a note on two convergence acceleration methods for ordinary continued fractions abstract in this note we compare two recent methods of convergence acceleration for ordinary continued fractions the first one introduced by thron and waadeland 13 and further developed by brezinski 1 the second one by jacobsen and waadeland 4 5

codes and equations on trees the objective of this paper is to study by new formal methods the notion of tree code introduced by m nivat in cite n92 in particular we introduce the notion of stability for sets of trees closed under concatenation this allows us to give a characterization of tree codes which is very close to the algebraic characterization of word codes in terms of free monoids we further define the stable hull of a set of trees and derive a defect theorem for trees which generalizes the analogous result for words as a consequence we obtain some properties of tree codes having two elements moreover we propose a new algorithm to test whether a finite set of trees is a tree code the running time of the algorithm is polynomial in the size of the input we also introduce the notion of tree equation as a complementary point of view to tree codes the main problem emerging in this approach is to decide the satisfiability of tree equations it is a special case of second order unification and it remains still open

general variational approach to the interpolation problem abstract the talmi and gilat variational approach to the interpolation problem in arbitrary dimension is presented together with the corresponding physical model the connection of this approach to some known spline methods is demonstrated and new interpolation functions are derived for one two and three dimensional cases they are designed to be flexible through the use of meaningful parameters and to give good approximations of both the function itself and its derivatives as well

logic with equality partisan corroboration and shifted pairing herbrand s theorem plays a fundamental role in automated theorem proving methods based on tableaux the crucial step in procedures based on such methods can be described as the corroboration problem or the herbrand skeleton problem where given a positive integer m called multiplicity and a quantifier free formula one seeks a valid disjunction of m instantiations of that formula in the presence of equality which is the case in this paper this problem was recently shown to be undecidable the main contributions of this paper are two theorems the first the partisan corroboration theorem relates corroboration problems with different multiplicities the second the shifted pairing theorem is a finite tree automata formalization of a technique for proving undecidability results through direct encodings of valid turing machine computations these theorems are used in the paper to explain and sharpen several recent undecidability results related to the corroboration problem the simultaneous rigid e unification problem and the prenex fragment of intuitionistic logic with equality

higher order unification via combinators abstract we present an algorithm for unification in the simply typed lambda calculus which enumerates complete sets of unifiers using a finitely branching search space in fact the types of terms may contain type variables so that a solution may involve type substitution as well as term substitution the problem is first translated into the problem of unification with respect to exstensional equality in combinatory logic and the algorithm is defined in terms of transformations on systems of combinatory terms these transformations are based on a new method itself based on systems for deciding extensional equality between typed combinatory logic terms

deciding context unification in first order term unification variables represent well formed terms over a given signature and we are to solve equations built using function symbols from the signature and such variables this problem is well known to be decidable in linear time in second order term unification the variables take arguments i e other terms and a substitution uses those arguments an arbitrary number of times for instance an equation f x c x c e x f c c has a solution x e b where b is a special symbol denoting the place in which the argument is substituted under this substitution both sides evaluate to f c c there are other solutions for instance x e f b b which evaluates both sides tof f c c f c c in general a solution that evaluates both sides to full binary tree of arbitrary height is easy to construct second order unification is in general undecidable context unification is a natural problem in between first and second order unification we deal with equations over terms the variables take arguments but we restrict the set of solutions the argument is used exactly once formally contexts are terms with exactly one occurrence of the special symbol b and in context unification we are given an equation over terms with variables representing contexts and ask about the satisfiability of this equation for instance when the aforementioned equation f x c x c e x f c c is treated as a context unification problem then it has exactly one solution x e b other substitutions that are solutions of it as an instance of the second unification problem say x e f b b are not valid as b is used more than once context unification also generalizes satisfiability of word equations which is decidable in pspace the decidability status of context unification remained unknown for almost two decades in this article we show that context unification is in pspace in exptime when tree regular constraints are also allowed those results are obtained by extending the recently developed recompression technique which was previously defined for strings and used to obtain a new pspace algorithm for satisfiability of word equations in this article the technique is generalized to trees and the corresponding algorithm is generalized from word equations to context unification the idea of recompression is to apply simple compression rules replacing pairs of neighboring function symbols to the solution of the context equation to this end we appropriately modify the equation without the knowledge of the actual solution so compressing the solution can be simulated by compressing parts of the equation it is shown that if the compression operations are appropriately chosen then the size of the instance is polynomial during the whole algorithm thus giving a pspace upper bound

on the existence of speed independent circuits abstract a circuit is called speed independent if its nontransient behavior does not depend on the size of the delays in the different components of the circuit we show that the class of speed independent circuits is relatively small in fact very many useful circuits cannot be realized in a speed independent design for example we show that there does not exist any speed independent mod k counter for any k 1 the results are derived using a very general model of a network which is applicable to both gate circuits and more modern mos switch level circuits furthermore the results are also robust with respect to different delay assumptions and definitions of speed independence

complexity of nilpotent unification and matching problems we consider the complexity of equational unification and matching problems where the equational theory contains a nilpotent function i e a function f satisfying f x x 0 where 0 is a constant we show that nilpotent unification and matching are np complete in the presence of associativity and commutativity the problems still remain np complete however when 0 is also assumed to be the unity for the function f the problems are solvable in polynomial time we also show that the problem remains in p even when a homomorphism is added an application of this result to a subclass of set constraints is illustrated second order matching modulo nilpotence is shown to be undecidable

equational theory of context unification is undecidable context unification is a particular case of second order unification in which all second order variables are unary and only linear functions are sought for as solutions its decidability is an intriguing open problem with only a very weak known np lower bound we present the simplest currently known undecidable quantified fragment of the theory of context unification by showing that the set of 5 quantified context equations i e sentences of the form wu v s g hs t is undecidable and in fact is co recursively enumerable hard i e every set with recursively enumerable complement is many one reducible to it

the undecidability of the second order unification problem abstract it is shown that there is no effective procedure for determining whether or not two terms of the language of second order logic have a common instance

conditional equational theories and complete sets of transformations abstract the idea of combining the advantages of function and logic programming has attracted many researchers their work ranges from the integration of existing languages over higher order logic to equational logic languages where logic programs are augmented with equational theories recently it has been proposed to handle those equational theories by complete sets of transformations these transformations are extensions of the rules introduced by herbrand and later used by martelli and montanari to compute the most general unifier of two expressions we generalize this idea to complete sets of transformations for arbitrary conditional equational theories the largest class of equational theories that admit a least herbrand model the completeness proof is based on the observation that each refutation with respect to linear paramodulation and reflection can be modelled by the transformations as certain conditions imposed on an equational theory restrict the search space generated by paramodulation and reflection we can easily refine our transformations due to the completeness proof if the conditional equational theory is ground confluent or canonical

a note on the relation between two convergence acceleration methods for ordinary continued fractions in this note we relate two methods of convergence acceleration for ordinary continued fractions the first one is due to lorentzen and waadeland jacobsen and waadeland 1980 1998 the second one to waadeland 1986 1987 1987 1987 1988

convergence acceleration of limit periodic continued fractions under asymptotic side conditions we present a method of convergence acceleration for limitk periodic continued fractionsk an 1 ork 1 bn satisfying certain asymptotic side conditions the method represents an improvement of the fixed point modification considered by thron and waadeland 8 under these conditions the regularc fraction expansions of hypergeometric functions2f1 a 1 c z and2f1 a b c z 2f1 a b 1 c 1 z are examples of continued fractions satisfying these conditions

the undecidability of simultaneous rigid e unification abstract simultaneous rigid e unification was introduced in 1987 by gallier raatz and snyder it is used in the area of automated reasoning with equality in extension procedures like the tableau method or the connection method many articles in this area assumed the existence of an algorithm for the simultaneous rigid e unification problem there were several faulty proofs of the decidability of this problem in this paper we prove that simultaneous rigid e unification is undecidable as a consequence we obtain the undecidability of the fragment of intuitionistic logic with equality

decidability of bounded second order unification it is well known that first order unification is decidable whereas second order and higher order unification is undecidable bounded second order unification is second order unification under the restriction that only a bounded number of bound variables in the instantiating terms for second order variables is permitted however the size of the instantiation is not restricted in this paper a decision algorithm for bounded second order unification is described this is the first nontrivial decidability result for second order unification where the signature is not restricted and there are no restrictions on the occurrences of variables this supports the claim that bounded second order unification is easier than context unification whose decidability is currently an open question

referential logic of proofs we introduce an extension of the propositional logic of single conclusion proofs by the second order variables denoting the reference constructors of the type the formula which is proved by x the resulting logic of proofs with references flpref is shown to be decidable and to enjoy soundness and completeness with respect to the intended provability semantics we show that flpref provides a complete test of admissibility of inference rules in a sound extension of arithmetic this paper may be regarded as a contribution to the theory of automated reasoning systems

higher order unification via explicit substitutions higher order unification is equational unification for conversion but it is not first order equational unification as substitution has to avoid capture thus the methods for equational unification such as narrowing built upon grafting i e substitution without renaming cannot be used for higher order unification which needs specific algorithms our goal in this paper is to reduce higher order unification to first order equational unification in a suitable theory this is achieved by replacing substitution by grafting but this replacement is not straightforward as it raises two major problems first some unification problems have solutions with grafting but no solution with substitution then equational unification algorithms rest upon the fact that grafting and reduction commute but grafting and reduction do not commute in calculus and reducing an equation may change the set of its solutions this difficulty comes from the interaction between the substitutions initiated by reduction and the ones initiated by the unification process two kinds of variables are involved those of conversion and those of unification so we need to set up a calculus which distinguishes between these two kinds of variables and such that reduction and grafting commute for this purpose the application of a substitution of a reduction variable to a unification one must be delayed until this variable is instantiated such a separation and delay are provided by a calculus of explicit substitutions unification in such a calculus can be performed by well known algorithms such as narrowing but we present a specialised algorithm for greater efficiency at last we show how to relate unification in calculus and in a calculus with explicit substitutions thus we come up with a new higher order unification algorithm which eliminates some burdens of the previous algorithms in particular the functional handling of scopes huet s algorithm can be seen as a specific strategy for our algorithm since each of its steps can be decomposed into elementary ones leading to a more atomic description of the unification process also solved forms in calculus can easily be computed from solved forms in calculus

higher order rewrite systems and their confluence abstract we study higher order rewrite systems hrss which extend term rewriting to terms hrss can describe computations over terms with bound variables we show that rewriting with hrss is closely related to undirected equational reasoning we define pattern rewrite systems prss as a special case of hrss and extend three confluence results from term rewriting to prss the critical pair lemma by knuth and bendix confluence of rewriting modulo equations a la huet and confluence of orthogonal prss

the complexity of monadic second order unification monadic second order unification is second order unification where all function constants occurring in the equations are unary here we prove that the problem of deciding whether a set of monadic equations has a unifier is np complete where we use the technique of compressing solutions using singleton context free grammars we prove that monadic second order matching is also np complete

improving a method for computing non dominant solutions of certain second order recurrence relations of poincare type in this paper we present a method of convergence acceleration for the calculation of non dominant solutions of second order linear recurrence relations for which the coefficients satisfy certain asymptotic conditions it represents an improvement of the method recently proposed by jacobsen and waadeland 3 4 for limit periodic continued fractions for continued fractions the method corresponds to a repeated application of the bauer muir transformation some examples and a generalization to non homogeneous recurrence relations are given

asymptotic cyclic expansion and bridge groups of formal proofs abstract formal proofs even simple ones may hide an unexpected intricate combinatorics we define a new combinatorial invariant the bridge group of a proof which encodes the cyclic structure of proofs in the sequent calculus we compute the bridge groups of two infinite families of proofs and identify them with the baumslag solitar and gersten groups we observe that the distortion of cyclic subgroups in these groups equals the asymptotic growth of the procedure of elimination of lemmas from the proofs

equational unification word unification and 2nd order equational unification for finite convergent term rewriting systems it is shown that the equational unification problem is recursively independent of the equational matching problem the word matching problem and the 2nd order equational matching problem apart from the latter these results are derived by considering term rewriting systems on signatures that contain unary function symbols only i e string rewriting systems also for this special case 2nd order equational matching is shown to be reducible to 1st order equational matching in addition we present some new decidability results for simultaneous equational matching and unification finally we compare the word unification problem to the 2nd order equational unification problem