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The tournament equilibrium set (TEQ), proposed by Schwartz in 1990, is a classical voting rule based on pairwise majority comparisons. Its definition is recursive: an alternative is judged by the alternatives that beat it, called its dominators, and by which of those dominators TEQ itself selects. Deciding whether a given alternative belongs to TEQ was known to be NP-hard and to lie in PSPACE. We close this gap by proving that the problem is PSPACE-complete. To our knowledge, this is the first PSPACE-completeness result for membership in a tournament solution. The hardness persists even when every nonempty subtournament has a unique minimal retentive set, the property conjectured by Schwartz to hold for all tournaments. Moreover, every such subtournament in our hard instances contains an alternative with at most three dominators. We prove that a single alternative with at most four dominators suffices both to guarantee uniqueness and to find a TEQ winner in $O(N^2)$ time. Thus, on our hard instances, finding one winner is efficient even though recognizing a specified winner is PSPACE-complete. We also study how the complexity changes with the dominator bound. If every nonempty subtournament has an alternative with at most one dominator, the entire TEQ can be computed in quadratic time; allowing two dominators makes membership NP-hard. Under the same condition with four dominators, all $t$ winners can be computed in $O(2^t N^2)$ time, where $N$ is the number of alternatives.

Kalah is a classic two-player game in the mancala family of sowing games. Computers have solved several standard-size starting positions, but the complexity of the game on boards of growing size has remained open. We study the natural generalization in which the number of pits per side is part of the input and an arbitrary position is given, with the stone counts given in unary. We show that deciding whether a designated player can force a win from such a position is PSPACE-complete. The hard positions have Alice to move first and both stores initially empty. The same positions are hard under either common convention for capture opposite an empty pit, and with or without the bonus rule, which gives a player another turn when the last stone is sown into their own store. We introduce Line Kalah as an intermediate problem, a game on a line in which stones that leave the line effectively disappear and Alice must save each of her target pits before Bob captures it, and we reduce quantified Boolean formulas to it. The reduction uses a set of sequential timer piles whose last stones approach a group of targets, forcing Alice to choose between two captures in order to set the truth value of a variable, and a final capture gadget to test a clause chosen by Bob (only one is needed adversarially). Since either player may move any pile at any time, the stages are placed so that premature timer firings protect earlier potential targets. For a true formula, Alice's strategy against arbitrary opposing play either gains enough time to save the remaining targets (after Bob deviates from the intended play) or reaches the final test with a correctly encoded, protected input. Finally, the line positions are embedded in Kalah with both stores empty. A separate Bob pit holds the score adjustment, so the winner takes a strict majority of the stones.

We study predictive properties of a reflective variant of Solomonoff induction formulated using reflective oracles (Fallenstein et al. 2015, Wyeth et al. 2025). We show that the resulting optimal predictor succeeds against computable predictive strategies that specify only part of the data-generating process. The argument relies on the fact that hypotheses computable from the predictor's own conditional probabilities remain within the reflective hypothesis class. In particular, the predictor learns incomplete models specified by computable interval restrictions on next-bit probabilities. We also establish weighted forms of unbiasedness and calibration. Finally, we discuss whether an optimal predictor's conditional probabilities suffice to compute every hypothesis in the reflective class.

We study exact evaluation of the arithmetized $k$-Orthogonal Vectors polynomial $FOV^k$ of Ball et al. (2017, 2018). For every fixed integer $k \geq 3$, set $d(n) = \left \lceil \log^2 n \right \rceil$. Assuming the basic decision version of Quantum Strong Exponential-Time Hypothesis (QSETH), for every constant $\varepsilon > 0$, no uniform quantum algorithm runs in $O(n^{k/2 - \varepsilon})$ time on every input and evaluates this polynomial correctly with probability at least $3 / 4$ over a uniformly random input in $\mathbb F_{p(n)}^{knd(n)}$ and the algorithm's internal randomness, for all sufficiently large $n$. This holds when $p(n)$ is the smallest prime greater than $n^{k - 1}$ and when it is the smallest prime greater than $n^k$. Our direct reduction combines classical polynomial self-correction with calls to a quantum average-case evaluator. For the smaller prime, random sampling finds an orthogonal tuple with constant probability whenever the count is large enough to wrap around modulo $p(n)$; otherwise, the exact residue determines the answer. For the larger prime, quantum minimum finding and an asymptotic prime-gap bound account for the cost of generating the prescribed field. In view of prior isolation-based reductions permitting small characteristics, we present these results as explicit reductions for the two canonical-prime families, with all setup costs included in the uniform quantum running time. The hardness guarantee concerns single-instance exact evaluation; a matching quantum evaluation algorithm and a quantum proof-of-work protocol remain open.

In this article, we show that the problem of deciding whether a finite relational structure $\mathbf{A}$ admits a Maltsev polymorphism is a computational problem, which can be solved in polynomial time. We provide a proof of this result by reducing the problem to deciding whether a binary relational structure, associated with $\mathbf{A}$ admits such a polymorphism. In our construction of such an algorithm, we make an extensive use of a property of binary relations, which is implied by the existence of a Maltsev polymorphism, the so-called rectangularity property.

Let $q$ be an odd prime. Let $ζ_q$ be a primitive $q$th root of unity, and put $K=\mathbb{Q}(ζ_q)$, with ring of integers $\mathcal{O}_K=\mathbb{Z}[ζ_q]$. As $q$ ranges over odd primes, we prove that the decision version of the Shortest Vector Problem ($\mathsf{SVP}$) in the canonical $\ell_2$-norm is $\mathsf{NP}$-complete on full-rank free submodules of $\mathcal{O}_K^2$ by giving a deterministic polynomial-time many-one reduction from Exact Cover by 3-Sets (X3C). The module rank is fixed at two. The main obstacle is closure under the action of $\mathcal{O}_K$. A module containing a nonzero vector also contains every scalar multiple of that vector by a nonzero element of $\mathcal{O}_K$, and some of these multiples may be shorter. Three ideas overcome this obstacle. First, we map the Bennett--Peikert Reed--Solomon lattice to a principal cyclotomic ideal and use Wan's point-count estimates to prove that a coset of this ideal contains many binary coefficient representatives. Second, we construct a checker whose canonical squared norm encodes the X3C equations, with a controlled nonnegative error term. Third, the checker and a second module coordinate combine with a separation bound for ideal cosets to rule out short vectors in NO instances. Each constructed instance consists of an odd prime $q$, two integral generators whose $2\times2$ generator matrix has nonzero determinant, and an integer squared threshold. The construction also gives $\mathsf{NP}$-hardness of search-$\mathsf{SVP}$ under deterministic polynomial-time Turing reductions with at most one oracle call.