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In this work, we introduce the notion of decisional width of a finite relational structure and the notion of decisional width of a regular class of finite structures. Our main result states that given a first-order formula ψ over a vocabulary τ , and a finite automaton F over a suitable alphabet B(Σ, w, τ ) representing a width-w regular-decisional class of τ -structures C, one can decide in time f (τ, Σ, ψ, w) • |F| whether some τ -structure in C satisfies ψ. Here, f is a function that depends on the parameters τ, Σ, ψ, w, but not on the size of the automaton F representing the class. Therefore, besides implying that the first-order theory of any given regular-decisional class of finite structures is decidable, it also implies that when the parameters τ , ψ, Σ and w are fixed, decidability can be achieved in linear time on the size of the input automaton F. Building on the proof of our main result, we show that the problem of counting satisfying assignments for a first-order logic formula in a given structure A of width w is fixed-parameter tractable with respect to w, and can be solved in quadratic time on the length of the input representation of A.

A fundamental drawback that arises when one is faced with the task of deterministically certifying solutions to computational problems in PSPACE is the fact that witnesses may have superpolynomial size, assuming that NP is not equal to PSPACE. Therefore, the complexity of such a deterministic verifier may already be super-polynomially lower-bounded by the size of a witness. In this work, we introduce a new symbolic framework to address this drawback. More precisely, we introduce a PSPACE-hard notion of symbolic constraint satisfaction problem where both instances and solutions for these instances are implicitly represented by ordered decision diagrams (i.e. read-once, oblivious, branching programs). Our main result states that given an ordered decision diagram D of length k and width w specifying a CSP instance, one can determine in time f(w,w')*k whether there is an ODD of width at most w' encoding a solution for this instance. Intuitively, while the parameter w quantifies the complexity of the instance, the parameter w' quantifies the complexity of a prospective solution. We show that CSPs of constant width can be used to formalize natural PSPACE hard problems, such as reachability of configurations for Turing machines working in nondeterministic linear space. For such problems, our main result immediately yields an algorithm that determines the existence of solutions of width w in time g(w)*n, where g:N->N is a suitable computable function, and n is the size of the input.

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