Sheila A. Greibach

Remarks on blind and partially blind one-way multicounter machines☆

We consider one-way nondeterministic machines which have counters allowed to hold positive or negative integers and which accept by final state with all counters zero. Such machines are called blind if their action depends on state and input alone and not on the counter configuration. They are partially blind if they block when any counter is negative (i.e., only nonnegative counter contents are permissible) but do not know whether or not any of the counters contain zero. Blind multicounter machines are equivalent in power to the reversal bounded multicounter machines of Baker and Book [1], and for both blind and reversal bounded multicounter machines, the quasirealtime family is as powerful as the full family. The family of languages accepted by blind multicounter machines is the least intersection closed semiAFL containing {anbn|n⩾0} and also the least intersection closed semiAFL containing the two-sided Dyck set on one letter. Blind multicounter machines are strictly less powerful than quasirealtime partially blind multicounter machines. Quasirealtime partially blind multicounter machines accept the family of computation state sequences or Petri net languages which is equal to the least intersection closed semiAFL containing the one-sided Dyck set on one letter but is not a principal semiAFL. For partially blind multicounter machines, as opposed to blind machines, linear time is more powerful than quasirealtime. Assuming that the reachability problem for vector addition systems is decidable [16], partially blind multicounter machines accept only recursive sets and do not accept even {anbn|n⩾0∗, and quasirealtime partially blind multicounter machines are less powerful than general quasirealtime multicounter machines.

One-way stack automata

A number of operations which either preserve sets accepted by one-way stack automata or preserve sets accepted by deterministic one-way stack automata are presented. For example, sequential transduction preserves the former; set complementation, the latter. Several solvability questions are also considered.

Abstract families of languages

The notion of an abstract family of languages (AFL) as a family of sets of words satisfying certain properties common to many types of formal languages is introduced. Operations preserving AFL are then considered. The concept of an abstract family of acceptors (AFA) is also introduced and shown to give rise to an AFL. A necessary and sufficient condition on an AFL is presented in order that the AFL come from some AFA. Finally, abstract families of transducers (AFA with output) are discussed.

Quasi-realtime languages

The quasi-realtime languages are seen to be the languages accepted by nondeterministic multitape Turing machines in real time. The family of quasi-realtime languages forms an abstract family of languages closed under intersection, linear erasing, and reversal. It is identical with the family of languages accepted by nondeterministic multitape Turing machines in linear time. Every quasi-realtime language can be accepted in real time by a nondeterministic one stack, one pushdown store machine, and can be expressed as the length-preserving homomorphic image of the intersection of three context-free languages.

Scattered context grammars

Scattered context grammars are defined and the closure properties of the family of languages generated are considered. This family of languages is contained in the family of context sensitive languages and contains all languages accepted by linear time nondeterministic Turing machines.

Stack automata and compiling

Compilation consists of two parts, recognition and translation. A mathematical model is presented which embodies salient features of many modern compiling techniques. The model, called the stack automaton, has the desirable feature of being deterministic in nature. This deterministic device is generalized to a nondeterministic device (nondeterministic stack automaton) and particular instances of this more general device are noted. Sets accepted by nondeterministic stack automata are recursive. Each set accepted by a deterministic linear bounded automaton is accepted by some nonerasing stack automaton. Each context-sensitive language is accepted by some (deterministic) stack automaton.

Chains of full AFL's

If a full AFLℒ is not closed under substitution, thenℒ ô ℒ, the result of substituting members ofℒ intoℒ, is not substitution closed and henceℒ generates an infinite hierarchy of full AFLs. Ifℒ1 andℒ2 are two incomparable full AFLs, then the least full AFL containingℒ1 andℒ2 is not substitution closed. In particular, the substitution closure of any full AFL properly contained in the context-free languages is itself properly contained in the context-free languages. If any set of languages generates the context-free languages, one of its members must do so. The substitution closure of the one-way stack languages is properly contained in the nested stack languages. For eachn, there is a class of full context-free AFLs whose partial ordering under inclusion is isomorphic to the natural partial ordering onn-tuples of positive integers.

Ambiguity in Graphs and Expressions

A regular expression is called unambiguous if every tape in the event can be generated from the expression in one way only. The flow-graph technique for constructing an expression is shown to preserve ambiguities of the graph, and thus, if the graph is that of a deterministic automaton, the expression is unambiguous. A procedure for generating a nondeterministic automaton which preserves the ambiguities of the given regular expression is described. Finally, a procedure for testing whether a given expression is ambiguous is given.

Full AFLs and nested iterated substitution

A superAFL is a family of languages closed under union with unitary sets, intersection with regular sets, and nested iterated substitution and containing at least one nonunitary set. Every superAFL is a full AFL containing all context-free languages. If L is a full principal AFL, then Ŝ∞(L), the least superAFL containing L, is full principal. If L is not substitution closed, the substitution closure of L is properly contained in Ŝ∞(L). The indexed languages form a superAFL which is not the least superAFL containing the one-way stack languages. If L has a decidable emptiness problem, so does Ŝ∞(L). IfDs is an AFA, L = L(Ds) and Dw is the family of machines whose data structure is a pushdown store of tapes of Ds, then L(Ds) = Ŝ∞(L) if, and only if, Ds is nontrivial. If Ds is uniformly erasable and L(Ds) has a decidable emptiness problem, then it is decidable if a member of Dw is finitely nested.

Time- and tape-bounded turing acceptors and AFLs

Complexity classes of formal languages defined by time- and tape-bounded Turing acceptors are studied. Sufficient conditions for these classes to be AFLs are given. Further, it is shown that a time-bounded nondeterministic Turing acceptor need have only two storage tapes.