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Cutting Boards. The concrete computation of these solutions can thus be postponed or omitted. Pre-unification is utilized in all the resolution based theorem provers mentioned in Section 4. Pattern unification refers a small subset of unification problems, first studied by Miller , whose identification has been important for the construction of practical systems.
Thus, existentially quantified variables cannot be applied to general terms but a very restricted set of bound variables. Pattern unification, like first-order unification, is decidable and most general unifiers exist for solvable problems. The Unifying Principle was introduced in Smullyan see also Smullyan as a tool for deriving a number of basic metatheorems about first-order logic in a uniform way. We outline these extensions in some more detail below. We first give a definition and then state the principle. Here is a typical application of the Unifying Principle.
A primary motivation was to support refutational completeness investigations for the proof calculi underlying the emerging higher-order automated theorem provers see Section 4 below. Analogous to before, a distinction has been made between extensional type theory with defined equality as in Section 1. A first attempt towards a Unifying Principle for extensional type theory with primitive equality is presented in Kohlhase The conditions given there, which are still incomplete [ 1 ] , were subsequently modified and complemented as follows:.
The saturation condition However, since this condition is related to the proof theoretic notion of cut-elimination, it limits the utility of the principle in completeness proofs for machine-oriented calculi.
One option is to introduce a reflexivity and substitutivity condition. An alternative is to combine a reflexivity condition with a condition connecting primitive with defined equality, so that the substitutivity condition follows. The principle is applied to obtain completeness proofs for an associated range of natural deduction calculi. The saturation condition is still used in this work. The two weaker conditions, termed mating and decomposition, are easier to demonstrate than saturation in completeness proofs for machine-oriented calculi.
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They are omitting some type information in the second one and abusing notation :. Moreover, the notions of cut-simulation and cut-strong axioms are introduced in this work, and the need for omitting defined equality and for eliminating cut-strong axioms such as extensionality, description, choice and induction in machine-oriented calculi e. An expansion proof is a generalization of the notion of a Herbrand expansion of a theorem of first-order logic; it provides a very elegant, concise, and nonredundant representation of the relationship between the theorem and a tautology which can be obtained from it by appropriate instantiations of quantifiers and which underlies various proofs of the theorem.
In Brown and , this concept is generalized to that of an extensional expansion proof to obtain an analogous theorem involving type theory with extensionality. Since type theory includes first-order logic, it is no surprise that most systems of type theory are undecidable.
However, one may look for solvable special cases of the decision problem. Information about some very special cases of this decision problem may be found in Andrews , and we now summarize this. By contrast, the corresponding decision problem in first-order logic with function symbols is known to be solvable Maslov Compared, e. Obviously, the challenges are also much bigger than in first-order logic. Moreover, remember that unification, which constitutes a very important control and filter mechanism in first-order theorem proving, is undecidable in general in type theory; see Section 3.
On the positive side, however, there is a chance to find significantly shorter proofs than in first-order logic. This is well illustrated with a small, concrete example in Boolos The challenges include. For the time being, however, we provide only a selection of historical and more recent references for the interested reader see also Section 5 below :. See Other Internet References section below for links to further info on these and other provers mentioned later. The majority of the above systems focused at least initially on interactive proof and provided rather limited support for additional proof automation.
Full proof automation was pioneered, in particular, by the TPS project. Progress was made in the nineties, when other projects started similar activities, respectively, enforced theirs. However, the resource investments and achievements were lacking much behind those seen in first-order theorem proving. The TPS prover Andrews et al. When searching for a proof automatically, TPS first searches for an expansion proof Miller or an extensional expansion proof Brown , of the theorem.
Part of this process involves searching for acceptable matings Andrews , Bishop The behavior of TPS is controlled by sets of flags, also called modes. A simple scheduling mechanism is employed in the latest versions of TPS to sequentially run a about fifty modes for a limited amount of time. LEO was the first prover to implement calculus rules for extensionality to avoid cut-simulation effects.
LEO-II inherits and adapts them, and provides additional calculus rules for description and choice. Therefore, LEO-II progressively launches time limited calls with these clauses to a first-order theorem prover, and when the first-order prover reports a refutation, LEO-II also terminates. Parts of these ideas were already implemented in the predecessor LEO.
An initial tableau branch is formed from the assumptions of a conjecture and negation of its conclusion. From that point on, Satallax tries to determine unsatisfiability or satisfiability of this branch. Satallax progressively generates higher-order formulas and corresponding propositional clauses. Satallax uses the SAT solver MiniSat as an engine to test the current set of propositional clauses for unsatisfiability. If the clauses are unsatisfiable, the original branch is unsatisfiable.
Satallax provides calculus rules for extensionality, description and choice. If there are no quantifiers at function types, the generation of higher-order formulas and corresponding clauses may terminate. In that case, if MiniSat reports the final set of clauses as satisfiable, then the original set of higher-order formulas is satisfiable by a standard model in which all types are interpreted as finite sets.
However, in order to ease user interaction several automatic proof tactics have been added over the years. The agsyHOL prover is based on a generic lazy narrowing proof search algorithm. Backtracking is employed and a comparably small search state is maintained. The prover outputs proof terms in sequent style which can be verified in the Agda system. The system outputs proof terms which are accepted as proofs after the addition of a few definitions by the Coq proof assistant. The prover employs axioms for functional extensionality, choice, and excluded middle. Boolean extensionality is not supported.
In addition to axioms, a small library of basic lemmas is employed. The system has put an emphasis on the implementation of an efficient set of underlying data structures, on simplification routines and on heuristic rewriting. In the tradition of its predecessors, Leo-III cooperates with first-order reasoning tools using translations to many-sorted first-order logic.
The prover accepts every common TPTP syntax dialect and is thus very widely applicable. Recently, the prover has also been extended to natively supports almost every normal higher-order modal logic. Zipperposition Bentkamp et al. The most advanced finite model finding support is currently realized in the systems Nitpick, Nunchaku and Refute. These tools have been integrated with the Isabelle proof assistant. The systems are particularly valuable for exposing errors and misconceptions in problem encodings, and for revealing bugs in the THF theorem provers.
Pioneering work on this was done by Richard Montague.