Showing papers presented at "Conference on Computability in Europe in 2015"

Weihrauch Degrees of Finding Equilibria in Sequential Games

Abstract: We consider the degrees of non-computability (Weihrauch degrees) of finding winning strategies (or more generally, Nash equilibria) in infinite sequential games with certain winning sets (or more generally, outcome sets). In particular, we show that as the complexity of the winning sets increases in the difference hierarchy, the complexity of constructing winning strategies increases in the effective Borel hierarchy.

New Bounds on Optimal Sorting Networks

Abstract: We present new parallel sorting networks for \(17\) to \(20\) inputs. For \(17, 19,\) and \(20\) inputs these new networks are faster (i.e., they require fewer computation steps) than the previously known best networks. Therefore, we improve upon the known upper bounds for minimal depth sorting networks on \(17, 19,\) and \(20\) channels. Furthermore, we show that our sorting network for \(17\) inputs is optimal in the sense that no sorting network using less layers exists. This solves the main open problem of [D. Bundala & J. Zavodný. Optimal sorting networks, Proc. LATA 2014].

Iterative Forcing and Hyperimmunity in Reverse Mathematics

Abstract: The separation between two theorems in reverse mathematics is usually done by constructing a Turing ideal satisfying a theorem P and avoiding the solutions to a fixed instance of a theorem Q. Lerman, Solomon and Towsner introduced a forcing technique for iterating a computable non-reducibility in order to separate theorems over omega-models. In this paper, we present a modularized version of their framework in terms of preservation of hyperimmunity and show that it is powerful enough to obtain the same separations results as Wang did with his notion of preservation of definitions.

Towards the Effective Descriptive Set Theory

Abstract: We prove effective versions of some classical results about measurable functions and derive from this extensions of the Suslin-Kleene theorem, and of the effective Hausdorff theorem for the computable Polish spaces (this was established in [2] with a different proof) and for the computable \(\omega \)-continuous domains (this answers an open question from [2]).

Design Techniques for Planning Navigational Systems in 3-D Video Games

Abstract: Navigation is an essential element of many high-budget games (known as AAA titles). In such products, players are expected to walk through and interact with aesthetically rich 3-D spaces. Therefore, designers should provide meaningful information to guide the users within a challenging environment. While there has been much research on both games and 3-D environments, there is very little research investigating design techniques used to guide players through 3-D game worlds. This paper is focused on proposing a set of navigational patterns or techniques currently used in commercial 3-D action-adventure titles. These design techniques are composed of [a] 21 patterns used to aid navigation, [b] three level design choices affecting navigation, and [c] eight game mechanics related to navigation. We uncovered these design techniques through a detailed analysis of 21 3-D action-adventure games. This contribution has several important facets. First, the set of design techniques and terminology proposed here can be used as a training construct to teach 3-D game and environment design. Second, it can also be used as a toolset for designers. Third, it will provide an important start for a formal vocabulary that can be used by designers and researchers discussing navigation in 3-D games.

Completely Regular Bishop Spaces

Abstract: Bishop’s notion of a function space, here called a Bishop space, is a constructive function-theoretic analogue to the classical set-theoretic notion of a topological space. Here we introduce the quotient, the pointwise exponential and the completely regular Bishop spaces. For the latter we present results which show their correspondence to the completely regular topological spaces, including a generalized version of the Tychonoff embedding theorem for Bishop spaces. All our proofs are within Bishop’s informal system of constructive mathematics \(\mathrm {BISH}\).

A note on the computable categoricity of l^p spaces

Abstract: Suppose that \(p\) is a computable real and that \(p \ge 1\) We show that in both the real and complex case, \(\ell ^p\) is computably categorical if and only if \(p = 2\) The proof uses Lamperti’s characterization of the isometries of Lebesgue spaces of \(\sigma \)-finite measure spaces

P Systems with Parallel Rewriting for Chain Code Picture Languages

Abstract: Chain code pictures are composed of unit lines in the plane, drawn according to a sequence of instructions left, right, up, down codified by words over \(\varSigma = \{ {l}, r, u, d \}\). P systems to generate such languages have been considered in previous work with sequential rewriting in the membranes. We consider here parallel rewriting, with the advantage of reducing the number of membranes. We also consider the problem of generating the finite approximations of space-filling curves, the Hilbert curve and the Peano curve.

Hyperprojective Hierarchy of qcb 0 -Spaces.

Abstract: We extend the Luzin hierarchy of qcb0-spaces introduced in [ScS13] to all countable ordinals, obtaining in this way the hyperprojective hierarchy of qcb0-spaces. We generalize all main results of [ScS13] to this larger hierarchy. In particular, we extend the Kleene-Kreisel continuous functionals of finite types to the continuous functionals of countable types and relate them to the new hierarchy. We show that the category of hyperprojective qcb0-spaces has much better closure properties than the category of projective qcb0-space. As a result, there are natural examples of spaces that are hyperprojective but not projective.

Base-Complexity Classifications of QCB$$_$$-Spaces

Abstract: We define and study new classifications of qcb\(_0\)-spaces based on the idea to measure the complexity of their bases. The new classifications complement those given by the hierarchies of qcb\(_0\)-spaces introduced in [7, 8] and provide new tools to investigate non-countably based qcb\(_0\)-spaces. As a by-product, we show that there is no universal qcb\(_0\)-space and establish several apparently new properties of the Kleene-Kreisel continuous functionals of countable types.

Degrees of Unsolvability: A Tutorial

Abstract: Given a problem \(P\), one associates to \(P\) a degree of unsolvability, i.e., a quantity which measures the amount of algorithmic unsolvability which is inherent in \(P\). We focus on two degree structures: the semilattice of Turing degrees, \(\mathcal {D}_\mathrm {T}\), and its completion, \(\mathcal {D}_\mathrm {w}=\widehat{\mathcal {D}_\mathrm {T}}\), the lattice of Muchnik degrees. We emphasize specific, natural degrees and their relationship to reverse mathematics. We show how Muchnik degrees can be used to classify tiling problems and symbolic dynamical systems of finite type. We describe how the category of sheaves over \(\mathcal {D}_\mathrm {w}\) forms a model of intuitionistic mathematics, known as the Muchnik topos. This model is a rigorous implementation of Kolmogorov’s nonrigorous 1932 interpretation of intuitionism as a “calculus of problems”.

Weighted Automata on Infinite Words in the Context of Attacker-Defender Games

Abstract: We consider several infinite-state Attacker-Defender games with reachability objectives. The results of the paper are twofold. Firstly we prove a new language-theoretic result for weighted automata on infinite words and show its encoding into the framework of Attacker-Defender games. Secondly we use this novel concept to prove undecidability for checking existence of a winning strategy in several low-dimensional mathematical games including vector reachability games, word games and braid games.

On Computability of Navier-Stokes’ Equation

Abstract: We approach the question of whether the Navier-Stokes Equation admits recursive solutions in the sense of Weihrauch’s Type-2 Theory of Effectivity: A suitable encoding (“representation”) is carefully constructed for the space of solenoidal vector fields in the \(L_q\) sense over the \(d\)-dimensional open unit cube with zero boundary condition. This is shown to render both the Helmholtz projection and the semigroup generated by the Stokes operator uniformly computable in the case \(q=2\).

Error and Predicativity

Abstract: The article surveys ideas emerging within the predicative tradition in the foundations of mathematics, and attempts a reading of predicativity constraints as highlighting different levels of understanding in mathematics. A connection is made with two kinds of error which appear in mathematics: local and foundational errors. The suggestion is that ideas originating in the predicativity debate as a reply to foundational errors are now having profound influence to the way we try to address the issue of local errors. Here fundamental new interactions between computer science and mathematics emerge.

Prefix and Right-Partial Derivative Automata

Abstract: Recently, Yamamoto presented a new method for the conversion from regular expressions (REs) to non-deterministic finite automata (NFA) based on the Thompson \(\varepsilon \)-NFA (\(\mathcal {A}_\mathsf {T}\)). The \(\mathcal {A}_\mathsf {T}\) automaton has two quotients discussed: the suffix automaton \(\mathcal {A}_\mathsf {suf}\) and the prefix automaton, \(\mathcal {A}_\mathsf {pre}\). Eliminating \(\varepsilon \)-transitions in \(\mathcal {A}_\mathsf {T}\), the Glushkov automaton (\(\mathcal {A}_{\mathsf {pos}}\)) is obtained. Thus, it is easy to see that \(\mathcal {A}_\mathsf {suf}\) and the partial derivative automaton (\(\mathcal {A}_\mathsf {pd})\) are the same. In this paper, we characterise the \(\mathcal {A}_\mathsf {pre}\) automaton as a solution of a system of left RE equations and express it as a quotient of \(\mathcal {A}_{\mathsf {pos}}\) by a specific left-invariant equivalence relation. We define and characterise the right-partial derivative automaton (\(\overleftarrow{\mathcal {A}}_\mathsf {pd}\)). Finally, we study the average size of all these constructions both experimentally and from an analytic combinatorics point of view.

AgriVillage: A Game to Foster Awareness of the Environmental Impact of Agriculture

Abstract: Agriculture, while of uttermost importance for society, may also have a strong negative impact on the environment. Hence we propose a game that offers players the opportunity to experience the effects of different styles of agriculture on the environment. The game was built with the purpose of promoting the awareness of agriculture issues, such as, (1) the impact of fertilizers in sources of fresh water, (2) the problems related to deforestation and impact on the weather, and (3) the importance of balancing environmental and economic perspectives in order to produce food of good quality with low impact on the environment—and at the same time keep the activity sustainable. To make players care about these issues, we added a direct impact of the players' actions on a population of non-player characters, the villagers, whose simple autonomous behaviors resemble that of living entities. The game was implemented in the multi-user online three-dimensional (3-D) virtual world platform Open-Simulator, which supports an immersive user experience and high accessibility. An experiment was performed and showed that the game improved players' knowledge about agriculture and their awareness of the environmental impact of agriculture.

A New Approach to the Paperfolding Sequences

Abstract: In this paper we show how to re-derive known results about the paperfolding sequences, and obtain new ones, using a new approach using a decision method and some machine computation. We also obtain exact expressions for the recurrence and appearance function of the paperfolding sequences, and solve an open problem of Rampersad about factors shared in common between two different paperfolding sequences.

Newton’s Forward Difference Equation for Functions from Words to Words

Abstract: Newton’s forward difference equation gives an expression of a function from \({\mathbb {N}}\) to \({\mathbb {Z}}\) in terms of the initial value of the function and the powers of the forward difference operator. An extension of this formula to functions from \(A^*\) to \({\mathbb {Z}}\) was given in 2008 by P. Silva and the author. In this paper, the formula is further extended to functions from \(A^*\) into the free group over \(B\).

Local Compactness for Computable Polish Metric Spaces is \(\varPi ^1_1\)-complete

Abstract: We show that the property of being locally compact for computable Polish metric spaces is \(\varPi ^1_1\) complete. We verify that local compactness for Polish metric spaces can be expressed by a sentence in \(L_{\omega _1, \omega }\).

Intuitionistic Provability versus Uniform Provability in \mathsf{RCA}

Abstract: We provide an exact formalization of uniform provability in \(\mathsf {RCA}\) and show that for any \(\Pi _{2}^{1}\) sentence of some syntactical form, it is intuitionistically provable if and only if it is uniformly provable in \(\mathsf {RCA}\).

Sound Localization on Tabletop Computers: A Comparison of Two Amplitude Panning Methods

Abstract: Tabletop computers (also known as surface computers and smart tables) have been growing in popularity for the past decade and are poised to make inroads into the consumer market, opening up a new market for the games industry. But before tabletop computers become widely accepted, there are many questions with respect to sound production and reception for these devices that need to be explored, particularly when it comes to multimedia consumption on the devices. For example, which loudspeaker setups should be used to take into consideration the multi-user nature of tabletop computers, and which panning method(s) maximize the spatial localization abilities of the user(s)q Previous work suggests that a quadraphonic diamond-shaped loudspeaker configuration—whereby a loudspeaker is placed at each of the four sides of the tabletop computer—leads to more accurate localization results when compared with a traditional quadraphonic loudspeaker configuration—whereby a loudspeaker is placed at each of the four corners of the tabletop computer. Given this preference for a diamond loudspeaker configuration, we examine two amplitude-panning methods (bilinear interpolation and inverse distance) for spatializing a sound on the (horizontal) surface of the table-computer with a diamond loudspeaker configuration. Results from the study detailed in this paper indicate that there are no significant differences between the two methods and that both methods are prone to error.

Turing Jumps Through Provability

Abstract: Fixing some computably enumerable theory \(T\), the Friedman-Goldfarb-Harrington (FGH) theorem says that over elementary arithmetic, each \({\varSigma }_1\) formula is equivalent to some formula of the form \(\Box _T \varphi \) provided that \(T\) is consistent. In this paper we give various generalizations of the FGH theorem. In particular, for \(n>1\) we relate \({\varSigma }_{n}\) formulas to provability statements \([n]^{\mathsf{True}}_T\varphi \) which are a formalization of “provable in \(T\) together with all true \({\varSigma }_{n+1}\) sentences”. As a corollary we conclude that each \([n]^{\mathsf{True}}_T\) is \({\varSigma }_{n+1}\)-complete. This observation yields us to consider a recursively defined hierarchy of provability predicates \([n+1]^\Box _T\) which look a lot like \([n+1]^{\mathsf{True}}_T\) except that where \([n+1]^{\mathsf{True}}_T\) calls upon the oracle of all true \({\varSigma }_{n+2}\) sentences, the \([n+1]^\Box _T\) recursively calls upon the oracle of all true sentences of the form \(\langle n \rangle _T^\Box \phi \). As such we obtain a ‘syntax-light’ characterization of \({\varSigma }_{n+1}\) definability whence of Turing jumps which is readily extended beyond the finite. Moreover, we observe that the corresponding provability predicates \([n+1]_T^\Box \) are well behaved in that together they provide a sound interpretation of the polymodal provability logic \({\mathsf {GLP}} _\omega \).

On the Computational Content of Termination Proofs

Abstract: Given that a program has been shown to terminate using a particular proof, it is natural to ask what we can infer about its complexity. In this paper we outline a new approach to tackling this question in the context of term rewrite systems and recursive path orders. From an inductive proof that recursive path orders are well-founded, we extract an explicit realiser which bounds the derivational complexity of rewrite systems compatible with these orders. We demonstrate that by analysing our realiser we are able to derive, in a completely uniform manner, a number of results on the relationship between the strength of path orders and the bounds they induce on complexity.

Kalmár and Péter: Undecidability as a Consequence of Incompleteness

Abstract: Laszlo Kalmar and Peter Rozsa “proved that the existence of (...) undecidable problems follows from Godel’s Theorem on relatively undecidable problems” ([6], p. vii). Unfortunately, the only available document of their joint work is Kalmar’s sketch of the proof in his [3]. In the following, I assemble a paper from Kalmar’s manuscripts on this issue.

Randomness and Differentiability of Convex Functions

Abstract: We study first and second derivatives of computable convex functions on \(\mathbb {R}^n\). The main result of the paper is an effective form of Aleksandrov’s Theorem: we show that computable randomness implies twice-differentiability of computable convex functions.

Prime Model with No Degree of Autostability Relative to Strong Constructivizations

Abstract: We build a decidable structure \(\mathcal {M}\) such that \(\mathcal {M}\) is a prime model of the theory \(Th(\mathcal {M})\) and \(\mathcal {M}\) has no degree of autostability relative to strong constructivizations.

Nonexistence of Minimal Pairs in L[{\mathbf d}]

Abstract: For a d.c.e. set \(D\) with a d.c.e. approximation \(\{D_s\}_{s\in \omega }\), the Lachlan set of \(D\) is defined as \(L(D) = \{ s: \exists x \in D_{s} - D_{s-1} \ \hbox {and} \ x ot \in D\}.\) For a d.c.e. degree \({\mathbf d}\), \(L[\mathbf{d}]\) is defined as the class of c.e. degrees of those Lachlan sets of d.c.e. sets in \(\mathbf{d}\). In this paper, we prove that for any proper d.c.e. degree \(\mathbf{d}\), no two elements in \(L[\mathbf{d}]\) can form a minimal pair. This result gives another solution to Ishmukhametov’s problem, which asks whether for any proper d.c.e. degree \(\mathbf{d}\), \(L[\mathbf{d}]\) always has a minimal element. A negative answer to this question was first given by Fang, Wu and Yamaleev in 2013.

Some Results on Interactive Proofs for Real Computations

Abstract: We study interactive proofs in the framework of real number complexity theory as introduced by Blum, Shub, and Smale. Shamir’s famous result characterizes the class IP as PSPACE or, equivalently, as PAT and PAR in the Turing model. Since space resources alone are known not to make much sense in real number computations the question arises whether IP can be similarly characterized by one of the latter classes. Ivanov and de Rougemont [9] started this line of research showing that an analogue of Shamir’s result holds in the additive Blum-Shub-Smale model of computation when only Boolean messages can be exchanged. Here, we introduce interactive proofs in the full BSS model. As main result we prove an upper bound for the class \(\mathrm{IP}_{{\mathbb R}}\). It gives rise to the conjecture that a characterization of \(\mathrm{IP}_{{\mathbb R}}\) will not be given via one of the real complexity classes \(\mathrm{PAR}_{{\mathbb R}}\) or \(\mathrm{PAT}_{{\mathbb R}}\). We report on ongoing approaches to prove as well interesting lower bounds for \(\mathrm{IP}_{{\mathbb R}}\).

Covering the recursive sets

Abstract: We give solutions to two of the questions in a paper by Brendle, Brooke-Taylor, Ng and Nies. Our examples derive from a 2014 construction by Khan and Miller as well as new direct constructions using martingales.

Rice’s Theorem in Effectively Enumerable Topological Spaces

Abstract: In the framework of effectively enumerable topological spaces, we investigate the following question: given an effectively enumerable topological space whether there exists a computable numbering of all its computable elements. We present a natural sufficient condition on the family of basic neighborhoods of computable elements that guarantees the existence of a principal computable numbering. We show that weakly-effective \(\omega \)–continuous domains and the natural numbers with the discrete topology satisfy this condition. We prove weak and strong analogues of Rice’s theorem for computable elements.