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如何在您知道某些变化的情况下推断未知变量的状态



存在一个博弈，其状态由布尔变量p_1、p_2、...、p_n n量化。假设0 <= n <= 10 .游戏的目的是跟踪尽可能多的变量的状态，以及它们何时会改变状态(见下文(。

一次只能观察到变量的子集。当一个变量的状态从true变为false时，我们总是被告知，但只有当所讨论的变量被观察到时，我们才会被告知哪个变量被改变了。

如果某些p_i在t时间将状态从true更改为false，那么它将在t + 60时间将状态从false更改回true。

下面是一些示例情况和所需的行为。假设有三个变量，p_1、p_2和p_3。

  • 所有变量都是可观察的。变量改变状态，通过观察，它是p_1 。我们知道p_1会在 t + 60 时改变状态。
  • p_1和p_2是可观察到的。变量更改状态。应该推断它是p_3.我们知道p_3将在t + 60时改变状态。
  • p_1是可观察到的，并且已知p_2在t - 30时将状态更改为false。变量更改状态。它没有通过观察p_1，p_2只会在t + 30时将状态更改为true，因此它被p_3。
  • 已知p_1 t - 30时将状态更改为false，p_2 t - 75时将状态更改为false，p_3在t - 80时将状态更改为假。变量更改状态。在t + 1 p_2观察到的值为 true 。因为p_1仍然是false的，并且p_2被观测到true，所以应该推断p_3是false的，并且会在t + 60时将状态改变回true。
  • p_1和p_2是可观察到的。众所周知，p_3在t - 70时将状态更改为false。没有变量更改状态。应该推断p_3是正确的，因为它在t - 10时会从false变为true，如果它从true变为false，我们将被告知变量变化的状态。

我尝试过的一种方法涉及在每次变量状态更改通知上迭代每个变量，并根据以下条件排除可能的匹配项：

  • 它是否可见且未更改状态
  • 它的最后一次状态更改是否已知，并且过去的时间还不够远，无法更改回来。

然后，对于状态已知的剩余变量，根据变量在特定状态下最后一次观察到的时间、变量可能已更改状态的时间(从 false 到 true 返回(等来确定其最后一个变量的上限和下限。然后，如果该边界内仅存在一个变量更改事件，则它应该是与该变量对应的事件。结果太麻烦了，我没有继续。每一行都是一个负担，因为似乎可能的状态组合数量太大，我不会考虑一些事情。

鉴于问题陈述，上述方法是否合理？有没有一种通用的方法来对这样的问题进行建模，以便更优雅地解决它们？

在@NicoSchertler的建议下，我想出了一个解决方案，通过基于Observations和Hypotheses序列创建状态集来处理不确定性。观测值是特定(观察到的(变量的已知状态，而假设是状态的通知，但没有关于它适用于哪个变量的信息。我们能够假设假设不能应用于正在观察的变量。此解决方案不处理启动状态未知(尚(的情况。

有一个单一的起始状态对应于每个变量都true的情况。当提供hypothesis时，通过假设每个未观测的变量是hypothesis的主体，可能会产生多个(n(后继状态。导致矛盾的继承国被抛弃。当提供observation时，每个当前状态都会生成一个后继状态。任何导致矛盾的状态都被抛弃。通过这种方式，假设和观察的序列导致变量可能处于的一组可能状态。

出于我的特定目的，我想知道基于这些状态可以知道什么(而不是例如是否存在有效的解决方案(。将状态组合在一起，并返回一个结果，如果该变量在所有状态中具有相同的状态，则给出每个变量的状态。

考虑到n状态和m通知，最坏情况的复杂性是n^m这可能非常有限，但对于我有限的应用程序来说应该没问题。

这是 JavaScript 实现和测试代码。

求解器.js 

// Time for state to change back.
var STATE_CHANGE = 6e4;
// Possible notification lag.
var EPSILON = 2e3;
// Comparison operations.
function lt(a, b) {
  return a - b < EPSILON;
}
function gt(a, b) {
  return b - a < EPSILON;
}
function eq(a, b) {
  return Math.abs(a - b) < EPSILON;
}
// Object clone.
function clone(obj) {
  return JSON.parse(JSON.stringify(obj));
}
module.exports = Solver;
/**
 * Solver solves boolean dynamic state.
 * @param {Array<string>} variables - array of variable names.
 */
function Solver(variables) {
  this.variables = {};
  this.states = [];
  this._time = null;
  var state = {};
  var time = Date.now();
  var self = this;
  // TODO: Handle unknown or variable start.
  variables.forEach(function (variable) {
    self.variables[variable] = {
      observed: false
    };
    state[variable] = {
      state: true,
      intervals: [{
        state: true,
        start: time,
        observed: false,
        end: null
      }]
    };
  });
  this.states.push(state);
}
// Set subset of variables as observed, the rest assumed not.
Solver.prototype.setObserved = function(variables) {
  var unobserved_variables = Object.keys(this.variables).filter(function (variable) {
    return variables.indexOf(variable) === -1;
  });
  var self = this;
  variables.forEach(function (variable) {
    self.variables[variable].observed = true;
  });
  unobserved_variables.forEach(function (variable) {
    self.variables[variable].observed = false;
  });
};
// Hypothesis has time, state.
Solver.prototype.addHypothesis = function(h) {
  this.updateVariables();
  var states = [];
  for (var i = 0; i < this.states.length; i++) {
    var newStates = this.applyHypothesis(this.states[i], h);
    if (newStates)
      Array.prototype.push.apply(states, newStates);
  }
  this.states = states;
};
// Observation has time, state, variable.
Solver.prototype.addObservation = function(o) {
  this.updateVariables();
  var states = [];
  for (var i = 0; i < this.states.length; i++) {
    var newState = this.applyObservation(this.states[i], o);
    if (newState)
      states.push(newState);
  }
  this.states = states;
};
// Get set of possible states.
Solver.prototype.getStates = function() {
  this.updateVariables();
  return this.states.slice();
};
// Get consolidated state.
// Each variable has state (true|false|null), change (if false). change
// is number or array (if there is disagreement)
Solver.prototype.getState = function() {
  this.updateVariables();
  // Construct output.
  var out = {};
  var state = this.states[0];
  for (var name in state) {
    var variable = state[name];
    if (variable.state) {
      out[name] = {
        state: variable.state
      };
    } else {
      var time = variable.intervals[variable.intervals.length - 1].end;
      out[name] = {
        state: variable.state,
        time: time
      };
    }
  }
  // Compare results across all states.
  return this.states.slice(1).reduce(function (out, state) {
    for (var name in out) {
      var out_variable = out[name],
          variable = state[name];
      // Check for matching states.
      if (out_variable.state === variable.state) {
        // Falsy check time.
        if (!out_variable.state) {
          // TODO: check undefined in case interval not updated?
          var change = variable.intervals[variable.intervals.length - 1].end;
          if (out_variable.time instanceof Array) {
            if (out_variable.time.indexOf(change) === -1) {
              out_variable.push(change);
            }
          } else if (out_variable.time !== change) {
            var times = [out_variable.time, change];
            out_variable.time = times;
          } // Else matches, so no problem.
        }
      } else {
        // Conflicted states.
        out_variable.state = null;
        // In case it was set.
        delete out_variable.time;
      }
    }
    return out;
  }, out);
};
// Update `false` state variables based on false end
// time, if present.
Solver.prototype.updateVariables = function() {
  var time = this._time || Date.now();
  for (var i = 0; i < this.states.length; i++) {
    var state = this.states[i];
    for (var name in state) {
      var variable = state[name];
      // Update changeback.
      if (!variable.state) {
        if (variable.intervals.length > 0) {
          var last = variable.intervals[variable.intervals.length - 1];
          if (last.end && last.end <= time) {
            // update to true.
            variable.state = true;
            variable.intervals.push({
              state: true,
              start: time,
              end: null
            });
          }
        }
      }
    }
  }
};
// Return state with observation applied or null if invalid.
Solver.prototype.applyObservation = function(state, observation) {
  var variable = state[observation.variable];
  if (variable.state && !observation.state) {
    // Change in observed variable true -> false
    variable.state = observation.state;
    variable.intervals.push({
      state: variable.state,
      start: observation.time,
      end: observation.time + STATE_CHANGE
    });
    return state;
  } else if (variable.state && observation.state) {
    // Expected state.
    return state;
  } else if (!variable.state && observation.state) {
    // Potentially updating variable.
    var time = variable.intervals[variable.intervals.length - 1];
    if (eq(time, observation.time)) {
      // update state.
      variable.state = observation.state;
      variable.intervals.push({
        state: observation.state,
        start: observation.time,
        end: null
      });
      return state;
    } else {
      // Could not update this variable.
      return null;
    }
  } else if (!variable.state && !observation.state) {
    // Expected state.
    return state;
  }
};
// Returns multiple states or null if invalid
Solver.prototype.applyHypothesis = function(state, hypothesis) {
  hypothesis = clone(hypothesis);
  var states = [];
  for (var name in state) {
    // Skip observed variables, no guessing with them.
    if (this.variables[name].observed)
      continue;
    var newState = clone(state);
    var variable = newState[name];
    // Hypothesis is always false.
    if (variable.state) {
      // Change in observed variable true -> false
      variable.state = hypothesis.state;
      variable.intervals.push({
        state: variable.state,
        start: hypothesis.time,
        end: hypothesis.time + STATE_CHANGE
      });
    } else {
      newState = null;
    }
    if (newState !== null) {
      states.push(newState);
    }
  }
  if (states.length === 0) {
    return null;
  } else {
    return states;
  }
};

测试求解器.js

var Solver = require('./solver');
var p_1 = "p_1",
    p_2 = "p_2",
    p_3 = "p_3";
var solver = new Solver([p_1, p_2, p_3]);
var t = Date.now();
solver.setObserved([p_1, p_2, p_3]);
solver._time = t + 5e3;
solver.addObservation({
  variable: p_1,
  state: false,
  time: t + 5e3
});
var result = solver.getState();
if (!result[p_1].state && result[p_1].time === t + 65e3 &&
    result[p_2].state &&
    result[p_3].state) {
  console.log("PASS: Test 1.");
} else {
  console.log("FAIL: Test 1.");
}
solver = new Solver([p_1, p_2, p_3]);
solver.setObserved([p_1, p_2]);
solver._time = t + 5e3;
solver.addHypothesis({
  state: false,
  time: t + 5e3
});
result = solver.getState();
if (result[p_1].state &&
    result[p_2].state &&
    !result[p_3].state && result[p_3].time === t + 65e3) {
  console.log("PASS: Test 2.");
} else {
  console.log("FAIL: Test 2.");
}
solver = new Solver([p_1, p_2, p_3]);
solver.setObserved([p_1]);
solver._time = t - 30e3;
solver.addObservation({
  variable: p_2,
  time: t - 30e3,
  state: false
});
solver._time = t;
solver.addHypothesis({
  state: false,
  time: t
});
var result = solver.getState();
if (result[p_1].state &&
    !result[p_2].state && result[p_2].time === t + 30e3 &&
    !result[p_3].state && result[p_3].time === t + 60e3) {
  console.log("PASS: Test 3.");
} else {
  console.log("FAIL: Test 3.");
}
solver = new Solver([p_1, p_2, p_3]);
solver._time = t - 80e3;
solver.addObservation({
  variable: p_3,
  time: t - 80e3,
  state: false
});
solver._time = t - 75e3;
solver.addObservation({
  variable: p_2,
  time: t - 75e3,
  state: false
});
solver._time = t - 30e3;
solver.addObservation({
  variable: p_1,
  time: t - 30e3,
  state: false
});
solver._time = t;
solver.addHypothesis({
  state: false,
  time: t
});
result = solver.getState();
if (!result[p_1].state && result[p_1].time === t + 30e3 &&
    result[p_2].state === null &&
    result[p_3].state === null) {
  console.log("PASS: Test 4.");
} else {
  console.log("FAIL: Test 4.");
}
solver._time = t + 1;
solver.addObservation({
  variable: p_2,
  time: t + 1,
  state: true
});
var result = solver.getState();
if (!result[p_1].state && result[p_1].time === t + 30e3 &&
    result[p_2].state &&
    !result[p_3].state && result[p_3].time === t + 60e3) {
  console.log("PASS: Test 5.");
} else {
  console.log("FAIL: Test 5.");
}

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