Considering non-deterministic PDAs, the superposition of states is possible by definition. However, non-deterministic PDAs have only one stack which cannot be in multiple states simultaneously. How is this possible?

/ / Published in Cybersecurity, EITC/IS/CCTF Computational Complexity Theory Fundamentals, Pushdown Automata, Equivalence of CFGs and PDAs

To address the question regarding non-deterministic pushdown automata (PDAs) and the apparent paradox of state superposition with a single stack, it is essential to consider the fundamental principles of non-determinism and the operational mechanics of PDAs.

A pushdown automaton is a computational model that extends the capabilities of finite automata by incorporating an auxiliary storage medium known as a stack. This stack provides the automaton with the ability to store an unbounded amount of information, albeit in a last-in, first-out (LIFO) manner, which is important for recognizing context-free languages. Non-deterministic pushdown automata (NPDA), in particular, enhance this model by allowing multiple possible transitions for a given state and input symbol, akin to the concept of non-determinism in finite automata.

The notion of non-determinism in the context of PDAs is not directly related to the quantum mechanical concept of superposition. Instead, it refers to the capability of the automaton to simultaneously explore multiple computational paths. This is achieved by allowing the automaton to make arbitrary choices among available transitions. When an NPDA encounters a choice point, it can "branch" into multiple computational paths, each representing a different sequence of state transitions and stack operations.

The stack, however, remains a singular entity within each computational path. It does not exist in multiple states simultaneously across these paths. Rather, each branch of computation maintains its own independent version of the stack. This independence is important for the NPDA to correctly simulate multiple potential computations concurrently. When visualizing the operation of an NPDA, one can think of it as maintaining a tree-like structure of computations, where each node represents a unique configuration of state, input position, and stack content.

Consider an NPDA designed to recognize the language of balanced parentheses. Suppose the automaton is in a state where it has read an opening parenthesis and must decide whether to push it onto the stack or transition to another state without pushing. In a non-deterministic fashion, the NPDA can "choose" both options simultaneously, effectively creating two branches of computation. In one branch, the stack contains the opening parenthesis, while in the other, it does not. Each branch proceeds independently based on its initial choice, with the stack contents evolving according to the specific sequence of operations in that branch.

This branching capability enables NPDAs to explore multiple hypotheses about the input string's structure in parallel. If at least one branch of computation leads to an accepting state with an empty stack, the NPDA accepts the input. This non-deterministic branching is a powerful feature that allows NPDAs to recognize a broader class of languages than deterministic PDAs, specifically all context-free languages.

The concept of non-determinism in PDAs can be further elucidated by examining the formal definition of a non-deterministic pushdown automaton. An NPDA is typically defined as a 7-tuple:

    \[ (Q, \Sigma, \Gamma, \delta, q_0, Z_0, F) \]

where:
– Q is a finite set of states.
– \Sigma is the input alphabet.
– \Gamma is the stack alphabet.
– \delta is the transition function, which maps Q \times (\Sigma \cup \{\epsilon\}) \times \Gamma to a finite subset of Q \times \Gamma^*.
– q_0 is the initial state.
– Z_0 is the initial stack symbol.
– F is the set of accepting states.

The transition function \delta is the core of non-determinism in PDAs. It allows for multiple possible transitions for a given state, input symbol, and stack top symbol. These transitions can involve moving to a new state, consuming an input symbol, and modifying the stack by pushing or popping symbols. The presence of \epsilon-transitions (transitions that do not consume an input symbol) further enhances the flexibility of NPDAs by allowing them to change states and manipulate the stack without reading the input.

To illustrate, consider a simple NPDA designed to recognize the language L = \{ a^n b^n | n \geq 0 \}. This language consists of strings with an equal number of a's followed by b's. The NPDA operates as follows:
1. It starts in an initial state q_0 with the initial stack symbol Z_0.
2. For each a read from the input, it pushes an X onto the stack, transitioning to state q_1.
3. Upon encountering a b, it pops an X from the stack, transitioning to state q_2.
4. The NPDA accepts if it reaches an accepting state with the stack empty after processing the entire input.

The non-deterministic aspect allows the NPDA to handle cases where the input string does not conform to the expected pattern. For instance, if the input string contains more b's than a's, the stack will become empty before the end of the input, leading to a rejection. Alternatively, if there are more a's than b's, the stack will not be empty after processing the input, resulting in rejection.

The key takeaway is that non-determinism in PDAs enables the automaton to explore multiple computational paths without requiring the stack to be in multiple states simultaneously. Each path maintains its own stack configuration, allowing the NPDA to simulate different potential computations concurrently. This capability is what allows NPDAs to recognize context-free languages effectively.

In essence, the single stack in an NPDA is not a limitation but a feature that supports the non-deterministic exploration of computational paths. By maintaining separate stack configurations for each branch of computation, the NPDA can evaluate multiple hypotheses about the input string's structure, ultimately determining whether the string belongs to the language recognized by the automaton.

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