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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 54 (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.
On the Undecidability of SecondOrder Unification
 INFORMATION AND COMPUTATION
, 2000
"... ... this paper, and it is the starting point for proving some novel results about the undecidability of secondorder unification presented in the rest of the paper. We prove that secondorder unification is undecidable in the following three cases: (1) each secondorder variable occurs at most t ..."
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Cited by 33 (16 self)
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... this paper, and it is the starting point for proving some novel results about the undecidability of secondorder unification presented in the rest of the paper. We prove that secondorder unification is undecidable in the following three cases: (1) each secondorder variable occurs at most twice and there are only two secondorder variables; (2) there is only one secondorder variable and it is unary; (3) the following conditions (i)#(iv) hold for some fixed integer n: (i) the arguments of all secondorder variables are ground terms of size <n, (ii) the arity of all secondorder variables is <n, (iii) the number of occurrences of secondorder variables is #5, (iv) there is either a single secondorder variable or there are two secondorder variables and no firstorder variables.
Unification and Matching modulo Nilpotence
 In Proc. CADE13, volume 1104 of LNCS
, 1996
"... . We consider equational unification and matching problems where the equational theory contains a nilpotent function, i.e., a function f satisfying f(x;x) = 0 where 0 is a constant. Nilpotent matching and unification are shown to be NPcomplete. In the presence of associativity and commutativity, t ..."
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Cited by 8 (0 self)
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. We consider equational unification and matching problems where the equational theory contains a nilpotent function, i.e., a function f satisfying f(x;x) = 0 where 0 is a constant. Nilpotent matching and unification are shown to be NPcomplete. In the presence of associativity and commutativity, the problems still remain NPcomplete. But when 0 is also assumed to be the unity for the function f , the problems are solvable in polynomial time. We also show that the problem remains in P even when a homomorphism is added. Secondorder matching modulo nilpotence is shown to be undecidable. Subject area: MECHANISMS: unification 1 Introduction Equational unification is an important computational problem in automated theorem proving. Its usefulness derives from the ability to `build in' many proof steps into the pattern matching algorithm, possibly shortening the search for a proof. As a new practical application, we define a class of set constraints and show that problems in this class ca...
Referential logic of proofs
 Theoretical Computer Science
"... We introduce an extension of the propositional logic of singleconclusion proofs by the second order variables denoting the reference constructors of the type “the formula which is proved by x. ” The resulting Logic of Proofs with References, FLPref, is shown to be decidable, and to enjoy soundness ..."
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Cited by 7 (0 self)
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We introduce an extension of the propositional logic of singleconclusion proofs by the second order variables denoting the reference constructors of the type “the formula which is proved by x. ” The resulting Logic of Proofs with References, FLPref, is shown to be decidable, and to enjoy soundness and completeness with respect to the intended provability semantics. We show that FLPref provides a complete test of admissibility of inference rules in a sound extension of arithmetic. Key words: proof theory, explicit modal logic, single conclusion logic of proofs, proof term, reference, unification, admissible inference rule. 1
THE COMPLEXITY OF MONADIC SECONDORDER UNIFICATION ∗
, 1113
"... Abstract. Monadic secondorder unification is secondorder unification where all function constants occurring in the equations are unary. Here we prove that the problem of deciding whether a set of monadic equations has a unifier is NPcomplete, where we use the technique of compressing solutions us ..."
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Cited by 4 (1 self)
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Abstract. Monadic secondorder unification is secondorder unification where all function constants occurring in the equations are unary. Here we prove that the problem of deciding whether a set of monadic equations has a unifier is NPcomplete, where we use the technique of compressing solutions using singleton contextfree grammars. We prove that monadic secondorder matching is also NPcomplete.
and
, 2012
"... Under consideration for publication in J. Functional Programming 1 How to make ad hoc proof automation less ad hoc ..."
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Under consideration for publication in J. Functional Programming 1 How to make ad hoc proof automation less ad hoc
and
, 2013
"... Under consideration for publication in J. Functional Programming 1 How to make ad hoc proof automation less ad hoc ..."
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Under consideration for publication in J. Functional Programming 1 How to make ad hoc proof automation less ad hoc