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We have developed a catalytic synthetic method to prepare phenoxasilins. A borane-catalyzed double sila-Friedel–Crafts reaction between amino group-containing diaryl ethers and dihydrosilanes can be used to prepare a variety of phenoxasilin derivatives in good to excellent yields. The optimized reaction conditions were also applicable for diaryl thioethers to afford their corresponding six-membered silacyclic products. The gram-scale synthesis of a representative bis(dimethylamino)phenoxasilin and the transformation of its amino groups have also been demonstrated.

In the dependency pair framework for proving termination of rewriting systems, polynomial interpretations are used to transform dependency chains into bounded decreasing sequences of integers, and they play an important role for the success of proving termination, especially for constrained rewriting systems. In this paper, we show sufficient conditions of linear polynomial interpretations for transforming dependency chains into bounded monotone (i.e., decreasing or increasing) sequences of integers. Such polynomial interpretations transform rewrite sequences of the original system into decreasing or increasing sequences independently of the transformation of dependency chains. When we transform rewrite sequences into increasing sequences, polynomial interpretations have non-positive coefficients for reducible positions of marked function symbols. We propose four DP processors parameterized by transforming dependency chains and rewrite sequences into either decreasing or increasing sequences of integers, respectively. We show that such polynomial interpretations make us succeed in proving termination of the McCarthy 91 function over the integers.

A parameterised Boolean equation system (PBES) is a set of equations that defines sets satisfying the equations as the least and/or greatest fixed-points. Thus this system is regarded as a declarative program defining predicates, where a program execution returns whether a given ground atomic formula holds or not. The program execution corresponds to the membership problem of PBESs, which is however undecidable in general. This paper proposes a subclass of PBESs which expresses universal-quantifiers free formulas, and studies a technique to solve the problem on it. We use the fact that the membership problem is reduced to the problem whether a proof graph exists. To check the latter problem, we introduce a so-called dependency space which is a graph containing all of the minimal proof graphs. Dependency spaces are, however, infinite in general. Thus, we propose some conditions for equivalence relations to preserve the result of the membership problem, then we identify two vertices as the same under the relation. In this sense, dependency spaces possibly result in a finite graph. We show some examples having infinite dependency spaces which are reducible to finite graphs by equivalence relations. We provide a procedure to construct finite dependency spaces and show the soundness of the procedure. We also implement the procedure using an SMT solver and experiment on some examples including a downsized McCarthy 91 function.

In this paper, we show that the SR transformation, a computationally equivalent transformation proposed by Serbanuta and Rosu, is a sound structure-preserving transformation for weakly-left-linear deterministic conditional term rewriting systems. More precisely, we show that every weakly-left-linear deterministic conditional term rewriting system can be converted to an equivalent weakly-left-linear and ultra-weakly-left-linear deterministic conditional term rewriting system and prove that the SR transformation is sound for weakly-left-linear and ultra-weakly-left-linear deterministic conditional term rewriting systems. Here, soundness for a conditional term rewriting system means that reduction of the transformed unconditional term rewriting system creates no undesired reduction sequence for the conditional system.

A parameterised Boolean equation system (PBES) is a set of equations that defines sets as the least and/or greatest fixed-points that satisfy the equations. This system is regarded as a declarative program defining functions that take a datum and returns a Boolean value. The membership problem of PBESs is a problem to decide whether a given element is in the defined set or not, which corresponds to an execution of the program. This paper introduces reduced proof graphs, and studies a technique to solve the membership problem of PBESs, which is undecidable in general, by transforming it into a reduced proof graph. A vertex X(v) in a proof graph represents that the data v is in the set X, if the graph satisfies conditions induced from a given PBES. Proof graphs are, however, infinite in general. Thus we introduce vertices each of which stands for a set of vertices of the original ones, which possibly results in a finite graph. For a subclass of disjunctive PBESs, we clarify some conditions which reduced proof graphs should satisfy. We also show some examples having no finite proof graph except for reduced one. We further propose a reduced dependency space, which contains reduced proof graphs as sub-graphs if a proof graph exists. We provide a procedure to construct finite reduced dependency spaces, and show the soundness and completeness of the procedure.

An inductive theorem proving method for constrained term rewriting systems, which is based on rewriting induction, needs a decision procedure for reduction-completeness of constrained terms. In addition, the sufficient complete property of constrained term rewriting systems enables us to relax the side conditions of some inference rules in the proving method. These two properties can be reduced to intersection emptiness problems related to sets of ground instances for constrained terms. This paper proposes a method to construct deterministic, complete, and constraint-complete constrained tree automata recognizing ground instances of constrained terms.

Unravelings are transformations from a conditional term rewriting system (CTRS, for short) over an original signature into an unconditional term rewriting systems (TRS, for short) over an extended signature. They are not sound w.r.t. reduction for every CTRS, while they are complete w.r.t. reduction. Here, soundness w.r.t. reduction means that every reduction sequence of the corresponding unraveled TRS, of which the initial and end terms are over the original signature, can be simulated by the reduction of the original CTRS. In this paper, we show that an optimized variant of Ohlebusch's unraveling for a deterministic CTRS is sound w.r.t. reduction if the corresponding unraveled TRS is left-linear or both right-linear and non-erasing. We also show that soundness of the variant implies that of Ohlebusch's unraveling. Finally, we show that soundness of Ohlebusch's unraveling is the weakest in soundness of the other unravelings and a transformation, proposed by Serbanuta and Rosu, for (normal) deterministic CTRSs, i.e., soundness of them respectively implies that of Ohlebusch's unraveling.