Decidable Word Research Articles

FINITELY GENERATED INFINITE SIMPLE GROUPS OF INFINITE SQUARE WIDTH AND VANISHING STABLE COMMUTATOR LENGTH

It is shown that there exist finitely generated infinite simple groups of infinite commutator width and infinite square width on which there exists no stably unbounded conjugation-invariant norm, and in particular stable commutator length vanishes. Moreover, a recursive presentation of such a group with decidable word and conjugacy problems is constructed.

FINITELY GENERATED INFINITE SIMPLE GROUPS OF INFINITE COMMUTATOR WIDTH

It is shown that there exists a finitely generated infinite simple group of infinite commutator width, and that the commutator width of a finitely generated infinite boundedly simple group can be arbitrarily large. Besides, such groups can be constructed with decidable word and conjugacy problems.

Decidability and Complexity Analysis by Basic Paramodulation

It is shown that for sets of Horn clauses saturated underbasic paramodulationthe word and unifiability problems are in NP, and the number of minimal unifiers is simply exponential (i). For Horn sets saturated w.r.t. a special ordering under the more restrictive inference rule ofbasic superposition, the word and unifiability problems are still decidable and unification is finitary (ii). These two results are applied to the following languages. Forshallowpresentations (equations with variables at depth at most one) we show that the closure under paramodulation can be computed in polynomial time. Applying result (i), it follows that shallow unifiability is in NP, which is optimal since unifiability in ground theories is already NP-hard. The shallow word problem is even shown to be polynomial. Generalizing shallow theories to the Horn case, we obtain (two versions of) a language we callCatalog, a natural extension of Datalog to include functions and equality. The closure under paramodulation is finite for Catalog sets, hence (i) still applies. For Catalog sets S the decidability of the full first-order theory of T(F)=Sis shown as well. Finally we definestandard theories, which include and significantly extendshallowtheories. Standard presentations can be finitely closed under superposition and result (ii) applies, thus obtaining a new fundamental class with decidable word and unifiability problems and where unification is finitary.