Algebraic theory of minimal nondeterministic finite automata with applications

Since the 1950s, the theory of deterministic and nondeterministic finite automata (DFAs and NFAs, respectively) has been a cornerstone of theoretical computer science. In this dissertation, our main object of study is minimal NFAs. In contrast with minimal DFAs, minimal NFAs are computationally challenging: first, there can be more than one minimal NFA recognizing a given language; second, the problem of converting an NFA to a minimal equivalent NFA is NP-hard, even for NFAs over a unary alphabet. Our study is based on the development of two main theories, inductive bases and partials, which in combination form the foundation for an incremental algorithm, ibas, to find minimal NFAs. An inductive basis is a collection of languages with the property that it can generate (through union) each of the left quotients of its elements. We prove a fundamental characterization theorem which says that a language can be recognized by an n-state NFA if and only if it can be generated by an n-element inductive basis. A partial is an incompletely-specified language. We say that an NFA recognizes a partial if its language extends the partial, meaning that the NFA’s behavior is unconstrained on unspecified strings; it follows that a minimal NFA for a partial is also minimal for its language. We therefore direct our attention to minimal NFAs recognizing a given partial. Combining inductive bases and partials, we generalize our characterization theorem, showing that a partial can be recognized by an n-state NFA if and only if it can be generated by an n-element partial inductive basis. We apply our theory to develop and implement ibas, an incremental algorithm that finds minimal partial inductive bases generating a given partial. In the case of unary languages, ibas can often find minimal NFAs of up to 10 states in about an hour of computing time; with brute-force search this would require many trillions of years.

Forecasting enrollment in differential assessment programs using cellular automata

Urban growth models have been used for decades to forecast urban development in metropolitan areas. Since the 1990s cellular automata, with simple computational rules and an explicitly spatial architecture, have been heavily utilized in this endeavor. One such cellular-automata-based model, SLEUTH, has been successfully applied around the world to better understand and forecast not only urban growth but also other forms of land-use and land-cover change, but like other models must be fed important information about which particular lands in the modeled area are available for development. Some of these lands are in categories for the purpose of excluding urban growth that are difficult to quantify since their function is dictated by policy. One such category includes voluntary differential assessment programs, whereby farmers agree not to develop their lands in exchange for significant tax breaks. Since they are voluntary, today’s excluded lands may be available for development at some point in the future. Mapping the shifting mosaic of parcels that are enrolled in such programs allows this information to be used in modeling and forecasting. In this study, we added information about California’s Williamson Act into SLEUTH’s excluded layer for Tulare County. Assumptions about the voluntary differential assessments were used to create a sophisticated excluded layer that was fed into SLEUTH’s urban growth forecasting routine. The results demonstrate not only a successful execution of this method but also yielded high goodness-of-fit metrics for both the calibration of enrollment termination as well as the urban growth modeling itself.

Algebraic Theory of Minimal Nondeterministic Finite Automata with Applications

Since the 1950s, the theory of deterministic and nondeterministic finite automata (DFAs and NFAs, respectively) has been a cornerstone of theoretical computer science. In this dissertation, our main object of study is minimal NFAs. In contrast with minimal DFAs, minimal NFAs are computationally challenging: first, there can be more than one minimal NFA recognizing a given language; second, the problem of converting an NFA to a minimal equivalent NFA is NP-hard, even for NFAs over a unary alphabet. Our study is based on the development of two main theories, inductive bases and partials, which in combination form the foundation for an incremental algorithm, ibas, to find minimal NFAs. An inductive basis is a collection of languages with the property that it can generate (through union) each of the left quotients of its elements. We prove a fundamental characterization theorem which says that a language can be recognized by an n-state NFA if and only if it can be generated by an n-element inductive basis. A partial is an incompletely-specified language. We say that an NFA recognizes a partial if its language extends the partial, meaning that the NFA's behavior is unconstrained on unspecified strings; it follows that a minimal NFA for a partial is also minimal for its language. We therefore direct our attention to minimal NFAs recognizing a given partial. Combining inductive bases and partials, we generalize our characterization theorem, showing that a partial can be recognized by an n-state NFA if and only if it can be generated by an n-element partial inductive basis. We apply our theory to develop and implement ibas, an incremental algorithm that finds minimal partial inductive bases generating a given partial. In the case of unary languages, ibas can often find minimal NFAs of up to 10 states in about an hour of computing time; with brute-force search this would require many trillions of years.

The inclusion of differentially assessed lands in urban growth model calibration: a comparison of two approaches using SLEUTH

This research sought to understand the role that differentially assessed lands (lands in the United States given tax breaks in return for their guarantee to remain in agriculture) play in influencing urban growth. Our method was to calibrate the SLEUTH urban growth model under two different conditions. The first used an excluded layer that ignored such lands, effectively rendering them available for development. The second treated those lands as totally excluded from development. Our hypothesis was that excluding those lands would yield better metrics of fit with past data. Our results validate our hypothesis since two different metrics that evaluate goodness of fit both yielded higher values when differentially assessed lands are treated as excluded. This suggests that, at least in our study area, differential assessment, which protects farm and ranch lands for tenuous periods of time, has indeed allowed farmland to resist urban development. Including differentially assessed lands also yielded very different calibrated coefficients of growth as the model tried to account for the same growth patterns over two very different excluded areas. Excluded layer design can greatly affect model behavior. Since differentially assessed lands are quite common through the United States and are often ignored in urban growth modeling, the findings of this research can assist other urban growth modelers in designing excluded layers that result in more accurate model calibration and thus forecasting.