Results 1  10
of
72
PROOFS IN HIGHERORDER LOGIC
, 1983
"... Expansion trees are defined as generalizations of Herbrand instances for formulas in a nonextensional form of higherorder logic based on Church’s simple theory of types. Such expansion trees can be defined with or without the use of skolem functions. These trees store substitution terms and either ..."
Abstract

Cited by 71 (13 self)
 Add to MetaCart
Expansion trees are defined as generalizations of Herbrand instances for formulas in a nonextensional form of higherorder logic based on Church’s simple theory of types. Such expansion trees can be defined with or without the use of skolem functions. These trees store substitution terms and either critical variables or skolem terms used to instantiate quantifiers in the original formula and those resulting from instantiations. An expansion tree is called an expansion tree proof (ETproof) if it encodes a tautology, and, in the form not using skolem functions, an “imbedding ” relation among the critical variables be acyclic. The relative completeness result for expansion tree proofs not using skolem functions, i.e. if A is provable in higherorder logic then A has such an expansion tree proof, is based on Andrews ’ formulation of Takahashi’s proof of the cutelimination theorem for higherorder logic. If the occurrences of skolem functions in instantiation terms are restricted appropriately, the use of skolem functions in place of critical variables is equivalent to the requirement that the imbedding relation is acyclic. This fact not only resolves the open question of what
Controlled Integrations of the Cut Rule into Connection Tableau Calculi
"... In this paper techniques are developed and compared which increase the inferential power of tableau systems for classical firstorder logic. The mechanisms are formulated in the framework of connection tableaux, which is an amalgamation of the connection method and the tableau calculus, and a genera ..."
Abstract

Cited by 61 (3 self)
 Add to MetaCart
In this paper techniques are developed and compared which increase the inferential power of tableau systems for classical firstorder logic. The mechanisms are formulated in the framework of connection tableaux, which is an amalgamation of the connection method and the tableau calculus, and a generalization of model elimination. Since connection tableau calculi are among the weakest proof systems with respect to proof compactness, and the (backward) cut rule is not suitable for the firstorder case, we study alternative methods for shortening proofs. The techniques we investigate are the folding up and the folding down operation. Folding up represents an efficient way of supporting the basic calculus, which is topdown oriented, with lemmata derived in a bottomup manner. It is shown that both techniques can also be viewed as controlled integrations of the cut rule. In order to remedy the additional redundancy imported into tableau proof procedures by the new inference rules, we develop and apply an extension of the regularity condition on tableaux and the mechanism of antilemmata which realizes a subsumption concept on tableaux. Using the framework of the theorem prover SETHEO, we have implemented three new proof procedures which overcome the deductive weakness of cutfree tableau systems. Experimental results demonstrate the superiority of the systems with folding up over the cutfree variant and the one with folding down.
SemanticsBased Translation Methods for Modal Logics
, 1991
"... 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 po ..."
Abstract

Cited by 40 (1 self)
 Add to MetaCart
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 firstorder manysorted logic with built in equality. Therefore the ‘source logic’ may also be firstorder manysorted 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
A FirstOrder Logic DavisPutnamLogemannLoveland Procedure
"... The DavisPutnamLogemannLoveland procedure (DPLL) was introduced in the early ..."
Abstract

Cited by 38 (6 self)
 Add to MetaCart
The DavisPutnamLogemannLoveland procedure (DPLL) was introduced in the early
A Local System for Classical Logic
 of Lecture Notes in Artificial Intelligence
, 2001
"... The calculus of structures is a framework for specifying logical systems, which is similar to the onesided sequent calculus but more general. We present a system of inference rules for propositional classical logic in this new framework and prove cut elimination for it. The system enjoys a deco ..."
Abstract

Cited by 30 (6 self)
 Add to MetaCart
The calculus of structures is a framework for specifying logical systems, which is similar to the onesided sequent calculus but more general. We present a system of inference rules for propositional classical logic in this new framework and prove cut elimination for it. The system enjoys a decomposition theorem for derivations that is not available in the sequent calculus. The main novelty of our system is that all the rules are local : contraction, in particular, is reduced to atomic form. This should be interesting for distributed proofsearch and also for complexity theory, since the computational cost of applying each rule is bounded.
A connection based proof method for intuitionistic logic
 TH WORKSHOP ON THEOREM PROVING WITH ANALYTIC TABLEAUX AND RELATED METHODS, LNAI 918
, 1995
"... We present a proof method for intuitionistic logic based on Wallen’s matrix characterization. Our approach combines the connection calculus and the sequent calculus. The search technique is based on notions of paths and connections and thus avoids redundancies in the search space. During the proof s ..."
Abstract

Cited by 29 (19 self)
 Add to MetaCart
We present a proof method for intuitionistic logic based on Wallen’s matrix characterization. Our approach combines the connection calculus and the sequent calculus. The search technique is based on notions of paths and connections and thus avoids redundancies in the search space. During the proof search the computed firstorder and intuitionistic substitutions are used to simultaneously construct a sequent proof which is more human oriented than the matrix proof. This allows to use our method within interactive proof environments. Furthermore we can consider local substitutions instead of global ones and treat substitutions occurring in different branches of the sequent proof independently. This reduces the number of extra copies of formulae to be considered.
On transforming intuitionistic matrix proofs into standardsequent proofs
 TABLEAUX–95, LNAI 918
, 1995
"... We present a procedure transforming intuitionistic matrix proofs into proofs within the intuitionistic standard sequent calculus. The transformation is based on L. Wallen’s proof justifying his matrix characterization for the validity of intuitionistic formulae. Since this proof makes use of Fitting ..."
Abstract

Cited by 26 (15 self)
 Add to MetaCart
We present a procedure transforming intuitionistic matrix proofs into proofs within the intuitionistic standard sequent calculus. The transformation is based on L. Wallen’s proof justifying his matrix characterization for the validity of intuitionistic formulae. Since this proof makes use of Fitting‘s nonstandard sequent calculus our procedure consists of two steps. First a nonstandard sequent proof will be extracted from a given matrix proof. Secondly we transform each nonstandard proof into a standard proof in a structure preserving way. To simplify the latter step we introduce an extended standard calculus which is shown to be sound and complete.
Tstringunification: unifying prefixes in nonclassical proof methods
 5 TH TABLEAUX WORKSHOP, LNAI 1071
, 1996
"... For an efficient proof search in nonclassical logics, particular in intuitionistic and modal logics, two similar approaches have been established: Wallen’s matrix characterization and Ohlbach’s resolution calculus. Beside the usual termunification both methods require a specialized stringunificat ..."
Abstract

Cited by 23 (12 self)
 Add to MetaCart
For an efficient proof search in nonclassical logics, particular in intuitionistic and modal logics, two similar approaches have been established: Wallen’s matrix characterization and Ohlbach’s resolution calculus. Beside the usual termunification both methods require a specialized stringunification to unify the socalled prefixes of atomic formulae (in Wallen’s notation) or worldpaths (in Ohlbach’s notation). For this purpose we present an efficient algorithm, called TStringUnification, which computes a minimal set of most general unifiers. By transforming systems of equations we obtain an elegant unification procedure, which is applicable to the intuitionistic logic J and the modal logic S4. With some modifications we are able to treat the modal logics D, K, D4, K4, S5, and T. We explain our method by an intuitive graphical presentation, prove correctness, completeness, minimality, and termination and investigate its complexity.
Normalisation control in deep inference via atomic flows
, 2008
"... Abstract. We introduce ‘atomic flows’: they are graphs obtained from derivations by tracing atom occurrences and forgetting the logical structure. We study simple manipulations of atomic flows that correspond to complex reductions on derivations. This allows us to prove, for propositional logic, a n ..."
Abstract

Cited by 23 (11 self)
 Add to MetaCart
Abstract. We introduce ‘atomic flows’: they are graphs obtained from derivations by tracing atom occurrences and forgetting the logical structure. We study simple manipulations of atomic flows that correspond to complex reductions on derivations. This allows us to prove, for propositional logic, a new and very general normalisation theorem, which contains cut elimination as a special case. We operate in deep inference, which is more general than other syntactic paradigms, and where normalisation is more difficult to control. We argue that atomic flows are a significant technical advance for normalisation theory, because 1) the technique they support is largely independent of syntax; 2) indeed, it is largely independent of logical inference rules; 3) they constitute a powerful geometric formalism, which is more intuitive than syntax. 1.
Proof Transformations in HigherOrder Logic
, 1987
"... We investigate the problem of translating between different styles of proof systems in higherorder logic: analytic proofs which are well suited for automated theorem proving, and nonanalytic deductions which are well suited for the mathematician. Analytic proofs are represented as expansion proofs, ..."
Abstract

Cited by 22 (5 self)
 Add to MetaCart
We investigate the problem of translating between different styles of proof systems in higherorder logic: analytic proofs which are well suited for automated theorem proving, and nonanalytic deductions which are well suited for the mathematician. Analytic proofs are represented as expansion proofs, H, a form of the sequent calculus we define, nonanalytic proofs are represented by natural deductions. A nondeterministic translation algorithm between expansion proofs and Hdeductions is presented and its correctness is proven. We also present an algorithm for translation in the other direction and prove its correctness. A cutelimination algorithm for expansion proofs is given and its partial correctness is proven. Strong termination of this algorithm remains a conjecture for the full higherorder system, but is proven for the firstorder fragment. We extend the translations to a nonanalytic proof system which contains a primitive notion of equality, while leaving the notion of expansion proof unaltered. This is possible, since a nonextensional equality is definable in our system of type theory. Next we extend analytic and nonanalytic proof systems and the translations between them to include extensionality. Finally, we show how the methods and notions used so far apply to the problem of translating expansion proofs into natural deductions. Much care is taken to specify this translation in a