Pushdown automaton

In the theory of computation, a branch of theoretical computer science, a pushdown automaton (PDA) is 
a type of automaton that employs a stack.
Pushdown automata are used in theories about what can be computed by machines. They are more capable than finite-state machines but less capable than Turing machines (see below).
Deterministic pushdown automata can recognize all deterministic context-free languages while nondeterministic ones can recognize all context-free languages, with the former often used in parser design.
The term "pushdown" refers to the fact that the stack can be regarded as being "pushed down" like a tray dispenser at a cafeteria, since the operations never work on elements other than the top element. A stack automaton, by contrast, does allow access to and operations on deeper elements. Stack automata can recognize a strictly larger set of languages than pushdown automata.
A nested stack automaton allows full access, and also allows stacked values to be entire sub-stacks rather than just single finite symbols.
Informal description.
A finite-state machine just looks at the input signal and the current state: it has no stack to work with. It chooses a new state, the result of following the transition. A pushdown automaton (PDA) differs from a finite state machine in two ways:
A pushdown automaton reads a given input string from left to right. In each step, it chooses a transition by indexing a table by input symbol, current state, and the symbol at the top of the stack. A pushdown automaton can also manipulate the stack, as part of performing a transition. The manipulation can be to push a particular symbol to the top of the stack, or to pop off the top of the stack. The automaton can alternatively ignore the stack, and leave it as it is. 
Put together: Given an input symbol, current state, and stack symbol, the automaton can follow a transition to another state, and optionally manipulate (push or pop) the stack.
If, in every situation, at most one such transition action is possible, then the automaton is called a deterministic pushdown automaton (DPDA). In general, if several actions are possible, then the automaton is called a general, or nondeterministic, PDA. A given input string may drive a nondeterministic pushdown automaton to one of several configuration sequences; if one of them leads to an accepting configuration after reading the complete input string, the latter is said to belong to the "language accepted by the automaton".
Formal definition.
We use standard formal language notation: formula_1 denotes the set of finite-length strings over alphabet formula_2 and formula_3 denotes the empty string.
A PDA is formally defined as a 7-tuple:
formula_4
where
An element formula_13 is a transition of formula_14. It has the intended meaning that formula_14, in state formula_16, on the input formula_17 and with formula_18 as topmost stack symbol, may read formula_19, change the state to formula_20, pop formula_21, replacing it by pushing formula_22. The formula_23 component of the transition relation is used to formalize that the PDA can either read a letter from the input, or proceed leaving the input untouched.
In many texts the transition relation is replaced by an (equivalent) formalization, where
Here formula_27 contains all possible actions in state formula_28 with formula_21 on the stack, while reading formula_19 on the input. One writes for example formula_31 precisely when formula_32 because formula_33. Note that "finite" in this definition is essential.
 "Computations" 
In order to formalize the semantics of the pushdown automaton a description of the current situation is introduced. Any 3-tuple formula_34 is called an instantaneous description (ID) of formula_14, which includes the current state, the part of the input tape that has not been read, and the contents of the stack (topmost symbol written first). The transition relation formula_8 defines the step-relation formula_37 of formula_14 on instantaneous descriptions. For instruction formula_13 there exists a step formula_40, for every formula_41 and every formula_42.
In general pushdown automata are nondeterministic meaning that in a given instantaneous description formula_43 there may be several possible steps. Any of these steps can be chosen in a computation.
With the above definition in each step always a single symbol (top of the stack) is popped, replacing it with as many symbols as necessary. As a consequence no step is defined when the stack is empty.
Computations of the pushdown automaton are sequences of steps. The computation starts in the initial state formula_44 with the initial stack symbol formula_45 on the stack, and a string formula_46 on the input tape, thus with initial description formula_47. 
There are two modes of accepting. The pushdown automaton either accepts by final state, which means after reading its input the automaton reaches an accepting state (in formula_48), or it accepts by empty stack (formula_3), which means after reading its input the automaton empties its stack. The first acceptance mode uses the internal memory (state), the second the external memory (stack).
Formally one defines
Here formula_55 represents the reflexive and transitive closure of the step relation formula_56 meaning any number of consecutive steps (zero, one or more).
For each single pushdown automaton these two languages need to have no relation: they may be equal but usually this is not the case. A specification of the automaton should also include the intended mode of acceptance. Taken over all pushdown automata both acceptance conditions define the same family of languages.
Theorem. For each pushdown automaton formula_14 one may construct a pushdown automaton formula_58 such that formula_59, and vice versa, for each pushdown automaton formula_14 one may construct a pushdown automaton formula_58 such that formula_62
Example.
The following is the formal description of the PDA which recognizes the language formula_63 by final state:
formula_64, where
The transition relation formula_8 consists of the following six instructions:
In words, the first two instructions say that in state any time the symbol is read, one is pushed onto the stack. Pushing symbol on top of another is formalized as replacing top by (and similarly for pushing symbol on top of a ).
The third and fourth instructions say that, at any moment the automaton may move from state to state .
The fifth instruction says that in state , for each symbol read, one is popped.
Finally, the sixth instruction says that the machine may move from state to accepting state only when the stack consists of a single .
There seems to be no generally used representation for PDA. Here we have depicted the instruction formula_77 by an edge from state to state labelled by formula_78 (read ; replace by formula_79).
Understanding the computation process.
The following illustrates how the above PDA computes on different input strings. The subscript from the step symbol formula_80 is here omitted.
PDA and context-free languages.
Every context-free grammar can be transformed into an equivalent nondeterministic pushdown automaton. The derivation process of the grammar is simulated in a leftmost way. Where the grammar rewrites a nonterminal, the PDA takes the topmost nonterminal from its stack and replaces it by the right-hand part of a grammatical rule ("expand"). Where the grammar generates a terminal symbol, the PDA reads a symbol from input when it is the topmost symbol on the stack ("match"). In a sense the stack of the PDA contains the unprocessed data of the grammar, corresponding to a pre-order traversal of a derivation tree.
Technically, given a context-free grammar, the PDA has a single state, 1, and its transition relation is constructed as follows.
The PDA accepts by empty stack. Its initial stack symbol is the grammar's start symbol.
For a context-free grammar in Greibach normal form, defining (1,γ) ∈ δ(1,"a","A") for each grammar rule "A" → "a"γ also yields an equivalent nondeterministic pushdown automaton.
The converse, finding a grammar for a given PDA, is not that easy. The trick is to code two states of the PDA into the nonterminals of the grammar.
Theorem. For each pushdown automaton formula_14 one may construct a context-free grammar formula_86 such that formula_87.
The language of strings accepted by a deterministic pushdown automaton is called a deterministic context-free language. Not all context-free languages are deterministic. As a consequence, the DPDA is a strictly weaker variant of the PDA 
A finite automaton with access to two stacks is a more powerful device, equivalent in power to a Turing machine. A linear bounded automaton is a device which is more powerful than a pushdown automaton but less so than a Turing machine.
PDA and Turing machines.
A pushdown automaton is computationally equivalent to a 'restricted' Turing Machine (TM) with two tapes which is restricted in following manner- On the first tape, the TM can only read the input and move from left to right (it cannot make changes). On the second tape, it can only 'push' and 'pop' data. Or equivalently, it can read, write and move left and right with the restriction that the only action it can perform at each step is to either delete the left most character in the string (pop) or add an extra character left to the left most character in the string (push).
That a PDA is weaker than a TM can be brought down to the fact that the procedure 'pop' deletes some data. In order to make a PDA as strong as a TM, we need to save somewhere the data lost through 'pop'. We can achieve this by introducing a second stack. In the TM model of PDA of last paragraph, this is equivalent to a TM with 3 tapes, where the first tape is the read only input tape, and the 2nd and the 3rd tape are the 'push and pop' (stack) tapes. In order for such a PDA to simulate any given TM, we give the input of the PDA to the first tape, while keeping both the stacks empty. It then goes on to push all the input from the input tape to the first stack. When the entire input is transferred to the 1st stack, now we proceed like a normal TM, where moving right on the tape is the same as popping a symbol from the 1st stack and pushing a (possibly updated) symbol into the second stack, and moving left corresponds to popping a symbol from the 2nd stack and pushing (a possibly updated) symbol into the first stack. We hence have a PDA with 2 stacks that can simulate any TM.
Generalized pushdown automaton (GPDA).
A GPDA is a PDA which writes an entire string of some known length to the stack or removes an entire string from the stack in one step.
A GPDA is formally defined as a 6-tuple: 
where formula_89, and are defined the same way as a PDA.
is the transition function.
Computation rules for a GPDA are the same as a PDA except that the formula_92's and formula_93's are now strings instead of symbols.
GPDA's and PDA's are equivalent in that if a language is recognized by a PDA, it is also recognized by a GPDA and vice versa.
One can formulate an analytic proof for the equivalence of GPDA's and PDA's using the following simulation:
Let formula_94 be a transition of the GPDA
where formula_95.
Construct the following transitions for the PDA:
Stack automaton.
As a generalization of pushdown automata, Ginsburg, Greibach, and Harrison (1967) investigated stack automata, which may additionally step left or right in the input string (surrounded by special endmarker symbols to prevent slipping out), and step up or down in the stack in read-only mode. 
A stack automaton is called "nonerasing" if it never pops from the stack. The class of languages accepted by nondeterministic, nonerasing stack automata is "NSPACE"("n"2), which is a superset of the context-sensitive languages. The class of languages accepted by deterministic, nonerasing stack automata is "DSPACE"("n"⋅log("n")).
Alternating pushdown automata.
An alternating pushdown automaton (APDA) is a pushdown automaton with a state set
States in formula_99 and formula_100 are called "existential" resp. "universal". In an existential state an APDA nondeterministically chooses the next state and accepts if "at least one" of the resulting computations accepts. In a universal state APDA moves to all next states and accepts if "all" the resulting computations accept.
The model was introduced by Chandra, Kozen and Stockmeyer. Ladner, Lipton and Stockmeyer proved that this model is equivalent to EXPTIME i.e. a language is accepted by some APDA if, and only if, it can be decided by an exponential-time algorithm.
Aizikowitz and Kaminski introduced "synchronized alternating pushdown automata" (SAPDA) that are equivalent to conjunctive grammars in the same way as nondeterministic PDA are equivalent to context-free grammars.