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One of the fundamental ways to construct De Bruijn sequences is by using a shift-rule. A shift-rule receives a word as an argument and computes the symbol that appears after it in the sequence. An optimal shift-rule for an (n, k)-De Bruijn sequence runs in time O(n). We propose an extended notion we name a generalized-shift-rule, which receives a word, w, and an integer, c, and outputs the c symbols that comes after w. An optimal generalized-shift-rule for an (n, k)-De Bruijn sequence runs in time O(n + c). We show that, unlike in the case of a shiftrule, a time optimal generalized-shift-rule allows to construct the entire sequence efficiently. We provide a time optimal generalized-shift-rule for the well-known prefer-max and prefer-min De Bruijn sequences.

In the Adapter Design Pattern, a programmer implements a Target interface by constructing an Adapter that accesses an existing Adaptee code. In this work, we present a reactive synthesis interpretation to the adapter design pattern, wherein an algorithm takes an Adaptee and a Target transducers, and the aim is to synthesize an Adapter transducer that, when composed with the Adaptee, generates a behavior that is equivalent to the behavior of the Target. One use of such an algorithm is to synthesize controllers that achieve similar goals on different hardware platforms. While this problem can be solved with existing synthesis algorithms, current state-of-the-art tools fail to scale. To cope with the computational complexity of the problem, we introduce a special form of specification format, called Separated GR(k), which can be solved with a scalable synthesis algorithm but still allows for a large set of realistic specifications. We solve the realizability and the synthesis problems for Separated GR(k), and show how to exploit the separated nature of our specification to construct better algorithms, in terms of time complexity, than known algorithms for GR(k) synthesis. We then describe a tool, called SGR(k), that we have implemented based on the above approach and show, by experimental evaluation, how our tool outperforms current state-of-the-art tools on various benchmarks and test-cases.

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