# Logic for Computer Science and Artificial Intelligence

# Logic for Computer Science and Artificial Intelligence

## Ricardo Caferra

Language: English

Pages: 544

ISBN: 1848213018

Format: PDF / Kindle (mobi) / ePub

Logic and its components (propositional, first-order, non-classical) play a key role in Computer Science and Artificial Intelligence. While a large amount of information exists scattered throughout various media (books, journal articles, webpages, etc.), the diffuse nature of these sources is problematic and logic as a topic benefits from a unified approach. Logic for Computer Science and Artificial Intelligence utilizes this format, surveying the tableaux, resolution, Davis and Putnam methods, logic programming, as well as for example unification and subsumption. For non-classical logics, the translation method is detailed.

Logic for Computer Science and Artificial Intelligence is the classroom-tested result of several years of teaching at Grenoble INP (Ensimag). It is conceived to allow self-instruction for a beginner with basic knowledge in Mathematics and Computer Science, but is also highly suitable for use in traditional courses. The reader is guided by clearly motivated concepts, introductions, historical remarks, side notes concerning connections with other disciplines, and numerous exercises, complete with detailed solutions, The title provides the reader with the tools needed to arrive naturally at practical implementations of the concepts and techniques discussed, allowing for the design of algorithms to solve problems.

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B) ⇒ ((B ⇒ C) ⇒ (A ⇒ C)) i) (A ⇒ (B ⇒ C)) ⇒ (B ⇒ (A ⇒ C)) R EMARK 3.27.– Note that the hypotheses of (b), (c), (d), and (e) have been distinguished typographically and correspond to names that are given to particular formulas (and not to meta-variables that denote arbitrary formulas). A, B, C each denote a formula. The reason for this is that, for example, in (b), if instead of B we had B as a hypothesis, then we could directly prove the formula using B ← A ⇒ C. Nevertheless, the same

between (operators of) set theory and connectives in PL is given in the following table: ∪→∨ ∩→∧ − →¬ U → 1(T) ∅ → 0(F) More generally, every Boolean algebra is isomorphic to a non-empty family (of subsets of a set) that is closed for the union, intersection, and complement operations21. Before giving the deﬁnition of a lattice based on the notion of orderings, we recall the deﬁnitions of a partial order, sup, and inf. D EFINITION 3.23.– (order). – A partially ordered set or poset < A, > is a

science, we quote an opinion of one of the greatest mathematicians of the 20th Century (S. Mac Lane): . . . the sciences do not try to explain, they hardly try to interpret, they mainly make models. . . The justiﬁcation (of a model) is solely and precisely that it is expected to work . . . Furthermore, it must satisfy certain aesthetic criteria – that is, in relation to how much it describes, it must be simple. 5.2.1. The notions of truth and satisfaction What is truth? Pilate (Gospel according

the fact that the free variables that were introduced can be replaced by any number of terms. To keep this property, we introduce renamings of the free variables that we shall note: x −→ X1 , X1 −→ X2 , X2 −→ X3 , . . . – The problem with rule δ is that, when there will be, say, n free variables X1 , X2 , . . . , Xn that have been introduced in the branch corresponding to formula F , where ∃y appears in the scope of X1 , . . . , Xn , as usual, ∃y will be erased, and y will be replaced by f (X1 ,

exception. Here is another one: i) The sentence below is false. ii) The sentence above is true. If (i) is T then (ii) is F hence (i) is F. If (i) is F then (ii) is T hence (i) is T. If (ii) is F then (i) is T hence (ii) is T. If (ii) is T then (i) is F hence (ii) is F. 2.1.2. Paradoxes and set theory Perhaps those paradoxes that had the most impact are those that involve set theory: probably because of the importance of set theory in mathematics, and particularly in the foundations of