First-order Logic and Numeration Systems

The Büchi-Bruyère theorem states that a multidimensional set of non-negative integers is b-recognizable if and only if it is b-definable. This result is a powerful tool for showing that many properties of b-automatic sequences are decidable. Going a step further, first-order logic can be used to show that many enumeration problems of b-automatic sequences can be described by b-regular sequences. The latter sequences can be viewed as a generalization of b-automatic sequences to integer-valued sequences. These techniques were extended to two wider frameworks: U-recognizable multidimensional sets of non-negative integers and multidimensional beta-recognizable sets of reals. In the second case, real numbers are represented by infinite words, and hence, the notion of beta-recognizability is defined by means of Büchi automata. Again, logic-based characterizations of $U$-recognizable (resp. beta-recognizable) sets allows us to obtain various decidability results. The aim of this chapter is to present a survey of this very active research domain

Correction: Chemopreventive Effect of PSP Through Targeting of Prostate Cancer Stem Cell-Like Population

Recent evidence suggested that prostate cancer stem/progenitor cells (CSC) are responsible for cancer initiation as well as disease progression. Unfortunately, conventional therapies are only effective in targeting the more differentiated cancer cells and spare the CSCs. Here, we report that PSP, an active component extracted from the mushroom Turkey tail (also known as Coriolus versicolor), is effective in targeting prostate CSCs. We found that treatment of the prostate cancer cell line PC-3 with PSP led to the down-regulation of CSC markers (CD133 and CD44) in a time and dose-dependent manner. Meanwhile, PSP treatment not only suppressed the ability of PC-3 cells to form prostaspheres under non-adherent culture conditions, but also inhibited their tumorigenicity in vivo, further proving that PSP can suppress prostate CSC properties. To investigate if the anti-CSC effect of PSP may lead to prostate cancer chemoprevention, transgenic mice (TgMAP) that spontaneously develop prostate tumors were orally fed with PSP for 20 weeks. Whereas 100% of the mice that fed with water only developed prostate tumors at the end of experiment, no tumors could be found in any of the mice fed with PSP, suggesting that PSP treatment can completely inhibit prostate tumor formation. Our results not only demonstrated the intriguing anti-CSC effect of PSP, but also revealed, for the first time, the surprising chemopreventive property of oral PSP consumption against prostate cancer