Reflective Equilibrium MOdel in MAthematica (REMoMa)

This repository complements the philosophical paper "Making Reflective Equilibrium Precise. A Formal Model" by C. Beisbart, G. Betz and G. Brun, to-be-published with ERGO. It makes available all the code used to generate the computational results presented and discussed therein.

Simulation Results of RE Processes

Results of RE process simulations (equilibration) are stored in the folder results. The repo contains four ensembles:

• four-cases: RE process simulations of the four illustrative cases discussed in the paper with default parameter settings.
• ini-com_sample: RE process simulations with random initial commitments, parameters as in four_cases.
• par-narrow_sample: RE process simulations with small random variations of the parameter values used in four_cases; initial commitments as in four_cases. This ensemble us used to check robustness of the precise RE process as simulated in four_cases.
• par-broad_sample: RE process simulations with full random variations of the parameter values used in four_cases; initial commitments as in four_cases. This ensemble us used to check robustness of the general claim that RE processes lead to global optima / full RE states.

Analysis and Visualization

Various notebooks are used to analyze and visualize simulation results (folder notebooks). When running these Notebooks, please make sure to adjust file-paths to your system.

• VisualizingResults is used to plot every step of a RE equilibration process.
• GlobalOptimization checks whether the equilibration end points in the four-cases are global optima; and visualizes, in addition, various properties of global optima over the entire parameter space.
• LocalAndGlobalOptima_RandomSampleIniComs calculates how many equilibration fixed points in an ensemble with random initial commitments are global optima.
• LocalAndGlobalOptima_RandomSampleParams calculates how many equilibration fixed points in an ensemble with random parameter combinations (weights of the achievement function) are global optima.
• ParameterRobustness checks the robustness of the simulation of the four illustrative cases to small parameter perturbations.

Running the Simulations

We provide three scripts to run simulations and generate your own results.

• run_4simus.m has been used to simulate four RE processes and to generate the four illustrative cases.
• run_simus_random_sample_param.m has been used to generate the two ensemble with parameter perturbations.
• run_simus_random_sample_inicom.m has been used to generate the ensemble with random initial commitments.

Technical Caveats

Formal Representation of Positions as Integers

We model theories and commitments as positions, that is as subsets of the sentence pool, for example: {1,-4,6} or {2,3,-6,-7}. Such (minimally consistent) positions are represented -- in the code -- as integers. The idea is the following. Every integer n can be written as a ternary number, i.e.

n = (c_0 * 3^0) + (c_1 * 3^1) + (c_2 * 3^2) + ...

with 0 ≤ c_i ≤ 2. We now interpret the coefficients c_i for n-1 (i.e., the digits of the ternary representation of n-1) as an index function over the non-negative sentence pool which determines whether the position accepts / negates / suspends judgment wrt. a given sentence.

As you see for example in run_4simus.m, the illustrative initial commitments are represented as {118,121,1090,1333}. Now,

118-1 = 81 + 27 + 9  
      = (1 * 3^4) + (1 * 3^3) + (1 * 3^2)      

So, written as a ternary number, 118-1 equals 0011100 (with two 0 padded). 118 hence picks 3,4,5 from the sentence pool. Likewise

1333-1  = 729 + 486 + 81 + 27 + 9
        = (1 * 3^6) + (2 * 3^5) + (1 * 3^4) + (1 * 3^3) + (1 * 3^2)

Therefore, the integer 1333 represents the position {3,4,5,-6,7}.

Such ternary representation of positions allows for efficient coding.

Parameters alpha and beta

The simulation runs are initialized with parameters alpha and beta. These parameters determine the relative weight of

• Account vs Systematicity (alpha) in step theory choice,
• Account vs Faithfulness (beta) in step commitment adjustment.

The higher the parameter values, the more weight is given to account.

These parameters alpha and beta are related to the weights w_a, w_s, and w_f of the achievement (as presented in the paper) as follows:

WeightAccount[alpha_, beta_] := 
  (alpha*beta)/(alpha + beta - alpha*beta);
WeightSystematicity[alpha_, beta_] := 
  (beta - alpha*beta)/(alpha + beta - alpha*beta);
WeightCloseness[alpha_, beta_] := 
  1 - (WeightAccount[alpha, beta] + WeightSystematicity[alpha, beta]);