INF 329, Fall 2012, Plan of Lectures






Nr.DateWeekTopicsLit.





1.24.08.1234 - format and content of the course
- short history of Algebra
- discussion of behaviour and processes
- streams as completions of lists wit prefix order





2.28.08.1235 - 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"
3.31.08.1235 - 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





4.04.09.1236 - 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)
5.07.09.1236 - 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





6.11.09.1237 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
7.14.09.1237 Lecture Eirik: Functors - motivation, definition and examples
- product categories
- constant functors
- diagonal functor
- product and sum functor
- covariant power set functor
- function space functor





8.18.09.1238 - T-algebras and their homomorphisms: category Alg(T)
- categories Alg(Σ) and Alg(TΣ) are isomorphic
- lists as parametrized T-algebras
- limits of T-algebras are "for free" - example product
9.21.09.1238 - Initial T-algebras
- structure map is iso - selectors and recursion
- example natural numbers and lists
- construction by categorical fix point





10.25.09.1239 - 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
11.28.09.1239 - 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)





12.02.10.1240 - categorical construction of final T-coalgebras
- examples for T = X, A x X
- no final coalgebras for ℘(X)
13.05.10.1240 - Final coalgebras for partial automata, i.e., for T = [A→X]
- discussion that this gives exactly the deterministic CSP processes





14.09.10.1241 Lecture Truls: Modal Logic
- Motivation and syntax
- Kripke frames and Kripke models
12.10.1241 no lecture (Truls has duties and Barnehage stengt)





16.10.1242 no lecture (Arrangements by the Faculty)
15.19.10.1242 Lecture Truls: Modal Logic
- Homomorphisms, strong and bounded homomorphisms
- Equivalence and bisimilarity
- Kripke frames and models as coalgebras





23.10.1243 no lecture (Uwe in Germany)
16.26.10.1243 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





17.30.10.1244 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
02.11.1244 no lecture (workshop NWPT)





18.06.11.1245 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
19.09.11.1245 Lecture Erik: from basic modal logic to coalgebraic modal logic
- Kripke functor
- Bisimulation via relation lifting
- Cover modality ∇
- Coalgebraic modal logic





20.13.11.1246 Lecture Erik: Modal μ-calculus
- Syntax of modal μ-calculus
- Game semantics
- Fixpoints and algebraic semantics
21.16.11.1246 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





20.11.1247 No more lectures










10./11.12.1250 oral Exam