On the State Complexity of
Ultrametric Finite
Automata
KASPARS BALODIS, A NDA BERIŅA,
KRISTĪNE CĪPOLA, M AKSIMS DMITRIJEVS,
JĀNIS IRAIDS, KĀRLIS JĒRIŅŠ, VLADIMIRS
KACS, JĀNIS KALĒJS, RIHARDS
KRIŠLAUKS, KĀRLIS LUKSTIŅŠ, R EINHOLDS
RAUMANIS, NATĀLIJA SOMOVA, IRINA
ŠČEGUĻNAJA, A NNA VANAGA, R ŪSIŅŠ
FREIVALDS
p-adic absolute values
 Any rational non-zero number can be expressed as
𝛼 = ±2𝛼2 ∗ 3𝛼3 ∗ 5𝛼5 ∗ 7𝛼7 ∗ 11𝛼11 ∗ ⋯, where all 𝛼𝑖
are integers
 From this, the p-adic absolute value (also called pnorm) of 𝛼 for a prime p is defined as 𝛼 𝑝 =
0, 𝑖𝑓 𝛼 = 0
𝑝−𝛼𝑝 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Ultrametric finite automata (UFA)
 Generalization of probabilistic finite automata
 Instead of transition probabilities, use transition
amplitudes – p-adic numbers
 When determining whether or not to accept a
word, the p-norms of all accepting state amplitudes
are calculated and added together
 If the sum is at least (or at most, depending on the
automaton) a given treshold, the automaton
accepts the word
Language 𝐿𝑘,𝑚
 Let 𝑤 = 𝑤1 , 𝑤2 , … , 𝑤𝑚 ∈ 0,1, … 𝑘 − 1 𝑚
 Consider two operations on w:
a)
b)


Cyclic shift: 𝑓𝑎 𝑤 = 𝑤𝑚 , 𝑤1 , 𝑤2 , … , 𝑤𝑚−1
Increasing first element:
𝑓𝑏 𝑤 = 𝑤1 + 1 𝑚𝑜𝑑 𝑘, 𝑤2 , … , 𝑤𝑚
Let 𝑥 ∈ 𝑎, 𝑏 𝑛 . Define
𝑓𝑥1 𝑥2 …𝑥𝑛 𝑤 = 𝑓𝑥𝑛 ⋯ 𝑓𝑥2 𝑓𝑥1 𝑤
𝐿𝑘,𝑚 = 𝑥 ∈ 𝑎, 𝑏
∗
𝑓𝑥 0𝑚 = 0𝑚
⋯
Recognizing 𝐿𝑘,𝑚
 Deterministic finite automaton – at least 𝑘 𝑚 states
are necessary
 For any prime p, there is a p-ultrametric finite
automaton that can recognize 𝐿𝑘,𝑚 with 𝑘 ∗ 𝑚
states
 For every prime p, if p > m, there is a p-UFA that
can recognize 𝐿𝑝,𝑚 with 𝑚 + 1 states
 Visit us at the poster session to see proofs of the
above 