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This paper outlines an implemented system named PROVERB that transforms and abstracts machine-found proofs to natural deduction style proofs at an adequate level of abstraction and then verbalizes them in natural language. The abstracted proofs, originally employed only as an intermediate representation, also prove to be useful for proof planning and proving by analogy.
MUltlog is a system which takes as input the specification of a finitely-valued first-order logic and produces a sequent calculus, a natural deduction system, and a calculus for transforming a many-valued formula to clauses suitable for many-valued resolution. All generated rules are optimized regarding their branching degree. The output is in the form of a scientific paper, written in LATEX.
The CtCoq system is a graphical user-interface using a distributed architecture adapted to the Coq proof system [3]. Basic features provided by this graphical interface are direct manipulation of formulas and commands using the mouse, mathematical notations with an extended character set and colors, menus for guiding users in their manipulations of commands and formulas. More advanced features also...
We establish that there is no polynomial-time general combination algorithm for unification in finitary equational theories, unless the complexity class #P of counting problems is contained in the class FP of function problems solvable in polynomial-time. The prevalent view in complexity theory is that such a collapse is extremely unlikely for a number of reasons, including the fact that the containment...
We consider equational unification and matching problems where the equational theory contains a nilpotent function, i.e., a function f satisfying f(x,x)=0 where 0 is a constant. Nilpotent matching and unification are shown to be JVP-complete. In the presence of associativity and commutativity, the problems still remain NP-complete. But when 0 is also assumed to be the unity for the function f, the...
The first-order theories of finite and rational, constructor and feature trees possess complete axiomatizations and are decidable by quantifier elimination [15, 13, 14, 5, 10, 3, 20, 4, 2]. By using the uniform inseparability lower bounds techniques due to Compton and Henson [6], based on representing large binary relations by means of short formulas manipulating with high trees, we prove that all...
The INKA system is a first-order theorem prover with induction based on the explicit induction paradigm. Since 1986 when a first version of the INKA system was developed there have been many improvements. In this description we will give a short overview of the current system state and its abilities.
XRay is a theorem prover for default logics. Its deductive power is primarily due to our approach of integrating default reasoning into existing model elimination based provers using the well-known PTTP approach. We conceived and integrated a number of enhancements, such as lemma handling, regularity-based truncations of underlying search spaces and a model-based approach to consistency checking.
Many implementations of model elimination perform proof search by iteratively increasing a bound on the total size of the proof. We propose an optimized version of this search mode using a simple divide-and-conquer refinement. Optimized and unoptimized modes are compared, together with depth-bounded and best-first search, over the entire TPTP problem library. The optimized size-bounded mode seems...
It is known that a fixed computation rule, such as used in most logic programming systems, does not suffice for normal logic programs. For instance, SLS resolution, which evaluates programs according to the well-founded semantics, makes use of an oracle to determine the next literal to select in [5], and of ideal parallelism in [8]. Given these limits, it is natural to define a subclass of normal...
In this paper the decidability of unification in pseudo-linear sort theories is shown. In contrast to standard unification, variables range over subsets of the domain described by sorts. The denotations of sorts are fixed by declarations in a sort theory. Then two terms are unifiable, if they are unifiable in the standard sense and the assignments of the unifier satisfy the restrictions on the domain...
Let G be a group on generators A. We investigate C(G,A), the class of all t-complete rewrite systems for G over A, that is convergent inter-reduced rewrite systems for G which can be proved terminating with a total division ordering on A*. Given G, A and an element > of Tot(A), the total division orderings over A, there is a unique convergent inter-reduced rewrite system for G which...
We introduce a new technique for proving termination of term rewriting systems. The technique, a specialization of Zantema's semantic labelling technique, is especially useful for establishing the correctness of transformation methods that attempt to prove termination by transforming term rewriting systems into systems whose termination is easier to prove. We apply the technique to modularity, distribution...
Cancellative abelian monoids encompass abelian groups, but also such ubiquitous structures as the natural numbers or multisets. Both the AC axioms and the cancellation law are difficult for a general purpose theorem prover, as they create many variants of clauses which contain sums. We describe a refined superposition calculus for cancellative abelian monoids which requires neither explicit inferences...
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