# deterministic pushdown automaton

A pushdown automaton $M=(Q,\Sigma,\Gamma,T,q_{0},\bot,F)$ is usually called “non-deterministic” because the image of the transition function $T$ is a subset of $Q\times\Gamma^{*}$, which may possibly contain more than one element. In other words, the transition from one configuration to the next is not uniquely determined. When there is uniqueness, $M$ is called “deterministic”.

Formally, a deterministic pushdown automaton, or DPDA for short, is a non-deterministic pushdown automaton $M=(Q,\Sigma,\Gamma,T,q_{0},\bot,F)$ where the transition function $T$ has the following properties: for any $p\in Q$, $a\in\Sigma$, and $A\in\Gamma$,

1. 1.

$T(p,a,A)\cup T(p,\lambda,A)$ is at most a singleton,

2. 2.

$T(p,a,A)\cap T(p,\lambda,A)=\varnothing$.

The properties can be interpreted as follows: given any configuration of $M$, if there is a transition to the next configuration, the transition must be unique. The second property just insures that $T(p,a,A)\neq T(p,\lambda,A)$, so that when a $\lambda$-transition is possible for a given $(p,A)$, no other transitions are possible for the same $(p,A)$.

The way a DPDA works is exactly the same as an NPDA, with several modes of acceptance: acceptance on final state, acceptance on empty stack, and acceptance on final state and empty stack. However, unlike a NPDA, these acceptance methods are not equivalent. It can be shown that the set $\mathscr{E}$ of languages accepted on empty stack is a proper subset of the set $\mathscr{F}$ of languages determined on final state. In fact, every language in $\mathscr{E}$ is prefix-free, while some languages in $\mathscr{F}$ are not.

Nevertheless, any regular language can be accepted by a DPDA on empty stack, and any language accepted by a DPDA on final state is unambiguous, and, as a result, $\mathscr{F}$ is a proper subset of the family of all context-free languages. This is quite unlike the case for finite automata: every non-deterministic finite automaton is equivalent to a deterministic finite automaton. A language in $\mathscr{F}$ called a deterministic language.

Some examples: the set of palindromes $\{u\in\Sigma^{*}\mid u=\operatorname{rev}(u)\}$ is unambiguous, but not deterministic. The language $\{a^{m}b^{n}\mid m\geq n\geq 0\}$ is deterministic, but not prefix-free, and hence can not be accepted by any DPDA on empty stack. The language $\{a^{n}b^{n}\mid n\geq 0\}$ can be accepted by a DPDA on empty stack, but is not regular.

Any formal grammar that generates a deterministic language is said to be deterministic context-free. A deterministic context-free grammar can be described by what is known as the $LR(k)$ (http://planetmath.org/LRk) grammars.

The family of deterministic languages is closed under complementation, intersection with a regular language, but not arbitrary (finite) intersection, and hence not union.

Remark. The reason why $\mathscr{E}\neq\mathscr{F}$ can be traced back to the definition of a DPDA: it allows for the following possibilities for a DPDA $M$:

• $M$ completely stops reading an input word because either there are no available transitions from one configuration to the next:

 $T(p,a,A)\cup T(p,\lambda,A)=\varnothing,$

or the stack is emptied before the last input symbol is read: a configuration $(p,u,\lambda)$ is reached and $u$ is not empty.

• $M$ consumes the last input symbol, and continues processing because of $\lambda$-transitions.

Some authors consider these imperfections of $M$ as being “non-deterministic”, and put additional constraints on $M$, such as making sure $T$ is a total function, the stack is never empty, and delimiting input strings.

## References

• 1 A. Salomaa Computation and Automata, Encyclopedia of Mathematics and Its Applications, Vol. 25. Cambridge (1985).
• 2 S. Ginsburg, The Mathematical Theory of Context-Free Languages, McGraw-Hill, New York (1966).
• 3 D. C. Kozen, Automata and Computability, Springer, New York (1997).
• 4 J.E. Hopcroft, J.D. Ullman, Formal Languages and Their Relation to Automata, Addison-Wesley, (1969).
 Title deterministic pushdown automaton Canonical name DeterministicPushdownAutomaton Date of creation 2013-03-22 18:56:00 Last modified on 2013-03-22 18:56:00 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 14 Author CWoo (3771) Entry type Definition Classification msc 03D10 Classification msc 68Q42 Classification msc 68Q05 Synonym DPDA Related topic ContextFreeLanguage Related topic AmbiguousGrammar Defines deterministic Defines deterministic language Defines deterministic context-free