The analysis showed no significant differences in performance between the experimental groups at baseline in any of the tasks: ACT, SCT, or CRTT (all p > 0.05, for more details please refer to Supplementary Table 1, Supplementary Figs. 1–7, and Fig. 2A). The different task conditions had the expected effect on the performance in all tasks (Fig. 1C). In ACT, in accordance with our previous studies (Marko et al. 2019, 2021, 2023), delivering semantically unrelated words (dissociations) required longer time than delivering related words (associations), and switching from associative to dissociative production further increased response latencies. In SCT, finishing unpredictable sentences took longer than finishing unpredictable sentences, and in CRTT, responses were longer in incongruent trials compared with congruent trials.

A The speed of word retrieval in the associated fixed condition of ACT. B The effect of cerebellar tDCS on performance in all ACT conditions. Plotted are difference RT (post-tDCS minus baseline) in each group estimated from the LMEM. Error bars: ± SE; Holm adjusted p-values: *p < .05, **p < .01, ***p < .001

The full-factorial LMEM indicated a significant interaction of stimulation, block and condition, F(6,20,575.3) = 3.45, p = 0.002, suggesting that the cerebellar tDCS affected the ACT performance in a specific way (a complete report of the LMEM results is provided in Supplementary Table 2). To disclose this complex effect, we separately analyzed each ACT condition (Table 2). A significant effect of block was present in all but Associative Alternating condition, reflecting that responses were overall faster in the second block. Importantly, in the Associative Fixed condition, a significant block x stimulation interaction was revealed (F(2, 5052.1) = 12.30, p < 0.001, Fig. 2A). To analyze the effects of stimulation in more detail, post-test minus baseline differences (ΔRT) were estimated from the LMEM and compared across groups. As depicted in Fig. 2B, the acceleration in fluency was significantly higher following anodal stimulation as compared with sham (group difference in ΔRT = − 318 ms, 95% CI [− 553, − 84], p = 0.001) and cathodal tDCS (group difference in ΔRT = − 449 ms, 95% CI [− 680, − 217], p < 0.001; also see Supplementary Table 3). The effect of cathodal tDCS was in the opposite direction than that of anodal stimulation, but the difference to sham stimulation was not significant (group difference in ΔRT =  + 130 ms, 95% CI [− 102, + 363], p = 0.161). The interaction between block and stimulation was not significant in any other ACT condition (Table 2, Supplementary Figs. 1–3).

The analysis showed significant effects of block, F(1, 15,554) = 39.0, p < 0.001, and condition, F(1, 15,554.1) = 13,963.8, p < 0.001, reflecting overall faster responses during the post-tDCS vs. baseline block (ΔRT = − 3 ms, 95% CI [− 69, – 36], p < 0.001), and in predictable vs. unpredictable sentences (ΔRT = − 998 ms, 95% CI [−1001, − 982], p < 0.001). However, these effects were independent of stimulation (block x condition x stimulation, F(2, 15,553.9) = 1.21, p = 0.298), indicating that tDCS had no effect on the latency of sentence completion. For complete results of the analysis and depiction of the experimental effects please refer to Supplementary Tables 4–6 and Supplementary Figs. 4–5.

The effects of block, F(1, 63,322) = 6381.78, p < 0.001, and condition, F(1, 63,321) = 32,980.16, p < 0.001, were significant, reflecting shorter RTs in the post-tDCS vs. baseline block, and in congruent vs. incongruent trials. Furthermore, there was a significant interaction between stimulation, block, and condition, F(2, 63,320) = 5.95, p = 0.003. Separate analyses for the two task conditions revealed a significant block x stimulation interaction in congruent trials, F(2, 47,766) = 7.33, p < 0.001. To analyze the effects of stimulation in more detail, post-test minus baseline differences (ΔRT) were estimated from the LMEM and compared across groups. While all three groups had shorter RTs in the post-tDCS block vs. baseline, when compared with the sham group, this improvement in RT was significant only in the anodal group (group difference in ΔRT = − 5 ms, 95% CI [– 9, 1], p = 0.019), but not in the cathodal group (group difference in ΔRT =  + 2 ms, 95% CI [– 2 + 6], p = 0.255), and the anodal and the cathodal group also differed significantly (group difference in ΔRT = − 7 ms, 95% CI [– 11, − 2], p < 0.001). For incongruent trials, the block x stimulation interaction was not significant, F(2, 15,426) = 2.07, p = 0.126. For complete results of the analysis and depiction of the experimental effects please refer to Supplementary Tabs. 7–9 and Supplementary Figs. 6–7.