Abstract
BDDs and their algorithms implement a decision procedure for Quantified Propositional Logic. BDDs are a kind of acyclic automata. But unrestricted automata (recognizing unbounded strings of bit vectors) can be used to decide monadic second-order logics, which are more expressive. Prime examples are WS1S, a number-theoretic logic, or the string-based logical notation of introductory texts. One problem is that it is not clear which one is to be preferred in practice. For example, it is not known whether these two logics are computationally equivalent to within a linear factor, that is, whether a formula ϕ of one logic can be transformed to a formula %phis;′ of the other such that %phis;′ is true if and only if ϕ is and such that ϕ′ is decided in time linear in that of the time for ϕ.
Another problem is that first-order variables in either version are given automata-theoretic semantics according to relativizations, which are syntactic means of restricting the domain of quantification of a variable. Such relativizations lead to technical arbitrations that may involve normalizing each subformula in an asymmetric manner or may introduce spurious state space explosions.
In this paper, we investigate these problems through studies of congruences on strings. This algebraic framework is adapted to language-theoretic relativizations, where regular languages are intersected with restrictions. The restrictions are also regular languages. We introduce ternary and sexpartite characterizations of relativized regular languages. From properties of the resulting congruences, we are able to carry out detailed state space analyses that allow us to address the two problems.
We report briefly on practical experiments that support our results. We conclude that WS1S with first-order variables can be robustly implemented in a way that efficiently subsumes string-based notations.
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Dedicated to the memory of Bob Paige and his contributions to automata algorithms
Some of the material in this paper appeared in Computer Aided Verification, CAV ‘99, LNCS 1633, 1999, under the title “A theory of restrictions for logics and automata.” This work was carried out while the author was with AT&T Labs–Research; itwas also supported in part by grant CCR-0341658 from the National Science Foundation.
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Klarlund, N. Relativizations for the Logic-Automata Connection. Higher-Order Symb Comput 18, 79–120 (2005). https://doi.org/10.1007/s10990-005-7007-4
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DOI: https://doi.org/10.1007/s10990-005-7007-4