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Unification under a mixed prefix
 Journal of Symbolic Computation
, 1992
"... Unification problems are identified with conjunctions of equations between simply typed λterms 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 pr ..."
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Cited by 128 (13 self)
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Unification problems are identified with conjunctions of equations between simply typed λterms 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 preunifiers described by Huet is easily extended to the mixed prefix setting, although solving flexibleflexible 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.
Natural Deduction as HigherOrder Resolution
 Journal of Logic Programming
, 1986
"... An interactive theorem prover, Isabelle, is under development. In LCF, each inference rule is represented by one function for forwards proof and another (a tactic) for backwards proof. In Isabelle, each inference rule is represented by a Horn clause. ..."
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Cited by 55 (8 self)
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An interactive theorem prover, Isabelle, is under development. In LCF, each inference rule is represented by one function for forwards proof and another (a tactic) for backwards proof. In Isabelle, each inference rule is represented by a Horn clause.
Birewrite systems
, 1996
"... In this article we propose an extension of term rewriting techniques to automate the deduction in monotone preorder theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a birewrite system 〈R⊆, R⊇〉, that is, a pair of rewrite relations ..."
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Cited by 29 (9 self)
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In this article we propose an extension of term rewriting techniques to automate the deduction in monotone preorder theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a birewrite system 〈R⊆, R⊇〉, that is, a pair of rewrite relations −−− → R ⊆ and −−− → R ⊇ , and seek a common term c such that a −−−→ R ⊆ c and b −−−→
Extensional higherorder resolution
 In Kirchner and Kirchner [KK98
, 1998
"... Abstract. In this paper we present an extensional higherorder resolution calculus that is complete relative to Henkin model semantics. The treatment of the extensionality principles – necessary for the completeness result – by specialized (goaldirected) inference rules is of practical applicabilit ..."
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Abstract. In this paper we present an extensional higherorder resolution calculus that is complete relative to Henkin model semantics. The treatment of the extensionality principles – necessary for the completeness result – by specialized (goaldirected) inference rules is of practical applicability, as an implentation of the calculus in the LeoSystem shows. Furthermore, we prove the longstanding conjecture, that it is sufficient to restrict the order of primitive substitutions to the order of input formulae. 1
HigherOrder Automated Theorem Proving
, 1998
"... 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 ..."
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Cited by 6 (2 self)
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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 bhnormal 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...
unknown title
"... We develop an efficient representation and a preunification algorithm in the style of Huet for the linear λcalculus λ→−◦& ⊤ which includes intuitionistic functions (→), linear functions (−◦), additive pairing (&), and additive unit (⊤). Applications lie in proof search, logic programming, a ..."
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We develop an efficient representation and a preunification algorithm in the style of Huet for the linear λcalculus λ→−◦& ⊤ which includes intuitionistic functions (→), linear functions (−◦), additive pairing (&), and additive unit (⊤). Applications lie in proof search, logic programming, and logical frameworks based on linear type theories. We also show that, surprisingly, a similar preunification algorithm does not exist for certain sublanguages. 1
Birewrite Systems † JORDI LEVY ‡ § AND JAUME AGUST Í¶�
, 1995
"... In this article we propose an extension of term rewriting techniques to automate the deduction in monotone preorder theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a birewrite system 〈R⊆,R⊇〉, that is, a pair of rewrite relations − ..."
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In this article we propose an extension of term rewriting techniques to automate the deduction in monotone preorder theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a birewrite system 〈R⊆,R⊇〉, that is, a pair of rewrite relations −−−→ R ⊆ and −−−→ R ⊇ , and seek a common term c such that a −−−→ R c and b −−−→ R c. Each component of the birewrite system −−−→
CHRISTOPH BENZMÜLLER COMPARING APPROACHES TO RESOLUTION BASED HIGHERORDER THEOREM PROVING
"... ABSTRACT. We investigate several approaches to resolution based automated theorem proving in classical higherorder logic (based on Church’s simply typed λcalculus) and discuss their requirements with respect to Henkin completeness and full extensionality. In particular we focus on Andrews ’ higher ..."
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ABSTRACT. We investigate several approaches to resolution based automated theorem proving in classical higherorder logic (based on Church’s simply typed λcalculus) and discuss their requirements with respect to Henkin completeness and full extensionality. In particular we focus on Andrews ’ higherorder resolution (Andrews 1971), Huet’s constrained resolution (Huet 1972), higherorder Eresolution, and extensional higherorder resolution (Benzmüller and Kohlhase 1997). With the help of examples we illustrate the parallels and differences of the extensionality treatment of these approaches and demonstrate that extensional higherorder resolution is the sole approach that can completely avoid additional extensionality axioms. 1.
On the Limits of SecondOrder Unification
"... SecondOrder Unification is a problem that naturally arises when applying automated deduction techniques with variables denoting predicates. The problem is undecidable, but a considerable effort has been made in order to find decidable fragments, and understand the deep reasons of its complexity. Tw ..."
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SecondOrder Unification is a problem that naturally arises when applying automated deduction techniques with variables denoting predicates. The problem is undecidable, but a considerable effort has been made in order to find decidable fragments, and understand the deep reasons of its complexity. Two variants of the problem, Bounded SecondOrder Unification and Linear SecondOrder Unification –where the use of bound variables in the instantiations is restricted–, have been extensively studied in the last two decades. In this paper we summarize some decidability/undecidability/complexity results, trying to focus on those that could be more interesting for a wider audience, and involving less technical details. 1