make sure that a pair f : U → V and F : Y → X in that set, satisﬁes our dialectica condition: ∀u ∈ U, y ∈ Y , uRFy → fuSy This give us an object (V U × XY , U × Y , eval) where eval: V U × XY × (U × Y ) → 2 is the ‘relation’ that evaluates the pair (f , F) on the pair (u, y) and checks the dialectica implication between relations. Valeria de Paiva ASL 2014 – Boulder, CO
39. 39. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic The Right Structure! Because it’s fun, let us calculate the “reverse engineering” necessary for a model of Linear Logic.. A ⊗ B → C if and only if A → [B −◦ C] U × V (R ⊗ S)XV × Y U U R X ⇓ ⇓ W f ? T T 6 (g1, g2) W V × Y Z ? (S −◦ T)V × Z 6 Valeria de Paiva ASL 2014 – Boulder, CO
40. 40. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Dialectica Categories Applications In CS: models of Petri nets (more than 2 phds), non-commutative version for Lambek calculus (linguistics), it has been used as a model of state (Correa et al) and even of quantum groups. Generic models of Linear Logic (with Schalk04) and for Linguistics Analysis of the syntax-semantics interface for Natural Language, the Glue Approach (Dalrymple, Lamping and Gupta). Recently: Bodil Biering ‘Copenhagen Interpretation’ (ﬁrst ﬁbrational version), P. Hofstra. ”The dialectica monad and its cousins”. Also ”The Compiler Forest”Budiu, Galenson and Plotkin (2012) and P. Hyvernat. “A linear category of polynomial diagrams”. Most recent:Tamara Von Glehn ”polynomials”/containers (2014?). Piedrot (2014) Krivine machine interpretation... Valeria de Paiva ASL 2014 – Boulder, CO
41. 41. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic My ‘newest’ Application Blass (1995) Dialectica categories, or rather category PV as a tool for proving inequalities between cardinalities of the continuum. Blass realized that my model of Linear Logic was also used by Peter Votj´as for set theory, proving inequalities between cardinal invariants and wrote Questions and Answers A Category Arising in Linear Logic, Complexity Theory, and Set Theory (1995). Four years ago I learnt from Samuel Gomes da Silva about his and Charles Morgan’s work using Blass/Votj´as’ ideas and we started working together. Valeria de Paiva ASL 2014 – Boulder, CO
42. 42. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Goal Blass (1995) It is an empirical fact that proofs between cardinal characteristics of the continuum usually proceed by representing the characteristics as norms of objects in PV and then exhibiting explicit morphisms between those objects. Why? so far only tiny calculation of natural numbers object in Dialectica categories. (de Paiva, Morgan and da Silva, Natural Number Objects in Dialectica Categories, LFSA 2013, to appear in ENTCS) Valeria de Paiva ASL 2014 – Boulder, CO
43. 43. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Conclusions Introduced you to the under-appreciated Curry-Howard correspondence. Hinted at its importance for interdisciplinarity: Categorical Proof Theory Described one example: Dialectica categories Dial2(Sets), Illustrated one easy, but essential, theorem in categorical logic. Hinted at Blass and Votj´as use for mapping cardinal invariants. Much more explaining needed... Valeria de Paiva ASL 2014 – Boulder, CO
44. 44. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Take Home Working in interdisciplinary areas is hard, but rewarding. The frontier between logic, computing, linguistics and categories is a fun place to be. Mathematics teaches you a way of thinking, more than speciﬁc theorems. Barriers: over-specialization, lack of open access and unwillingness to ‘waste time’ on formalizations Enablers: international scientiﬁc communities, open access, growing interaction between ﬁelds?... Handsome payoﬀ expected Fall in love with your ideas and enjoy talking to many about them.. Valeria de Paiva ASL 2014 – Boulder, CO
45. 45. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Thank you! Valeria de Paiva ASL 2014 – Boulder, CO
46. 46. Introduction Edwardian Proofs Futuristic Programs Categorical Proof Theory Back to the future: Linear Logic Some References A.Blass, Questions and Answers: A Category Arising in Linear Logic, Complexity Theory, and Set Theory, Advances in Linear Logic (ed. J.-Y. Girard, Y. Lafont, and L. Regnier) London Math. Soc. Lecture Notes 222 (1995). 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) P. Vojt´aˇs, Generalized Galois-Tukey-connections between explicit relations on classical objects of real analysis. In: Set theory of the reals (Ramat Gan, 1991), Israel Math. Conf. Proc. 6, Bar-Ilan Univ., Ramat Gan (1993), 619–643. Valeria de Paiva ASL 2014 – Boulder, CO