Time to Reason: Scalable Neurosymbolic Learning for LTLf via Fuzzy Semantics

Riccardo Andreoni; Andrei Buliga; Alessandro Daniele; Paolo Felli; Chiara Ghidini; Marco Montali; Massimiliano Ronzani

cs.AIAug 17, 2026

Time to Reason: Scalable Neurosymbolic Learning for LTLf via Fuzzy Semantics

Authors: Riccardo Andreoni, Andrei Buliga, Alessandro Daniele, Paolo Felli, Chiara Ghidini, Marco Montali, Massimiliano Ronzani

Organizations: Fondazione Bruno Kessler, Via Sommarive, 18, Trento, 38123, Italy · Free University of Bozen-Bolzano, Via Bruno Buozzi, 1, Bolzano, 39100, Italy · Università di Bologna, Via Mura Anteo Zamboni 7, Bologna, 40126, Italy

Abstract

Neurosymbolic (NeSy) Artificial Intelligence aims to integrate Deep Learning (DL) architectures with symbolic reasoning. While initial NeSy approaches have targeted mainly symbolic reasoning in propositional and first-order logics, recent works have started to address the construction of neurosymbolic frameworks for Temporal Logics, and in particular for LTLf. These approaches have established temporal NeSy as a promising research direction, laying the foundations for learning under temporal constraints. Nonetheless, they leave many questions unanswered. From a theoretical perspective, several differentiable semantics for interpreting LTLf have been proposed but have not yet been formally and systematically defined within a unified framework. Moreover, existing approaches commonly rely on automata to represent temporal knowledge, resulting in limited scalability. Motivated by this research gap, this paper provides the following contributions: (i) formally defining different fuzzy semantics for LTLf, and systematically analysing theoretical properties regarding equivalences and dualities of temporal operators; (ii) showing how these semantics can be directly integrated within a novel NeSy framework, called DiffLTLf, enabling flexible and scalable learning without relying on the usage of automata; and (iii) introducing a novel evaluation protocol of increased complexity of learning tasks w.r.t. existing benchmarks. Our results show that the choice of fuzzy semantics has a significant impact on predictive performance. Moreover, DiffLTLf achieves performance on par with, and sometimes superior to, state-of-the-art probabilistic approaches while substantially improving scalability. Taken together, these results establish direct fuzzy interpretations as a competitive and scalable alternative to existing temporal NeSy frameworks.

Figures & tables

Figure 1: Illustrative example of the weakly supervised learning task on a simplified manufacturing scenario. Left: correct perception leads to a correct assessment of non-compliance, matching the ground-truth label. Right: incorrect perception leads to a wrong assessment of compliance, producing a corrective learning signal.

Panel of Figure 2 — Figure 2: Graphical depiction of an ltl f trace of length 5, and satisfaction of the two formulae $a\mathsf{U}b$ and $\mathsf{X}b$ in every instant of the trace.

declare Pattern	ltl f Template	Description
Response	$\mathsf{G}(A\rightarrow\mathsf{F}\,B)$	If $A$ occurs, $B$ must eventually follow
Not Succession	$\mathsf{G}(A\rightarrow\neg\mathsf{F}\,B)$	If $A$ occurs, $B$ must never follow
Chain Succession	$\mathsf{G}(A\rightarrow\mathsf{X}\,B)$	If $A$ occurs, $B$ must hold at the next timestep
Not Chain Succession	$\mathsf{G}(A\rightarrow\neg\mathsf{X}\,B)$	If $A$ occurs, $B$ must not hold at the next timestep
Precedence	$(\neg B\;\mathbin{\mathsf{U}}\;A)\lor\mathsf{G}(\neg B)$	$B$ cannot occur before $A$ has occurred
Chain Precedence	$\mathsf{G}(\mathsf{X}\,B\rightarrow A)$	Every occurrence of $B$ is immediately preceded by $A$

Table 1: declare pattern templates used in the benchmark.

A

and

B

denote stream-specific propositions or Boolean combinations of them.

Panel of Figure 3 — Figure 3: Graphical depiction of a fltl ${}_{f}^{x}$ trace of length 3, and satisfaction of the formula $a\mathsf{U}b$ in every instant of the trace, under all three semantics.

Figure 4: Overview of the $\partial$ LTL f framework. The perception module $f_{\theta}$ produces the fuzzy trace $\lambda_{\theta}$ (rows represent the propositions $p^{s}_{c}$ grouped by stream). The temporal logic module evaluates $\Phi$ under the fltl ${}_{f}^{x}$ semantics. The loss evaluates the predictions of the temporal logic module in comparison with the ground truth label.

Panel of Figure 5 — Figure 5: Computation of $v(\mathsf{G}(a\to\mathsf{F}b),\lambda)$ , from Example 3 .

Figure 6: Symbol grounding accuracy (%) across 10 independent runs for

\partial

LTL f instantiated with the three fuzzy semantics: Product ( fltl

{}_{f}^{P}

), Łukasiewicz ( fltl

{}_{f}^{L}

), and Gödel ( fltl

{}_{f}^{G}

). Each subplot corresponds to one experimental setting (formula set

\times

image dataset), with the number of classes

|\mathcal{C}|

on the horizontal axis.

	Img. Acc. (%) $\uparrow$				Seq. Acc. (%) $\uparrow$
$\|\mathcal{C}\|$	2	4	6	8	2	4	6	8
fltl ${}_{f}^{P}$	99.9	97.7	93.3	89.2	99.9	98.9	97.0	96.0
fltl ${}_{f}^{G}$	99.9	62.2	32.1	19.9	99.9	90.0	88.6	86.2
fltl ${}_{f}^{L}$	94.9	64.0	49.2	37.6	96.6	90.6	89.3	87.7

Table 2: Symbol grounding and sequence classification accuracy (%) of

\partial

LTL f under the three fuzzy semantics, averaged over the four experimental settings and 10 runs. Results in bold are significantly better than both other semantics (Wilcoxon,

p<0.05

		Img. Acc. ( $\%$ ) $\uparrow$			Seq. Acc. ( $\%$ ) $\uparrow$			Time (s) $\downarrow$
Setting	$\|\mathcal{C}\|$	$\partial$ LTL f	FuzzyA	NeSyA	$\partial$ LTL f	FuzzyA	NeSyA	$\partial$ LTL f	FuzzyA	NeSyA
MNIST Set1	2	99.9	99.9	99.9	99.8	99.8	99.8	0.04	0.17	0.17
	4	97.4	97.4	97.4	98.9	98.9	99.0	0.05	0.33	0.40
	6	95.7	95.6	91.3	98.3	98.3	97.6	0.05	0.63	0.99
	8	92.7	91.5	88.4	96.7	96.5	95.5	0.05	1.97	5.08
MNIST Set2	2	99.9	99.9	99.9	99.8	99.9	99.9	0.05	0.17	0.15

Table 3: Performance comparison varying formula complexity. Results are averaged over 10 independent runs. Img. Acc : Symbol Grounding Accuracy; Seq. Acc : Sequence Classification Accuracy; Time : Logic Module Time per Epoch. Results in bold are statistically significantly better than all other methods (Wilcoxon,

p<0.05

Figure 7: Comparison of symbol grounding accuracy (left) and logic module time per epoch (right) across number of classes

|\mathcal{C}|\in\{2,4,6,8\}

. Values are averaged across all four experimental settings (formula set

\times

image dataset).

		Img. Acc. ( $\%$ ) $\uparrow$			Seq. Acc. ( $\%$ ) $\uparrow$			Time (s) $\downarrow$
$\|\mathcal{C}\|$	Len	$\partial$ LTL f	FuzzyA	NeSyA	$\partial$ LTL f	FuzzyA	NeSyA	$\partial$ LTL f	FuzzyA	NeSyA
4	20	97.9	97.9	98.0	97.7	97.8	97.8	0.007	0.41	0.57
	40	97.9	98.0	98.0	98.0	98.1	98.1	0.007	0.74	1.14
	60	97.9	98.1	98.1	97.2	97.7	97.6	0.007	1.09	1.74
	80	98.0	97.9	97.9	98.0	98.2	98.2	0.007	1.46	2.33
	100	98.0	98.0	98.1	98.1	98.0	98.0	0.008	1.83	2.91

Table 4: Performance comparison varying sequence length. Results are averaged over 10 independent runs. Img. Acc : Symbol Grounding Accuracy; Seq. Acc : Sequence Classification Accuracy; Time : Logic Module Time per Epoch. Results in bold are statistically significantly better than all other methods (Wilcoxon,

p<0.05

Figure 8: Comparison of symbol grounding accuracy (top row) and logic module time per epoch (bottom row) across sequence lengths varying from 20 to 200. Left column: temporal specification with

|\mathcal{C}|=4

classes. Right column: temporal specification with

|\mathcal{C}|=8

classes. Lines represent the mean over 10 independent runs; shaded bands indicate the range between the minimum and maximum values. Time is shown on a logarithmic scale.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Set 1				Set 2
$\|\mathcal{C}\|$ :	2	4	6	8	2	4	6	8
Img. Acc. (%) $\uparrow$
$\partial$ LTL f (Product)	99.9	97.4	95.7	92.7	99.9	97.4	95.0	92.7
$\partial$ LTL f (Gödel)	99.9	60.8	32.7	22.8	99.9	59.2	29.6	16.3
$\partial$ LTL f (Łukasiewicz)	90.1	58.4	46.2	35.4	99.9	62.6	57.1	42.0
T-ILR	99.9	60.8	32.7	22.8	99.9	59.2	29.6	16.3

Appendix

Table 5: Full results varying formula complexity on MNIST, averaged over 10 runs. Img. Acc. : Symbol Grounding Accuracy; Seq. Acc. : Sequence Classification Accuracy; Time : Logic Module Time per Epoch. Results in bold are statistically significantly better than all other methods (Wilcoxon,

p<0.05

Method	Set 1				Set 2
$\|\mathcal{C}\|$ :	2	4	6	8	2	4	6	8
Img. Acc. (%) $\uparrow$
$\partial$ LTL f (Product)	100.0	97.8	91.7	86.0	99.9	98.0	90.9	85.1
$\partial$ LTL f (Gödel)	100.0	76.0	34.0	23.6	99.9	52.7	32.0	16.7
$\partial$ LTL f (Łukasiewicz)	89.9	55.9	47.6	40.0	99.9	79.2	45.8	33.0
T-ILR	100.0	76.0	34.0	23.6	99.9	52.7	32.0	16.7

Appendix

Table 6: Full results varying formula complexity on Fashion-MNIST, averaged over 10 runs. Img. Acc. : symbol grounding accuracy; Seq. Acc. : sequence classification accuracy; Time : Logic Module Time per Epoch. Results in bold are statistically significantly better than all other methods (Wilcoxon,

p<0.05

	Sequence length
Method	20	40	60	80	100	120	140	160	180	200
Img. Acc. (%) $\uparrow$
$\partial$ LTL f (Product)	97.9	97.9	97.9	98.0	98.0	97.9	98.1	97.7	97.9	97.7
$\partial$ LTL f (Gödel)	43.5	35.0	47.2	35.9	49.9	48.5	41.1	55.4	55.0	54.8
$\partial$ LTL f (Łukasiewicz)	72.9	92.7	87.8	92.1	85.5	78.6	82.6	86.6	91.1	81.2
T-ILR	43.5	35.0	47.2	35.9	49.9	48.5	41.1	55.4	55.0	54.8

Appendix

Table 7: Full results varying the sequence length,

|\mathcal{C}|=4

, averaged over 10 runs. Img. Acc. : Symbol Grounding Accuracy; Seq. Acc. : Sequence Classification Accuracy; Time : Logic Module Time per Epoch. Bold marks a value statistically significantly better than all other methods (Wilcoxon,

p<0.05

	Sequence length
Method	20	40	60	80	100	120	140	160	180	200
Img. Acc. (%) $\uparrow$
$\partial$ LTL f (Product)	95.8	96.0	95.6	95.7	95.6	95.5	96.1	95.8	95.8	95.6
$\partial$ LTL f (Gödel)	17.5	17.1	17.7	16.7	17.3	15.1	14.8	16.4	15.2	14.9
$\partial$ LTL f (Łukasiewicz)	14.6	14.6	14.6	14.6	14.5	14.5	14.5	14.5	14.6	14.6
T-ILR	17.5	17.1	17.7	16.7	17.3	15.1	14.8	16.4	15.2	14.9

Appendix

Table 8: Full results varying the sequence length,

|\mathcal{C}|=8

, averaged over 10 runs. Img. Acc. : Symbol Grounding Accuracy; Seq. Acc. : Sequence Classification Accuracy; Time : Logic Module Time per Epoch. Results in bold are statistically significantly better than all other methods (Wilcoxon,

p<0.05

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