Richard Kaye (Birmingham) "Separation of parameters and model theoretic proofs of witnessing theorems". For those of us (me included) who do not need detailed constructive proofs for converting proofs to algorithms and who find that the full machinery of sequent calculus and cut-elimination is too heavy and long winded and with possible potential for error (especially regarding the `trivial' rules that are often `supressed') I have devised a model-theoretic approach that gives the same results, and is sufficiently fine-grained to prove all known results of this type. (In this respect, the elegant methods from indicator theory fail, as they do not generalise to complexity theory.) I will outline these methods and sketch a proof of Buss's characterisation of P using log-induction on NP formulas.
Sean Walsh (Birkbeck, London) "Abstraction Principles and Theories of Arithmetic"
Abstract: In this talk we survey some results on the interpretability strength of fragments of arithmetic and abstraction principles with limited amounts of comprehension. Suppose that one is working in second-order logic and has at hand an equivalence relation E(X,Y) on second-order objects X,Y, and suppose further that second-order logic is expanded to include a new function symbol F from second-order objects to first-order objects. Then the statement that F(X)=F(Y) iff E(X,Y) is an example of an abstraction principle. One of the classical results here is that the abstraction principle associated to the equivalence relation E(X,Y) that says "X,Y are bijective" is mutually interpretable with second-order arithmetic. Another important result here is a result of Visser which describes how certain iterations of abstraction principles with limited amounts of comprehension align with $I\Delta_0+\mathrm{exp}$. Our goal here is simply to describe in overview some of the basic results, techniques, and open questions.
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