Abstract
This chapter provides an extended survey of inference mechanisms which are candidates for inclusion in the initial endowment of a proof-verifier based on set theory, and points up some efficiency considerations which limit the complexity of the sets of statements to which each inference mechanism can be applied.
In addition to discourse-manipulation mechanisms, the verifier depends critically on a collection of routines which work by combinatorial search. These are able to examine certain limited classes of logical and set-theoretic formulae and determine their logical validity or invalidity directly. Together they constitute the verifier’s inferential core. A variety of candidate algorithms of this kind is examined. While all of these are interesting in their own right, not all are worth including in the verifier’s initial endowment of deduction procedures: some are in fact too inefficient to be practical, while others are too specialized to serve often in ordinary mathematical discourse. The review of candidates begins with one of the most elementary but important decision procedures, the Davis–Putnam–Logemann–Loveland technique for deciding the validity of propositional formulae. The resolution principle and Knuth–Bendix’s method are also treated.
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Notes
- 1.
x∉p is short for ¬(x∈p).
- 2.
Strictly speaking, places p y should be denoted by \( p_{M(y)} \).
- 3.
Notice that in the context of additive arithmetic, integer multiplication is admitted only in terms of the form c⋅A, where c is a positive integer constant and A is a well-formed Presburger term; thus c⋅A can be considered as a short for \(\underbrace{A+ \cdots+ A}_{c \text{ times}}\).
- 4.
Throughout this section, ‘⋅’ denotes multiplication, often designated by ‘∗’ in the rest of the book.
- 5.
Thus to show that a given formula of RMCF + is a theorem, one can prove that its negation is unsatisfiable by any RMCF +-assignment.
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Schwartz, J.T., Cantone, D., Omodeo, E.G. (2011). A Survey of Inference Mechanisms. In: Computational Logic and Set Theory. Springer, London. https://doi.org/10.1007/978-0-85729-808-9_3
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