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Yes for regular languages but not for higher level ones. For example, deterministic context-free is a subset of context-free. For languages that are turing complete, the question is less about ability to compute and more about the speed at which something can be computed. This is an unsolved problem known as P vs. NP.
Deterministic Finite Automaton. It's a concept from automata theory which is a concept from theory of computation. Implementations of DFA's are how libraries like google's re2 or golang's regex are implemented. They're not as powerful as regex libs built with a non-regular language model but that's by design.
Fun trivia: DFAs and NFAs are computationally equivalent. Any NFA can be converted into an equivalent DFA.
https://en.wikipedia.org/wiki/Deterministic_finite_automaton
If this interests you, you'll probably enjoy reading Google's paper on Spanner. Cockroachdb was heavily influenced by it.