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#### 1Programming and Computation

Interpreting types as abstract values [The Abstract of the lecture notes] We expound a view of type checking as evaluation with `abstract values'. Whereas dynamic semantics, evaluation, deals with (dynamic) values like 0, 1, etc., static semantics, type checking, deals with approximations like int. A type system is sound if it correctly approximates the dynamic behavior and predicts its outcome: if the static semantics predicts that a term has the type int, the dynamic evaluation of the term, if it terminates, will yield an integer. As object language, we use simply-typed and let-polymorphic lambda calculi with integers and integer operations as constants. We use Haskell as a metalanguage in which to write evaluators, type checkers, type reconstructors and inferencers for the object language.
12 years ago by @draganigajic
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#### 1DoCon the Algebraic Domain Constructor

DoCon is a program for symbolic computation in mathematics - package of modules DoCon joins the categorial approach to the mathematical computation expressed via the Haskell type classes, and explicit processing of the domain description terms. It implements recently a good piece of commutative algebra: linear algebra, polynomial gcd, factorization, Groebner bases, and other functions. They are programmed under the very generic assumptions , like "over any Euclidean ring", over any GCD-ring, any field, and so on. DoCon also supports the constructions on domains: Fraction, Polynomial, Residue ring, and others. That is certain set of operations on a constructed domain is built automatically.
12 years ago by @draganigajic
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#### 1What Would the World Look Like if Everything was Computable?: An Introduction to Hyland's Effective Topos " XOR's Hammer

Suppose that we wanted to construct a mathematical universe where all objects were computable in some sense. How would we do it? Well, we could certainly allow the set \mathbb{N} into our universe: natural numbers are the most basic computational objects there are. (Notation: I’ll use N to refer to \mathbb{N} when we’re considering it as part of the universe we’ll building, and just \mathbb{N} when we’re talking about the set of natural numbers in the “real” world.) What should we take as our set of functions N^N from N to N? Since we want to admit only computable things, we should let N^N be the set of computable functions from \mathbb{N} to \mathbb{N}, which we can represent non-uniquely by their indices (i.e., by the programs which compute them).
12 years ago by @draganigajic
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#### 1Modelling computations: a 2-categorical framework

An introduction to 2-categories is given by illustrating how the structure of typed lambda calculus may naturally be viewed as a 2-category. In this vein. the structure of computations or conversions gives rise to notions of lax 2':"adjointness.
12 years ago by @draganigajic
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#### 1Computation and the Periodic Table

John Baez Algebraic Topological Methods in Computer Science 2008, University of Paris Diderot (Paris 7) July 7, 2008 Computation and the Periodic Table By now there is an extensive network of interlocking analogies between physics, topology, logic and computer science, which can be seen most easily by comparing the roles that symmetric monoidal closed categories play in each subject. However, symmetric monoidal categories are just the n = 1, k = 3 entry of a hypothesized "periodic table" of k-tuply monoidal n-categories. This raises the question of how these analogies extend. We present some thoughts on this question, focusing on how symmetric monoidal closed 2-categories might let us understand the lambda calculus more deeply. This is based on work in progress with Mike Stay. Click on this to see the transparencies of the talk:
12 years ago by @draganigajic
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