80 research outputs found
On complexity of Ehrenfeucht-Fraïssé games
In this paper we initiate the study of Ehrenfeucht-Fraïssé games for some standard finite structures. Examples of such standard structures are equivalence relations, trees, unary relation structures, Boolean algebras, and some of their natural expansions. The paper concerns the following question that we call Ehrenfeucht-Fraïssé problem. Given n ϵ ω as a parameter, two relational structures A and B from one of the classes of structures mentioned above. How efficient is it to decide if Duplicator wins the n-round EF game G_n (A,B)? We provide algorithms for solving the Ehrenfeucht-Fraïssé problem for the mentioned classes of structures. The running times of all the algorithms are bounded by constants. We obtain the values of these constants as functions of n
Computable categoricity of graphs with finite components
A computable graph is computably categorical if any two computable presentations of the graph are computably isomorphic. In this paper we investigate the class of computably categorical graphs. We restrict ourselves to strongly locally finite graphs; these are the graphs all of whose components are finite. We present a necessary and sufficient condition for certain classes of strongly locally finite graphs to be computably categorical. We prove that if there exists an infinite \Delta^0_2-set of components that can be properly embedded into infinitely many components of the graph then the graph is not computably categorical. We outline the construction of a strongly locally finite computably categorical graph with an infinite chain of properly embedded components
A Dichotomy Theorem for Polynomial Evaluation
A dichotomy theorem for counting problems due to Creignou and Hermann states
that or any nite set S of logical relations, the counting problem #SAT(S) is
either in FP, or #P-complete. In the present paper we show a dichotomy theorem
for polynomial evaluation. That is, we show that for a given set S, either
there exists a VNP-complete family of polynomials associated to S, or the
associated families of polynomials are all in VP. We give a concise
characterization of the sets S that give rise to "easy" and "hard" polynomials.
We also prove that several problems which were known to be #P-complete under
Turing reductions only are in fact #P-complete under many-one reductions
Unary automatic graphs: an algorithmic perspective
This paper studies infinite graphs produced from a natural
unfolding operation applied to finite graphs. Graphs produced via such
operations are of finite degree and can be described by finite automata
over the unary alphabet. We investigate algorithmic properties of such
unfolded graphs given their finite presentations. In particular, we ask
whether a given node belongs to an infinite component, whether two
given nodes in the graph are reachable from one another, and whether
the graph is connected. We give polynomial time algorithms for each
of these questions. Hence, we improve on previous work, in which nonelementary or non-uniform algorithms were foun
On computably enumerable structures
© 2014, Pleiades Publishing, Ltd. This is a short article based on the plenary lecture presented by the author at Algebra Computability and Logic conference in Kazan, June 1–6, 2014. The paper is a new approach that introduces effectiveness into the study of algebraic structures. In this approach the initial objects are the domains of the form ω/E, where E is an equivalence relation on ω. In this set-up one investigates effective algebraic structures that are admitted by domains of the formω/E. This lecture was motivated by the papers [2, 20, 21], open questions in [17], and is based on the papers [15, 16, 19, 23]
The Isomorphism Relation Between Tree-Automatic Structures
An -tree-automatic structure is a relational structure whose domain
and relations are accepted by Muller or Rabin tree automata. We investigate in
this paper the isomorphism problem for -tree-automatic structures. We
prove first that the isomorphism relation for -tree-automatic boolean
algebras (respectively, partial orders, rings, commutative rings, non
commutative rings, non commutative groups, nilpotent groups of class n >1) is
not determined by the axiomatic system ZFC. Then we prove that the isomorphism
problem for -tree-automatic boolean algebras (respectively, partial
orders, rings, commutative rings, non commutative rings, non commutative
groups, nilpotent groups of class n >1) is neither a -set nor a
-set
Solving infinite games on trees with back-edges
We study the computational complexity of solving the following problem: Given a game G played on a fi- nite directed graph G, output all nodes in G from which a specific player wins the game G. We pro- vide algorithms for solving the above problem when the games have Büchi and parity winning conditions and the graph G is a tree with back-edges. The running time of the algorithm for Büchi games is O(min{r·m,ℓ+m}) where m is the number of edges, ℓ is the sum of the distances from the root to all leaves and the parameter r is bounded by the height of the tree. The algorithm for parity has a running time of O(ℓ + m)
A polychromatic Ramsey theory for ordinals
The Ramsey degree of an ordinal α is the least number n such that any colouring of the edges of the complete graph on α using finitely many colours contains an n-chromatic clique of order type α. The Ramsey degree exists for any ordinal α < ω ω . We provide an explicit expression for computing the Ramsey degree given α. We further establish a version of this result for automatic structures. In this version the ordinal and the colouring are presentable by finite automata and the clique is additionally required to be regular. The corresponding automatic Ramsey degree turns out to be greater than the set theoretic Ramsey degree. Finally, we demonstrate that a version for computable structures fails
Automatic structures of bounded degree revisited
The first-order theory of a string automatic structure is known to be
decidable, but there are examples of string automatic structures with
nonelementary first-order theories. We prove that the first-order theory of a
string automatic structure of bounded degree is decidable in doubly exponential
space (for injective automatic presentations, this holds even uniformly). This
result is shown to be optimal since we also present a string automatic
structure of bounded degree whose first-order theory is hard for 2EXPSPACE. We
prove similar results also for tree automatic structures. These findings close
the gaps left open in a previous paper of the second author by improving both,
the lower and the upper bounds.Comment: 26 page
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