Separating Strings With Small Automata: Progress And Challenges
The problem of string separation
The formal problem of string separation involves designing an automaton that can accept or reject input strings to divide them into two disjoint sets. More precisely, given two finite sets of strings X and Y over some alphabet, the objective is to construct an automaton that accepts all strings in X while rejecting all strings in Y. This separation problem has important applications in many areas of computer science including formal verification, program analysis, and syntactic pattern recognition.
However, designing small automata that can perfectly separate two complex string sets poses significant algorithmic challenges. The number of states required in the minimal separating automaton may be exponentially larger than the sizes of the input string sets in the worst case. Therefore, an ongoing research goal is to establish tight bounds on the complexity of optimal separators and develop efficient algorithms and heuristics for constructing compact yet accurate automata.
Automata-based approaches
A wide range of automata models including deterministic finite automata (DFAs), nondeterministic finite automata (NFAs), and advanced variants have been studied for the string separation problem. We review some key automata-based approaches and their limitations next.
Finite automata that accept regular languages are among the simplest models. For example, consider the problem of separating the set X = {abc, def} from Y = {abd, aef}. This can be achieved using a DFA with 5 states that reads strings from left to right, transitions to an accept state only after abc or def, and rejects all other strings.
While such DFAs work well for some simple cases, their expressiveness is quite limited for separating complex string sets encountered in practice. Since DFAs require unique transitions from every state on each symbol, they may need up to an exponential number of states in the worst case. Minimizing DFA size for separation is also computationally expensive.
Advanced automata models
More sophisticated automata models can help alleviate DFA limitations using additional mechanisms. Nondeterministic finite automata (NFAs) allow multiple transition options from a given state on any symbol. Though NFAs can be more succinct, determining if an input string is accepted is more challenging.
Two-way and alternating automata augment NFAs with additional traversal modes and choices to gain further expressiveness. For instance, a two-way NFA can switch direction and move its input head back and forth to remember more context. Though separation may now be possible with fewer states, testing string membership remains non-trivial.
In general, while advanced automata can separate more complex string sets, they present challenges in analysis, construction, and usage. Quantitative trade-offs between separation accuracy and descriptonal complexity metrics like automata size remain active research questions.
Open problems
Many open questions remain on both applied and theoretical aspects of the separation problem. Obtaining stronger lower bound proofs on the minimum sizes of optimal separating automata for different string classes is an important direction. The development of efficient algorithms that can construct minimal or near-minimal solutions also remains challenging.
Additional work is needed to make automata-based separators practical for tasks like analyzing software and network packets. Interface and software tools that allow easy specification, training, and matching using compact separators need to be built for adoption in verification and monitoring systems.
Recent progress
Notwithstanding these challenges, promising progress has been made along several fronts. Researchers have derived improved succinctness bounds for restricted classes of NFAs and shown separation guarantees using limited nondeterminism. New results also quantify tradeoffs between description complexity and one-sided error when perfect precision is relaxed.
Specialized automata constructions for separating common string types like numeric constants and boilerplate code have also been developed. These leverage application domain structure and semantics to enable compact yet accurate classification. Connections between minimal separators and communication complexity have also yielded some improved theoretical limits.
Future directions
Looking ahead, several interesting avenues exist for advancing the state of the art. Quantum and probabilistic automata introduce uncertainty during computation to gain power using fewer states. Integrating learning mechanisms to infer concise models from training data is also promising.
Exploring supplements to pure syntax using semantic knowledge extracted from corpora and ontologies can further bolster separation capacity. Overall, the intersection of language theory, learning, and knowledge representation offers rich possibilities on developing the next generation of practical string separators.
