INF 329, Fall 2012, Plan of Lectures

Nr. | Date | Week | Topics | Lit. | |

1. | 24.08.12 | 34 | - format and content of the course - short history of Algebra - discussion of behaviour and processes - streams as completions of lists wit prefix order | ||

2. | 28.08.12 | 35 | - examples of algebras - signatures Σ: definition and examples - Σ-algebra with examples - Σ-subalgebras: definition, examples, closed under intersection - least Σ-subalgebra containing a given set G of "generators" | ||

3. | 31.08.12 | 35 | - Σ-homomorphisms: definition and examples - Σ-subalgebras as Σ-homomorphisms - composition of Σ-homomorphisms and identical Σ-homomorphisms - category Alg(Σ) - monomorphisms, epimorphims and isomorphisms in Alg(Σ) - homomorphic image and epi-mono factorization - terminal Σ-algebra - kernel of a map, kernels of maps are equivalences - kernels of Σ-homomorphisms are congruences - definition of congruences - quotient Σ-algebras: construction and natural homomorphism - representation algebras: construction and iso to quotient algebra | ||

4. | 04.09.12 | 36 | - congruence relations as Σ-algebras, projections are Σ-homomorphims - kernels as equalizers - quotient algebras as coequalizers - discussion: for algebras limits are easy (inherited from Set) but colimits involved - definition of (ground) Σ-terms - (ground) Σ-term algebra - recursion theorem - (ground) Σ-term algebra is initial in Alg(Σ) | ||

5. | 07.09.12 | 36 | - two sides of induction: existence and uniqueness of initial homomorphisms and initial subalgebra
has no proper subalgebra (with examples) - abstraction of Σ-algebras? - functors T:Set -> Set and definition of T-algebras | ||

6. | 11.09.12 | 37 | Lecture Piotr: Why and in what sense are the following concepts dual?- cartesian product and disjoint union - empty set and singleton set - injective and surjective | ||

7. | 14.09.12 | 37 | Lecture Eirik: Functors - motivation, definition and examples - product categories - constant functors - diagonal functor - product and sum functor - covariant power set functor - function space functor | ||

8. | 18.09.12 | 38 | - T-algebras and their homomorphisms: category Alg(T) - categories Alg(Σ) and Alg(T _{Σ}) are isomorphic - lists as parametrized T-algebras - limits of T-algebras are "for free" - example product | ||

9. | 21.09.12 | 38 | - Initial T-algebras - structure map is iso - selectors and recursion - example natural numbers and lists - construction by categorical fix point | ||

10. | 25.09.12 | 39 | - T-coalgebras and their homomorphisms - category CAlg(T) - state transition systems as coalgebras - step by step - examples: T = X, A x X, ℘(X) - partial automata as coalgebras - currying trick | ||

11. | 28.09.12 | 39 | - colimits of T-coalgebras are "for free" - What is the dual of a subalgebra? - final coalgebras - structure map is iso, i.e., there is also a T-algebra - examples for T = X, AxX - no final coalgebras for ℘(X) | ||

12. | 02.10.12 | 40 | - categorical construction of final T-coalgebras - examples for T = X, A x X - no final coalgebras for ℘(X) | ||

13. | 05.10.12 | 40 | - Final coalgebras for partial automata, i.e., for T = [A→X] - discussion that this gives exactly the deterministic CSP processes | ||

14. | 09.10.12 | 41 | Lecture Truls: Modal Logic- Motivation and syntax - Kripke frames and Kripke models | ||

| 12.10.12 | 41 | no lecture (Truls has duties and Barnehage stengt) | ||

| 16.10.12 | 42 | no lecture (Arrangements by the Faculty) | ||

15. | 19.10.12 | 42 | Lecture Truls: Modal Logic- Homomorphisms, strong and bounded homomorphisms - Equivalence and bisimilarity - Kripke frames and models as coalgebras | ||

| 23.10.12 | 43 | no lecture (Uwe in Germany)
| ||

16. | 26.10.12 | 43 | Cocongruences - Cocongruence := jointly epi cospan of coalgebra morphisms - Example: cospan of "bounded morphisms" - Cocongruence ≡ span of coalgebra morphisms (if T preserves weak pullbacks) - Discussion: span of coalgebra morphisms ≡ bisimulation | ||

17. | 30.10.12 | 44 | Coinductive definition of maps - Coinduction = existence of unique moprhism into final coalgebra - examples for existence: definition of maps merge, even, odd on streams - uniqueness as a tool to prove equalities between maps - example: (odd,even);merge=id | ||

| 02.11.12 | 44 | no lecture (workshop NWPT) | ||

18. | 06.11.12 | 45 | Coinductive proofs of properties - Coinduction = final coalgebra has no proper quotient - that is, any outgoing epimorphism is isomorphism - equivalently: bisimulation in final coalgebra is identitity (if T preserves weak pullback) - example: operations merge, even, odd on streams | ||

19. | 09.11.12 | 45 | Lecture Erik: from basic modal logic to coalgebraic modal logic- Kripke functor - Bisimulation via relation lifting - Cover modality ∇ - Coalgebraic modal logic | ||

20. | 13.11.12 | 46 | Lecture Erik: Modal μ-calculus - Syntax of modal μ-calculus - Game semantics - Fixpoints and algebraic semantics | ||

21. | 16.11.12 | 46 | Covariables ≡ propositional variables - set of variables V and Σ-term algebra over V - "set of variables" abstractly by addition of V - dualization by product with "set of colours" C - variables as pointers and covariables as maps into the set of Boolean values - Lattice of subsets vs. lattice of partitions | ||

| 20.11.12 | 47 | No more lectures |
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10./11.12.12 | 50 | oral Exam | |||