This is the WebPPL version of the original Church Plural Predication model. As written, it does not run (the box below is shown statically; see the diagnosis after it).
// helper functions
// error function
var erf = function(x) {
var a1 = 0.254829592;
var a2 = -0.284496736;
var a3 = 1.421413741;
var a4 = -1.453152027;
var a5 = 1.061405429;
var p = 0.3275911;
var sign = x < 0 ? -1 : 1
var z = Math.abs(x);
var t = 1.0/(1.0 + p*z);
var y = 1.0 - (((((a5*t + a4)*t) + a3)*t + a2)*t + a1)*t*Math.exp(-z*z);
var answer = sign*y
return answer
}
// check array identity
var arraysEqual = function(a1,a2) {
return JSON.stringify(a1)==JSON.stringify(a2);
}
// get ERP probabilities
var erpProbs = function(ERP,support) {
var scores = function(val) {
return Math.exp(ERP.score(val))
}
return map(scores,support)
}
// KL divergence
var KL = function(P,Q) {
var diverge = function(xp,xq) {
return xp == 0 ? 0 : (xp * Math.log(xp / xq) )
}
return sum(map2(diverge,P,Q))
}
// wrapper for plural predication model
var pluralPredication = function(numberObjects,
collectiveNoise,
knowledge,
thingSaid
) {
var utterances = [
"ambiguous-pos",
"each-pos",
"together-pos"
];
// ambiguous utterance twice as likely
var utterancePrior = function() {
return categorical([2,1,1],utterances)
};
// possible object sizes
var objects = [3,4];
var objectPrior = function() {
return uniformDraw(objects);
}
// build states with n many objects
var statePrior = function(nObjLeft,stateSoFar) {
var stateSoFar = stateSoFar == undefined ? [] : stateSoFar
if (nObjLeft == 0) {
return stateSoFar
} else {
var newObj = objectPrior()
var newState = stateSoFar.concat([newObj])
return statePrior(nObjLeft - 1,newState)
}
}
// threshold priors
var distThetaPrior = function(){return objectPrior()};
var collThetaPrior = function(){return uniformDraw([9,10,11,12])};
// noise variance
var noiseVariance = collectiveNoise == "no" ? 0.01 :
collectiveNoise == "low" ? 0.75 :
collectiveNoise == "mid" ? 1 : 1.25
// x > theta interpretations
var collInterpretationPos = function(state, collTheta,noise) {
var weight = 1 - (0.5 * (1 + erf((collTheta - sum(state)) /
(noise * Math.sqrt(2)))))
return flip(weight)
}
var distInterpretationPos = function(state, distTheta) {
return all(function(x){x > distTheta}, state) ? true : false
}
// meaning function
var meaning = function(utt,state,distThetaPos,collThetaPos,isCollective,noise) {
return utt == "each-pos" ? distInterpretationPos(state,distThetaPos) :
utt == "together-pos" ? collInterpretationPos(state,collThetaPos,noise) :
isCollective ? collInterpretationPos(state,collThetaPos,noise) :
distInterpretationPos(state,distThetaPos)
}
var alpha = 7
var literal = cache(function(utterance,distThetaPos,collThetaPos,isCollective) {
Infer({method:"enumerate"}, function(){
var state = statePrior(numberObjects);
var noise = noiseVariance
condition(meaning(utterance,state,distThetaPos,collThetaPos,isCollective,noise));
return state;
})
});
var speakerBelief = cache(function(state,speakerKnows) {
Infer({method:"enumerate"}, function(){
var obs = function(s) {
return speakerKnows ? s : sum(s)
}
var bState = statePrior(numberObjects)
condition(arraysEqual(obs(bState),obs(state)))
return bState
})
})
var speaker = cache(function(state,distThetaPos,collThetaPos,isCollective,speakerKnows) {
Infer({method:"enumerate"}, function(){
var utterance = utterancePrior()
var bDist = speakerBelief(state,speakerKnows)
var lDist = literal(utterance,distThetaPos,collThetaPos,isCollective)
factor(alpha * (-1 *
KL(erpProbs(bDist,bDist.support()),
erpProbs(lDist,bDist.support())
)))
return utterance
})
});
var listener = cache(function(utterance,speakerKnows) {
Infer({method:"enumerate"}, function(){
var state = statePrior(numberObjects);
var isCollective = flip(0.5)
var distThetaPos = distThetaPrior();
var collThetaPos = collThetaPrior();
factor(
speaker(state,distThetaPos,collThetaPos,isCollective,speakerKnows).score(utterance)
);
return isCollective
});
});
return listener(thingSaid,knowledge)
}
print(pluralPredication(3,"no",true,"ambiguous-pos"))
print(pluralPredication(3,"no",false,"ambiguous-pos"))
print(pluralPredication(3,"low",true,"ambiguous-pos"))
print(pluralPredication(3,"low",false,"ambiguous-pos"))
print(pluralPredication(3,"mid",true,"ambiguous-pos"))
print(pluralPredication(3,"mid",false,"ambiguous-pos"))
print(pluralPredication(3,"high",true,"ambiguous-pos"))
print(pluralPredication(3,"high",false,"ambiguous-pos"))
Editorial note (2026): the port originally had two separate problems. First, objectPrior was missing a return, so every sampled state was [undefined, ...] and all paths had probability zero (this is now fixed above and below). Second, with that fixed, the strict-semantics model still fails in WebPPL: when distThetaPos is 4, no state of 3s and 4s satisfies the each reading, so the literal listener’s Enumerate has empty support, which WebPPL treats as a hard error while Church’s enumeration tolerated it. That is the problem the rest of this page discusses.
The problem is the deterministic semantics in the distributive interpretation, which creates zero-probability events (that Church, for some reason, can handle…):
var distInterpretationPos = function(state, distTheta) {
return all(function(x){x > distTheta}, state) ? true : false
}
To get the model to run, we need to weaken the semantics:
var distInterpretationPos = function(state, distTheta) {
return all(function(x){x > distTheta}, state) ? flip(0.99) : flip(0.01)
}
But with the noisy distributive interpretation, now the model predicts exactly the opposite pattern (from what we expected), namely an increase in collective interpretations as interpretation noise increases.
///fold:
// helper functions
// error function
var erf = function(x) {
var a1 = 0.254829592;
var a2 = -0.284496736;
var a3 = 1.421413741;
var a4 = -1.453152027;
var a5 = 1.061405429;
var p = 0.3275911;
var sign = x < 0 ? -1 : 1
var z = Math.abs(x);
var t = 1.0/(1.0 + p*z);
var y = 1.0 - (((((a5*t + a4)*t) + a3)*t + a2)*t + a1)*t*Math.exp(-z*z);
var answer = sign*y
return answer
}
// check array identity
var arraysEqual = function(a1,a2) {
return JSON.stringify(a1)==JSON.stringify(a2);
}
// get ERP probabilities
var erpProbs = function(ERP,support) {
var scores = function(val) {
return Math.exp(ERP.score(val))
}
return map(scores,support)
}
// KL divergence
var KL = function(P,Q) {
var diverge = function(xp,xq) {
return xp == 0 ? 0 : (xp * Math.log(xp / xq) )
}
return sum(map2(diverge,P,Q))
}
// wrapper for plural predication model
var pluralPredication = function(numberObjects,
collectiveNoise,
knowledge,
thingSaid
) {
var utterances = [
"ambiguous-pos",
"each-pos",
"together-pos"
];
// ambiguous utterance twice as likely
var utterancePrior = function() {
return categorical([2,1,1],utterances)
};
// possible object sizes
var objects = [3,4];
var objectPrior = function() {
return uniformDraw(objects);
}
// build states with n many objects
var statePrior = function(nObjLeft,stateSoFar) {
var stateSoFar = stateSoFar == undefined ? [] : stateSoFar
if (nObjLeft == 0) {
return stateSoFar
} else {
var newObj = objectPrior()
var newState = stateSoFar.concat([newObj])
return statePrior(nObjLeft - 1,newState)
}
}
// threshold priors
var distThetaPrior = function(){return objectPrior()};
var collThetaPrior = function(){return uniformDraw([9,10,11,12])};
// noise variance
var noiseVariance = collectiveNoise == "no" ? 0.01 :
collectiveNoise == "low" ? 0.75 :
collectiveNoise == "mid" ? 1 : 1.25
// x > theta interpretations
var collInterpretationPos = function(state, collTheta,noise) {
var weight = 1 - (0.5 * (1 + erf((collTheta - sum(state)) /
(noise * Math.sqrt(2)))))
return flip(weight)
}
var distInterpretationPos = function(state, distTheta) {
return all(function(x){x > distTheta}, state) ? flip(0.99) : flip(0.01)
}
// meaning function
var meaning = function(utt,state,distThetaPos,collThetaPos,isCollective,noise) {
return utt == "each-pos" ? distInterpretationPos(state,distThetaPos) :
utt == "together-pos" ? collInterpretationPos(state,collThetaPos,noise) :
isCollective ? collInterpretationPos(state,collThetaPos,noise) :
distInterpretationPos(state,distThetaPos)
}
var alpha = 7
var literal = cache(function(utterance,distThetaPos,collThetaPos,isCollective) {
Infer({method:"enumerate"}, function(){
var state = statePrior(numberObjects);
var noise = noiseVariance
condition(meaning(utterance,state,distThetaPos,collThetaPos,isCollective,noise));
return state;
})
});
var speakerBelief = cache(function(state,speakerKnows) {
Infer({method:"enumerate"}, function(){
var obs = function(s) {
return speakerKnows ? s : sum(s)
}
var bState = statePrior(numberObjects)
condition(arraysEqual(obs(bState),obs(state)))
return bState
})
})
var speaker = cache(function(state,distThetaPos,collThetaPos,isCollective,speakerKnows) {
Infer({method:"enumerate"}, function(){
var utterance = utterancePrior()
var bDist = speakerBelief(state,speakerKnows)
var lDist = literal(utterance,distThetaPos,collThetaPos,isCollective)
factor(alpha * (-1 *
KL(erpProbs(bDist,bDist.support()),
erpProbs(lDist,bDist.support())
)))
return utterance
})
});
var listener = cache(function(utterance,speakerKnows) {
Infer({method:"enumerate"}, function(){
var state = statePrior(numberObjects);
var isCollective = flip(0.5)
var distThetaPos = distThetaPrior();
var collThetaPos = collThetaPrior();
factor(
speaker(state,distThetaPos,collThetaPos,isCollective,speakerKnows).score(utterance)
);
return isCollective
});
});
return listener(thingSaid,knowledge)
}
///
print(pluralPredication(3,"no",true,"ambiguous-pos"))
print(pluralPredication(3,"no",false,"ambiguous-pos"))
print(pluralPredication(3,"low",true,"ambiguous-pos"))
print(pluralPredication(3,"low",false,"ambiguous-pos"))
print(pluralPredication(3,"mid",true,"ambiguous-pos"))
print(pluralPredication(3,"mid",false,"ambiguous-pos"))
print(pluralPredication(3,"high",true,"ambiguous-pos"))
print(pluralPredication(3,"high",false,"ambiguous-pos"))
Curiously, if I weaken the semantics in the original Church model, I get the same (unexpected) qualitative predictions from the WebPPL model.
So, something Church does with handling zero-probability events got us our original results. The hope is that if I can understand what that thing is, we can start to understand how the model ought to be structured.