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Fragments of First-Order Logic (Hardcover)
Loot Price: R4,191
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Fragments of First-Order Logic (Hardcover)
Series: Oxford Logic Guides
Expected to ship within 12 - 19 working days
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A sentence of first-order logic is satisfiable if it is true in
some structure, and finitely satisfiable if it is true in some
finite structure. The question arises as to whether there exists an
algorithm for determining whether a given formula of first-order
logic is satisfiable, or indeed finitely satisfiable. This question
was answered negatively in 1936 by Church and Turing (for
satisfiability) and in 1950 by Trakhtenbrot (for finite
satisfiability).In contrast, the satisfiability and finite
satisfiability problems are algorithmically solvable for restricted
subsets--or, as we say, fragments--of first-order logic, a fact
which is today of considerable interest in Computer Science. This
book provides an up-to-date survey of the principal axes of
research, charting the limits of decision in first-order logic and
exploring the trade-off between expressive power and complexity of
reasoning. Divided into three parts, the book considers for which
fragments of first-order logic there is an effective method for
determining satisfiability or finite satisfiability. Furthermore,
if these problems are decidable for some fragment, what is their
computational complexity? Part I focusses on fragments defined by
restricting the set of available formulas. Topics covered include
the Aristotelian syllogistic and its relatives, the two-variable
fragment, the guarded fragment, the quantifier-prefix fragments and
the fluted fragment. Part II investigates logics with counting
quantifiers. Starting with De Morgan's numerical generalization of
the Aristotelian syllogistic, we proceed to the two-variable
fragment with counting quantifiers and its guarded subfragment,
explaining the applications of the latter to the problem of query
answering in structured data. Part III concerns logics
characterized by semantic constraints, limiting the available
interpretations of certain predicates. Taking propositional modal
logic and graded modal logic as our cue, we return to the
satisfiability problem for two-variable first-order logic and its
relatives, but this time with certain distinguished binary
predicates constrained to be interpreted as equivalence relations
or transitive relations. The work finishes, slightly breaching the
bounds of first-order logic proper, with a chapter on logics
interpreted over trees.
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