419 research outputs found
Graph Representations for Higher-Order Logic and Theorem Proving
This paper presents the first use of graph neural networks (GNNs) for
higher-order proof search and demonstrates that GNNs can improve upon
state-of-the-art results in this domain. Interactive, higher-order theorem
provers allow for the formalization of most mathematical theories and have been
shown to pose a significant challenge for deep learning. Higher-order logic is
highly expressive and, even though it is well-structured with a clearly defined
grammar and semantics, there still remains no well-established method to
convert formulas into graph-based representations. In this paper, we consider
several graphical representations of higher-order logic and evaluate them
against the HOList benchmark for higher-order theorem proving
Irreversibility, heat and information flows induced by non-reciprocal interactions
We study the thermodynamic properties induced by non-reciprocal interactions between
stochastic degrees of freedom in time- and space-continuous systems. We show that, under fairly
general conditions, non-reciprocal coupling alone implies a steady energy flow through the system,
i.e., non-equilibrium. Projecting out the non-reciprocally coupled degrees of freedom renders
non-Markovian, one-variable Langevin descriptions with complex types of memory, for which we
find a generalized second law involving information flow.We demonstrate that non-reciprocal
linear interactions can be used to engineer non-monotonic memory, which is typical for, e.g.,
time-delayed feedback control, and is automatically accompanied with a nonzero information flow
through the system. Furthermore, already a single non-reciprocally coupled degree of freedom can
extract energy from a single heat bath (at isothermal conditions), and can thus be viewed as a
minimal version of a time-continuous, autonomous ‘Maxwell demon’.We also show that for
appropriate parameter settings, the non-reciprocal system has characteristic features of active
matter, such as a positive energy input on the level of the fluctuating trajectories without global
particle transport
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