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Linear temporal logic over finite traces (LTLf) satisfiability checking is a fundamental and hard (PSPACE-complete) problem in the artificial intelligence community. We explore teaching end-to-end neural networks to check satisfiability in polynomial time. It is a challenge to characterize the syntactic and semantic features of LTLf via neural networks. To tackle this challenge, we propose LTLfNet, a recursive neural network that captures syntactic features of LTLf by recursively combining the embeddings of sub-formulae. LTLfNet models permutation invariance and sequentiality in the semantics of LTLf through different aggregation mechanisms of sub-formulae. Experimental results demonstrate that LTLfNet achieves good performance in synthetic datasets and generalizes across large-scale datasets. They also show that LTLfNet is competitive with state-of-the-art symbolic approaches such as nuXmv and CDLSC.

The objective of this study was to determine the effect of dietary supplementation of β-hydroxybutyric acid (BHBA) on performance, nutrient digestibility, organ development, and serum composition in early-weaned goat kids. Sixty-four goat kids at 30 d of age were assigned to 4 treatments in a completely randomized design: 1) control (basal diet); 2) low (basal diet with 3 g/d per head BHBA); 3) medium (basal diet with 6 g/d per head BHBA; and 4) high (basal diet with 9 g/d per head BHBA). Subsequently, 48 (6 kids per treatment) goat kids were randomly selected and slaughtered at 60 and 90 d of age. Compared with the control group, BHBA at low and high doses increased body weight (P < 0.05), average daily gain (P < 0.01), and average daily starter intake (P < 0.01). The BHBA improved organ development, especially at the lowest dose (P < 0.01). The digestibility of dry matter and crude protein increased with age (P < 0.05). However, BHBA did not affect nutrient digestibility. Compared with the control group, serum ceruloplasmin increased (P < 0.05) with high BHBA level at 90 d of age. However, the serum creatinine (P < 0.05) increased over time but was not affected by BHBA. The serum total antioxidant capacity and superoxide dismutase decreased with the high dose of BHBA at 90 d of age (P < 0.01). In contrast, the serum glutathione peroxidase and malondialdehyde increased with the high doses of BHBA (P < 0.01). Overall, low doses of BHBA were positive for growth performance, organ development, and health status against weaning stress. Whereas high doses of BHBA in the long term could negatively affect antioxidant status.