Abstract not found

Unification problems are identified with conjunctions of equations between simply typed λ-terms where free variables in the equations can be universally or existentially quantified. Two schemes for simplifying quantifier alternation, called Skolemization and raising (a dual of Skolemization), are presented. In this setting where variables of functional type can be quantified and not all types contain closed terms, the naive generalization of firstorder Skolemization has several technical problems that are addressed. The method of searching for pre-unifiers described by Huet is easily extended to the mixed prefix setting, although solving flexible-flexible unification problems is undecidable since types may be empty. Unification problems may have numerous incomparable unifiers. Occasionally, unifiers share common factors and several of these are presented. Various optimizations on the general unification search problem are as discussed. 1.

HOL is a computer system for generating proofs in a version of higher order logic derived from Church's simple theory of types. This paper is the original description of the logic underlying the HOL system. Since it was written the system has changed enormously, but the logic has been relatively stable. The main changes have been in the organisation of theories and some enhancements to the syntactic resources of the logic (e.g. records and userdefined mixfix syntax are now supported)...

The unification of simply typed λ-terms modulo the rules of β- and η-conversions is often called “higher-order ” unification because of the possible presence of variables of functional type. This kind of unification is undecidable in general and if unifiers exist, most general unifiers may not exist. In this paper, we show that such unification problems can be coded as a query of the logic programming language Lλ in a natural and clear fashion. In a sense, the translation only involves explicitly axiomatizing in Lλ the notions of equality and substitution of the simply typed λ-calculus: the rest of the unification process can be viewed as simply an interpreter of Lλ searching for proofs using those axioms. 1

We use the general scheme of building resolution calculi (also called the inverse method) originating from S.Maslov and G.Mints to design and implement a resolution theorem prover for intuitionistic logic. A number of search strategies is introduced and proved complete. The resolution method is shown to be a decision procedure for a new syntactically described decidable class of intuitionistic logic. We compare the search strategies suitable for the resolution method with strategies suitable for the tableau method. The performance of our prover is compared with the performance of a tableau prover for intuitionistic logic presented in [17].

Even though higher-order calculi for automated theorem proving are rather old, tableau calculi have not been investigated yet. This paper presents two free variable tableau calculi for higher-order logic that use higher-order unification as the key inference procedure. These calculi differ in the treatment of the substitutional properties of equivalences. The first calculus is equivalent in deductive power to the machineoriented higher-order refutation calculi known from the literature, whereas the second is complete with respect to Henkin's general models.

Abstract. In this paper we present an extensional higher-order resolution calculus that is complete relative to Henkin model semantics. The treatment of the extensionality principles – necessary for the completeness result – by specialized (goal-directed) inference rules is of practical applicability, as an implentation of the calculus in the Leo-System shows. Furthermore, we prove the long-standing conjecture, that it is sufficient to restrict the order of primitive substitutions to the order of input formulae. 1

This paper describes a tableau-based higher-order theorem prover Hot and an application to natural language semantics. In this application, Hot is used to prove equivalences using world knowledge during higher-order unification (HOU). This extended form of HOU is used to compute the licensing conditions for corrections. 1 Introduction Mechanized reasoning systems have many applications in Computational Linguistics. Based on the observation that some phenomena of natural language can be modeled as deductive processes, firstorder theorem provers or related inference systems have been used for instance in phonology [2], generation [17] and semantic analysis [22]. [11] describes an abductive framework for natural language understanding that includes world knowledge into the semantics construction process. Other approaches use higher-order logics and in particular fi-reduction and higher-order unification (HOU) as inference procedures. Following Montague [18] who has used the typed -calcul...

Consistency Class) Let Ñ S be a class of sets of propositions, then Ñ S is called an abstract consistency class, iff each Ñ S is closed under subsets, and satisfies conditions (1) to (8) for all sets F 2 Ñ S . If it also satisfies (9), then we call it extensional. 1. If A is atomic, then A = 2 F or :A = 2 F. 2. If A 2 F and if B is the bh-normal form of A, then B F 2 Ñ S 2 . 3. If ::A 2 F, then A F 2 Ñ S . 4. If AB2F, then F A 2 Ñ S or F B 2 Ñ S . 5. If :(AB) 2 F, then F :A :B2 Ñ S . 6. If P a A 2 F, then F AB 2 Ñ S for each closed formula B 2 wff a (S). 7. If :P a A 2 F, then F :(Aw a ) 2 Ñ S for any witness constant w a 2 W that does not occur in F. 8. If :(A = a!b B) 2 F, then F :(Aw a = Bw) 2 Ñ S for any witness constant w a 2 W that does not occur in F. 9. If :(A = o B) 2 F, then F[fA;:Bg 2 Ñ S or F[f:A;Bg 2 Ñ S . Here, we treat equality as an abbreviation for Leibniz definition. We call an abstract consistency class saturated, iff for all F 2 Ñ S and all...

Abstract. We present a set of problems that may support the development of calculi and theorem provers for classical higher-order logic. We propose to employ these test problems as quick and easy criteria preceding the formal soundness and completeness analysis of proof systems under development. Our set of problems is structured according to different technical issues and along different notions of semantics (including Henkin semantics) for higher-order logic. Many examples are either theorems or non-theorems depending on the choice of semantics. The examples can thus indicate the deductive strength of a proof system. 1 Motivation: Test Problems for Higher-Order Reasoning Systems Test problems are important for the practical implementation of theorem provers as well as for the preceding theoretical development of calculi, strategies and heuristics. If the test theorems can be proven (resp. the non-theorems cannot) then they ideally provide a strong indication for completeness (resp. soundness). Examples for early publications providing first-order test problems are [21,29,23]. For more than decade now the TPTP library [28] has been developed as a systematically structured electronic repository of