This book is meant for the serious practitioner-to-be of constructing intelligent machines. Machines that are aware of the world around them, that have goals to achieve, and the ability to imagine the future and make appropriate choices to achieve those goals. It is an introduction to a fundamental building block of artificial intelligence (AI). As the book shows, search is central to intelligence.

Clearly AI is not one monolithic algorithm but a collection of processes working in tandem, an idea espoused by Marvin Minsky in his book The Society of Mind (1986). Human problem solving has three critical components. The ability to make use of experiences stored in memory; the ability to reason and make inferences from what one knows; and the ability to search through the space of possibilities. This book focuses on the last of these. In the real world we sense the world using vision, sound, touch, and smell. An autonomous agent will need to be able to do so as well. Language, and the written word, is perhaps a distinguishing feature of the human species. It is the key to communication which means that human knowledge becomes pervasive and is shared with future generations. The development of mathematical sciences has sharpened our understanding of the world and allows us to compute probabilities over choices to take calculated risks. All these abilities and more are needed by an autonomous agent.

Can one massive neural network be the embodiment of AI? Certainly, the human brain as a seat of intelligence suggests that. Everything we humans do has its origin in activity in our brains, which we call the mind. Perched on the banks of a stream in the mountains we perceive the world around us and derive a sense of joy and well-being. In a fit of contented creativity, we may pen an essay or a poem using our faculty of language. We may call a friend on the phone and describe the scene around us, allowing the friend to visualize the serene surroundings. She may reflect upon her own experiences and recall a holiday she had on the beach. You might start humming your favourite song and then be suddenly jolted out of your reverie remembering that friends are coming over for dinner. You get up and head towards your home with cooking plans brewing in your head.

Knowledge is necessary for intelligence. Without knowledge, problem solving with search is blind. We saw this in the last chapter. In general, knowledge is that sword in the armoury of a problem solver that can cut through the complexity. Knowledge accrues over time, either distilled from our own experiences or assimilated from interaction with others – parents, teachers, authors, coaches, and friends. Knowledge is the outcome of learning and exists in diverse forms, varying from tacit to explicit. When we learn to ride a bicycle, we know it but are unable to articulate our knowledge. We are concerned with explicit knowledge. Most textbook knowledge is explicit, for example, knowing how to implement a leftist heap data structure.

In a well known incident from ancient Greece, it is said that Archimedes, considered by many to be the greatest scientist of the third century BC, ran naked onto the streets of Syracuse. King Hieron II was suspicious that a goldsmith had cheated him by adulterating a bar of gold given to him for making a crown. He asked Archimedes to investigate without damaging the crown. Stepping into his bathtub Archimedes noticed the water spilling out, and realized in a flash that if the gold were to be adulterated with silver, then it would displace more water since silver was less dense. This was his epiphany moment when he discovered what we now know as the Archimedes principle. And he ran onto the streets shouting ‘Eureka, eureka!’ We now call such an enlightening moment a Eureka moment!

We will adopt the overall goal of artificial intelligence (AI) to be ‘to build machines with minds, in the full and literal sense’ as prescribed by the Canadian philosopher John Haugeland (1985).

Not to create machines with a clever imitation of human-like intelligence. Or machines that exhibit behaviours that would be considered intelligent if done by humans – but to build machines that reason.

This book focuses on search methods for problem solving. We expect the user to define the goals to be achieved and the domain description, including the moves available with the machine. The machine then finds a solution employing first principles methods based on search. A process of trial and error. The ability to explore different options is fundamental to thinking.

As we describe subsequently, such methods are just amongst the many in the armoury of an intelligent agent. Understanding and representing the world, learning from past experiences, and communicating with natural language are other equally important abilities, but beyond the scope of this book. We also do not assume that the agent has meta-level abilities of being self-aware and having goals of its own. While these have a philosophical value, our goal is to make machines do something useful, with as general a problem solving approach as possible.

This and other definitions of what AI is do not prescribe how to test if a machine is intelligent. In fact, there is no clear-cut universally accepted definition of intelligence. To put an end to the endless debates on machine intelligence that ensued, the brilliant scientist Alan Turing proposed a behavioural test.

Can Machines Think?

Ever since the possibility of building intelligent machines arose, there have been raging debates on whether machine intelligence is possible or not. All kinds of arguments have been put forth both for and against the possibility. It was perhaps to put an end to these arguments that Alan Turing (1950) proposed his famous imitation game, which we now call the Turing Test. The test is simply this: if a machine interacts with a human using text messages and can fool human judges a sufficiently large fraction of times that they are chatting with another human, then we can say that the machine has passed the test and is intelligent.

We revert to graph search for the study of algorithms that guarantee optimal solutions. The task is to find a shortest path in a graph from a start node to a goal node. We have already studied algorithms BFS and DFID in Chapter 3. The key idea there was to extend that partial path which was the shortest. We begin with the same strategy. Except that now we add weights to edges in the graph. Without edge weights, the optimal or shortest path has the least number of edges in the path. With edge weights added, we modify this notion to the sum of the weights on the edges.

The question that remains is the definition of ‘best’. In DFS, the deepest node is the best node. In BestFirstSearch, the node that appears to be closest to the goal is the best. In BFS, the node closest to the start node is the best. We begin by extending the idea behind breadth first search.

We can generalize our common theme as follows. With every node N on OPEN, we associate a number that stands for the estimated cost of the final solution.

Douglas Hofstadter says that humans are not known to have a head for numbers (Hofstadter, 1996). For most of us, the numbers 3.2 billion and 5.3 million seem vaguely similar and big. A very popular book (Gamow, 1947) was titled One, Two, Three … Infinity. The author, George Gamow, talks about the Hottentot tribes who had the only numbers one, two, and three in their vocabulary, and beyond that used the word many. Bill Gates is famously reputed to have said, ‘Most people overestimate what they can do in one year and underestimate what they can do in ten years.’

So, how big is big? Why are computer scientists wary of combinatorial growth? In Table 2.1 we looked at the exponential function 2N and the factorial N!, which are respectively the sizes of search spaces for SAT and TSP, with N variables or cities. How long will take it to inspect all the states when N = 50?

For a SAT problem with 50 variables, 250 = 1,125,899,906,842,624. How big is that? Let us say we can inspect a million or 106 nodes a second. We would then need 1,125,899,906.8 seconds, which is about 35.7 years! There are N! = 3.041409320 × 1064 non-distinct tours (each distinct tour has 2N representations) of 50 cities.

In this chapter we lay the foundations of the search spaces that an agent would explore.

First, we imagine the space of possibilities. Next, we look at a mechanism to navigate this space. And then in the chapters that follow we figure out what search strategy an algorithm can use to do so efficiently.

Our focus is on creating domain independent solvers, or agents, which can be used to solve a variety of problems. We expect that the users of our solvers will implement some domain specific functions in a specified form that will create the domain specific search space for our domain independent algorithms to search in. In effect, these domain specific functions create the space, which our algorithm will view as a graph over which to search. But the graph is not supplied to the search algorithm upfront. Rather, it is constructed on the fly during search. This is done by the user supplied neighbourhood function that links a node in this graph to its neighbours, generating them when invoked. The neighbourhood function takes a node as an input and computes, or returns, the set of neighbours in the abstract graph for the search algorithm to search in.

Logic and mathematics are often considered to be synonymous. Both are concerned with truth of statements. In this chapter we confine ourselves to the family of classical logics, also known as mathematical logics, in which every sentence has exactly two possible truth values – true and false. Nothing in between. No fuzzy concepts like tall and dark. Is a person whose height is 176 centimetres tall? What about 175 then? And 174? When does she become not tall? Or modalities like maybe.

A constraint network R or a CSP is a triple,

where X is a set of variable names, D is a set of domains, one for each variable, and C is a set of constraints on some subsets of variables (Dechter, 2003). We will use the names X = ﹛x1, x2, …, xn﹜ where convenient with the corresponding domains D = ﹛D1, D2, …, Dn﹜. The domains can be different for each variable and each domain has values that the variable can take, Di = ﹛ai1, ai2, …, aik﹜. Let C = ﹛C1, C2, …, Cm﹜ be the constraints. Each constraint Ci has a scope Si R X and a relation Ri that is a subset of the cross product of the domains of the variables in Si. Based on the size of Si, we will refer to the constraints as unary, binary, ternary, and so on. A CSP is often depicted by a constraint graph and a matching diagram, as described in the examples to follow.

We will confine ourselves to finite domain CSPs, in which the domain of each variable is discrete and finite. We will also specify the relations in extensional form well suited for our algorithms.

An intelligent agent acts in the world to achieve its goals. Given the state of the world it is in, and given the goals it has, it has to choose an appropriate set of actions. The process of selecting those actions is called planning. Planning is the reasoning side of acting (Ghallab, Nau, and Traverso, 2004). Planning and acting do not happen in isolation. A third process is an integral part of intelligent agency – perceiving. An agent senses the world it is operating in, deliberates upon its goals to produce a plan, and executes the actions in the plan. This is often referred to as the sense–deliberate–act cycle. The entire process may need to be monitored by the agent. Since the world may be changing owing to other agencies, it may even have to modify its plans on the fly.

There has been considerable work on autonomous agents that plan their activity. This became necessary in space applications where communication with Earth takes too long, necessitating autonomy. This was the case with the Mars rovers experiments by NASA, and even ten years after landing on Mars the rover Curiosity is still active.

In the last chapter we looked at the notion of search spaces. Search spaces, as shown in Figure 2.2, are trees corresponding to the different traversals possible in the state space or the solution space. In this chapter we begin by constructing the machinery, viz. algorithms, for navigating this space. We begin our study with the corresponding tiny state space shown in Figure 3.1.

The tiny search problem has seven nodes, including the start node S, the goal node G, and five other nodes named A, B, C, D, and E. Without any loss of generality, let us assume that the nodes are states in a state space. The algorithms apply to the solution space as well. The left side of the figure describes the MoveGen function with the notation Node → (list of neighbours). On the right side is the corresponding graph which, remember, is implicit and not given upfront. The algorithm itself works with the MoveGen function and also the GoalTest function. The latter, for this example, simply knows that state G is the goal node. For configuration problems like the N-queens, it will need to inspect the node given as the argument.

The search space that an algorithm explores is implicit. It is generated on the fly by the MoveGen function, as described in Algorithm 2.1. The candidates generated are added to what is traditionally called OPEN, from where they are picked one by one for inspection. In this chapter we represent OPEN as a list data structure.

Chess has long fascinated humankind as a game of strategy and skill. It was probably invented in India in the sixth century in the Gupta empire when it was known as chaturanga. A comprehensive account of its history was penned in 1913 by H.J.R. Murray (2015). The name refers to the four divisions an army may have. The infantry includes the pawns, the knights make up the cavalry, the rooks correspond to the chariotry, and the bishops the elephantry (though the Hindi word for the piece calls it a camel). In Persia the name was shortened to chatrang. This in turn transformed to shatranj as exemplified in the 1924 story by Munshi Premchand (2020) and the film of the same name by Satyajit Ray, Shatranj Ke Khiladi (The Chess Players). It became customary to warn the king by uttering shāh (the Persian word for king) which became check, and the word mate came from māt which means defeated. Checkmate is derived from shāh māt which says that the king has been vanquished.

Table 8.1 lists the names of the chess pieces in Sanskrit, Persian, Arabic, and English (Murray, 2015). In Hindi users often say oont (camel) for bishop and haathi (elephant) for rook.

From India the game spread to Persia, and then to Russia, Europe, and East Asia around the ninth century.

The search algorithms we have studied so far take a holistic view of a state representing the given situation. In practice, states are represented in some language in which the different constituents are described. The state description is essentially a set of statements. As the importance of knowledge for problem solving became evident, using rules to spot patterns in the description and proposing actions emerged as a problem solving strategy.

Pattern Directed Inference Systems

An approach to problem solving that was developed in the mid-1970s was called pattern directed inference systems (Waterman and Hayes-Roth, 1978). The basic idea is that patterns in a given state are associated with actions.