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This note addresses the issue as to which ceers can be realized by word problems of computably enumerable (or, simply, c.e.) structures (such as c.e. semigroups, groups, and rings), where being realized means to fall in the same reducibility degree (under the notion of reducibility for equivalence relations usually called computable reducibility), or in the same isomorphism type (with the isomorphism induced by a computable function), or in the same strong isomorphism type (with the isomorphism induced by a computable permutation of the natural numbers). We observe for instance that every ceer is isomorphic to the word problem of some c.e. semigroup, but (answering a question of Gao and Gerdes) not every ceer is in the same reducibility degree of the word problem of some finitely presented semigroup, nor is it in the same reducibility degree of some non-periodic semigroup. We also show that the ceer provided by provable equivalence of Peano Arithmetic is in the same strong isomorphism type as the word problem of some non-commutative and non-Boolean c.e. ring.
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Algorithmic issues concerning Elliott local semigroups are seldom considered in the literature, although these combinatorial structures completely classify AF algebras. In general, the addition operation of an Elliott local semigroup is {it partial}, but for every AF algebra $mathfrak B$ whose Murray-von Neumann order of projections is a lattice, this operation is uniquely extendible to the addition of an involutive monoid $E(mathfrak B)$. Let $mathfrak M_1$ be the Farey AF algebra introduced by the present author in 1988 and rediscovered by F. Boca in 2008. The freeness properties of the involutive monoid $E(mathfrak M_1)$ yield a natural word problem for every AF algebra $mathfrak B$ with singly generated $E(mathfrak B)$, because $mathfrak B$ is automatically a quotient of $mathfrak M_1$. Given two formulas $phi$ and $psi$ in the language of involutive monoids, the problem asks to decide whether $phi$ and $psi$ code the same equivalence of projections of $mathfrak B$. This mimics the classical definition of the word problem of a group presented by generators and relations. We show that the word problem of $mathfrak M_1$ is solvable in polynomial time, and so is the word problem of the Behnke-Leptin algebras $mathcal A_{n,k}$, and of the Effros-Shen algebras $mathfrak F_{theta}$, for $thetain [0,1]setminus mathbb Q$ a real algebraic number, or $theta = 1/e$. We construct a quotient of $mathfrak M_1$ having a Godel incomplete word problem, and show that no primitive quotient of $mathfrak M_1$ is Godel incomplete.
We consider the operation of sum on Kripke frames, where a family of frames-summands is indexed by elements of another frame. In many cases, the modal logic of sums inherits the finite model property and decidability from the modal logic of summands. In this paper we show that, under a general condition, the satisfiability problem on sums is polynomial space Turing reducible to the satisfiability problem on summands. In particular, for many modal logics decidability in PSPACE is an immediate corollary from the semantic characterization of the logic.
Altenbernd, Thomas and Wohrle have considered in [ATW02] acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with the usual acceptance conditions, such as the Buchi and Muller ones, firstly used for infinite words. Many classical decision problems are studied in formal language theory and in automata theory and arise now naturally about recognizable languages of infinite pictures. We first review in this paper some recent results of [Fin09b] where we gave the exact degree of numerous undecidable problems for Buchi-recognizable languages of infinite pictures, which are actually located at the first or at the second level of the analytical hierarchy, and highly undecidable. Then we prove here some more (high) undecidability results. We first show that it is $Pi_2^1$-complete to determine whether a given Buchi-recognizable languages of infinite pictures is unambiguous. Then we investigate cardinality problems. Using recent results of [FL09], we prove that it is $D_2(Sigma_1^1)$-complete to determine whether a given Buchi-recognizable language of infinite pictures is countably infinite, and that it is $Sigma_1^1$-complete to determine whether a given Buchi-recognizable language of infinite pictures is uncountable. Next we consider complements of recognizable languages of infinite pictures. Using some results of Set Theory, we show that the cardinality of the complement of a Buchi-recognizable language of infinite pictures may depend on the model of the axiomatic system ZFC. We prove that the problem to determine whether the complement of a given Buchi-recognizable language of infinite pictures is countable (respectively, uncountable) is in the class $Sigma_3^1 setminus (Pi_2^1 cup Sigma_2^1)$ (respectively, in the class $Pi_3^1 setminus (Pi_2^1 cup Sigma_2^1)$).
Finite-domain constraint satisfaction problems are either solvable by Datalog, or not even expressible in fixed-point logic with counting. The border between the two regimes coincides with an important dichotomy in universal algebra; in particular, the border can be described by a strong height-one Maltsev condition. For infinite-domain CSPs, the situation is more complicated even if the template structure of the CSP is model-theoretically tame. We prove that there is no Maltsev condition that characterizes Datalog already for the CSPs of first-order reducts of (Q;<); such CSPs are called temporal CSPs and are of fundamental importance in infinite-domain constraint satisfaction. Our main result is a complete classification of temporal CSPs that can be expressed in one of the following logical formalisms: Datalog, fixed-point logic (with or without counting), or fixed-point logic with the Boolean rank operator. The classification shows that many of the equivalent conditions in the finite fail to capture expressibility in Datalog or fixed-point logic already for temporal CSPs.
We use the framework of reverse mathematics to address the question of, given a mathematical problem, whether or not it is easier to find an infinite partial solution than it is to find a complete solution. Following Flood, we say that a Ramsey-type variant of a problem is the problem with the same instances but whose solutions are the infinite partial solutions to the original problem. We study Ramsey-type variants of problems related to Konigs lemma, such as restrictions of Konigs lemma, Boolean satisfiability problems, and graph coloring problems. We find that sometimes the Ramsey-type variant of a problem is strictly easier than the original problem (as Flood showed with weak Konigs lemma) and that sometimes the Ramsey-type variant of a problem is equivalent to the original problem. We show that the Ramsey-type variant of weak Konigs lemma is robust in the sense of Montalban: it is equivalent to several perturbations of itself. We also clarify the relationship between Ramsey-type weak Konigs lemma and algorithmic randomness by showing that Ramsey-type weak weak Konigs lemma is equivalent to the problem of finding diagonally non-recursive functions and that these problems are strictly easier than Ramsey-type weak Konigs lemma. This answers a question of Flood.
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