The concept of a computable function is quite a well-studied one, however, it is possible to capture certain important properties of computability categorically. A special type of category used for this purpose is called a Turing category. This thesis starts with a brief overview of Turing categories, followed by a study of additional categorical structure they may contain, based on the types of structure found in the world of computable functions, and how this is reflected in the underlying combinatorial structures.

The concept of a computable function is quite a well-studied one, however, it is possible to capture certain important properties of computability categorically. A special type of category used for this purpose is called a Turing category. This thesis starts with a brief overview of Turing categories, followed by a study of additional categorical structure they may contain, based on the types of structure found in the world of computable functions, and how this is reflected in the underlying combinatorial structures.

The concept of a computable function is quite a well-studied one, however, it is possible to capture certain important properties of computability categorically. A special type of category used for this purpose is called a Turing category. This thesis starts with a brief overview of Turing categories, followed by a study of additional categorical structure they may contain, based on the types of structure found in the world of computable functions, and how this is reflected in the underlying combinatorial structures.

The concept of a computable function is quite a well-studied one, however, it is possible to capture certain important properties of computability categorically. A special type of category used for this purpose is called a Turing category. This thesis starts with a brief overview of Turing categories, followed by a study of additional categorical structure they may contain, based on the types of structure found in the world of computable functions, and how this is reflected in the underlying combinatorial structures.

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The main theme of this thesis is studying classes of structures with respect to various measurements of complexity. We will briefly discuss the notion of computable dimension, while the breadth of the paper will focus on calculating the Turing ordinal and the back-and-forth ordinal of various classes, along with an exploration of how these two ordinals are related in general. Computable structure theorists study which computable dimensions can be realized by structures from a given class. Using a structural characterization of the computably categorical equivalence structures due to Calvert, Cenzer, Harizanov and Morozov, we prove that the only possible computable dimension of an equivalence structure is 1 or ω. In 1994, Jockusch and Soare introduced the notion of the Turing ordinal of a class of structures. It was unknown whether every computable ordinal was the Turing ordinal of some class. Following the work of Ash, Jocksuch and Knight, we show that the answer is yes, but, as one might expect, the axiomatizations of these classes are complex. In 2009, Montalban defined the back-and-forth ordinal of a class using the back-and-forth relations. Montalban, following a result of Knight, showed that if the back-and-forth ordinal is n+1, then the Turing ordinal is at least n. We will prove a theorem stated by Knight that extends the previous result to all computable ordinals and show that if the back-and-forth ordinal is α (infinite) then the Turing ordinal is at least α. It is conjectured at present that if a class of structures is relatively nice then the Turing ordinal and the back-and-forth ordinal of the class differ by at most 1. We will present many examples of classes having axiomatizations of varying complexities that support this conjecture; however, we will show that this result does not hold for arbitrary Borel classes. In particular, we will prove that there is a Borel class with infinite Turing ordinal but finite back-and-forth ordinal and show that, for each positive integer d, there exists a Borel class of structures such that the Turing ordinal and the back-and-forth ordinal of the class are both finite and differ by at least d.

The main theme of this thesis is studying classes of structures with respect to various measurements of complexity. We will briefly discuss the notion of computable dimension, while the breadth of the paper will focus on calculating the Turing ordinal and the back-and-forth ordinal of various classes, along with an exploration of how these two ordinals are related in general. Computable structure theorists study which computable dimensions can be realized by structures from a given class. Using a structural characterization of the computably categorical equivalence structures due to Calvert, Cenzer, Harizanov and Morozov, we prove that the only possible computable dimension of an equivalence structure is 1 or ω. In 1994, Jockusch and Soare introduced the notion of the Turing ordinal of a class of structures. It was unknown whether every computable ordinal was the Turing ordinal of some class. Following the work of Ash, Jocksuch and Knight, we show that the answer is yes, but, as one might expect, the axiomatizations of these classes are complex. In 2009, Montalban defined the back-and-forth ordinal of a class using the back-and-forth relations. Montalban, following a result of Knight, showed that if the back-and-forth ordinal is n+1, then the Turing ordinal is at least n. We will prove a theorem stated by Knight that extends the previous result to all computable ordinals and show that if the back-and-forth ordinal is α (infinite) then the Turing ordinal is at least α. It is conjectured at present that if a class of structures is relatively nice then the Turing ordinal and the back-and-forth ordinal of the class differ by at most 1. We will present many examples of classes having axiomatizations of varying complexities that support this conjecture; however, we will show that this result does not hold for arbitrary Borel classes. In particular, we will prove that there is a Borel class with infinite Turing ordinal but finite back-and-forth ordinal and show that, for each positive integer d, there exists a Borel class of structures such that the Turing ordinal and the back-and-forth ordinal of the class are both finite and differ by at least d.

This study investigates general equilibrium effects of an alternative social security policy in Indonesia. The study aims to analyse some financial issues of the proposed policy using a dynamic CGE model. The focus is investigating possible tax scenarios to finance the proposed policy and their impacts on the economy. The simulation results suggest that the consumption tax base should be used as the main financing method. This is because based on various simulations the selected consumption taxes have less negative impacts on the economy than the selected income taxes. Those selected consumption taxes more equitably distribute tax burden and improve income inequality in the long run. However, the increasing price because of this policy selection should also be considered seriously. The simulations also include the study of the demographic transition in Indonesia. A view that is common in the literature is that the rapid increase of labor force in the next three decades could raise the proportion of skilled workers in the labor force and enhance the economic growth. Instead the simulations suggest contrary results. When we repeat the tax/transfer simulations with the demographic transition, real GDP per capita and consumption per capita fall further below the baseline projections. Further simulations are conducted to investigate possible policy actions to mitigate the effects of this demographic transition. This study also covers possible allocation decision trade-offs surrounding the proposed social security policy. That is, the trade-offs between universal social pension insurance and universal social health insurance, and between universal tax-financed social security programs and other important development programs. Given the limitation of our study, that all stakeholders have agreed to develop a universal tax-financed social security program, we conclude that universal tax-financed social health insurance should be given more priority than universal tax-financed social pension insurance. The study concludes with some remarks regarding important areas for future research.

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