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Succinctness is a natural measure for comparing the strength of different logics. Intuitively, a logic L_1 is more succinct than another logic L_2 if all properties that can be expressed in L_2 can be expressed in L_1 by formulas of (approximately) the same size, but some properties can be expressed in L_1 by (significantly) smaller formulas.
We study the succinctness of logics on linear orders. Our first theorem is concerned with the finite variable fragments of first-order logic. We prove that:
(i) Up to a polynomial factor, the 2- and the 3-variable fragments of first-order logic on linear orders have the same succinctness. (ii) The 4-variable fragment is exponentially more succinct than the 3-variable fragment. Our second main result compares the succinctness of first-order logic on linear orders with that of monadic second-order logic. We prove that the fragment of monadic second-order logic that has the same expressiveness as first-order logic on linear orders is non-elementarily more succinct than first-order logic.
Background: We analyzed whether co-occurring mutations influence the outcome of systemic therapy in ALK-rearranged non-small-cell lung cancer (NSCLC).
Patients and methods: ALK-rearranged stage IIIB/IV NSCLC patients were analyzed with next-generation sequencing and fluorescence in situ hybridization analyses on a centralized diagnostic platform. Median progression-free survival (PFS) and overall survival (OS) were determined in the total cohort and in treatment-related sub-cohorts. Cox regression analyses were carried out to exclude confounders.
Results: Among 216 patients with ALK-rearranged NSCLC, the frequency of pathogenic TP53 mutations was 23.8%, while other co-occurring mutations were rare events. In ALK/TP53 co-mutated patients, median PFS and OS were significantly lower compared with TP53 wildtype patients [PFS 3.9 months (95% CI: 2.4–5.6) versus 10.3 months (95% CI: 8.6–12.0), P < 0.001; OS 15.0 months (95% CI: 5.0–24.9) versus 50.0 months (95% CI: 22.9–77.1), P = 0.002]. This difference was confirmed in all treatment-related subgroups including chemotherapy only [PFS first-line chemotherapy 2.6 months (95% CI: 1.3–4.1) versus 6.2 months (95% CI: 1.8–10.5), P = 0.021; OS 2.0 months (95% CI: 0.0–4.6) versus 9.0 months (95% CI: 6.1–11.9), P = 0.035], crizotinib plus chemotherapy [PFS crizotinib 5.0 months (95% CI: 2.9–7.2) versus 14.0 months (95% CI: 8.0–20.1), P < 0.001; OS 17.0 months (95% CI: 6.7–27.3) versus not reached, P = 0.049] and crizotinib followed by next-generation ALK-inhibitor [PFS next-generation inhibitor 5.4 months (95% CI: 0.1–10.7) versus 9.9 months (95% CI: 6.4–13.5), P = 0.039; OS 7.0 months versus 50.0 months (95% CI: not reached), P = 0.001).
Conclusions: In ALK-rearranged NSCLC co-occurring TP53 mutations predict an unfavorable outcome of systemic therapy. Our observations encourage future research to understand the underlying molecular mechanisms and to improve treatment outcome of the ALK/TP53 co-mutated subgroup.