Quiz 2 Example Questions PDF

Explain Cantor’s Diagonalisation method and how it proves that real numbers are not enumerable. Suppose the set of real numbers (ℝ) is enumerable, meaning we can list all real numbers between 0 and 1. Construct a new number by tweaking the diagonal digits of a list of real numbers. For instance, if...

Explain Cantor’s Diagonalisation method and how it proves that real numbers are not enumerable. Suppose the set of real numbers (ℝ) is enumerable, meaning we can list all real numbers between 0 and 1. Construct a new number by tweaking the diagonal digits of a list of real numbers. For instance, if a digit is 0, make it 1, otherwise make it 0. This new number will differ from each number in the list at some position, meaning it’s not on the list. This is a diagonal contradiction, proving that the real numbers are not enumerable. How does Cantor’s Diagonal Argument relate to Turing’s concept of computability? Cantor’s Diagonal Argument proves that the set of real numbers is not enumerable by constructing a number that differs from each in a list at some position. Turing used similar reasoning to show that not all real numbers are computable by a Turing Machine. Specifically, Turing Machines can only handle a subset of numbers (the computable ones), while many real numbers cannot be computed algorithmically. This underscores that certain types of problems (e.g., generating every real number) are beyond the capabilities of mechanical computation. Discuss Richard’s Paradox about definable numbers and responses to it. (What is your view on this?) English phrases are enumerable. Suppose we list all meaningful English phrases (by length and alphabetical order) that describe a decimal number between 0 and 1. Using Cantorian tweaking, we can construct a diagonal number that differs from every number in the list. This number cannot be in the list so it cannot be defined by any English number. Yet, in defining how to construct it, we have done it in English. A paradox! The notion of a "meaningful English phrase" that succeeds in specifying a number is not well-defined. There is also the vicious circle principle whereby defining a number by referencing a collection of which it is a part leads to contradictions. This paradox reflects the limitations of language and systems when describing infinities or defining sets. They demonstrate how self-reference can lead to contradictions, a recurring theme in logic and computation. Discuss consistency, completeness and decidability (and soundness). A logical system is consistent if it is not possible to derive a contradiction within it. That is, there is no formula φ such that both φ and ㄱφ are provable within the system. A logical system is syntactically complete if for any well-formed formula φ, either φ or ㄱφ is provable within the system. A system is semantically complete if every semantically valid formula is provable within the system. A logical system is decidable if there exists some “algorithm” that can determine, in a finite time, whether any given formula is provable within the system. A logical system is sound if it is not possible to prove falsehoods within the system. A rule of inference is sound if it cannot lead from true premises to a false conclusion. Explain the Entscheidungsproblem and Turing’s resolution of it. By following through the logic of the Richardian would-be paradox regarding computable numbers, Turing discovered an uncomputable decision problem. Hilbert’s “Decision Problem” asks whether there is an effective method for calculating whether any given formula of predicate logic is provable or not. To answer this, Turing focused on a property of computing machines which is easy to represent in predicate logic: “There can be no machine E which, when supplied with the standard description of an arbitrary machine M, will determine whether it prints a given symbol.” This can be proven by reductio ad absurdum. If E were possible, then a diagonally impossible machine (a “circle-free” testing machine) would also be possible. A contradiction! Turing illustrates how G, the supposed machine that can decide whether M prints “0”, can be constructed from E. Variations of M (M1, M2, etc.) modify the number of “0”s printed by M by replacing the first few “0”s with another symbol. Then, machine G uses E to test each of these modified machines in sequence. G will output a “0” if and only if E determines that a machine will not print “0”. G will print “0” an infinite number of times, unless M prints “0” infinitely, in which case G will never print “0”. We can now test G itself using E to find out whether it ever prints “0”. That is, we have a test for whether or not M prints “0” infinitely often i.e. whether or not M is a circle-free machine. If such a machine E could exist, we could determine whether a machine is circle-free. Since this is impossible, no machine E can exist to decide this question. Entscheidungsproblem is unsolvable because it’s impossible to determine if a formula is provable, just like it’s impossible to determine if a machine is circle-free. Discuss the Turing Machine and how it works. The Turing Machine is a simple abstract machine that manipulates symbols on a tape based on a set of rules. It consists of tape divided into squares where symbols can be printed or erased; a read/write head that can move left or right, print or erase any symbols and read them; a set of states that is the machine’s active memory; instructions in the form of a “Machine Table” which tells the machine what to do in each possible circumstance. The Turing Machine starts in the initial state, with the read/write head positioned on the leftmost cell of the input on the tape. Based on the current state and the symbol under the read/write head, the machine looks up the appropriate instruction in the machine table. The machine then: writes a symbol on the tape, moves the head left or right and changes its state. This process repeats, with the machine continually reading from the tape, modifying the tape, and moving the head according to the rules in the machine table. The machine stops when it reaches a halting state. The symbols remaining on the tape at this point represent the output of the computation. Discuss the Turing Machine as a model for human computation. Turing described computation as the process of writing symbols on paper (or a tape divided into squares). The computer is viewed as a person following instructions, with an operation performed determined by the state of mind of the computer and the observed symbols. In particular, they determine the state of mind of the computer after the operation is carried out. The machine simulates human calculation by executing simple steps like reading, writing, and changing states. Discuss how to construct a paradox of computable numbers. A computable number is one whose digits can be generated by a machine following a set of instructions. Computable numbers do not include all definable numbers i.e. a definable number may not be computable. Suppose there is some systematic way of defining real numbers between 0 and 1 by specifying methods for computing them, calculating and printing their digits. If these methods are enumerable, they can be listed and ordered, allowing us to compute the digits of each number. By applying Cantor's diagonalization, a new number can be constructed by altering the nth digit of the nth number in the list. This newly created diagonal number would differ from every number on the list in at least one digit, meaning it cannot be part of the original set. This leads to a paradox: while the diagonal number should be uncomputable (since it isn't on the list), the process of constructing it suggests it is computable. Discuss the Church-Turing thesis. Turing proved his machines equivalent to Church’s lambda calculus and Gödel’s general recursion, thus establishing the Church-Turing Thesis. Any function that can be computed by a human following a set of instructions can also be computed by a Turing machine. If something is impossible for a Turing Machine, it’s impossible for any other computer or algorithm. This thesis essentially defines the limits of what can be computed. While it remains unproven, it forms the basis of much of modern computational theory. Discuss the Universal Turing Machine and where the paradox breaks down. The Universal Turing Machine (U) can simulate any other Turing Machine (M). U reads M’s encoded description and mimics its operations, generating the same sequence of digits - and the same computable number - as M would. A paradox arises when U tweaks the diagonal digit of the number it computes, seemingly generating an uncomputable number. However, this paradox breaks down because it is impossible to systematically determine if a machine is circle-free. Discuss the implications of Turing's work on the Halting Problem. Turing’s work led to the Halting Problem, which asks if there is a method to determine whether a computer program will halt or run indefinitely. Turing proved by contradiction that no such method exists. If a program could decide if it halts or loops, it would result in a paradox: a program that halts if it doesn't and loops if it does. Thus, the Halting Problem is undecidable. Discuss the Turing Test and its interpretations. The Turing Test measures intelligence based on a machine’s ability to engage in conversation that is indistinguishable from that of a human. The machine has to try and pretend to be a man, by answering questions put to it, and it will only pass if the pretence is reasonably convincing. A considerable proportion of a jury, who should not be expert about machines, must be taken in by the pretence. The Jury’s questions can be used to elicit the computer’s “knowledge” about almost any of the fields of human endeavour. The machine would be permitted all sorts of tricks so as to appear more man-like, such as waiting a bit before giving the answer, or making spelling mistakes. Each jury has to judge quite a number of times, and sometimes they really are dealing with a man and not a machine. This will prevent them saying ‘It must be a machine’ every time without proper consideration. Turing restricts his test to digital computers (machines that can carry out any operations which could be done by a human computer). He discusses what an abstract digital computer could do in principle. The “extreme exemplar” interpretation: The Turing Test could be seen as an existence proof of the possibility of an intelligent machine, given the universality of digital computers. If a machine could handle complex conversation across a wide range of topics with human-like sophistication, it would be unreasonable to withhold the label of intelligence. This interpretation provides a reasonable criterion for machine intelligence, where the machine’s behaviour, not its resemblance to human thought, determines its intelligence. Discuss the argument of consciousness against the Turing Test and Turing’s response. This argument asserts that true intelligence requires consciousness, which machines cannot possess. Turing responds that this is a solipsistic view and that we cannot judge consciousness in others either, so we should focus on external behaviour. The argument entails two different lines of thought, which Turing would have done well to distinguish. Jefferson denies the validity of the Turing Test because it does not test for genuine consciousness, and genuine consciousness (rather than artificial signalling) is necessary for intelligence. Artificial signalling of apparent emotions is unworthy of being deemed intelligent because it is an easy contrivance. Turing would have been better to say: if a machine can mimic human behaviour convincingly enough, there would be reason to call it intelligent irrespective of whether or not it has genuine feelings; intelligence need not require consciousness. Discuss the merits of the Turing Test. The Turing Test is a practical way to assess machine intelligence. It measures whether a machine can convincingly imitate a human in conversation. Turing’s test avoids defining “thinking” and instead focuses on performance. Avoiding anti-machine bias At first glance, the Turing Test seems biased in favour of human intelligence because a machine has to do the difficult task of imitating human behaviour to pass. However, Turing designed the test to avoid anti-machine bias by the interrogator. While humans might easily pass, the test is more about ensuring that machines aren’t unfairly judged for their non-human characteristics. The goal is to focus on whether machines can imitate intelligent behaviour, not whether they think in the same way humans do. Passing as a Sufficient Criterion only Turing suggested that passing the Turing Test is sufficient to prove intelligence but not a necessary condition for it. In other words, a machine doesn’t need to imitate humans perfectly to be intelligent. This distinction acknowledges that machine intelligence could be different from human intelligence. A machine might not pass the Turing Test but still exhibit intelligent behaviour in other ways, such as solving complex problems or learning. Fair Play for the Machines Turing advocates for fair play for the machines. Machines shouldn’t be judged more harshly than humans. For example, humans make mistakes, and machines should be allowed to as well, especially when trying new techniques. This idea is essential to ensuring that machines are judged on their abilities rather than on human-centric expectations. If machines are expected to be flawless, they will always fail the intelligence test. Irrelevance of Human Imitation The role of human imitation in Turing’s setup is not so much to make that a criterion of intelligence, as to ensure a fair test that focuses on the relevant behaviour and prevents the interrogator from judging by prejudice about the agent. With an unbiased interrogator, perhaps human-likeness need not be a criterion of success. Discuss the problems of the Turing Test. Unreliable Criterion Fooling an average interrogator is relatively easy to achieve, but not by techniques that plausibly involve genuinely intelligent information processing. Joseph Weizenbaum’s ELIZA demonstrated that simple pattern recognition can simulate conversational intelligence, fooling people into believing they are talking to a human therapist. So, fooling “an average interrogator” 30% of the time in a 5-minute casual conversation is not enough. Test is too Easy or too Difficult Turing acknowledged that passing his Test wasn’t a necessary condition for intelligence. But unless interpreted fairly rigorously, it’s not sufficient either, because it’s too easy to pass due to human lack of critical discernment! But if interpreted rigorously, the test is too demanding to be useful: indistinguishability from human conversation isn’t a sensible AI goal. Passing the Test by Brute Force Machines could theoretically pass the test by brute force, storing vast amounts of conversational data without actually understanding the content, as illustrated by the “Blockhead” thought experiment. Such a mindless machine, working by such a crude mechanism, surely cannot be labelled as intelligent. How does the Tutoring Test differ from the Turing Test, and why might it be a better measure of intelligence? The Tutoring Test differs from the Turing Test in that it fon the revelation of information processing in a teaching task in a specific domain rather than imitating human conversation. There is no need to pretend, and no expectation of indistinguishability. It is a better measure of intelligence as it assesses intelligence in action, rather than relying on deception. It provides a clearer measure of a machine's capability to handle complex knowledge and reasoning tasks, which is more relevant to real-world AI applications. Discuss Blockhead. Ned Block attacked the Turing Test on the grounds that it can be passed by a mindless machine. “Blockhead” works by storing every possible “sensible” conversation of a given length, and choosing its responses accordingly. Something working so mindlessly, by a mechanism that involves no understanding, surely cannot be called intelligent, but it will always generate sensible conversation! Blockhead emphasises that mere behavioural performance is not enough for true intelligence. Our “intuitive” judgments in response to Blockhead suggest that our notion of intelligence involves not just impressive behaviour, but impressive behaviour generated “cleverly” with limited resources. Thought-experiments can float free of any plausible reality, but in practice, appropriately flexible and timely responses to a rapidly changing situation are achievable only by clever processing. Both Block and Turing’s thought experiments postulate an algorithmic system capable of generating conversation that is indistinguishable in quality from that of an intelligent native speaker; But they draw opposite conclusions, by focusing on different aspects of the situation and appealing to contrary “intuitions”. Discuss the relevance of thought experiments. Classical intuition pumps are designed to illuminate one thing by harnessing our familiar understanding of something else. We are encouraged to see the one thing as relevantly similar to the other. Likewise, thought experiments serve as intuition pumps that encourage us to consider the implications of a hypothesis or theory. However, many of our concepts are open-textured: it is not clear in advance how we would apply them to all possible cases. When revising our concepts, we have every right to ignore crazy thought experiments as open-texture cuts in two directions. Firstly, we cannot expect our concepts to be prepared in advance for all new eventualities: they may have to be revised or “tuned” to new contexts. Secondly, we needn’t accept any requirement, when re-tuning concepts, to make them immune to future revision: we don’t have to take all future possibilities – let alone all logical possibilities – into account. “Intuition pumps” can tempt us into misleading or simplistic comparisons, based on familiar “intuitive” assumptions and drowning out unexpected differences. Practical confrontation with new realities can be far more vivid, and far more surprising, forcing us to seriously revise our thinking by making it impossible to ignore things that are unintuitive. Real novelty beats imagined similarity. Discuss the aim of Searle’s Chinese Room. There is a theoretical error in supposing that a conversational AI system – however excellent its performance – could properly be considered “intelligent”, on the grounds that its internal processing fails to have an genuine understanding or semantic grasp of the topics that it purports to be discussing. In Searle’s scenario, a man inside a room receives cards with questions written in Chinese. Using rule books, he manipulates the symbols on the cards based on their shape and structure (syntax), producing responses that seem meaningful to the outside observer. However, the man has no comprehension of the meaning (semantics) of the symbols he processes. Programs, like the rules followed by the man in the room, are purely formally specifiable i.e. they have no semantic content. Computers, like the man in the room, may follow syntactic rules without truly understanding the content they process. Thus, Searle argues that mere syntactic manipulation of symbols cannot invest them with semantic content or genuine intentionality. He denies that digital machines can have intentionality (genuine understanding or semantic content) and argues that they lack mental states, consciousness, or cognitive processes. Contrary to Turing, “strong AI” is impossible: digital computers cannot think. Searle’s argument is a strong challenge to computational theories of mind. Understanding might involve more than algorithmic symbol manipulation — it may require consciousness or intentionality, a realm that machines have yet to reach. Discuss the objections to Searle’s Chinese Room. The System Reply Although the man in the room does not understand Chinese, the system of which he is a part does (as shown by its intelligent responses). Searle responds that there is no way that the system can get from the syntax to the semantics. The man, the central processing unit, has no way of figuring out what any of these symbols means; but then neither does the whole system. Copeland counters this: As a matter of logic, the man’s lack of understanding does not prove that the system of which he is a part does not understand Chinese. This would be like claiming that because a single employee doesn't sell pyjamas, the entire company doesn’t either. The Robot Reply If the system were embedded in the real world, like in a robot that can sense and act, it might develop semantic understanding. Searle responds that the man in the room has no understanding of any such inputs and outputs. As long as we suppose that the robot has only a computer for a brain, there is still no way of getting from the syntax to the semantics of Chinese. What are some possible responses to Searle? Intelligence Without Intentionality Even if we agree with Searle that even the most powerful LLM has no intrinsic, genuinely-system-understood, intentionality, we might still insist that its processing of information is “intelligent”, on the basis of its impressive processing of relevant information even if it exhibits only “derived intentionality” (in line with the “extreme exemplar” interpretation). Intrinsic Intentionality in Abstract Domains Searle’s denial that syntactic processing can generate intrinsic intentionality is more convincing when applied to physical objects, but less so when considering “thought” about abstract entities. In these cases, systems like a chess-playing program, while not conscious, respond appropriately to logical relations. This suggests they exhibit a form of intelligence based on their responsiveness to abstract structures. Going “Concrete” through Robotic Interaction? (Robot Reply) If we believe intentionality requires interaction with the physical world, robotic AI can literally “reach out” semantically to things in the world by detecting them and affecting them. If its symbolic manipulations are thus responsive to real things, and interact with them accordingly, doesn’t that achieve intrinsic intentionality? For example, a robotic crane that intelligently cuts trees seems to have real intentionality: its internal states are responsive to physical things, and impact causally on them. Combining Robot and System Replies Suppose we agree with Searle that the man in the Chinese cabin has no “semantic” grasp of what is going on – no idea that he is controlling a robotic lumberjack crane. As Copeland insists, this does not imply that the system as a whole lacks “semantics” – that would require another argument. It is also plausible that the information processing of the crane system achieves “semantic content” through its relationship to the sensors and motors: the man’s unawareness of this is irrelevant. What do Machines Lack? Searle will deny that the robotic lumberjack’s internal states have “semantic content”, despite having these real-world causal relations. But what do they lack? His Chinese-Room-style arguments suggest “understanding” in the sense of conscious awareness is the crucial factor. Discuss the relation between intelligence and consciousness. Turing's Perspective To support machine intelligence, Turing must take machine consciousness to be as reasonable as believing that other people are conscious. There would be reason to call the machine ‘intelligent’ irrespective of whether or not it has genuine feelings. Intelligent thinking need not require consciousness (nor even – contra Searle – potential consciousness). Turing emphasised that the focus should be on behaviour rather than internal states. This view aligns with the "extreme exemplar" interpretation of intelligence, where behaviour serves as a primary criterion for assessing intelligence, independent of consciousness. Searle's Argument Without a conscious awareness of semantics, a machine's intelligent behaviour is merely a product of syntactic manipulation, lacking true understanding or intentionality. Thus, consciousness is a crucial factor in evaluating intelligence. Potential for Non-Conscious Intelligence For instance, machine learning systems, including large language models (LLMs), can generate impressive outputs based on patterns in data without any conscious comprehension of their content. Can we attribute intelligence to systems that demonstrate sophisticated processing capabilities without consciousness? If we distinguish sophistication of information processing from phenomenology, then it’s clear that intelligence is far more a measure of the former than the latter. In our new world of unconscious – but highly sophisticated – information processors, it makes sense to allow our concept of “intelligence” to evolve accordingly. Are machines conscious? Consciousness affects our behaviour – it is a causally active process. This is both evident, and evolutionarily required. Although the behaviour of computers can be very complicated, their fundamental causal processes are extremely well understood, and abstract away from any physical “mysteries”. So there is no room for some other causally active process to make a difference to how they behave. We do not understand our consciousness, but we do understand enough about how computers work to rule out their being conscious. Discuss computer machinery and intelligence. The Turing Test provides a context in which a machine’s intelligence can be judged without bias, on the basis of performance alone. Under the “extreme exemplar” interpretation, the aim is to make a point about the potential performance of imagined machines, rather than actual performance. It’s an interesting question whether, thus understood, Large Language Models have passed the threshold of deserving the accolade of “intelligence”. This issue is best separated from that of consciousness, accepting that our concepts might need to change in the light of new realities. We can (and perhaps should) accept the possibility of genuine machine intelligence, without being drawn into fantasies about conscious machines.