Algorithm 如果您知道发生了变化，如何推断未知变量的状态？

存在一个游戏，其状态由

n

布尔变量

p_1

，

p_2

，…，

p_n

量化。假设

0根据@NicoSchertler的建议，我提出了一个解决方案，通过基于一系列
观测值
和
假设
创建一组状态来处理不确定性。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。我们能够做出假设，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（尚未）的情况

每个变量均为
true
的情况对应一个启动状态。当提供一个
假设
时，通过假设每个未观察到的变量都是
假设
的主体，可能会产生多个（
n
）后续状态。导致矛盾的继承国被抛弃。当提供
观察时
，将为每个当前状态生成一个后续状态。任何导致矛盾的状态都将被丢弃。通过这种方式，假设和观察序列将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）。如果每个变量在所有状态中具有相同的状态，则将组合这些状态并返回一个结果，该结果将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

下面是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;
  }
};

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.");
}

我建议将该过程建模为一组假设。对于每个通知，创建变量已更改的假设，并将其与现有假设相结合。检查hyptohesis的有效性应该是直接的（对于每个假设，跟踪你知道变量状态的时间间隔）。如果你发现一个无效的，只需删除它。这可能会产生成倍多的有效假设（但这就是结果）。如果您只需要一个解决方案，那么通过假设空间进行深度优先遍历（基本上是回溯）可能是一个好主意。并且可能会根据通知的时间对其进行排序。我认为这可能会奏效。我想象一棵树，其中级别
I+1
的节点对应于
I
通知后的可能状态。边缘是假设。我的想法是，通知在“收到时”进行处理，所有这些都是对当前状态进行评分的重要因素，因此我认为只需要最低级别的通知。在一个新的假设或观察中，相互矛盾的叶子可以被剪掉，剩下的叶子可以产生后续的状态。对于可能的状态数而言，最坏的情况不是将是
n^n
？是的，最坏情况的复杂性是n_变量^n_通知。但在最坏的情况下，这是有效解决方案的数量。所以你再好不过了。除非您只需要一个有效的解决方案。那么，跟踪所有有效的部分解决方案是没有必要的。在我的例子中，我感兴趣的是将各州合并在一起，看看它们都同意什么。我发布了我的解决方案，作为受您建议启发的答案。谢谢你的帮助！