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Abstract:
A general theory of deduction systems is presented. The theory is illustrated with deduction systems based on the resolution calculus, in particular with clause graphs.
This theory distinguishes four constituents of a deduction system:
\begin{itemize}
\item
the logic, which establishes a notion of semantic entailment;
\item
the calculus, whose rules of inference provide the syntactic counterpart of entailment;
\item
the logical state transition system, which determines the representation of formulae or sets of formulae together
with their interrelationships, and also may allow additional operations reducing the search space;
\item
the control, which comprises the criteria used to choose the most promising from among all applicable inference
steps.
\end{itemize}
Much of the standard material on resolution is presented in this framework.
For the last two levels many alternatives are discussed. Appropriately adjusted notions of soundness, completeness, confluence, and Noetherianness are introduced in order to characterize the properties of particular deduction systems. For more complex deduction systems, where logical and topological phenomena interleave, such properties can be far from obvious.
