Limitations of Logic - The Search Space Problem by Wolfgang Ertel

In the search for a proof there are almost always many (depending on the calculus, potentially infinitely many) possibilities for the application of inference rules at every step. The result is the aforementioned explosive growth of the search space. In the worst case, all of these possibilities must be tried in order to find the proof, which is usually not possible in a reasonable amount of time.

If we compare automated provers or inference systems with mathematicians or human experts who have experience in special domains, we make interesting observations. For one thing, experienced mathematicians can prove theorems which are far out of reach for automated provers. On the other hand, automated provers perform tens of thousands of inferences per second. A human in contrast performs maybe one inference per second. Although human experts are much slower on the object level (that is, in carrying out inferences), they apparently solve difficult problems much faster.

There are several reasons for this. We humans use intuitive calculi that work on a higher level and often carry out many of the simple inferences of an automated prover in one step. Furthermore, we use lemmas, that is, derived true formulas that we already know and therefore do not need to re-prove them each time. Meanwhile there are also machine provers that work with such methods. But even they cannot yet compete with human experts.

A further, much more important advantage of us humans is intuition, without which we could not solve any difficult problems. The attempt to formalize intuition causes problems. Experience in applied AI projects shows that in complex domains such as medicine or mathematics, most experts are unable to formulate this intuitive meta-knowledge verbally, much less to formalize it. Therefore we cannot program this knowledge or integrate it into calculi in the form of heuristics. Heuristics are methods that in many cases can greatly simplify or shorten the way to the goal, but in some cases (usually rarely) can greatly lengthen the way to the goal. Heuristic search is important not only to logic, but generally to problem solving in AI.

An interesting approach, which has been pursued since about 1990, is the application of machine learning techniques to the learning of heuristics for directing the search of inference systems, which we will briefly sketch now. A resolution prover has, during the search for a proof, hundreds or more possibilities for resolution steps at each step, but only a few lead to the goal. It would be ideal if the prover could ask an oracle which two clauses it should use in the next step to quickly find the proof. There are attempts to build such proof-directing modules, which evaluate the various alternatives for the next step and then choose the alternative with the best rating. In the case of resolution, the rating of the available clauses could be computed by a function that calculates a value based on the number of positive literals, the complexity of the terms, etc., for every pair of resolvable clauses.

How can this function be implemented? Because this knowledge is "intuitive", the programmer is not familiar with it. Instead, one tries to copy nature and uses machine learning algorithms to learn from successful proofs.

The attributes of all clause pairs participating in successful resolution steps are stored as positive, and the attributes of all unsuccessful resolutions are stored as negative. Then, using this training data and a machine learning system, a program is generated which can rate clause pairs heuristically.

A different, more successful approach to improving mathematical reasoning is followed with interactive systems that operate under the control of the user. Here one could name computer algebra programs such as Mathematica, Maple, or Maxima, which can automatically carry out difficult symbolic mathematical manipulations. The search for the proof, however, is left fully to the human. The aforementioned interactive prover Isabelle provides distinctly more support during the proof search. There are at present several projects, such as Omega and MKM, 1 for the development of systems for supporting mathematicians during proofs.

In summary, one can say that, because of the search space problem, automated provers today can only prove relatively simple theorems in special domains with few axioms.

About the Author

Dr. Wolfgang Ertel is a professor at the Institute for Artificial Intelligence at the Ravensburg-Weingarten University of Applied Sciences, Germany.

This accessible and engaging textbook presents a concise introduction to the exciting field of artificial intelligence (AI). The broad-ranging discussion covers the key subdisciplines within the field, describing practical algorithms and concrete applications in the areas of agents, logic, search, reasoning under uncertainty, machine learning, neural networks, and reinforcement learning. Fully revised and updated, this much-anticipated second edition also includes new material on deep learning.

Topics and features:

Presents an application-focused and hands-on approach to learning, with supplementary teaching resources provided at an associated website
Contains numerous study exercises and solutions, highlighted examples, definitions, theorems, and illustrative cartoons
Includes chapters on predicate logic, PROLOG, heuristic search, probabilistic reasoning, machine learning and data mining, neural networks and reinforcement learning
Reports on developments in deep learning, including applications of neural networks to generate creative content such as text, music and art
Examines performance evaluation of clustering algorithms, and presents two practical examples explaining Bayes theorem and its relevance in everyday life
Discusses search algorithms, analyzing the cycle check, explaining route planning for car navigation systems, and introducing Monte Carlo Tree Search
Includes a section in the introduction on AI and society, discussing the implications of AI on topics such as employment and transportation

Ideal for foundation courses or modules on AI, this easy-to-read textbook offers an excellent overview of the field for students of computer science and other technical disciplines, requiring no more than a high-school level of knowledge of mathematics to understand the material.

A reader says,"The book has a substantial introduction to logic that was very helpful. Overall the selection of topics was perfect for a first course in AI."

A reader says,"An excellent book for beginners. Would like to have seen code examples in python. I realize python is not much historic for AI but it is lingua franca for data science and deep learning today. Good treatment of many topics on search etc. Definitely a great resource for those just getting started in AI."

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