algorithm - 鉴于您知道发生了某些变化，如何推断未知变量的状态？

Question

存在一个游戏，其状态由 n 量化 bool 变量 p_1 , p_2 , ..., p_n .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)。

一次只能观察变量的一个子集。当变量的状态从 true 发生变化时，我们总是会收到通知至 false , 但只有在观察到相关变量时，我们才会知道哪个变量发生了变化。

如果一些p_i从 true 更改状态至 false在时间t , 然后它将状态从 false 改回至 true在时间t + 60 .

以下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。一个变量改变状态，通过观察它是 p_1 .我们知道 p_1将在 t + 60 改变状态.
• p_1和 p_2是可观察的。变量改变状态。应该推断是p_3 .我们知道 p_3将在 t + 60 改变状态.
• p_1是可观察的，并且已知 p_2状态更改为 false在时间t - 30 .变量改变状态。它不是 p_1通过观察和p_2只会将状态更改为 true在 t + 30 ，因此它是 p_3 .
• 没有可观察到的变量已知p_1状态更改为 false在时间t - 30 , p_2状态更改为 false在 t - 75 , 和 p_3在 t - 80 处将状态更改为 false .变量改变状态。在 t + 1 p_2观察到值为 true .因为p_1还是false , 和 p_2已被观察到 true , 应该推断出 p_3是false , 并将状态更改回 true在 t + 60 .
• p_1和 p_2是可观察的。众所周知p_3状态更改为 false在时间t - 70 .没有变量改变状态。应该推断出p_3是真的，因为它会从 false 改变至 true在 t - 10 , 如果它从 true 改变了至 false ，我们会被告知变量的状态变化。

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

• 它是否可见并且没有改变状态
• 它的最后一次状态变化是否已知并且在过去还不够远以至于它已经变回来了。

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，变量可能已经改变状态(返回，从false 到 true ), 等等。那么如果在这个范围内只有一个变量变化事件，那么它应该是该变量对应的事件。结果太麻烦了，没继续。每一行都是一个负担，因为状态的可能组合似乎太多了，我不会考虑任何事情。

考虑到问题陈述，上述方法是否合理？是否有一种通用方法可以对此类问题进行建模，从而更优雅地解决这些问题？

Answer 1

根据@NicoSchertler 的建议，我提出了一种解决方案，它通过根据一系列Observations 和Hypotheses 创建状态集来处理不确定性。观察是特定(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。此解决方案不处理开始状态未知(尚)的情况。

只有一个起始状态对应于每个变量都为 true 的情况。当提供一个hypothesis 时，可能会生成多个 (n) 后继状态，方法是假设每个未观察到的变量都是 hypothesis 的主题。导致矛盾的后继状态被丢弃。当提供 observation 时，将为每个当前状态生成一个后继状态。任何导致矛盾的状态都会被丢弃。以这种方式，假设和观察的顺序导致变量可能处于的一组可能状态。

为了我的特定目的，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)。合并状态并返回一个结果，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。

给定 n 状态和 m 通知，最坏情况下的复杂度是 n^m 这可能非常有限，但它应该是适合我的有限应用。

这里是JavaScript实现和测试代码。

solver.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;
  }
};

test-solver.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.");
}