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Philipp Schlicht

Publications and source records attributed to Philipp Schlicht.

43 records · Page 3Linked to original sources
arXiv2014

Generalized Choquet spaces

We introduce an analog to the notion of Polish space for spaces of weight $\leqκ$, where $κ$ is an uncountable regular cardinal such that $κ^{<κ}=κ$. Specifically, we consider spaces in which player II has a winning strategy in a variant of the strong Choquet game which runs for $κ$ many rounds. After discussing the basic theory of these games and spaces, we prove that there is a surjectively universal such space and that there are exactly $2^κ$ many such spaces up to homeomorphism. We also establish a Kuratowski-like theorem that under mild hypotheses, any two such spaces of size $>κ$ are isomorphic by a $κ$-Borel function. We then consider a dynamic version of the Choquet game and show that in this case the existence of a winning strategy for player II implies the existence of a winning tactic, that is, a strategy that depends only on the most recent move. We also study a generalization of Polish ultrametric spaces where the ultrametric is allowed to take values in a set of size $κ$. We show that in this context, there is a family of universal Urysohn-type spaces, and we give a characterization of such spaces which are hereditarily $κ$-Baire.

math.LO↗
arXiv2014

Infinite computations with random oracles

We consider the following problem for various infinite time machines. If a real is computable relative to large set of oracles such as a set of full measure or just of positive measure, a comeager set, or a nonmeager Borel set, is it already computable? We show that the answer is independent from ZFC for ordinal time machines (OTMs) with and without ordinal parameters and give a positive answer for most other machines. For instance, we consider, infinite time Turing machines (ITTMs), unresetting and resetting infinite time register machines (wITRMs, ITRMs), and α-Turing machines for countable admissible ordinals α.

math.LO↗
arXiv2013

Structures Without Scattered-Automatic Presentation

Bruyere and Carton lifted the notion of finite automata reading infinite words to finite automata reading words with shape an arbitrary linear order L. Automata on finite words can be used to represent infinite structures, the so-called word-automatic structures. Analogously, for a linear order L there is the class of L-automatic structures. In this paper we prove the following limitations on the class of L-automatic structures for a fixed L of finite condensation rank 1+α. Firstly, no scattered linear order with finite condensation rank above ω^(α+1) is L-α-automatic. In particular, every L-automatic ordinal is below ω^ω^(α+1). Secondly, we provide bounds on the (ordinal) height of well-founded order trees that are L-automatic. If αis finite or L is an ordinal, the height of such a tree is bounded by ω^{α+1}. Finally, we separate the class of tree-automatic structures from that of L-automatic structures for any ordinal L: the countable atomless boolean algebra is known to be tree-automatic, but we show that it is not L-automatic.

cs.FL↗
arXiv2012

Wadge-like reducibilities on arbitrary quasi-Polish spaces

The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well-ordered), but for many other natural non-zero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called Δ^0_α-reductions, and try to find for various natural topological spaces X the least ordinal α_X such that for every α_X \leq β< ω_1 the degree-structure induced on X by the Δ^0_β-reductions is simple (i.e. similar to the Wadge hierarchy on the Baire space). We show that α_X \leq ω for every quasi-Polish space X, that α_X \leq 3 for quasi-Polish spaces of dimension different from \infty, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its powers.

math.LO↗
arXiv2012

The mate-in-n problem of infinite chess is decidable

Infinite chess is chess played on an infinite edgeless chessboard. The familiar chess pieces move about according to their usual chess rules, and each player strives to place the opposing king into checkmate. The mate-in-n problem of infinite chess is the problem of determining whether a designated player can force a win from a given finite position in at most n moves. A naive formulation of this problem leads to assertions of high arithmetic complexity with 2n alternating quantifiers---there is a move for white, such that for every black reply, there is a counter-move for white, and so on. In such a formulation, the problem does not appear to be decidable; and one cannot expect to search an infinitely branching game tree even to finite depth. Nevertheless, the main theorem of this article, confirming a conjecture of the first author and C. D. A. Evans, establishes that the mate-in-n problem of infinite chess is computably decidable, uniformly in the position and in n. Furthermore, there is a computable strategy for optimal play from such mate-in-n positions. The proof proceeds by showing that the mate-in-n problem is expressible in what we call the first-order structure of chess, which we prove (in the relevant fragment) is an automatic structure, whose theory is therefore decidable. Indeed, it is definable in Presburger arithmetic. Unfortunately, this resolution of the mate-in-n problem does not appear to settle the decidability of the more general winning-position problem, the problem of determining whether a designated player has a winning strategy from a given position, since a position may admit a winning strategy without any bound on the number of moves required. This issue is connected with transfinite game values in infinite chess, and the exact value of the omega one of chess is not known.

math.LO↗
arXiv2010

Non-permutation invariant Borel quantifiers

Every permutation invariant Borel subset of the space of countable structures is definable in $\La_{ω_1ω}$ by a theorem of Lopez-Escobar. We prove variants of this theorem relative to fixed relations and fixed non-permutation invariant quantifiers. Moreover we show that for every closed subgroup $G$ of the symmetric group $S_{\infty}$, there is a closed binary quantifier $Q$ such that the $G$-invariant subsets of the space of countable structures are exactly the $\La_{ω_1ω}(Q)$-definable sets.

math.LO↗