Abstract
The present research work discusses a possible formalisation in the Matita Interactive Theorem Prover, and initial results on the development of an decisional Turing Machine within Herbert Simon’s bounded rationality and satisficing decision theoretical framework. Specifically, it implements the decision procedure in static and dynamic ways. The present research has the purpose of being a potential preliminary step towards the creation of a future investigation the Reverse Complexity Program immersed in the classical behavioural economic framing, which would uncover part of the real mechanisms that an economic agent might face during the decision process.
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Notes
- 1.
This paper relies on this epistemological setting. According to [1], exists a distinction between orthodox economic theory and Turing’s economics percept that could be extrapolated from [2]’s last sentence: “[...] these, and some other results of mathematical logic may be regarded as going some way towards a demonstration, within mathematics itself, of the inadequacy of reason by common sense [...]” ([2], 1954, p. 23).
- 2.
The notion of bounded rationality might be summerised as follow: (i) focus on the limited computational capacities and restricted access to information, features of real time human decision process; and (ii) limitation of the information processing capacities. Specifically, this assumption is based on two essential restrictions: the limits of the information both gathered and processed and the limit of computational capacity itself when agents face complex situations.
- 3.
According to the author, this connection is possible because (i) the presence of logic and limited computational features lead to Simon’s bounded rationality notion; (ii) rational process may be represented in a symbolic form and with tacit procedures for dealing with unpredicted events; and (iii) in bounded rationality, the agent has limited information and limited computational capacities.
- 4.
- 5.
Setting this definition, Simon’s computational model could be assimilated into a Turing Machine subjected to the Church-Turing Thesis.
- 6.
Specifically, according to [6], the decisional process comes into light when a problem requires a solution. Consequently, the problem-solving procedures are activated and agents treat complex problems consecutively.
- 7.
The reader might have noted that Simon does not always assume that the human problem solver is endowed with the full and powerful model of computation instruments (always subjected to the Church-Turing Thesis) because they are limited by various psychological and neurological factors. Furthermore, the satisficing phenomenon comes together with the desire to approach the problem with a computational strategy.
- 8.
This Turing Machine is indexed by f when it has been effectively encoded.
- 9.
They perform decisions procedures both in synchronical and asynchronical ways.
- 10.
In particular, the present approach is based on the [11] work for what concerns the formal Turing Machine repository into Complexity Theory.
- 11.
The origin of the Entscheidungsproblem can be attributed to Gottfried Leibniz, who, after having constructed a successful mechanical calculating machine, dreamt of building a machine with a formal language that could manipulate symbols in order to determine the true values of mathematical statements. Furthermore, in 1928, [12] posed the question in modern terms. According to them, by the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms. In this context, in 1936, Church and Turing published three independent papers ([13, 14] and [15]) in which they demonstrated that a general solution to the Entscheidungsproblem is impossible. Instead, Turing assumed the existence of a general method able to decide whether any given Turing Machine halts or not (or equivalently, by those expressible in the lambda calculus). This assumption is now known as the Church–Turing Thesis.
- 12.
The state of the machine or M-configuration as it was defined by Turing himself. From this definition, the aim of a Turing Machine is to formalise the notion of the effective procedure.
- 13.
This configuration could be assimilated to a automatic machine or a-machine in which, at any given moment, its behaviour is completely determined by the current state and symbol (or configuration): the so-called determinacy condition. Specifically, from this condition it is possible to oppose the choice machines for which the next stage is centred on the decision of an external device or operator.
- 14.
The initial machine state starts from a blank tape called \(T_{simple}\). What follows from all these settings is the unsolubility of the so-called halting problem in which it is not possible to determine a priori if a Turing Machine would halt or not. For a more extensive review of the argument, see [16] and [15].
- 15.
In the Matita language, the FinSet library permits the description of every function f between two sets A and B by means of a finite graph of pairs \( \ll a,f \gg \).
- 16.
In this particular machine, the swapping operation requires an auxiliary memory cell due to the finite length of the tape. Moreover, in this machine, it is possible to execute the following four essential steps:
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\(swap_{0}\) = read the current symbol, save and register it and move to the right;
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\(swap_{1}\) = swap the current symbol with the register content and move back to the left;
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\(swap_{2}\) = write the register content at the current position;
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\(swap_{3}\) = stop;
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- 17.
The reader might have noted that a common issue of managing with tape machine mechanism is the storage of intermediate results since it requires sorting the whole tape back and forth, which requires close attention to not overwrite critical information. To solve this problem, the if-then-else result is a boolean, which makes it easier to store into a machine state.
This part of the formalisation is based on the [11] work.
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Cialfi, D. (2021). Beyond the Turing Test: A Formalisation of a Decisional Turing Machine. In: Bucciarelli, E., Chen, SH., Corchado, J.M., Parra D., J. (eds) Decision Economics: Minds, Machines, and their Society. DECON 2020. Studies in Computational Intelligence, vol 990. Springer, Cham. https://doi.org/10.1007/978-3-030-75583-6_1
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