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A closed expression refers to a formula which has no free variables [1]. This is also called sentence. In a logic system you have a set of axioms which are sentences and rules which state how to derive a sentence from this [2]. If a sentence can be derived from the axioms, this means that the axioms entail this sentence. If a sentence is not derivable, it is ...


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I want to preface this by saying that the distinction is not clear. Nevertheless, I'll tell you what I know about this, and I will attempt to make the further clarification: The Structure of rule-based agents is: Take input from environment, pass through condition-based rules, and perform the action through actuators or anything which creates some ...


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Mostly because traditional "knowledge based systems" are based purely on deductive logic and that's just the way deduction works. It only deals with what consequences must follow from the premises. Traditionally these systems didn't deal with probabilistic knowledge or other less strict forms of reasoning, like abduction. That said, it is possible to ...


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I don't think that the "try all the numbers" approach is very representative, because I'm not sure whether or not the agent that uses that approach can be considered by any means AI. There is no "intelligence" in just checking numbers to try to prove the statement. An agent that is considered to be intelligent should apply a more intelligent approach. This ...


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The statement is "for all $x$, there exists a value of $y$ such that for all $z$, $z\neq y$ if and only if $z \neq f(x)$". This can be simplified: $$\begin{align} & & \forall x \exists y \forall z (z\neq y \iff z \neq f(x))\\ &\implies & \forall x \exists y \forall z (z=y \iff z = f(x))\\ &\implies & \forall x \exists y \forall z (...


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I might misunderstand your question, but there seem to be different levels of logic at play here. Computing logic, whereby any computational process is based on processor logic. In this case, any computing is involving logic, as boolean logic drives any processing. Linguistic logic, where there is a logic in the sequencing of sentences within a text. A ...


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I think the answer here lies in that the dictionary definition of randomness you have is not the one used in statistics, ML, or mathematics. We define randomness to mean there exists a distribution with generally greater than 0 uncertainty. Depending on who you talk to, we live in a random universe (the way we define quantum mechanics depends on a wave ...


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First, you need to consider what are the "parameters" of this "optimization algorithm" that you want to "optimize". Let's take the most simple case, a SGD without momentum. The update rule for this optimizer is: $$ w_{t+1} \leftarrow w_{t} - a \cdot \nabla_{w_{t}} J(w_t) = w_{t} - a \cdot g_t $$ where $w_t$ are the weights at iteration $t$, $J$ is the cost ...


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I will first recapitulate the key concepts which you need to know in order to understand the answer to your question (which will be very simple, because I will just try to clarify what is given as a "definition"). In logic, a formula is e.g. $f$, $\lnot f$, $f \land g$, where $f$ can be e.g. the proposition (or variable) "today it will rain". So, in a (...


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I'll take a shot at answering this, though I'm no expert in Neural Nets or Deep Learning. Given that practical thought vectors (TVs) don't yet exist, and may be impractical or impossible, I think answering your question will require a lot of conjecture and speculation. So here goes... For thought vectors to be useful in or outside NNs, the vector values ...


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According to the Wikipedia entry on Uniqueness Quantification your lecturer is correct. There is no size requirement expressed in the FOL expression. The point about the implication is that it can be true if the antecedent is false. So, there is a house in area1 (which we call x). And all houses in area1 which are smaller than 200 are the same as x. But if ...


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Some of the work on descriptive logics and modal logics was done within the context of artificial intelligence from a research funding perspective. Some was part of the normal academic apparatus of mathematics departments. Furtherance of these fields in the AI context has been hindered by historically low return on investment. Although first order logic, ...


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When we state in English that "All As are Bs", this means that we gain information as soon as we observe an A, we can immediately deduce that it must also be a B. These are the kinds of situations where we use an implication. So, this would be written in formal logic as: $$\forall X \left( A(X) \rightarrow B(X) \right)$$ When we state in English that "Some ...


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If I understand what you are asking, I think the simple answer would be that AI is nowhere near having demonstrated sentience, thus they do not qualify for any type of rights. We won't have to "cross this bridge" until an AI demonstrates self-awareness and human-level-or-beyond intelligence, but it sure is interesting to think about! (Also, the UDHR dates ...


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We usually optimize with respect to something. For example, you can train a neural network to locate cats in an image. This operation of locating cats in an image can be thought of as a function: given an image, a neural network can be trained to return the position of the cat in the image. In this sense, we can optimize a neural network with respect to this ...


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Let me add an example from machine learning that shows that resorting to randomness is the optimal way, sometimes. When working on the whole data is not tractable (computation cost, data does not fit in memory), working on random samples can be an optimal way to train a machine learning algorithm. One of the most used optimization technique in those cases ...


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Previous answers are very well written. I just wanted to supplement the thread by giving a simple example. The example shows how a logical function can be computed without errors using noisy components. Taken verbatim from Neural Networks by Raul Rojas. An excellent book: an example of a network built using four units. Assume that the first three units ...


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In certain games, random selection is the optimal strategy. See: Matching Pennies Strategy is essentially a plan of action utilized to achieve a goal. If random choice can be a strategy, it seems that it must be a form of logic, even if the nature of the stochastic process is counter to all forms of formal logic. This seems paradoxical, in that the ...


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This has to do with the fact that you can define arithmetic inside the axiomatic system or not. In description logic you cannot speak about arithmetic sentences and in first order logic you do. if you look at the proof of incompleteness you will understand this in depth. This demonstration depends on an arithmetic coding of statements, and this ...


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There are several problems with this, which is why people have been working on tasks like that for about 50 years without getting very far. As you rightly notice, it has been tried in restricted domains, where it works reasonably well. Reason being, there is less ambiguity. Human language is full of vagueness and ambiguity. We generally have few problems ...


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Artificial Intelligence at Google — Our Principles Objectives for AI Applications Be socially beneficial. Avoid creating or reinforcing unfair bias. Be built and tested for safety. Be accountable to people. Incorporate privacy design principles. Uphold high standards of scientific excellence. Be made available for uses that accord with these ...


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The provided result is a non-sense for this input. In a closed world paradigm (ie: prolog), where all non-provable facts are assumed as false, from "p2->p3 and p3->p1", p2 is false, and the program result should be "p2 ? No". In an open world, as it is not possible to proof "p2" nor "not p2", the result must be "p2 ? unknown" "p2 -> p3 and p3 -> p1" doesn'...


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If you prune k and L then you could miss the optimal solution. Assume L=9, if you prune L then the value of the tree is 8. If you don't prune L then the value of the tree is 9. Now I will try and address what I think your actual question is But no matter what, the decision of the max(root) will not change, the max will choose the right side no matter what ...


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This is a very dicey question. Logic functions can be thought of as mapping multiple inputs to a single output. Now each logic function create its own boundary. So if you are using a complex logical equation it is actually very hard to approximate the underlying function. Here I am treating the input Booleans as the input features. From practical experience:...


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Sequential programming would not be suitable for this kind of problem, but an algorithm could be implemented in a declarative programming language. I would suggest using Answer Set Programming, a language that is designed for logic axioms.


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Rule-based systems cover a wide range of systems. Some make use of boolean if/then/else rules, others may use weighting or even probabilistic inference. Some operate on frames, some on java objects, some on propositions that can be formulated in predicate logic. An example of a popular rule system is Drools. Some rule systems can be expressed as a subset of ...


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It seems that they are stating that a knowledge base is consistent if and only if it never asserts the truth of both the truth and the negation of a particular P. In other words, a knowledge base is consistent if it never contradicts itself. Their definition allows incomplete knowledge bases to be considered consistent; by their definition, an empty ...


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Excellent question! I'm currently working on the pre-Socratics as the most basic philosophies for first principles (These philosophers are intriguing for their simplicity and universality, and the "dawn of consciousness", in some conceptions, may be ascribed to the Classical Era. Linguistically, ancient Greek is fundamental to meaning in the West.) ...


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You need some sort of interpretation abstraction before your mathematical reasoning. While the text might read "123", you need to parse this into a literal of type Natural Number or Integer. Similarly, "x" could be a member variable. Then your deduction becomes, is literal 123 a Natural Number? Yes. As for the second statement, you should hopefully be able ...


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