The atnext/atprevious hierarchy on the starfree languages

The temporal logic operators atnext and atprevious are alternatives for the operators until and since. P atnext Q has the meaning: at the next position in the future where Q holds it holds P. We define an asymmetric but natural notion of depth for the expressions of this linear temporal logic. The sequence of classes at_n of languages expressible via such depth-n expressions gives a parametrization of the starfree regular languages which we call the atnext/atprevious hierarchy, or simply at hierarchy. It turns out that the at hierarchy equals the hierarchy given by the n-fold weakly iterated block product of DA. It is shown that the at hierarchy is situated properly between the until/since and the dot-depth hierarchy

APERIODICITY, STAR-FREENESS, AND FIRST-ORDER LOGIC DEFINABILITY OF OPERATOR PRECEDENCE LANGUAGES

A classic result in formal language theory is the equivalence among non-counting, or aperiodic, regular languages, and languages defined through star-free regular expressions, or first-order logic. Past attempts to extend this result beyond the realm of regular languages have met with difficulties: for instance it is known that star-free tree languages may violate the non-counting property and there are aperiodic tree languages that cannot be defined through first-order logic. We extend such classic equivalence results to a significant family of deterministic context-free languages, the operator-precedence languages (OPL), which strictly includes the widely investigated visibly pushdown, alias input-driven, family and other structured context-free languages. The OP model originated in the ’60s for defining programming languages and is still used by high performance compilers; its rich algebraic properties have been investigated initially in connection with grammar learning and recently completed with further closure properties and with monadic second order logic definition. We introduce an extension of regular expressions, the OP-expressions (OPE) which define the OPLs and, under the star-free hypothesis, define first-order definable and non-counting OPLs. Then, we prove, through a fairly articulated grammar transformation, that aperiodic OPLs are first-order definable. Thus, the classic equivalence of star-freeness, aperiodicity, and first-order definability is established for the large and powerful class of OPLs. We argue that the same approach can be exploited to obtain analogous results for visibly pushdown languages too

Counting with Counterfree Automata

We study the power of balanced and unbalanced regular leaf-languages. First, we investigate (i) regular languages that are polylog-time reducible to languages in dot-depth 1/2 and (ii) regular languages that are polylog-time decidable. For both classes we provide • forbidden-pattern characterizations, and • characterizations in terms of regular expressions. Both classes are decidable. The intersection of class (i) with its complement is exactly class (ii). We apply our observations and obtain three consequences. 1. Gap theorems for balanced regular-leaf-language definable classes C and D: (a) Either C is contained in NP, orC contains coUP. (b) Either D is contained in P,orD contains UP or coUP. Also we extend both theorems such that no promise classes are involved. Formerly, such gap theorems were known only for the unbalanced approach. 2. Polylog-time reductions can tremendously decrease dot-depth complexity (despite that they cannot count). We exploit a weak type of counting which can be done by counterfree automata, and construct languages of arbitrary dot-depth that are reducible to languages in dot-depth 1/2. 3. Unbalanced starfree leaf-languages can be much stronger than balanced ones. We construct starfree regular languages Ln such that the balanced leaf-language class of L n is contained in NP, but the unbalanced leaf-language class of L n contains level n of the unambiguous alternation hierarchy. This demonstrates the power of unbalanced computations.