Finite Signatures Research Articles

Finite algebras with Hom-sets of polynomial size

We provide an internal characterization of those finite algebras (i.e., algebraic structures) A \mathbf {A} such that the number of homomorphisms from any finite algebra X \mathbf {X} to A \mathbf {A} is bounded from above by a polynomial in the size of X \mathbf {X} . Namely, an algebra A \mathbf {A} has this property if, and only if, no subalgebra of A \mathbf {A} has a nontrivial strongly abelian congruence. We also show that the property can be decided in polynomial time for algebras in finite signatures. Moreover, if A \mathbf {A} is such an algebra, the set of all homomorphisms from X \mathbf {X} to A \mathbf {A} can be computed in polynomial time given X \mathbf {X} as input. As an application of our results to the field of computational complexity, we characterize inherently tractable constraint satisfaction problems over fixed finite structures, i.e., those that are tractable and remain tractable after expanding the fixed structure by arbitrary relations or functions.

Metric spaces are universal for bi-interpretation with metric structures

In the context of metric structures introduced by Ben Yaacov, Berenstein, Henson, and Usvyatsov [3], we exhibit an explicit encoding of metric structures in countable signatures as pure metric spaces in the empty signature, showing that such structures are universal for bi-interpretation among metric structures with positive diameter. This is analogous to the classical encoding of arbitrary discrete structures in finite signatures as graphs, but is stronger in certain ways and weaker in others. There are also certain fine grained topological concerns with no analog in the discrete setting.

Profinite algebras and affine boundedness

We prove a characterization of profinite algebras, i.e., topological algebras that are isomorphic to a projective limit of finite discrete algebras. In general profiniteness concerns both the topological and algebraic characteristics of a topological algebra, whereas for topological groups, rings, semigroups, and distributive lattices, profiniteness turns out to be a purely topological property as it is equivalent to the underlying topological space being a Stone space.Condensing the core idea of those classical results, we introduce the concept of affine boundedness for an arbitrary universal algebra and show that for an affinely bounded topological algebra over a compact signature profiniteness is equivalent to the underlying topological space being a Stone space. Since groups, semigroups, rings, and distributive lattices are indeed affinely bounded algebras over finite signatures, all these known cases arise as special instances of our result. Furthermore, we present some additional applications concerning topological semirings and their modules, as well as distributive associative algebras. We also deduce that any affinely bounded simple compact algebra over a compact signature is either connected or finite. Towards proving the main result, we also establish that any topological algebra is profinite if and only if its underlying space is a Stone space and its translation monoid is equicontinuous.

New Ramsey Classes from Old

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be strong amalgamation classes of finite structures, with disjoint finite signatures $\sigma$ and $\tau$. Then $\mathcal{C}_1 \wedge \mathcal{C}_2$ denotes the class of all finite ($\sigma\cup\tau$)-structures whose $\sigma$-reduct is from $\mathcal{C}_1$ and whose $\tau$-reduct is from $\mathcal{C}_2$. We prove that when $\mathcal{C}_1$ and $\mathcal{C}_2$ are Ramsey, then $\mathcal{C}_1 \wedge \mathcal{C}_2$ is also Ramsey. We also discuss variations of this statement, and give several examples of new Ramsey classes derived from those general results.

Automata for the verification of monadic second-order graph properties

The model-checking problem for monadic second-order logic on graphs is fixed-parameter tractable with respect to tree-width and clique-width. The proof constructs finite automata from monadic second-order sentences. These automata recognize the terms over fixed finite signatures that define graphs satisfying the given sentences. However, this construction produces automata of hyper-exponential sizes, and is thus impossible to use in practice in many cases. To overcome this difficulty, we propose to specify the transitions of automata by programs instead of tables. Such automata are called fly-automata. By using them, we can check certain monadic second-order graph properties with limited quantifier alternation depth, that are nevertheless interesting for Graph Theory. We give explicit constructions of automata relative to graphs of bounded clique-width, and we report on experiments.

Parametric groups of definable automorphisms of strongly constructive models

We study the groups of definable automorphisms of constructive models. We introduce parametric groups of definable automorphisms and obtain a full description of all possible parametric groups of definable automorphisms of strongly constructive models, as well as corollaries for models in finite signatures.

A comment on Bermudez concerning the definability of identity

José Luis Bermúdez's article ‘Indistinguishable elements and mathematical structuralism’ (2007) is a contribution to the debate about the ‘identity problem for structuralism’ (see Burgess 1999 and Keränen 2001) and includes comments on Ladyman 2005 and Ketland 2006.1 Bermúdez writes that Ketland 2006‘contains some puzzling claims’ and, in particular, that one specific result (given without proof) ‘seems badly wrong’. My comment is simply that the result that Bermúdez disputes is correct (as are the others). Below, a proof is given. After all, results concerning the definability of identity in relational structures may be of independent philosophical and logical interest.2 Bermúdez writes, Ketland's paper contains some puzzling claims about the first-order definability of identity. He claims that ‘one can show that if the identity relation is first-order definable in a structure at all, then it is defined by the indiscernibility formula’ (Ketland 2006: 307), where the identity formula is essentially Quine's trick for eliminating the identity predicate. This seems badly wrong, since Quine's trick only applies to structures with finite signatures. ... any structure which contains the identity predicate and has an infinite signature is a counter-example to his claim. (Bermúdez 2007: 113, n. 2)

Model Representation over Finite and Infinite Signatures

Computationally adequate representation of models is a topic arising in various forms in logic and AI. Two fundamental decision problems in this area are: (1) to check whether a given clause is true in a represented model, and (2) to decide whether two representations of the same type represent the same model. ARMs, contexts and DIGs are three important examples of model representation formalisms. The complexity of the mentioned decision problems has been studied for ARMs only for finite signatures, and for contexts and DIGs only for infinite signatures, so far. We settle the remaining cases. Moreover we show that, similarly to the case for infinite signatures, contexts and DIGs allow one to represent the same classes of models also over finite signatures; however DIGs may be exponentially more succinct than all equivalent contexts.

Derivation lengths and order types of Knuth–Bendix orders

The derivation length function of a finite term rewriting system terminating via a Knuth–Bendix order is shown to be bounded by the Ackermann function applied to a single exponential function. This result is essentially optimal as there are rewrite systems with such derivation lengths. In a second part the order types of Knuth–Bendix orders over finite signatures are classified within the ordinals up to ω ω .

The Complexity of Modellability in Finite and ComputableSignatures of a Constraint Logic for Head-Driven PhraseStructure Grammar

The SRL (speciate re-entrant logic) of King (1989) is a sound, complete and decidable logic designed specifically to support formalisms for the HPSG (head-driven phrase structure grammar) of Pollard and Sag (1994). The SRL notion of modellability in a signature is particularly important for HPSG, and the present paper modifies an elegant method due to Blackburn and Spaan (1993) in order to prove that – modellability in each computable signature is Π01, – modellability in some finite signature is Π01-hard (hence not decidable), and – modellability in some finite signature is decidable. Since each finite signature is a computable signature, we conclude that Π01-completeness is the least upper bound on the complexity of modellability both in finite signatures and in computable signatures, though not a lower bound in either.

Computability of String Functions Over Algebraic Structures Armin Hemmerling

AbstractWe present a model of computation for string functions over single‐sorted, total algebraic structures and study some basic features of a general theory of computability within this framework. Our concept generalizes the Blum‐Shub‐Smale setting of computability over the reals and other rings. By dealing with strings of arbitrary length instead of tuples of fixed length, some suppositions of deeper results within former approaches to generalized recursion theory become superfluous. Moreover, this gives the basis for introducing computational complexity in a BSS‐like manner. Relationships both to classical computability and to Friedman's concept of eds computability are established. Two kinds of nondeterminism as well as several variants of recognizability are investigated with respect to interdependencies on each other and on properties of the underlying structures. For structures of finite signatures, there are universal programs with the usual characteristics. For the general case of not necessarily finite signature, this subject will be studied in a separate, forthcoming paper.

FAPKC3: A new finite automaton public key cryptosystem

This paper deals with finite automaton public key cryptosystem and digital signatures. A new system FAPKC3 is proposed which can be used for encryption and implementing digital signatures as well. Some performances of a software implementation of FAPKC3 are presented and its security is discussed.

Simple termination of rewrite systems

In this paper we investigate the concept of simple termination. A term rewriting system is called simply terminating if its termination can be proved by means of a simplification order. The basic ingredient of a simplification order is the subterm property, but in the literature two different definitions are given: one based on (strict) partial orders and another one based on preorders (or quasi-orders). We argue that there is no reason to choose the second one as the first one has certain advantages. Simplification orders are known to be well-founded orders on terms over a finite signature. This important result no longer holds if we consider infinite signatures. Nevertheless, well-known simplification orders like the recursive path order are also well-founded on terms over infinite signatures, provided the underlying precedence is well-founded. We propose a new definition of simplification order, which coincides with the old one (based on partial orders) in case of finite signatures, but which is also well-founded over infinite signatures and covers orders like the recursive path order. We investigate the properties of the ensuing class of simply terminating systems.

Total termination of term rewriting

We investigate proving termination of term rewriting systems by a compositional interpretation of terms in a total well-founded order. This kind of termination is calledtotal termination. Equivalently total termination can be characterized by the existence of an order on ground terms which is total, wellfounded and closed under contexts. For finite signatures, total termination implies simple termination. The converse does not hold. However, total termination generalizes most of the usual techniques for proving termination (including the well-known recursive path order). It turns out that for this kind of termination the only interesting orders below ɛ0 are built from the natural numbers by lexicographic product and the multiset construction. By examples we show that both constructions are essential. Most of the techniques used are based on ordinal arithmetic.

Total Termination of Term Rewriting

We investigate proving termination of term rewriting systems by a compositional interpretation of terms in a total well-founded order. This kind of termination is calledtotal termination. Equivalently total termination can be characterized by the existence of an order on ground terms which is total, wellfounded and closed under contexts. For finite signatures, total termination implies simple termination. The converse does not hold. However, total termination generalizes most of the usual techniques for proving termination (including the well-known recursive path order). It turns out that for this kind of termination the only interesting orders below ɛ0 are built from the natural numbers by lexicographic product and the multiset construction. By examples we show that both constructions are essential. Most of the techniques used are based on ordinal arithmetic.