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Admissibility and refutation: some characterisations of intermediate logics

2014
*
Archive for Mathematical Logic
*

We show that this gives a characterisation

doi:10.1007/s00153-014-0388-5
fatcat:goalirjsafalfdwngu6s45b77i
*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. ...##
###
A resolution framework for finitely-valued first-order logics

1992
*
Journal of symbolic computation
*

In this paper we propose a resolution proof framework on

doi:10.1016/s0747-7171(10)80002-1
fatcat:uiicseul3vf5jk6psgzcje5ode
*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. ...##
###
A Gentzen Calculus for Nothing but the Truth

2015
*
Journal of Philosophical Logic
*

But a sequent calculus can be given

doi:10.1007/s10992-015-9385-3
fatcat:xzr57erwarepvmnyi63gdjycai
*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. ...##
###
Inference rules and proof procedures for inequations

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 ...

##
###
An Introduction to Logic Programming
[chapter]

2006
*
Studies in Computational Intelligence
*

In its pure form it can be viewed as

doi:10.1007/978-3-540-33459-0_1
fatcat:s5uhe3y5vjfgblfgeb5tj7qhf4
*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. ...##
###
Tree-Like Proof Systems for Finitely-Many Valued Non-deterministic Consequence Relations

2020
*
Logica Universalis
*

I provide a couple

doi:10.1007/s11787-020-00263-0
fatcat:ykwpukvjsjcktk5tq2ez4gbmye
*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 ...##
###
Non-clausal reasoning with propositional definite theories
[chapter]

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 . ...

##
###
Analytic cut trees

2000
*
Logic Journal of the IGPL
*

Like tableaux analytic cut

doi:10.1093/jigpal/8.6.733
fatcat:lhklbinfsvf5fhpj6z6lexxtwu
*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. ...##
###
Hybrid Reasoning with Forest Logic Programs
[chapter]

2009
*
Lecture Notes in Computer Science
*

In this paper, we provide a sound, complete,

doi:10.1007/978-3-642-02121-3_27
fatcat:rz6rqg5bmbfprlqe62driwvwoi
*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 |*n*∈*N** } 1 , successors*of*x.*The*set A T = {(x, y) | x, y ∈ T, ∃*n*∈*N** : y = x ·*n*} is*the*set*of*arcs*of*a*tree*T . ...##
###
An informational view of classical logic

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 ...

##
###
A clausal resolution method for branching-time logic ECTL

2004
*
Proceedings. 11th International Symposium on Temporal Representation and Reasoning, 2004. TIME 2004.
*

ECTL · is strictly more expressive than

doi:10.1109/time.2004.1314431
dblp:conf/time/BolotovB04
fatcat:qqvk7h6rhjca7jypdpp6kt7z3u
*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 . ...##
###
A clausal resolution method for branching-time logic ECTL+

2006
*
Annals of Mathematics and Artificial Intelligence
*

ECTL · is strictly more expressive than

doi:10.1007/s10472-006-9018-1
fatcat:zixokqfv3bburoaiksupeyyspu
*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 . ...##
###
Bounded Nondeterminism of Logic Programs

2004
*
Annals of Mathematics and Artificial Intelligence
*

A program

doi:10.1023/b:amai.0000038175.53999.3a
fatcat:wjcrs6gqu5fv5n4bezvyi3tggy
*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 ...##
###
Representing knowledge with theories about theories

1990
*
The Journal of Logic Programming
*

(iii)

doi:10.1016/0743-1066(90)90035-4
fatcat:ci35lpj5njc3llxusiwbn74lb4
*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). ...##
###
Logic Program Specialisation
[chapter]

1999
*
Lecture Notes in Computer Science
*

Section 2 starts out from

doi:10.1007/3-540-47018-2_5
fatcat:tinunyvfo5bihgojzrjhkeulbm
*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 . ...
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