Decidability and Complexity of Finitely Closable Linear Equational Theories (original) (raw)

The Complexity of Monadic Second-Order Unification

SIAM Journal on Computing, 2008

Monadic second-order unification is second-order unification where all function constants occurring in the equations are unary. Here we prove that the problem of deciding whether a set of monadic equations has a unifier is NP-complete, where we use the technique of compressing solutions using singleton context-free grammars. We prove that monadic second-order matching is also NP-complete.

Decidability of Monadic Theories

Mathematical Foundations of Computer Science, Praha, Czechoslovakia, September 3–7, 1984. Proceedings. Lecture Notes in Computer Science. V. 176., 1984

In this survey article the author describes recent results on monadic theories. The first part concerns the monadic theory of < N, ≤, f > MTf, where N is the set of nonnegative integers, ≤ is the usual order on N, and f is a function from N into N with a finite range. f can be thought of as an ω-word in some finite alphabet. The author gives a general theorem characterizing the decidability of MTf. In the case where f is almost periodic one gets MTF is decidable if and only if f is computable and effectively almost periodic. Results on the monadic theory of < Z, ≤, f > where Z is the set all integers are also presented and their relation to symbolic dynamics indicated. The notion of a minimal transducer is described and a uniformization theorem for the monadic theory of < N, ≤> is given in terms of these automata. The second part of the paper describes a proof due to Muchnik of Rabin’s theorem on the decidability of the monadic theory of S2S. Muchnik’s proof uses a notion of automaton but avoids transfinite induction. The paper ends with brief remarks on weak monadic theories.

On the Undecidability of Second-Order Unification

Information and Computation/information and Control, 2000

There is a close relationship between word unification and secondorder unification. This similarity has been exploited, for instance, in order to prove decidability of monadic second-order unification and decidability of linear second-order unification when no second-order variable occurs more than twice. The attempt to prove the second result for (nonlinear) second-order unification failed and led instead to a natural reduction from simultaneous rigid E-unification to this second-order unification. This reduction is the first main result of this paper, and it is the starting point for proving some novel results about the undecidability of second-order unification presented in the rest of the paper. We prove that second-order unification is undecidable in the following three cases: (1) each secondorder variable occurs at most twice and there are only two second-order variables; (2) there is only one second-order variable and it is unary; (3) the following conditions (i) (iv) hold for some fixed integer n: (i) the arguments of all second-order variables are ground terms of size <n, (ii) the arity of all second-order variables is <n, (iii) the number of occurrences of second-order variables is 5, (iv) there is either a single second-order variable or there are two second-order variables and no first-order variables. ]

On the complexity of computing minimal unsatisfiable LTL formulas

We show that (1) the Minimal False QCNF search-problem (MF-search) and the Minimal Unsatisfiable LTL formula search problem (MU-search) are FPSPACE complete because of the very expressive power of QBF/LTL, (2) we extend the PSPACE-hardness of the MF decision problem to the MU decision problem. As a consequence, we deduce a positive answer to the open question of PSPACE hardness of the inherent Vacuity Checking problem. We even show that the Inherent Non Vacuous formula search problem is also FPSPACEcomplete.

Non-Computability of the Equational Theory of Polyadic Algebras

2001

In [3] Daigneault and Monk proved that the class of (! dimensional) representable polyadic algebras (RPA! for short) is axiomatizable by finitely many equationschemas. However, this result does not imply that the equational theory of RPA! would be recursively enumerable; one simple reason is that the language of RPA! contains a continuum of operation symbols. Here we prove the following. Roughly, for any reasonable generalization of computability to uncountable languages, the equational theory of RPA! remains non-recursively enumerable, or non-computable, in the generalized sense. This result has some implications on the non-computational character of Keisler’s completeness theorem for his “infinitary logic” in Keisler [6] as well.

On the combinatorial and algebraic complexity of quantifier elimination

Journal of the ACM, 1996

In this paper, a new algorithm for performing quantifier elimination from first order formulas over real closed fields in given. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this data. A new feature of this algorithm is that the role of the algebraic part (the dependence on the degrees of the imput polynomials) and the combinatorial part (the dependence on the number of polynomials) are sparated. Another new feature is that the degrees of the polynomials in the equivalent quantifier-free formula that is output, are independent of the number of input polynomials. As special cases of this algorithm new and improved algorithms for deciding a sentence in the first order theory over real closed fields, and also for solving the existential problem in the first order theory over real closed fields, are obtained.