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Register a new userAdventures in Mathematical Reasoning
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Toby Walsh
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Authors
Toby Walsh
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Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere. W.S. Anglin, the Mathematical Intelligencer, 4 (4), 1982.
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In this paper, we study isogeny graphs of supersingular elliptic curves. Supersingular isogeny graphs were introduced as a hard problem into cryptography by Charles, Goren, and Lauter for the construction of cryptographic hash functions [CGL06]. These are large expander graphs, and the hard problem is to find an efficient algorithm for routing, or path-finding, between two vertices of the graph. We consider four aspects of supersingular isogeny graphs, study each thoroughly and, where appropriate, discuss how they relate to one another. First, we consider two related graphs that help us understand the structure: the `spine $mathcal{S}$, which is the subgraph of $mathcal{G}_ell(overline{mathbb{F}_p})$ given by the $j$-invariants in $mathbb{F}_p$, and the graph $mathcal{G}_ell(mathbb{F}_p)$, in which both curves and isogenies must be defined over $mathbb{F}_p$. We show how to pass from the latter to the former. The graph $mathcal{S}$ is relevant for cryptanalysis because routing between vertices in $mathbb{F}_p$ is easier than in the full isogeny graph. The $mathbb{F}_p$-vertices are typically assumed to be randomly distributed in the graph, which is far from true. We provide an analysis of the distances of connected components of $mathcal{S}$. Next, we study the involution on $mathcal{G}_ell(overline{mathbb{F}_p})$ that is given by the Frobenius of $mathbb{F}_p$ and give heuristics on how often shortest paths between two conjugate $j$-invariants are preserved by this involution (mirror paths). We also study the related question of what proportion of conjugate $j$-invariants are $ell$-isogenous for $ell = 2,3$. We conclude with experimental data on the diameters of supersingular isogeny graphs when $ell = 2$ and compare this with previous results on diameters of LPS graphs and random Ramanujan graphs.
Conditionals are useful for modelling, but are not always sufficiently expressive for capturing information accurately. In this paper we make the case for a form of conditional that is situation-based. These conditionals are more expressive than classical conditionals, are general enough to be used in several application domains, and are able to distinguish, for example, between expectations and counterfactuals. Formally, they are shown to generalise the conditional setting in the style of Kraus, Lehmann, and Magidor. We show that situation-based conditionals can be described in terms of a set of rationality postulates. We then propose an intuitive semantics for these conditionals, and present a representation result which shows that our semantic construction corresponds exactly to the description in terms of postulates. With the semantics in place, we proceed to define a form of entailment for situated conditional knowledge bases, which we refer to as minimal closure. It is reminiscent of and, indeed, inspired by, the version of entailment for propositional conditional knowledge bases known as rational closure. Finally, we proceed to show that it is possible to reduce the computation of minimal closure to a series of propositional entailment and satisfiability checks. While this is also the case for rational closure, it is somewhat surprising that the result carries over to minimal closure.
Multi-hop reasoning has been widely studied in recent years to obtain more interpretable link prediction. However, we find in experiments that many paths given by these models are actually unreasonable, while little works have been done on interpretability evaluation for them. In this paper, we propose a unified framework to quantitatively evaluate the interpretability of multi-hop reasoning models so as to advance their development. In specific, we define three metrics including path recall, local interpretability, and global interpretability for evaluation, and design an approximate strategy to calculate them using the interpretability scores of rules. Furthermore, we manually annotate all possible rules and establish a Benchmark to detect the Interpretability of Multi-hop Reasoning (BIMR). In experiments, we run nine baselines on our benchmark. The experimental results show that the interpretability of current multi-hop reasoning models is less satisfactory and is still far from the upper bound given by our benchmark. Moreover, the rule-based models outperform the multi-hop reasoning models in terms of performance and interpretability, which points to a direction for future research, i.e., we should investigate how to better incorporate rule information into the multi-hop reasoning model. Our codes and datasets can be obtained from https://github.com/THU-KEG/BIMR.
Functional dependencies restrict the potential interactions among variables connected in a probabilistic network. This restriction can be exploited in qualitative probabilistic reasoning by introducing deterministic variables and modifying the inference rules to produce stronger conclusions in the presence of functional relations. I describe how to accomplish these modifications in qualitative probabilistic networks by exhibiting the update procedures for graphical transformations involving probabilistic and deterministic variables and combinations. A simple example demonstrates that the augmented scheme can reduce qualitative ambiguity that would arise without the special treatment of functional dependency. Analysis of qualitative synergy reveals that new higher-order relations are required to reason effectively about synergistic interactions among deterministic variables.
Probability trees are one of the simplest models of causal generative processes. They possess clean semantics and -- unlike causal Bayesian networks -- they can represent context-specific causal dependencies, which are necessary for e.g. causal induction. Yet, they have received little attention from the AI and ML community. Here we present concrete algorithms for causal reasoning in discrete probability trees that cover the entire causal hierarchy (association, intervention, and counterfactuals), and operate on arbitrary propositional and causal events. Our work expands the domain of causal reasoning to a very general class of discrete stochastic processes.
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