Assignment # 1. Farrukh Jabeen COMP 581 Mathematical Methods in AI Due Date: September 7, 2009

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1 Farrukh Jabeen COMP 581 Mathematical Methods in AI Due Date: September 7, 2009 Assignment # 1 1. a) What is the "Boolean Satisfiability Problem (SAT)"? b)why is this an important problem in the theory of algorithms? Explain your answers and give supporting examples. Answer: Satisfiability is the problem of determining if the variables of a given Boolean formula can be assigned in such a way as to make the formula evaluate to TRUE. Equally important is to determine whether no such assignments exist, which would imply that the function expressed by the formula is identically FALSE for all possible variable assignments. In this latter case, we would say that the function is unsatisfiable; otherwise it is satisfiable. To emphasize the binary nature of this problem, it is frequently referred to as Boolean or propositional satisfiability. The shorthand "SAT" is also commonly used to denote it, with the implicit understanding that the function and its variables are all binary-valued. The Boolean satisfiability(sat) problem plays a central role in combinatorial optimization and, in particular, in NP completeness. Any NP completeness, such as the max cut problem, the graph coloring problem, and the TSP, can be translated in polynomial time into a SAT problem. The SAT problem plays a central role in solving large-scale computational problems, such as planning and scheduling, integrated circuit design, computer architecture design, computer graphics, image processing and finding the folding state of protein. Propositional satisfiability (SAT) is a prototypical combinatorially hard problem which is as expressive as a general constraint satisfaction problem. The importance of SAT in artificial intelligence is clear from several practical applications and from several workshops, international competitions, and special issues of journals devoted to this problem. The practical applications of SAT include circuit design, diagnosis and verification, job scheduling and planning. Several combinatorial problems have been encoded as SAT and solved efficiently. These include map coloring and Steiner tree finding. Classical planning problem can be solved efficiently, when encoded and solved as a SAT problem. Local search-based methods have gained a significant attention in the last decade. These are iterative improvement methods. They are incomplete. These methods have solved many SAT and constraint satisfaction problems (CSPs) orders of magnitude faster than the global searchbased methods. Local search methods start with an initial assignment to variables and choose assignments that maximize the increase in the number of constraints satisfied or increase the number of satisfied constraints. The methods often choose assignments that do not lead to any improvement either to do simulated annealing or escape from local maximum. The next assignment which improves the value of the score function is generally chosen by examining several candidate assignments. These candidates are obtained by assigning different values to the same variable or assigning different values to multiple variables. In both cases, any candidate assignment

2 differs from the current assignment in the value of only one variable. Several local search-based methods have been reported. All of these generate candidate assignments by changing the value of only one variable. For example, given an n-variable SAT instance, n candidate assignments can be generated, each by flipping the value of one variable. Flipping the values of multiple variables on an iteration to generate the next assignment can be useful for several reasons. Multiple flips can avoid local maximum. Evaluating the effects of multiple flips allows a solver to look ahead. Multiple flips can reduce the number of iterations needed to solve a problem and/or solving time. There are different formulations for the SAT problem, but the most common one(which we discuss next), consists of two components. A set of n Boolean variables{x 1,x 2,..x n }, representing statements that can either be TRUE(=1) or FALSE(=0). The negation (the logical NOT) of a variable x is denoted by x. For example, TRUE= FALSE. A variable or its negatation is called a literal. A set of m distinct clauses{c 1,C 2, C m ) of the form C i = z i 1 z i 2.z ik, where z s are the literals and the denotes the logical OR operator. For example 0 V 1=1 The binary vector X=(x 1,x 2.,x n ) is called a truth assignment, or simply an assignment. Thus,x i =1 assigns truth to xi and xi=0 assigns truth to xi for each i=1..,n. The simplest SAT problem can now be formulated as: find a truth assignment X such that all clauses are true. Denoting the logical AND operator by, we can represent the above SAT problem via a single formula as F 1 = C 1,C 2, C m ) Where the {Ck} consists of literals connected with only operators. The SAT formula is said to be in conjunctive normal form (CNF). An alternative SAT formulation concerns formulas of the type F 2 = C 1 C 2 C m Where the clauses are of the C i = z i 1 z i 2.z ik. Such a SAT problem is then said to be in disjunctive normal form(dnf). In this case, a truth assignment x is sought that satisfies at least one of the clauses, which is usually a much simpler problem. Example As an illustration of SAT problem and the corresponding SAT counting problem, consider the following toy example of coloring the nodes of the graph of the graph in following figure. Is it possible to color the nodes either black or white in such a way that no two adjacent nodes have the same color.? If so how many colorings are there?

3 Can the graph be colores with two colors so that two adjacent nodes have the same color? We can translate this graph coloring problem into a SAT problem in the following way : Let x j be the Boolean variable representing the statement the j-th node is colored black. Obviously, x j is either TRUE or FALSE, and we wish to assign truth to either x j or x j for each j=1..,5. The restriction that adjacent nodes cannot have same color can be translated into number of clauses that must all hold. For example node 1 and node 3 cannot both be black can be translated as clause C 1 = x 1 x3. Similarly, the statement at least one of node 1 and node 2must be black. Is translated as C 2 = x 1 x3. The same holds for all other pairs of adjacent nodes. The clauses can now be summarized as in the following table. Here in the left hand table, for each clause C i a 1 in column j means that the clause contains x j, a -1 means that the clause contains the negation x j : and a 0 means that the clause does not contain either of them. Now call the corresponding matrix A=(aij) the clause matrix. For example, a76=-1 and a42= =0. An alternative representation of the clause matrix is to list for each clause only the indices of all Boolean variables present in that clause. In addition, each index that corresponds to a negation of a variable is preceded by a minus sign A SAT table and an alternative representationn of clause matrix.

4 Now let x= (x 1,x 2.,x 5 ) be a truth assignment t. The question is whether there exists an x such that all clauses {C k } are satisfied. To see if a single clause C k is satisfied, one must compare the truth assignment for each assignment in that clause with the values 1,-1 and 0 in the clause matrix A, which indicates that if literal corresponds to the variable, its negation,or that neither appears in the clause. If for example x j =0 and aij=-1, then the literal x j is TRUE. The entire clause is TRUE if it contains at least one true literal. Define the clause value C i (x)=1 if clause C i is TRUE with truth assignment x and C i (x)=0 if it is FALSE. Then it is easy to see that C i (x) = max{0,(2xj-1)aij}, j assuming that at least one aij is nonzero for clause C i (otherwise, the clause can be deleted). For example, for truth assignment(0,1,0,1,0) the corresponding clause values are given in the rightmost column of the lefthand table in the Table. We see that second and forth clauses are violated. However the assignment(1,1,0,1,0) does indeed yield all clauses true, and this therefore gives a way in which the nodes can be colored 1=black, 2= black 3= white, 4=black, 5= white. It is easy to see that (0,0 1,0,1) is the only assignment that renders all the clauses true. The problem of deciding whether there exists a valid assignment, and indeed providing such a vector is called the SAT assignment problem. Finding a coloring in the above example is a particular instance of the SAT assignment problem. A SAT assignment problem in which each clause contains exactly K literals is called a KSAT problem. It is well known that 2-SAT problems are easy and can be solve in polunomial time, while K-SAT problems for k 3 are NP hard. A more difficult problem is to find the maximum number of clauses that can be satisfied by one truth assignment. This is called Max-SAT problem. 2. What is the "Frame Problem" in Artificial Intelligence? Explain your answer and give specific examples Answer: The term "artificial intelligence" was coined by John.While there does not appear to be any consensus about the definition of AI, we can perhaps categorize the various definitions under three headings: Strong AI AI has as its goal the creation of machines with conscious intelligence. Weak AI AI has as its goal the creation of machines which behave as if they were intelligent Experimental Cognition AI is the study of human intellectual capabilities through the use of mental models implemented on a computer. In artificial intelligence, the frame problem was initially formulated as the problem of expressing a dynamical domain in logic without explicitly specifying which conditions are not affected by an action. The name "frame problem" derives from a common technique used by animated cartoon makers called framing where the currently moving parts of the cartoon are superimposed on the "frame," which depicts

5 the background of the scene, which does not change. In the logical context, actions are typically specified by what they change, with the implicit assumption that everything else (the frame) remains unchanged. In another words we can say that, when we carry out an action, the world changes. In fact some aspects of the world change, but others stay the same. Determining which stay the same is known as the frame problem. An effect axioms states what changes when the robot carries out a particular action in a particular situation. It does not make any statement about what does not change. For example when the robot carries out a Take action, it does not find itself in a different room. This kind of rule can be expressed in a frame axiom, such as the following Many of the frame axioms are needed if we are to describe all of the effects that do not result from carrying out a particular action. The problem of having enormous numbers of frame axioms is known as representational frame problem. In any complex situation, we have to frame the problem the right way in order to achieve a solution. "Framing" the problem means identifying what is relevant and what is irrelevant, and determining which bits of knowledge in our extensive repository can be brought to bear to arrive at a solution. But to properly frame the problem, we must understand the problem. As human beings, we routinely solve complex problems. Our success in that process is so frequent and so seemingly effortless, in fact, that it is easy to take the ability for granted. It is only when we begin trying to capture that expertise in automated systems (e.g. robots), and try to give them the intelligence to deal with new situations, that we discover how truly remarkable that capability is. In all of these cases, people are effectively reorganizing their problem-solving heuristics in order to rapidly hit on the "best" problem-solving strategy in any given situation. Currently, of course, our robotic systems tend to be "frozen", than "self-organizing". Once programmed, the program tends to operate in exactly the same way as time goes -- the situations may change, and the results generated by the program may differ as a result, but the program never changes at all. So clearly, a program that could "organize itself" would have the ability to address the frame problem. In other words, it could get smarter over time. But that, in itself, does not solve the problem, because any one attempt to find a solution can still take an inordinately long time. The "Robot Example" illustrates that problem nicely. Robot, R1, is created. It's task is to go into a room and get it's power supply. It goes into the room, finds its power supply in a wagon, and pulls it out of the room. Unfortunately, there is also a bomb in the wagon. BOOM! The wagon and robot go up in smoke. That's not good, so a "Robot Decision-maker", R1D1, is constructed. It is able to think out the consequences of it's actions in advance, so it won't pull out the bomb with the wagon. The new robot goes into the room, and immediately engages it's deductive engines. It deduces that the floor is under the wagon, and the wagon is under the roof, so therefore the floor is under the roof. Then it starts deducing that...boom! The bomb goes off before the program can complete it's exhaustive set of deductions. That's not good either, so a "Relevancy-detecting Robot Decision-maker", R2D1 is produced. This time, the robot won't spend time on useless deductions, but will focus on what's important! So it, too, is told to enter the room and retrieve it's power supply. This version sits outside the room without moving for an awfully long time, until finally... BOOM! The power supply is destroyed by the bomb. After

6 debugging the output, it turns out that the program had figured out that the position of the floor wasn't relevant, and the color of the drapes was relevant, and that was about as far is got in the time available. At least this time the robot was saved, but the "analysis paralysis" induced by the attempt to determine relevancy prevented the robot from taking any sort of action. The robot we need to solve the problem, of course, is the "R2D2" -- a Relevancy-detecting Robot Decisionmaker that acts Decisively! frame problem suggests that R2D2 is impossible to construct. Impossible, that is, if R2D2 must stand or fail on his own.. 3. Explain the following methods of reasoning: a) deduction; b) induction (note: this is not mathematical induction, but inductive reasoning); c) abduction. Abduction means determining the precondition Consider this example. It is logical to assert that It rains; if it rains, the floor is wet; hence, the floor is wet. But any reasonable person can see the problem in making statements like: The floor is wet; if it rains, the floor is wet; hence, it rains.. In order to make inferences to the best explanation, the researcher must need a set of plausible explanations, and thus, abduction is usually formulated in the following mode: The surprising phenomenon, X, is observed. Among hypotheses A, B, and C, A is capable of explaining X. Hence, there is a reason to pursue A. Applications of abduction Abduction can be well applied to quantitative research, especially Exploratory Data Analysis (EDA) and Exploratory statistics (ES), such as factor rotation in Exploratory Factor Analysis and path searching in Structural Equation Modeling.The whole notion of a controlled experiment is covertly based on the logic of abduction. In a controlled experiment, the researchers control alternate explanations and test the condition generated from the most plausible hypothesis. However, abduction shares more common ground with EDA than with controlled experiments. In EDA, after observing some surprising facts, we exploit them and check the predicted values against the observed values and residuals.although there may be more than one convincing pattern, we "abduct" only those that are more plausible for subsequent controlled experimentation. Since experimentation is hypothesis-driven and EDA is data-driven, the logic behind them are quite different. The abductive reasoning of EDA goes from data to hypotheses while inductive reasoning of experimentation goes from hypothesis to expected data. By the same token, in Exploratory Factor Analysis and Structural Equation Modeling, there might be more than one possible way to achieve a fit between the data and the model; again, the researcher must abduct a plausible set of variables and paths for modeling building. In EDA, the role of the researcher is to explore the data in as many ways as possible until a plausible "story" of the data emerges. EDA is not fishing significant results from all possible angles during research: it is not trying out everything. There are millions of possible explanations to a phenomenon. Due to the economy of research, we cannot afford to falsify every possibility. We don't have to know everything to know something. By the same token, we don't have to screen every false thing to dig out the authentic one. During the process of abduction, the researcher should be guided by the elements of generality to extract a proper mode of perception. Deduction means determining the conclusion

7 Deduction involves drawing logical consequences from premises. An inference is endorsed as deductionaly valid when the truth of all premises guarantees the truth of conclusion. For instance, First premise: All the beans from the bag are white (True). Second premise: These beans are from this bag (True). Conclusion: Therefore, these beans are white (True). Limitations of deduction There are several limitations of deductive logic. First, deductive logic confines the conclusion to a answer (True/False). A typical example is the rejection or failure of rejection of the null hypothesis. Second, this kind of reasoning cannot lead to the discovery of knowledge that is not already embedded in the premise. In some cases the premise may even be tautological--true by definition. Brown (1963) illustrated this weakness by using an example in economics: An entrepreneur seeks maximization of profits. The maximum profits will be gained when marginal revenue equals marginal cost. An entrepreneur will operate his business at the equilibrium between marginal cost and marginal revenue. The above deduction simply tells that a rational man would like to make more money. There is a similar example in cognitive psychology: Human behaviors are rational. One of several options is more efficient in achieving the goal. A rational human will take the option that directs him to achieve his goal (Anderson, 1990). The above two deductive inferences simply provide examples that a rational man will do rational things. The specific rational behaviors have been included in the bigger set of generic rational behaviors. Since deduction facilitates analysis based upon existing knowledge rather than generating new knowledge, Josephson and Josephson (1994) viewed deduction as truth preserving and abduction as truth producing. Russell and Whitehead (1910) attempted to develop a self-sufficient logical-mathematical system. In their view, not only can mathematics be reduced to logic, but also logic is the foundation of mathematics. Mathematical logic relies on many unproven premises and assumptions. Statistical conclusions are considered true only given that all premises and assumptions that are applied are true. Deduction alone is a necessary condition, but not a sufficient condition of knowledge. Peirce (1934/1960) warned that deduction is applicable only to the ideal state of things. In other words, deduction alone can be applied to a well-defined problem, but not an ill-defined problem, which is more likely to be encountered by researchers. Nevertheless, deduction performs the function of clarifying the relation of logical implications. When well-defined categories result from abduction, premises can be generated for deductive reasoning. Induction means determining the rule. It is learning the rule after numerous examples of the conclusion following the precondition. Example: "The grass has been wet every time it has rained. Thus, when it rains, the grass gets wet." Scientists are commonly associated with this style of reasoning. It allows inferring some a from multiple instantiations of b when a entails b. Induction is the process of inferring probable antecedents as a result of observing multiple consequents. An inductive statement requires perception for it to be true. For example, the statement, "it is snowing outside" is invalid until one looks or goes outside to see whether it is true or not. Induction requires sense experience. Limitations of induction Hume (1777/1912) argued that things are inconclusive by induction because in infinity

8 there are always new cases and new evidence. Induction can be justified, if and only if, instances of which we have no experience resemble those of which we have experience. Thus, the problem of induction is also known as the skeptical problem about the future (Hacking, 1975). We never know when a regression line will turn flat, go down, or go up. Even inductive reasoning using numerous accurate data and high power computing can go wrong, because predictions are made only under certain specified conditions. Take the modern economy as another example. Due to American economic problems in the early '80s, quite a few reputable economists made gloomy predictions about the U.S. economy such as the takeover of American economic and technological throne by Japan. By the end of the decade, Roberts (1989) concluded that those economists were wrong; contrary to those forecasts, I n the 80 s the U.S. enjoyed the longest economic expansion in its history. In the 1990s, the economic positions of the two nations changed: Japan experienced recession while America experienced expansion. Induction suggests the possible outcome in relation to events in long run. This is not definable for an individual event. To make a judgment for a single event based on probability like "your chance to survive this surgery is 75 percent" is nonsense. In actuality, the patient will either live or die. In a single event, not only the probability is indefinable, but also the explanatory power is absent. Induction yields a general statement that explains the event of observing, but not the facts observed. If we observe thousands of stones, trees and flowers, we never reach a point at which we observe a molecule. After we heat many iron bars, we can conclude the empirical fact that metals will bend when they are heated. But we will never discover the physics of expansion coefficients in this way. Indeed, superficial empirical-based induction could lead to wrong conclusions. For example, by repeated observations, it seems that heavy bodies (e.g. metal, stone) fall faster than lighter bodies (paper, feather). This Aristotelian belief had misled European scientists for over a thousand years. Galileo argued that indeed both heavy and light objects fall at the same speed. We don't know the real probability due to our finite existence. However, given a large number of cases, we can approximate the actual probability. We don't have to know everything to know something. Also, we don't have to know every case to get an approximation. This approximation is sufficient to fix our beliefs and lead us to further inquiry. 4. Explain the strengths and weaknesses of using formal logic as a knowledge representation method. Give specific examples. Knowledge representation and reasoning plays a central role in Artificial Intelligence. Research in Artificial Intelligence started off by trying to identify the general mechanisms responsible for intelligent behavior. However, it quickly became obvious that general and powerful methods are not enough to get the desired result, namely, intelligent behavior. Almost all tasks a human can perform which are considered to require intelligence are also based on a huge amount of knowledge. For instance, understanding and producing natural language heavily relies on knowledge about the language, about the structure of the world, about social relationships etc. One way to address the problem of representing knowledge and reasoning about it is to use some form of logic. Two perspectives on logic are possible. The first perspective, taken by McCarthy (1968), is that logic should be used to represent knowledge. That is, we use logic as the representational and reasoning tool inside the computer. Newell (1982) on the other hand proposed in his seminal paper on the knowledge level to use logic as a formal tool to analyze knowledge. Of course, these two views are not incompatible. Furthermore, once we accept that formal logic should be used as a tool for analyzing knowledge, it is a natural consequence to use logic for representing knowledge and for reasoning about it as well. Saying that logic is used as the main formal tool does not say which kind of logic is used. In fact, a large variety of logics have been employed or developed in order to solve knowledge representation and reasoning problems. Often, one started with a non clear specified problem, developed some kind knowledge representation formalism without a formal semantics, and only later started to provide a formal semantics.

9 Using this semantics, one could then analyze the complexity of the reasoning problems and develop sound and complete reasoning algorithms. This called as the logical method, which proved to be very fruitful in the past and has a lot of potential for the future. One good example for the evolution of knowledge representation formalisms is the development of description logics, which have their roots in so-called structured inheritance networks formalisms.these networks were originally developed in order represent word meanings. A concept node connects to other concept nodes using roles. Moreover, the roles could be structured as well. Determination of decidability and complexity as well as the design of decision algorithms are based on the rigorous formalization of the initial ideas. In particular, it is not just one logic that it is used to derive these results, but it is the logical method that led to the success. One starts with a specification of how expressions of the language or formalism have to be interpreted in formal terms. Based on that one can specify when a set of formulae logically implies a formula. Then one can start to find similar formalisms (e.g. modal logics) and prove equivalences and/or one can specify a method to derive logically entailed sentences and prove them to be correct and complete. Another interesting area where the logical method has been applied is the development of the so-called non-monotonic logics. These are based on the intuition that sometimes a logical consequence should be retracted if new evidence becomes known. For example, we may assume that our car will not be moved by somebody else after we have parked it. However, if new information becomes known, such as the fact that the car is not at the place where we have parked it, we are ready to drop the assumption that our car has not been moved. This general reasoning pattern was used quite regularly in early AI systems, but it took a while before it was analyzed from a logical point of view. In 1980, a special issue of the Artificial Intelligence journal appeared, presenting different approaches to non-monotonic reasoning, in particular Reiter s (1980) default logic and McCarthy s (1980) circumscription approach. A disappointing fact about nonmonotonic logics appears to be that it is very difficult to formalize a domain such that one gets the intended conclusions. In particular, in the area of reasoning about actions, McDermott (1987) has demonstrated that the straightforward formalization of an easy temporal projection problem (the Yale shooting problem ) does not lead to the desired consequences. However, it is possible to get around this problem. Once all underlying assumptions are spelled out, this and other problem can be solved (Sandewall 1994). It took more than a decade before people started to analyze the computational complexity (of the propositional versions) of these logics. As it turned out, these logics are usually somewhat more difficult than ordinary propositional logic (Gottlob 1992). This, however, seems tolerable since we get much more conclusions than in standard propositional logic. Right at the same time, the tight connection between nonmonotonic logic and belief revision (G ardenfors 1988) was noticed. Belief revision modeling the evolution of beliefs over time is just one way to describe how the set of nonmonotonic consequences evolve over time, which leads to a very tight connection on the formal level for these two forms of nonmonotonicity (Nebel 1991). Again, all these results and insights are mainly based on the logical method to knowledge representation. As mentioned, it is the idea of providing knowledge representation formalisms with formal (logical) semantics that enables us to communicate their meaning, to analyze their formal properties, to determine their computational complexity, and to devise reasoning algorithms. While the research area of knowledge representation is dominated by the logical approach, this does not mean that all approaches to knowledge representation must be based on logic. Probabilistic (Pearl 1988) and decision theoretic approaches, for instance, have become very popular lately. Nowadays a number of approaches aim at unifying decision theoretic and logical accounts by introducing a qualitative version of decision theoretic concepts. Other approaches aim at tightly integrating decision theoretic concepts such as Markov decision processes with logical approaches, for instance. Although this is not pure logic, the two latter approaches demonstrate the generality of the logical method: specify the formal meaning and analyze!

10 5. Research the "MIU Puzzle" (originally defined in D. Hofstadter's book "Gödel, Escher, Bach"). a) Answer the question: Can MU be produced from MI using the given rules? Explain your answer. b) In what sense is this a "typographical" system? c) What is Gödel's Incompleteness Theorem, and what does the MIU system have to do with it? Suppose there are the symbols M, I, and U which can be combined to produce strings of symbols called "words". The MU puzzle asks one to start with the "axiomatic" word MI and transform it into the word MU using in each step one of the following transformation rules: 1. Add a U to the end of any string ending in I. For example: MI to MIU. 2. Double any string after the M (that is, change Mx, to Mxx). For example: MIU to MIUIU. 3. Replace any III with a U For example: MUIIIU to MUUU. 4. Remove any UU. For example: MUUU to MU. Using these four rules is it possible to change MI into MU in a finite number of steps? The production rules can be written in a more schematic way. Suppose x and y behave as variables (standing for strings of symbols). Then the production rules can be written as: 1. xi xiu 2. Mx Mxx 3. xiiiy xuy 4. xuuy xy Is it possible to obtain the word MU using these rules? Solution The puzzle's solution is no. It is impossible to change the string MI into MU by repeatedly applying the given rules. In this case, one can look at the total number of I in a string. Only the second and third rules change this number. In particular, rule two will double it while rule three will reduce it by 3. Now, the invariant property is that the number of I is not divisible by 3: In the beginning, the number of Is is 1 which is not divisible by 3. Doubling a number that is not divisible by 3 does not make it divisible by 3. Subtracting 3 from a number that is not divisible by 3 does not make it divisible by 3 either. Thus, the goal of MU with zero I cannot be achieved because 0 is divisible by 3. In the language of modular arithmetic, the number n of I obeys the congruence where a counts how often the second rule is applied. b) In what sense is this a "typographical" system? The MIU-puzzle is in fact merely a puzzle about natural numbers in typographical disguise. If we could only find a way to transfer it to the domain of number theory, we might be able to solve it. If we try counting the numbers of I's contained in theorems, we will soon notice that it seems never to be 0. In other words, it seems that no matter how much lengthening and shortening is involved, we can never work in such a way that all I's are eliminated. Let us call the number of I's in any string the "I-count" of

11 that string. Note that the I-count of the axiom MI is 1. We can do more than show that the I-count can't be 0 -- we can show that the I-count can never be any multiple of 3. To begin with, notice that rules 1 and 4 (see ABOVE ) leave the I-count totally undisturbed. Therefore we need only think about rules 2 and 3. As far as rule 3 is concerned, it diminishes the I-count by exactly three. After an application of this rule, the I-count of the output might conceivably be a multiple of 3 -- but only if the I-count of the input was also. Rule 3, in short, never creates a multiple of 3 from scratch. It can only create one when it began with one. The same holds for rule 2, which doubles the I-count. The reason is that if 3 divides 2n, then -- because 3 does not divide 2 -- it must divide n (a simple fact from the theory of numbers). Neither rule 2 nor rule 3 can create a multiple of 3 from scratch. But this is the key to the MU-puzzle! Here is what we know : The I-count begins at 1 (the axiom MI ), and that is not a multiple of 3. Two of the rules do not affect the I-count at all. The two remaining rules which do affect the I-count do so in such a way as never to create a multiple of 3 unless given one initially. The conclusion -- and a typically hereditary one it is, too -- is that the I-count can never become any multiple of 3. In particular, 0 is a forbidden value of the I-count [because 0 is a multiple of 3, -- 0 x 3 = 0]. Hence, MU is not a theorem of the MIU-system. Notice that, even as a puzzle about I-counts, this problem was still plagued by the crossfire of lengthening and shortening rules. Zero became the goal. I-counts could increase (rule 2 ), could decrease (rule 3 ). Until we analyzed the situation, we might have thought that, with enough switching back and forth between the rules, we might eventually hit 0. Now, according to a simple number-theoretical argument, we know that that is impossible. Not all problems of the type which the MU-puzzle symbolizes are so easy to solve as this one. But we have seen that at least one such puzzle could be embedded within, and solved within, number theory [When we speak of "number theory" (which we often will indicate by "N") we mean its non-formalized version. The language in this version is thus plain English or some other spoken language. Its formalized version is the "Theoria Numerorum Typographica", or "TNT" for short (without quotation marks). Number theory is the "meaning" or (meaningful) interpretation of TNT, that is, TNT describes number theory. Both number theory and TNT concern the positive integers and zero (which are called "natural numbers") and their properties. When setting up TNT, all symbols, operations, etc. in number theory are translated into special TNT signs, in such a way that a minimum of such TNT signs is obtained.]. We are now going to see that there is a way to embed all problems about any formal system, in number theory. To illustrate it, we will use the MIU-system. We begin by considering the notation of the MIU-system. We shall map each symbol onto a new symbol : M <==> 3 I <==> 1 U <==> 0 The correspondence was chosen arbitrarily. The only reason to it is that each symbol looks a little like the one it is mapped onto. Each number is called the Gödel number of the corresponding letter. Now MU <==> 30 MIIU <==> 3110 MUU <==> 300 etc. Let us now take a look at a typical derivation in the MIU-system, written simultaneously in both notations : (1) MI axiom (2) MII rule (3) MIIII rule (4) MUI rule (5) MUIU rule (6) MUIUUIU rule (7) MUIIU rule

12 The left-hand column is obtained by applying our four familiar typographic rules. The right-hand column, too, could be thought of as having been generated by a similar set of typographic rules. Yet the right-hand column has a dual nature. Now we explain what this means. Seeing Things Both Typographically and Arithmetically We could say of the fifth string ('3010') that it was made from the fourth ('301'), by appending a '0' on the right. On the other hand we could equally well view the transition as caused by an arithmetical operation -- multiplication by 10, to be exact. When natural numbers are written in the decimal system, multiplication by 10 and putting a '0' on the right are indistinguishable operations. We can take advantage of this to write an arithmetical rule which corresponds to typographical rule I : Arithmetical Rule Ia : A number whose decimal expansion ends on the right in '1' can be multiplied by 10. We can eliminate the reference to the symbols in the decimal expansion by arithmetically describing the rightmost digit : Arithmetical Rule Ib : A number whose remainder when divided by 10 is 1, can be multiplied by 10. Now we could have stuck with a purely typographical rule (also here), such as the following one : Typographical Rule I : From any theorem whose rightmost symbol is '1' a new theorem can be made, by appending '0' to the right of that '1'. [Just using numerals instead of letters] They would have the same effect. This is why the right-hand column has a "dual nature" : It can be viewed either as a series of typographical operations changing one pattern of symbols into another, or as a series of arithmetical operations changing one magnitude into another. But there are powerful reasons for being more interested in the arithmetical version. Typographical rule which tells how certain digits are to be shifted, changed, dropped, or inserted, in any number represented decimally,. More briefly : Typographical rules for manipulating n u m e r a l s are actually arithmetical rules for operating on n u m b e r s. This simple observation is at the heart of Gödel's method, and it will have an absolutely shattering effect. It tells us that once we have a Gödel-numbering for any formal system, we can straightaway form a set of arithmetical rules which complete the Gödel isomorphism. The upshot is that we can transfer the study of any formal system -- in fact the study of all formal systems -- into number theory. What is Gödel's Incompleteness Theorem, and what does the MIU system have to do with it? Gödel's first incompleteness theorem states that there is no set of axioms of arithmetic that is both complete and consistent. In other words it means that there is at least one true statement in the system that cannot be derived within the system. Gödel s second incompleteness theorem states that if we have a set of axioms T, then T's consistency cannot be proven within T. To answer the question we will go back to MIU system. MIU-system Since this is a formal system, this system consists of some restriction or rules. Our formal system---miu system--- consists of only three letters of alphabet: M, I, U. The strings (which mean strings of letters) of the MIU-system are the strings that are composed of only those three letters. For example: MU UIM MUUMUU UIIUMIUUIMUIIUMIUUIMUIIU

13 are strings of the MIU-system. SYMBOLS: M, I, U AXIOM: MI RULES: (In the following, x is merely a variable) 1. If xi is a theorem, so is xiu. 2. If Mx is a theorem, so is Mxx. 3. In any theorem, III can be replace by U. 4. UU can be dropped from any theorem. Now the question is Can we produce a string, namely MU in this system? An axiom, namely MI is granted initially and we want to produce MU by using given axiom and rules as below: MI MU The reason we want to do is that we want to find out MU is derivable or not mathematically. 1) The I-count begins at 1 (not a multiple of 3). 2) Rules 1 and 4 do not affect the I-count at all. 3) Rules 2 and 3 affect the I-count in such a way that they never create a multiple of 3 unless given one initially. It follows that the I-count can never be a multiple of 3 and thus we can never derive MU from MI. In other words, MU is not a theorem of the MIU- system. As a result, we can see that MU actually exists in the system, but it is not derivable. So the system is incomplete. This corresponds to Gödel s first incompleteness theorem which says there is at least one true statement in any formal system, but it cannot be derived or proved. 6. Formal Logic is known for its use of symbolic representation of concepts, such as A B. In what sense are numbers symbols? Give examples. Formal logic : the branch of logic that examines patterns of reasoning to determine which ones necessarily result in valid, or formally correct, conclusions. In the history of formal logic, different symbols have been used at different times and by different authors In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was first used by Kurt Gödel for the proof of his incompleteness theorem. A Gödel numbering can be interpreted as an encoding in which a number is assigned to each symbol of a mathematical notation, after which a sequence of natural numbers can then represent a sequence of strings. These sequences of natural numbers can again be represented by single natural numbers, facilitating their manipulation in formal theories of arithmetic. Gödel used a system of Gödel numbering based on prime factorization. He first assigned a unique natural number to each basic symbol in the formal language of arithmetic he was dealing with.

14 To encode an entire formula, which is a sequence of symbols, Gödel used the following system. Given a sequence x1x2x3...xn of positive integers, the Gödel encoding of the sequence is the product of the first n primes raised to their corresponding values in the sequence: According to the fundamental theorem of arithmetic, any number obtained in this way can be uniquely factored into prime factors, so it is possible to recover the original sequence from its Gödel number (for any given number n of symbols to be encoded). Gödel specifically used this scheme at two levels: first, to encode sequences of symbols representing formulas, and second, to encode sequences of formulas representing proofs. This allowed him to show a correspondence between statements about natural numbers and statements about the provability of theorems about natural numbers, the key observation of the proof. There are more sophisticated (but more concise) ways to construct a Gödel numbering for sequences. Artificial intelligence systems are centrally concerned with representing and manipulating knowledge. Some knowledge, particularly in mathematics and sciences, is expressed as numbers and formulae expressions that consists of collection of numbers and arithmetic operations. Mathematics, when it involves more abstract reasoning process(such as proving, algebraic transformations and the symbolic solution of integral- differential equations), requires this more general and powerful language, which must be expressed in concepts and relationships are represented by symbols and string of symbols. In fact, numbers and formulae are really just a collection of symbols. Numbers are symbols whose properties are defined over the set of arithmetic operations and arithmetic operations are represented by symbols or strings of symbols( such as +,-,/,x). Thus the tools of artificial intelligence consist of languages, processes, and construct that allow the acquisition, representation, storing, transformation and other manipulation of concepts and relationships by information with the study of language theory, including higher order computer languages and computer compiler theory. Internal to a computer a symbol is just a sequence of bits that can be distinguished from other symbols. Some symbols have a fixed interpretation, e.g., symbols that represent numbers and symbols that represent characters. Symbols that do not have fixed meaning appear in many programming languages. Java, starting from Java 1.5, calls them enumeration types. Lisp calls them atoms. Usually, they are

15 implemented as indexes into a symbol table that gives the name to print out. The only operation needed on these symbols is equality to determine if two symbols are the same or not. 7. Do some research on the programming language called Prolog. Explain how Prolog uses backward chaining as its solution method. All programming languages have both declarative (definitional) and imperative (computational) components. Prolog is referred to as a declarative language because all program statements are definitional. In particular, a Prolog program consists of facts and rules which serve to define relations (in the mathematical sense) on sets of values. The imperative component of Prolog is its execution engine based on unification and resolution, a mechanism for recursively extracting sets of data values implicit in the facts and rules of a program. Prolog rules consist of: a consequent (the head), and an antecedent (the body) (connected by the :- symbol) The body of a rule may consist of one or more clauses. Each clause is a subgoal. The head is true if the subgoals making up the body are true. Example: "X is wierd if X is vegetarian and eats steak, or if X is a trainspotter" wierd(x) :- vegetarian(x), eats_steak(x). wierd(x) :- is_trainspotter(x). like facts, rules are also predicates. These 2 rules constitute the predicate "wierd/1") Prolog rules wierd(x) :- vegetarian(x), eats_steak(x). wierd(x) :- is_trainspotter(x). subgoals of a rule are usually separated by and (sometimes by or ) and terminated with a full stop.the scope of a variable is limited to that rule. There are no global variables in Prolog. All facts and rules have global scope Backward chaining Once knowledge is represented in a form that Prolog can use (facts and rules), a reasoning procedure is required to draw conclusions from the knowledge base. No need to program this ourselves - it is built into Prolog s inference engine, which - by default works by backward chaining. This means that we start with a hypothesis and reason backwards from the hypothesis trying to find facts that support it. For instance, in a database about grants for listed buildings we might want to ask:?- eligible_for_grant(jim). The Prolog interpreter will then attempt to prove the goal statement, given the assumptions in the KB(knowledge base). This may mean ascertaining that: Jim s property is listed Jim does not have huge funds at his disposal Ascertaining that Jim does not have huge funds at his disposal might entail making sure that: Jim does not have lots of investments, and

16 Jim does not earn > 75,000 a year Thus Prolog is working backwards from the goal statement to the facts. Prolog can be made to work in a forward-chaining fashion but this requires more programming effort. In Prolog implies is written as :- and is read from right to left. The universal quantifier is assumed for all variables and a comma is used for AND. Example: x Cat(x) Tame(X) Pet(x) would be written as pet(x) :- cat(x),tame(x). Sentences are apparently written 'backwards' to reflex the fact that Prolog uses backwards chaining. Thus, the above sentence could be read as to prove that X is a pet prove that X is a cat and that X is tame. (sentences in Prolog end in a period. Variables must start with a capital letter or an underscore and objects and relations begin with a lower-case letter.)a semicolon is used for OR. References: &f=false =hh5hsw04j4shuszvnt- 0OreXpt4&hl=en&ei=YDaQSpv6DoeQsgOruaAM&sa=X&oi=book_result&ct=result&resnum=1#v=onepage&q=frame%20problem%20artificial%20intelligenc e&f=false