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Handbook Of Practical Logic and Automated Reasoning

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Basically, we do not possess any picture of the animals, but only linguistic features. In this case, the animal recognition problem can nicely be formulated as inference in KB. For example, once the following animal features hair , hoofs , longneck , longlegs , tawny , darkspots are true, the quest for which animal is identified by these linguistic feature can be written in terms of truth of the following logic argument:. The inferential process to conclude whether the argument is true is based on the following steps:.

Notice that the above argument is equivalent to establishing the truth of the proposition. This formula, like all the formulas of the KB , is a Horn clause , which is special interesting logic fragment for which there exist polynomial algorithms for the automated proof. The KB of Table 6. This is referred to as intensional knowledge and represents an abstract definition of the categories.

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On the other hand, the explicit list of items belonging to a given category is referred to as extensional knowledge. For example, suppose we consider a set of animals and we name them all. This is simply represented by a list of unary predicates; for example,. Suppose we extend the KB of Table 6. Here, we assume that hantenv z is true if z is an environment which favors the hunt. If hantenv z then from these premises we can draw that conclusion eat Randy, Fajita , where we can see an inferential mechanism that relies on combining factual and abstract knowledge.

Now this corresponds with solving SAT , which is intractable.

Formal Methods for Everyone: Practical Tools for Computer-Aided Reasoning // Murphy Berzish

This is in fact the face of complexity recurrently emerging from the combinatorial search of satisfaction of Boolean constraints. In the following, we will see how the availability of related real-valued information from the environment leads to approximate reasoning processes that can be significantly more efficient than formal deduction.


What if some of the previous linguistic features are missing? Clearly, logic-based inferential processes are doomed to fail. More interestingly, suppose that instead of relying on the above collection of animal features, one is given a sentence in natural language that describes the animal. For example, one might want to answer the question:. We are missing some of the features that describe the giraffe and, in addition, in this case we must be able to process the sentence to draw conclusions.

Clearly, this is extremely critical, since we can easily reformulate the above question in such a way that the same features are present, whereas logic modifiers can indicate their absence, so as to indicate a different animal! As it will be shown in the next section, in other cases one might think of extending this animal identification problem to one in which, in addition to a linguistic query, we rely on the availability of a collection of animal pictures.

Fred A. Cummins, in Building the Agile Enterprise , Rule-driven processes use automated reasoning to determine the ad hoc ordering of activities based on requirements and circumstances encountered during the process. This might be viewed as managing a trip. As the traveler proceeds along the trip, different routes may be selected depending on road conditions, availability of overnight accommodations, and unforeseen problems.

A rule-driven process might be used where each customer order requires a unique product configuration that reflects consideration of dependencies between components such as configuration of a computer, an automobile, or an insurance policy. It also might be appropriate for field service operations where there may be a number of service requests outstanding and the work required for each may not be apparent until the technician arrives to diagnose the problem. In this case the focus of the process may be on managing the technician assignments to minimize travel time while ensuring a reasonably timely response to each service request.

Rule-driven processes are not common and require special tools, but they can be very effective where the operations are well defined but highly variable. Dov M. Gabbay, in Handbook of Automated Reasoning , The final approach towards translation-based automated reasoning for modal logic that we want to mention here is due to Montanasi, Policriti, and their colleagues and students.

In particular, the method is also applicable if the modal logic at hand is specified with Hilbert axioms only. The latter represents the set of states in the frame x in which the formula A holds. Instead of translating Hilbert axioms, one may use a semantics for L whenever it is available. Furthermore, the method is easily extended to poly-modal logics.

At the time of writing, experimental results comparing the approach to other translation-based approaches to automated reasoning in modal logic are not available. Van Benthem, D'Agostino, Montanari and Policriti [ ] extend the connection between modal logic and set theory outlined above to capture a larger part of the non r. The notion of validity on so-called general frames [ Blackburn et al.

Gregorz Malinowski, in Handbook of the History of Logic , Resolution is still among the main tools of Automated Reasoning. Recently, also the natural deduction becomes more and more acknowledged device in that context. The both, natural deduction and the resolution, are relatives of sequents and, to a certain degree, are sequent expressible. The method of natural deduction is a special formalization, which establishes relations between premises and a set of conclusions.

It operates with both kind of rules for all logical constants: rules of introduction and rules of elimination. Moreover, the set of conclusions may, in particular cases, consist of one formula.

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The history of natural deduction approach to many-valued logics is not very long. Essentially new systems using sequents are discussed in Baaz et al. The common feature of all systems of this kind is the property of conserving some logical values. Obviously, the most popular option is to save one logical value corresponding to the truth. On the other hand, the choice of many logic values at once, or a subset of the set of all values, of the logic at work, is also worth consideration.

Resolution is a refutation method organized on clauses i. A literal is a propositional variable positive literal or the negation of a propositional variable negative literal. The procedure starts with a set of clauses and the refutation ends with an empty clause. It operates with a single resolution rule , which is a form of the cut rule.

The pioneering works and their ancestrals have been based on special normal forms with multiple-valued literals, which used special unary connectives. The most recent outcome of investigation is an algebraic theory of resolution proof deductive proof systems developed by Stachniak []. The key idea on which the theory is based is that the refutational deductive proof systems based on a non-clausal resolution become finite algebraic structures, the so-called resolution algebras. In turn, the particular interpretation of the resolution principle shows it as the rule of consistency verification defined relative to an appropriate propositional logic.

In the classical case verifiers coincide with the formulas defining two standard truth values. The process of selecting verifiers for resolution counterparts of non-classical proof systems usually goes beyond the search for defining truth values. Thus, e. Mathematical theorems often contain the equality predicate and assume that it satisfies special equality axioms. The same is true for many areas of computer science where first-order logic is used. The necessity of handling equality in automated reasoning has been recognized already in the very early papers in the area [ Wang , Kanger , Wos, Robinson, Carson and Shalla , Robinson , Darlington , Robinson and Wos ].

Equality in first-order logic can be axiomatized by the following equality axioms :. However, this leads to a combinatorial explosion due to the universal applicability of equality axioms. Suppose that F contains a binary function symbol f. Thus, even the early methods of handling equality in automated reasoning tried to avoid the use of the equality axioms. In the rest of this section we will consider some of the main ideas developed in resolution-based theorem proving with equality.