Date/Time : 20 Oct 2021 16:00
Venue : via Zoom
Speaker : Yang Yue, National University of Singapore

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Meeting ID: 830 4925 8042
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Abstract:  This is a follow up of Liao Yuke’s talk last week. I will explain in detail his result which says that there exists a recursive coloring f: N -> {0,1} such that for all infinite subset X of N, if FS(X) is homogeneous for f, then X is not recursive in 0”. Here FS(X) is the set of all finite sums of distinct elements of X. Liao Yuke’s result improved Blass, Hirst and Simpson’s theorem in 1987 (from no 0′
recursive solutions to no 0” recursive ones).

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Date/Time : 13 Oct 2021 16:00
Venue : via Zoom
Speaker : Liao Yuke,

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Meeting ID: 830 4925 8042
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Abstract: Hindman’s theorem is a Ramsey type theorem related to finite sum while its proof-theoretic strength still has a huge gap. One form of the question is that if any computable coloring function would have an arithmetic solution (homogeneous set). We will construct a computable coloring functions that no Pi-3 set can be homogeneous.

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Date/Time : 06 Oct 2021 16:00
Venue : via Zoom
Speaker : Chong Chi Tat, National University of Singapore

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Meeting ID: 830 4925 8042
Passcode: 1729=x3+y3

Abstract:   In this talk we discuss the proof-theoretic strength, from the reverse mathematics perspective, of combinatorial principles concerning the coloring of binary trees and finite products of binary trees.  Beginning with the principle TT^1, which states that every finite coloring of the full binary tree has an isomorphic monochromatic subtree, we will cover its strengthening to the existence of a strong monochromatic subtree, and to the full generalization of the latter known as the Halpern-Laeuchli Theorem.

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Date/Time : 15 Sep 2021 16:00
Venue : via Zoom
Speaker : Bakhadyr Khoussainov, UESTC, Chengdu and The University of Auckland

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Meeting ID: 830 4925 8042
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Abstract: This talk belongs to the area of probabilistic logic semantics.

The first contribution of this work is the introduction of probability structures. Probability structures are the algebraic structures equipped with probability functions on the domains and the atomic predicates. These structures extend type 1 probability structures introduced by Halpern and Bacchus.

Type 1 probability structures contain probability functions on domains only. Our probability structures possess an additional statistical knowledge, – probability functions on atomic predicates.

We present a method that builds probability spaces for the first order logic formulas and prove that our semantics is sound.

The second contribution of this work is the introduction of smooth probability structures.

The smooth probability structures carefully refine probability structures so that we have a better control of the probability spaces defined by first order logic formulas. For these structures we initiate the study of first order probability logic (FOPLS), investigate axiomatizability of FOPLS, and address decidability and undecidability questions of the sets of valid formulas. We also study a few algorithmic questions on probability structures.

The work is joint with Mingyu Xiao (UESTC) and Jiamou Liu (The University of Auckland).

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Date/Time : 08 Sep 2021 16:00
Venue : via Zoom
Speaker : Bakhadyr Khoussainov / Frank Stephan, UESTC, Chengdu and The University of Auckland, New Zealand / NUS

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Meeting ID: 830 4925 8042
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Parity games are games where a marker is moved on a finite graph and each node is annotated with a natural number; the game runs forever and the largest number in an infinitely often visited node decides the winner, if it is even then player Anke wins else player Boris wins. Marcin Jurdzinski showed that this game is in UP intersected coUP and also provided the first not fully exponential algorithm for it; however, the exact time complexity remained unresolved. In 2017, Calude, Jain, Khoussainov, Li and Stephan found a quasipolynomial time algorithm which Jurdzinski and Lazic as well as Schewe and his collaborators improved to be in polynomial space as well. The talk provides the way this algorithm was found and the implications it has for the fixed-parameter-tracktability of parity games and related problems like coloured Muller games. Though now quite a number of quasipolynomial time algorithms are known and there is quite extensive research in this topic, the question on whether parity games can even be solved in polynomial time is still unresolved.

This talk is given by Bakhadyr Khoussainov and Frank Stephan jointly also on behalf of their coauthors Cristian Calude, Sanjay Jain and Wei Li.

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Date/Time : 01 Sep 2021 16:00
Venue : via Zoom
Speaker : Rupert Hoelzl, Bundeswehr University Munich

Join Zoom Meeting https://nus-sg.zoom.us/j/83049258042?pwd=UWViaWNvTFUrdFdhOHJCdEVydnVkdz09 Meeting ID: 830 4925 8042 Passcode: 1729=x3+y3 We investigates inductive inference from the perspective of reverse mathematics. Reverse mathematics is a framework that allows gauging the proof strength of theorems and axioms in many areas of mathematics. We apply its methods to basic notions of algorithmic learning theory such as Angluin’s tell-tale criterion and its variants for learning in the limit and for conservative learning, as well as to the more general scenario of partial learning. These notions are studied in the reverse mathematics context for uniformly and weakly represented families of languages. The results are stated in terms of axioms referring to induction strength and to domination of weakly represented families of functions. URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

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Date/Time : 25 Aug 2021 16:00
Venue : via Zoom
Speaker : Ng Keng Meng,

Join Zoom Meeting https://nus-sg.zoom.us/j/83049258042?pwd=UWViaWNvTFUrdFdhOHJCdEVydnVkdz09 Meeting ID: 830 4925 8042 Passcode: 1729=x3+y3 There has been a recent revival in the interest in sub-computable mathematics. One of these approaches is to consider “primitive recursive” or punctual structures. This has led to a greater understanding in the effective content of well-known objects and proofs in classical computability theory. When considering the punctual anaologies of classical computabilitiy we often obtain strange and surprising results. I will discuss some recent work in progress in this area, focussing particularly on structural results. URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html  

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Date/Time : 18 Aug 2021 16:00
Venue : via Zoom
Speaker : Yu Liang, Nanjing University

Join Zoom Meeting https://nus-sg.zoom.us/j/83049258042?pwd=UWViaWNvTFUrdFdhOHJCdEVydnVkdz09 Meeting ID: 830 4925 8042 Passcode: 1729=x3+y3 Besicovitch-Davis theorem says that the Hausdorff dimension of every analytic set can be approximated by its closed subset. But the Besicovitch-Davis theorem fails for co-analytic sets under the assumption V=L as observed by Slaman. We prove that the theorem holds for arbitrary sets under ZF+sTD. We also prove that the theorem holds for Sigma-1-2-sets under Martin’s axiom. This is joint work with Peng Yinhe and Wu Liuzhen.

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Date/Time : 11 Aug 2021 16:00
Venue : via Zoom
Speaker : Frank Stephan, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/83049258042?pwd=UWViaWNvTFUrdFdhOHJCdEVydnVkdz09 Meeting ID: 830 4925 8042 Password: 1729=x3+y3 Let a positive equivalence relation to be an r.e. equivalence relation on the set of natural numbers with infinitely many equivalence relations. Khoussainov initiated with coauthors a deep study of the following question: Given a positive equivalence relation eta, which structures from a given set of structures does this equivalence relation realise? Here realisation means that functions in the structure are recursive and relations are r.e. with the equality itself given by the equivalence relation eta. In other words, the given r.e. structure divided by eta is the structure realised by eta. Now questions studied by Khoussainov and his coworkers included questions like “What is the partial ordering on positive equivalence relations eta,rho where eta is below rho iff every structure of the given type realised by eta is also realised by rho? Besides algebraic structures and orders, it has also been studied how the learnability notions behave with respect to uniformly r.e. one-one families realised by positive equivalence relations.

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