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Admissibility and refutation: some characterisations of intermediate logics

Jeroen P. Goudsmit
2014 Archive for Mathematical Logic  
We show that this gives a characterisation of these logics in terms of their admissible rules. To illustrate the technique, we also provide a refutation system for Medvedev's logic.  ...  In this paper, we explore the close connection between refutation systems and admissible rules.  ...  Lemma For each natural n ≥ 2, the logic T n is sound and complete with respect to proper at most n-fold branching finite trees. Moreover, each of these trees is maximally separable.  ... 
doi:10.1007/s00153-014-0388-5 fatcat:goalirjsafalfdwngu6s45b77i

A resolution framework for finitely-valued first-order logics

Peter W. O'Hearn, Zbigniew Stachniak
1992 Journal of symbolic computation  
In this paper we propose a resolution proof framework on the basis of which automated proof systems for finitely -valued first-order logics (F FO logics) can be introduced and studied.  ...  \Ve define the notion of a first-order resolution proof system and we show that for every disjunctive F FO logic a refutationally complete resolution proof system can be constructed.  ...  Acknowledgments We thank the anonymous referees for their helpful comments and suggestions that led to many improvements in the presentation of the paper.  ... 
doi:10.1016/s0747-7171(10)80002-1 fatcat:uiicseul3vf5jk6psgzcje5ode

A Gentzen Calculus for Nothing but the Truth

Stefan Wintein, Reinhard Muskens
2015 Journal of Philosophical Logic  
But a sequent calculus can be given and in this paper we will show that a calculus for the Belnap-Dunn logic we have defined earlier can in fact be reused for the purpose of characterising ETL, provided  ...  In their paper Nothing but the Truth Andreas Pietz and Umberto Rivieccio present Exactly True Logic (ETL), an interesting variation upon the four-valued logic for first-degree entailment FDE that was given  ...  Acknowledgments We would like to thank the referee for encouraging words and helpful comments.  ... 
doi:10.1007/s10992-015-9385-3 fatcat:xzr57erwarepvmnyi63gdjycai

Inference rules and proof procedures for inequations

Chilukuri K. Mohan, Mandayam K. Srivas, Deepak Kapur
1990 The Journal of Logic Programming  
The negation of equality is an important relation that arises naturally in the study of equational programming languages and logic programming with equality.  ...  of the given inequation is a logical consequence of the given system. a  ...  (iii) For any Horn clause C such that C=A,+, we define 70(C) =/lb + ) where the predicate of A, is an n-ary predicate P (n 2 O), and the predicate of A6 is the n + 1-ary predicate P with the n arguments  ... 
doi:10.1016/0743-1066(90)90034-3 fatcat:5h4wz46nkfejnkghho4oiq4bom

An Introduction to Logic Programming [chapter]

2006 Studies in Computational Intelligence  
In its pure form it can be viewed as logic programming with the "left-first" selection rule and with the depth-first strategy for searching the empty node in an SLD-tree.  ...  Negation is implemented by means of the negation as failure rule.  ...  an assignment for each n-ary function/in L of a mapping Ji from Dn to D, d) an assignment for each n-ary relation r in L of an n-ary predicate r 1 on D, i.e. a subset of Dn.  ... 
doi:10.1007/978-3-540-33459-0_1 fatcat:s5uhe3y5vjfgblfgeb5tj7qhf4

Tree-Like Proof Systems for Finitely-Many Valued Non-deterministic Consequence Relations

Pawel Pawlowski
2020 Logica Universalis  
I provide a couple of examples of proof systems for well-known many-valued logics and prove the completeness of proof systems generated by the framework.  ...  The main goal of this paper is to provide an abstract framework for constructing proof systems for various many-valued logics.  ...  Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Vol. 14 (2020) Tree-Like Proof Systems for Finitely-Many 419  ... 
doi:10.1007/s11787-020-00263-0 fatcat:ykwpukvjsjcktk5tq2ez4gbmye

Non-clausal reasoning with propositional definite theories [chapter]

Zbigniew Stachniak
1998 Lecture Notes in Computer Science  
The notion of a definite formula is generic in the sense that it is available to any logical calculus.  ...  We argue that efficient automated reasoning techniques which utilize definite formula representation of knowledge (such as SLD-resolution) can be developed for classical and a variety of non-classical  ...  A semantic tree of X is a finite tree T X that represents all possible assignments of verifiers v 0 , . . . , v n to the variables p 1 , . . . , p k .  ... 
doi:10.1007/bfb0055921 fatcat:onfhgujs2rcpffkwtpgmhe5k2m

Analytic cut trees

C Cellucci
2000 Logic Journal of the IGPL  
Like tableaux analytic cut trees are a refutation system but unlike tableaux they have a single inference rule (a form of the analytic cut rule) and several branch closure rules.  ...  The main advantage of analytic cut trees over tableaux is efficiency: while analytic cut trees can simulate tableaux with an increase in complexity by at most a constant factor, tableaux cannot polynomially  ...  Acknowledgements Research for this paper was partially supported by CNR (Consiglio Nazionale delle Ricerche), grant 92.00477.CT12.  ... 
doi:10.1093/jigpal/8.6.733 fatcat:lhklbinfsvf5fhpj6z6lexxtwu

Hybrid Reasoning with Forest Logic Programs [chapter]

Cristina Feier, Stijn Heymans
2009 Lecture Notes in Computer Science  
In this paper, we provide a sound, complete, and terminating algorithm for satisfiability checking w.r.t. forest logic programs, a fragment where rules have a tree shape and allow for inequality atoms  ...  We note that f-hybrid knowledge bases do not require the usual (weakly) DL-safety of the rule component, providing thus a genuine alternative approach to hybrid reasoning.  ...  For x ∈ T , we call succ T (x) = {x · n ∈ T | nN * } 1 , successors of x. The set A T = {(x, y) | x, y ∈ T, ∃nN * : y = x · n} is the set of arcs of a tree T .  ... 
doi:10.1007/978-3-642-02121-3_27 fatcat:rz6rqg5bmbfprlqe62driwvwoi

An informational view of classical logic

Marcello D'Agostino
2015 Theoretical Computer Science  
The less natural rules, from the point of view of ordinary usage, namely the introduction rules for true conditionals and the elimination rules for false conditionals, are however faithful to the classical  ...  consequence relations", see Section 3.3. 18 Observe that the two-premise elimination rules for true disjunctions and false conjunctions correspond to time-honoured principles of inference: modus ponens  ...  The intelim rules generate intelim sequences, i.e., finite sequences ϕ 1 , . . . ϕ n of S-formulae such that, for every i = 0, . . . , n, either ϕ i is an assumption or it is the conclusion of the application  ... 
doi:10.1016/j.tcs.2015.06.057 fatcat:lm7ckkxi35fgxmacv7jr4cqosi

A clausal resolution method for branching-time logic ECTL

A. Bolotov, A. Basukoski
2004 Proceedings. 11th International Symposium on Temporal Representation and Reasoning, 2004. TIME 2004.  
ECTL · is strictly more expressive than the basic computation tree logic CTL and its extension, ECTL, as it allows Boolean combinations of fairness and single temporal operators.  ...  We expand the applicability of the clausal resolution technique to the branching-time temporal logic ECTL · .  ...  )Let Ì ´AE µ be a -ary infinite tree such that denotes the set ½ , of branching degrees of the states in Ì and ´× × µ × ¾ ¾ Ã .  ... 
doi:10.1109/time.2004.1314431 dblp:conf/time/BolotovB04 fatcat:qqvk7h6rhjca7jypdpp6kt7z3u

A clausal resolution method for branching-time logic ECTL+

Alexander Bolotov, Artie Basukoski
2006 Annals of Mathematics and Artificial Intelligence  
ECTL · is strictly more expressive than the basic computation tree logic CTL and its extension, ECTL, as it allows Boolean combinations of fairness and single temporal operators.  ...  We expand the applicability of the clausal resolution technique to the branching-time temporal logic ECTL · .  ...  )Let Ì ´AE µ be a -ary infinite tree such that denotes the set ½ , of branching degrees of the states in Ì and ´× × µ × ¾ ¾ Ã .  ... 
doi:10.1007/s10472-006-9018-1 fatcat:zixokqfv3bburoaiksupeyyspu

Bounded Nondeterminism of Logic Programs

Dino Pedreschi, Salvatore Ruggieri
2004 Annals of Mathematics and Artificial Intelligence  
A program and a query have bounded nondeterminism if there are finitely many refutations for them via any selection rule.  ...  We introduce the notion of bounded nondeterminism for logic programs and queries.  ...  P Note that, since SLD-trees are finitely branching, by König's Lemma, "every SLDderivation for P and Q via a selection rule s is finite" is equivalent to stating that the SLD-tree of P and Q via s is  ... 
doi:10.1023/b:amai.0000038175.53999.3a fatcat:wjcrs6gqu5fv5n4bezvyi3tggy

Representing knowledge with theories about theories

Edward P. Stabler
1990 The Journal of Logic Programming  
(iii) For any Horn clause C such that C=A,+, we define 70(C) =/lb + ) where the predicate of A, is an n-ary predicate P (n 2 O), and the predicate of A6 is the n + 1-ary predicate P with the n arguments  ...  Every n-ary predicate symbol p" is assigned a set of n-tuples in D' as follows: (a,,..., a,> EZ'( P"> iff for every 5, (Z(ai)*,..., Z(a,-i>*) +P"-'), and a, is a proof tree corresponding to a refutation  ...  Thus there is a l-step SLD refutation of S U {G) using C and mgu E. (2) Suppose the result holds for refutations of length n -1, and T,(G) has a refutation Corollary 1 (Nonintetierence).  ... 
doi:10.1016/0743-1066(90)90035-4 fatcat:ci35lpj5njc3llxusiwbn74lb4

Logic Program Specialisation [chapter]

Michael Leuschel
1999 Lecture Notes in Computer Science  
Section 2 starts out from the roots of logic programming in first-order logic and automated theorem proving and presents the syntax, semantics and proof theory of logic programs.  ...  In Section 3 the general idea of program specialisation, based on Kleene's S-M-N theorem, is introduced.  ...  An unfolding rule Í is a function which, given a program È and a goal , returns a finite and possibly incomplete SLDNF-tree for È .  ... 
doi:10.1007/3-540-47018-2_5 fatcat:tinunyvfo5bihgojzrjhkeulbm
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