Subsumption Algorithms for Three-Valued Geometric Resolution

In the implementation of geometric resolution, the most costly operation is subsumption (or matching). In the matching problem, one has to decide for a three-valued, geometric formula, whether this formula is false in a given interpretation. The formula contains only atoms with variables, equality, and existential quantifiers. The interpretation contains only atoms with constants, which are assumed to be distinct.

Because matching is not restricted by term structure, matching for geometric resolution is a hard problem. We translate the matching problem into a generalized constraint satisfaction problem, and give an algorithm that solves it efficiently. The algorithm uses learning teachings, similar to clause learning in propositional logic. After that, we adapt the algorithm in such a way that it finds solutions that use a minimal subset of the interpretation.

The techniques presented in this paper may also have applications in constraint solving.

The paper is accepted at IJCAR 2016. A preprint is available here. I also gave talk at our local seminar about this topic.


Theorem Proving for Classical Logic with Partial Functions by Reduction to Kleene Logic

Partial functions are abundant in mathematics and program specifications. Despite this, their importance has been mostly ignored in automated theorem proving. In this paper, we develop a theorem proving strategy for Partial Classical Logic (PCL). Proof search takes place in Kleene Logic. We show that PCL theories can be translated into equivalent sets of formulas in Kleene logic. For proof search in Kleene logic, we use a three-valued adaptation of geometric resolution. We prove that the procedure is sound and complete.

The paper has been accepted for publication in the Journal of Logic and Computation. This is a preprint.


Classical Logic with Partial Functions

In this paper, I introduce a semantics for classical logic with partial functions, in which ill-typed formulas are guaranteed to have no truth value, so that they cannot be used in any form of reasoning. The semantics makes it possible to mix reasoning about types and preconditions with reasoning about other properties. In this way, it is possible to treat partial functions that have preconditions of unlimited complexity. We show that, in spite of its increased complexity, the semantics is still a natural generalization of first-order logic with simple types. If one does not use the increased expressivity, the type system is not stronger than classical logic with simple types. I define two sequent calculi for the new semantics, and prove that they are sound and complete. The first calculus follows the semantics closely, and hence its completeness proof is fairly straightforward. The second calculus is further away from the semantics, but more suitable for practical use because it has better proof theoretic properties. Its completeness can be shown by proving that proofs from the first calculus can be translated.

This is a preprint of the paper. The final version was published in the Journal of Automated Reasoning, and can be obtained from Springer .


Classical Logic with Partial Functions

We introduce a semantics for classical logic with partial functions. We believe that this semantics is natural. When a formula contains a subterm in which a function is applied outside of its domain, our semantics ensures that the formula has no truth-value, so that it cannot be used for reasoning. The semantics relies on order of formulas. In this way, it is able to ensure that functions and predicates are properly declared before they are used. We define a sequent calculus for the semantics, and prove that this calculus is sound and complete for the semantics. The paper appeared in the proceedings of IJCAR 2010.

This is a preprint.


A small Framework for Proof Checking

We describe a small framework in which first-order theorem provers can be used for the verification of mathematical theories. The verification language is designed in such a way that the use of higher-order constructs is minimized. In this way, we expect to be able to take advantage of the first order theorem prover as much as possible.

ps or pdf .


Geometric Resolution: A Proof procedure based on Finite Model Search

We present a proof procedure that is complete for first-order logic, but which can also be used when searching for finite models. The procedure uses a normal form which is based on geometric formulas. For this reason we call the procedure geometric resolution. We expect that the procedure can be used as an efficient proof search procedure for first-order logic. In addition, the procedure can be implemented in such a way that it is complete for finding finite models. Download ps or pdf .

Using Resolution as a Decision Procedure (Habilitation Thesis)

This habilitation thesis is based on five papers that deal with resolution decision procedures. Resolution is a well-known technique for first-order theorem proving. It is complete, but it does not terminate in general, when there exists no proof. However, in many cases, resolution can be modified in such a way that it becomes a decision procedure for certain subclasses of first-order logic.

We have studied several aspects related to the use of resolution as decision procedure. The first two papers introduce resolution decision procedures for the guarded fragment with and without equality. The third paper builds on the decision procedure for the guarded fragment. Using the decision procedure, modal logics can be decided by translation into the guarded fragment with the relational translation. We have extended the standard relational translation of modal logics, so that more modal logics can be translated into the guarded fragment. In particular, modal logic S4 can now be translated. Before, only translations into extensions of the guarded fragment existed. The fourth paper shows that a standard refinement of resolution, which uses a liftable order, can be used to obtain a decison procedure for the E-plus class. This was posed as an open problem in (Resolution Methods for the Decision Problem, C. Fermueller, A. Leitsch, T. Tammet, N. Zamov, Springer Verlag 1993)

The fifth paper introduces a resolution-based decision procedure for the two-variable fragment with equality. The procedure consists of three stages: Saturation under resolution, elimination of equality, and one more time saturation under resolution.

Download ps or pdf .

Sonstige Arbeiten (Additional Works)

In addition to the main habilitation thesis, one has to submit 5 additional publications (sonstige Arbeiten). They are collected in a supplement, which you may download either as ps or pdf .


Deciding Regular Grammar Logics with Converse through First-Order Logic

(Joint work with Stephane Demri, LSV Paris, France)

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into $ \GF^2, $ which is the intersection of the guarded fragment and the $ 2 $-variable fragment of first-order logic. This translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. A consequence of the translation is that the general satisfiability problem for regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Using the same method, we show how some other modal logics can be naturally translated into $ \GF^2, $ including nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply $ \GF^2 $ without extra machinery such as fixed point-operators. Download as ps or pdf .

Implementing the Clausal Normal Form Transformation with Proof Generation

The paper describes how I intend to implement the clausal normal form transformation with proof generation described in the paper below. The paper introduces a convenient datastructure for sequent proofs, which allows easy proof checking and proof normalization. Download ps or pdf .

Translation of Resolution Proofs into Short First-Order Proofs without Choice Axioms

We present a way of transforming a resolution proof containing Skolemization into a natural deduction proof of the same formula but not using Skolemization. The size of the proof increases only moderately (polynomially). This makes it possible to translate the output of a resolution theorem prover into a purely first-order proof that is moderate in size. Download draft as ps or pdf .

Subsumption of concepts in FL0 for (cyclic) terminologies with respect to descriptive semantics is PSPACE-complete.

The paper solves an open problem concerning the complexities of the simple description logic FL0. FL0 is the description logic which allows only conjunction and universal value restriction. We show that subsumption in this logic is PSPACE-complete under the descriptive semantics. The descriptive semantics is the semantics that one obtains if one interprets definitions as simple first-order equivalences. (The other two possibilities are to interpret definitions as least fixed points, or greatest fixed points) Download ps or pdf .

Deciding the Guarded Fragments by Resolution

(joint work with Maarten de Rijke, University of Amsterdam)

We give resolution based decision procedures for both the strictly guarded fragment and the loosely guarded fragment of first-order logic. We prove that the decision procedures are in 2EXPTIME, which is theoretically optimal. Download ps or pdf . The article in the JSC contains a few crucial printer errors which are corrected here.


Automated Proof Construction in Type Theory using Resolution

(joint work with Marc Bezem, Dimitri Hendriks)

We provide techniques to integrate resolution logic with equality in type theory. The results may be rendered as follows:
  1. A clausification procedure in type theory, equipped with a correctness proof, all encoded using higher-order primitive recursion.
  2. A novel representation of clauses in minimal logic such that the $ \lambda $-representation of resolution steps is linear in the size of the premisses.
  3. Availability of the power of resolution theorem provers in interactive proof construction systems based on type theory
ps or pdf .

Extraction of Proofs from the Clausal Normal Form Transformation

We give techniques for extracting proofs from the Clausal Normal Form transformation. We discuss and solve three technical problems:
  1. How to handle the introduction of definitions and Skolem functions.
  2. How to generate short (linear size) proofs.
  3. How to handle optimized Skolemization. We reduce it to standard Skolemization.
Download as ps or pdf .


Splitting through New Proposition Symbols

This paper presents ways of simulating splitting through new propositional symbols. The contribution of the paper is that the method presented here does not loose backward redundancy. We give a general method of obtaining a calculus with splitting by adjoining splitting symbols to an arbitrary resolution refinement. We then prove a relative completeness theorem: Every saturated set of clauses of the resolution calculus with adjoined proposition symbols contains a saturated set of the original resolution refinement without adjoined symbols. Download as ps or pdf .


Datastructures for Resolution

(This is unpublished)

The paper presents the benchmark tests out of which Bliksem is grown. The first part compares various ways of implementing terms relative to unification and matching. The second part compares different ways of implementing substitutions. The third part compares three ways of implementing discrimination trees. The paper ends with a description of a subsumption algorithm. Download ps or pdf .


Translation of S4 and K5 into GF and 2VAR

(This is unpublished)

This short note gives a method of translating S4 and K5 into the intersection of GF and 2VAR through the introduction of new symbols. Download ps or pdf .

A Superposition Decision Procedure for the Guarded Fragment with Equality

We give a decision procedure for the guarded fragment with equality. The procedure is based on resolution with superposition. The relevance of the guarded fragment lies in the fact that many modal logics can be translated into it. In this way the guarded fragment acts as a framework explaining some of the nice properties of these modal logics. By constructing an implementable decision procedure for the guarded fragments we define an effective procedure for deciding these modal logics. It is surprising to see that one does not need any sophisticated simplification and redundancy elimination method to make superposition terminate on the class of clauses that is obtained from the clausification of guarded formulas. Yet the decision procedure obtained is optimal with regard to time complexity.


A Classification of Non-Liftable Orders for Resolution

This paper tries to solve the problem of completeness of resolution restricted by non-liftable orders that cannot be modelled by the resolution game. Various types of non-liftability are distinguished and combined with various corresponding types of subsumption. All but one of the meaningful combinations are solved. The remaining case is still open.

Download paper as pdf or ps.gz . The paper was published at CADE 97.


Thesis (October 1995, Delft University of Technology, the Netherlands)

The resolution game

Main contribution of the thesis is the resolution game . The resolution game is a game that can be played between two players, based on a set of ordered clauses. The opponent tries to derive the empty clause, using ordered resolution and factoring. The defender tries to disturb the opponent by changing the order at arbitrary moments. The main theorem states that under certain restrictions, the opponent has a winning strategy (i.e. can always derive the empty clause) if and only if the clause set is unsatisfiable.

Non-liftable Orders

The resolution game can be used for proving completeness of resolution with non-liftable orders. Non-liftable orders are orders that are not preserved by substition, i.e. they do not always satisfy A < B implies Theta(A) < Theta(B). The standard method of proving completeness (making use of lifting) does not work for such orders. With the resolution game, completeness can be proven for some such orders.

Resolution Based Decision Procedures

The main application of non-liftable orders is in resolution based decision procedures . These are resolution restrictions that are guaranteed to terminate on certain subsets of first-order logic. In my thesis, two decision procedures are developed, one for the E+ -class, and one for the two-variable fragment (without equality).

Maslov's Method

Another (smaller) contribution of the thesis is that it clarifies the relation between Maslov's inverse method and resolution. Most resolution calculi work top-down. They decompose a formula into clauses, resolving upon conflicts between clauses.

Using Maslov's method, it is very easy to obtain a resolution calculus for every logic that has a cut-free, complete sequent calculus. However, the resolution calculi that are obtained in this way are bottom-up, i.e. working from the axioms towards the goal.

In my thesis, I show that many sequent calculi have a reversal , which is obtained by exchanging axioms and goal, and reversing the direction of the rules. If one applies Maslov's method on the reversed sequent calculus, one obtains a top-down resolution calculus.

Download thesis as pdf or ps .


Resolution Games and Non-Liftable Resolution Orderings

This paper introduces the resolution game and proves the completeness of resolution with orders that are descending. An order < is descending if always A Theta < A, whenever A Theta is a real instance of A. The completeness proof of the resolution game in this paper is very rude. It is based on the analysis of saturated sets. The first completeness proof, which is given in my thesis, is uses intricate combinations of winning strategies, and is much nicer, but also more complicated.

Download paper as pdf or ps.gz . The paper was published at CSL 94.