Jens Otten

leanCoP: lean connection-based theorem proving

The Prolog program prove (M,I) : - append (Q, [C|R], M), +member (-_, C), append(Q,R,S), prove([!],[[-!|C] |S],[],I). prove ([],_,_,_). prove([L|C],M,P,I) :- (-N=L; -L=N) -> (member(N,P); append(Q,[D|R],M), copy_term(D,E), append(A,[N|B],E), append(A,B,F), (D==E -> append(R,Q,S); length(P,K), K<I, append(R,[D|Q],S)), prove(F,S,[L|P],I)), prove(C,M,P,I). implements a theorem prover for classical first-order (clausal) logic which is based on the connection calculus. It is sound and complete (provided that an arbitrarily large I is iteratively given), and demonstrates a comparatively strong performance.

A Connection Based Proof Method for Intuitionistic Logic

We present a proof method for intuitionistic logic based on Wallens matrix characterization. Our approach combines the connection calculus and the sequent calculus. The search technique is based on notions of paths and connections and thus avoids redundancies in the search space. During the proof search the computed first-order and intuitionistic substitutions are used to simultaneously construct a sequent proof which is more human oriented than the matrix proof. This allows to use our method within interactive proof environments. Furthermore we can consider local substitutions instead of global ones and treat substitutions occurring in different branches of the sequent proof independently. This reduces the number of extra copies of formulae to be considered.

T-String Unification: Unifying Prefixes in Non-classical Proof Methods

For an efficient proof search in non-classical logics, particular in intuitionistic and modal logics, two similar approaches have been established: Wallens matrix characterization and Ohlbachs resolution calculus. Beside the usual term-unification both methods require a specialized string-unification to unify the so-called prefixes of atomic formulae (in Wallens notation) or world-paths (in Ohlbachs notation). For this purpose we present an efficient algorithm, called T-String-Unification, which computes a minimal set of most general unifiers. By transforming systems of equations we obtain an elegant unification procedure, which is applicable to the intuitionistic logic J and the modal logic S4. With some modifications we are also able to treat the modal logics D, K, D4, K4, S5, and T. We explain our method by an intuitive graphical presentation, prove correctness, completeness, minimality, and termination and investigate its complexity.

A Uniform Proof Procedure for Classical and Non-Classical Logics

We present a proof procedure for classical and non-classical logics. The proof search is based on the matrix-characterization of validity where an emphasis on paths and connections avoids redundancies occur- ring in sequent or tableaux calculi. Our uniform path-checking algorithm operates on arbitrary (non-normal form) formulae and generalizes Bibels connection method for classical logic and formulae in clause-form. It can be applied to intuitionistic and modal logics by modifying the component for testing complementarity of connected atoms. Besides a short and el- egant path-checking procedure we present a specialized string-unification algorithm which is necessary for dealing with non-classical logics.

ileanTAP: An Intuitionistic Theorem Prover

We present a Prolog program that implements a sound and complete theorem prover for first-order intuitionistic logic. It is based on free-variable semantic tableaux extended by an additional string unification to ensure the particular restrictions in intuitionistic logic. Due to the modular treatment of the different logical connectives the implementation can easily be adapted to deal with other non-classical logics.

linTAP: A Tableau Prover for Linear Logic

linTAP is a tableau prover for the multiplicative and exponential fragment M?LL of Girards linear logic. It proves the validity of a given formula by constructing an analytic tableau and ensures the linear validity using prefix unification. We present the tableau calculus used by linTAP, an algorithm for prefix unification in linear logic, the linTAP implementation, and some experimental results obtained with linTAP.

A Multi-level Approach to Program Synthesis

We present an approach to a coherent program synthesis system which integrates a variety of interactively controlled and automated techniques from theorem proving and algorithm design at different levels of abstraction. Besides providing an overall view we summarize the individual research results achieved in the course of this development.

Guiding Program Development Systems by a Connection Based Proof Strategy

We present an automated proof method for constructive logic based on Wallens matrix characterization for intuitionistic validity. The proof search strategy extends Bibels connection method for classical predicate logic. It generates a matrix proof which will then be transformed into a proof within a standard sequent calculus. Thus we can use an efficient proof method to guide the development of constructive proofs in interactive proof/program development systems.

Matrix-based Constructive Theorem Proving

Formal methods for program verification, optimization, and synthesis rely on complex mathematical proofs, which often involve reasoning about computations. Because of that there is no single automated proof procedure that can handle all the reasoning problems occurring during a program derivation or verification. Instead, one usually relies on proof assistants like NuPRL (Constable et al., 1986), Coq (Dowek and et. al, 1991), Alf (Altenkirch et al., 1994) etc., which are based on very expressive logical calculi and support interactive and tactic controlled proof and program development. Proof assistants, however, suffer from a very low degree of automation, since all their inferences must eventually be based on sequent or natural deduction rules. Even proof parts that rely entirely on predicate logic can seldomly be found automatically, as there are no complete proof search procedures embedded into these systems. It is therefore desirable to extend the reasoning power of proof assistants by integrating well-understood techniques from automated theorem proving.

Connection-Based Proof Construction in Linear Logic

We present a matrix characterization of logical validity in the multiplicative fragment of linear logic. On this basis we develop a matrix-based proof search procedure for this fragment and a procedure which translates the machine-found proofs back into the usual sequent calculus for linear logic. Both procedures are straightforward extensions of methods which originally were developed for a uniform treatment of classical, intuitionistic and modal logics. They can be extended to further fragments of linear logic once a matrix characterization has been found.