Hans Jürgen Ohlbach

A Resolution Calculus for Modal Logics

A syntax transformation is presented that eliminates the modal logic operators from modal logic formulae by shifting the modal context information to the term level. The formulae in the transformed syntax can be brought into conjunctive normal form such that a clause based resolution calculus without any additional inference rule, but with special modal unification algorithms, can be defined. The method works for first-order modal logics with the two operators □ and ♦ and with constant-domain Kripke semantics where the accessibility relation is serial and may have any combination of the following properties: reflexivity, symmetry, transitivity. In particular the quantified versions of the modal systems T, S4, S5, B, D, D4 and DB can be treated. Extensions to non-serial and varying-domain systems are possible, but not presented here.

Semantics-Based Translation Methods for Modal Logics

A general framework for translating logical formulae from one logic into another logic is presented. The framework is instantiated with two different approaches to translating modal logic formulae into predicate logic. The first one, the well known ‘relational’ translation makes the modal logic’s possible worlds structure explicit by introducing a distinguished predicate symbol to represent the accessibility relation. In the second approach, the ‘functional’ translation method, paths in the possible worlds structure are represented by compositions of functions which map worlds to accessible worlds. On the syntactic level this means that every flexible symbol is parametrized with particular terms denoting whole paths from the initial world to the actual world. The ‘target logic’ for the translation is a first-order many-sorted logic with built in equality. Therefore the ‘source logic’ may also be first-order many-sorted with built in equality. Furthermore flexible function symbols are allowed. The modal operators may be parametrized with arbitrary terms and particular properties of the accessibility relation may be specified within the logic itself. Key worlds: Modal Logic; Translation of Logics; Calculi for Logics Journal of Logic and Computation, Vol. 1, no 6, 1991, pp 691-746

Translating Graded Modalities into Predicate Logic

From Minsky’s early frame systems, which were defined purely operationally, and Brachman’s kl-one knowledge representation system [4, 35] to the language AℒC of Schmidt-Schaus and Smolka’s [28] paper there has been a continuous trend in designing knowledge representation systems more and more according to logical principles with clear syntax and semantics and logical inferences as basic operations. AℒC in particular is a language with the usual logical connectives ⊓, ⊔, ⌝ and the additional constructs (all R C) and (some R C). For example, the following is an AℒC definition which defines a ‘concept’ proud-father as a father all of whose children are successful persons.

Link inheritance in abstract clause graphs

Clause graphs, as they were defined in the 1970s, are graphs representing first order formulas in conjunctive normal form: the nodes are labelled with literals and the edges (links) connect complementary unifiable literals, i.e. they provide an explicit representation of the resolution possibilities. This report describes a generalization of this concept, called abstract clause graphs. The nodes of abstract clause graphs are still labelled with literals, the links however connect literals that are ‘unifiable’ relative to a given relation between literals. This relation is not explicitely defined, only certain abstract properties are required. For instance the existence of a special purpose unification algorithm is assumed, which computes substitutions, the application of which makes the relation hold for two literals.When instances of already existing literals are added to the graph (e.g. due to resolution or factoring), the links to the new literals are derived from the links of their ancestors. An inheritance mechanism for such links is presented which operates only on the attached substitutions and does not have to unify the literals. The definition of abstract clause graphs and the theory about link inheritance is general enough to provide a framework so that as new ideas are proposed for graph based theorem provers, the methodology for both implementing links and proving their properties will be readily available.

A note on assumptions about Skolem functions

Skolemization is not an equivalence preserving transformation. For the purposes of refutational theorem proving it is sufficient that skolemization preserves satisfiability and unsatisfiability. Therefore there is sometimes some freedom in interpreting Skolem functions in a particular way. We show that in certain cases it is possible to exploit this freedom for simplifying formulae considerably. Examples for cases where this occurs systematically are the relational translation from modal logics to predicate logic and the relativization of first-order logics with sorts.

Reduction rules for resolution-based systems

Abstract Inference rules for resolution-based systems can be classified into deduction rules, which add new objects, and reduction rules, which remove objects. Traditional reduction rules like subsumption do not actively contribute to a solution, but they help to avoid redundancies in the search space. We present a number of advanced reduction rules, which can cope with high degrees of redundancy and play a distinctly active part because they find trivial solutions on their own and thus relieve the control component for the deduction rules from low level tasks. We describe how these reduction rules can be implemented with reasonable efficiency in a clause graph resolution system, but they are not restricted to this particular representation.

SCAN - Elimination of Predicate Quantifiers

Some higher-order formulas are equivalent to formulas of first-order predicate logic or even propositional logic. In applications where formulas of higherorder predicate logic occur naturally it is very useful to determine whether the given formula is in fact equivalent to a simpler formula of first-order or prepositional logic. Typical applications where this occurs are predicate minimization by circumscription, correspondence theory in non-classical logics and simple versions of set theory. In these areas we are faced with formulas of second-order predicate logic with existentially or universally quantified predicate variables and want to simplify them by computing equivalent first-order formulas.

Towards automating duality

Abstract Dualities between different theories occur frequently in mathematics and logic—between syntax and semantics of a logic, between structures and power structures, between relations and relational algebras, to name just a few. In this paper, we show for the case of structures and power structures how corresponding properties of the two related structures can be computed fully automatically by means of quantifier elimination algorithms and predicate logic theorem provers. We illustrate the method with some examples that were computed with the Otter theorem prover.

Computer support for the development and investigation of logics

The development and investigation of application{oriented logics comprises many aspects and problems. For a few of them some computer support is possible which frees the investigator from sometimes quite complex computations. This paper gives an overview about some developments in this area. In particular, we consider the correspondences between axiomatic and semantic speciications of a logic and the problem of nding one from the other by means of automated theorem provers and quantiier elimination algorithms. Other topics adressed in this paper are reasoning in Hilbert systems, the investigation of the expressiveness of a logic and the axiomatizability of semantic conditions. For the technical details of the methods and the proofs I refer to the original papers.

From a Hilbert Calculus to its Model Theoretic Semantics

There are different ways of constructing a logic. One possibility is to define a Hilbert calculus, i.e. a kind of grammar that produces all formulae to be considered true. A logic can also be defined by a model theoretic semantics for the logical connectives in the language. In this paper a general theory is presented for the transition from a Hilbert calculus to its model theoretic semantics such that soundness and completeness are automatically guaranteed.