9,614 research outputs found
The symmetry rule in propositional logic
AbstractThe addition of the symmetry rule to the resolution system sometimes allows considerable shortening in the length of refutations. We prove exponential lower bounds on the size of resolution refutations using two forms of a global symmetry rule. The paper also discusses the relationship of symmetry rules to the extension rule that allows the use of abbreviative definitions in proofs
Locality for Classical Logic
In this paper we will see deductive systems for classical propositional and
predicate logic in the calculus of structures. Like sequent systems, they have
a cut rule which is admissible. In addition, they enjoy a top-down symmetry and
some normal forms for derivations that are not available in the sequent
calculus. Identity axiom, cut, weakening and also contraction can be reduced to
atomic form. This leads to rules that are local: they do not require the
inspection of expressions of unbounded size
Deep Inference and Symmetry in Classical Proofs
In this thesis we see deductive systems for classical propositional and predicate logic which use deep inference, i.e. inference rules apply arbitrarily deep inside formulas, and a certain symmetry, which provides an involution on derivations. Like sequent systems, they have a cut rule which is admissible. Unlike sequent systems, they enjoy various new interesting properties. Not only the identity axiom, but also cut, weakening and even contraction are reducible to atomic form. This leads to inference rules that are local, meaning that the effort of applying them is bounded, and finitary, meaning that, given a conclusion, there is only a finite number of premises to choose from. The systems also enjoy new normal forms for derivations and, in the propositional case, a cut elimination procedure that is drastically simpler than the ones for sequent systems
Proving Craig and Lyndon Interpolation Using Labelled Sequent Calculi
We have recently presented a general method of proving the fundamental
logical properties of Craig and Lyndon Interpolation (IPs) by induction on
derivations in a wide class of internal sequent calculi, including sequents,
hypersequents, and nested sequents. Here we adapt the method to a more general
external formalism of labelled sequents and provide sufficient criteria on the
Kripke-frame characterization of a logic that guarantee the IPs. In particular,
we show that classes of frames definable by quantifier-free Horn formulas
correspond to logics with the IPs. These criteria capture the modal cube and
the infinite family of transitive Geach logics
A minimal classical sequent calculus free of structural rules
Gentzen's classical sequent calculus LK has explicit structural rules for
contraction and weakening. They can be absorbed (in a right-sided formulation)
by replacing the axiom P,(not P) by Gamma,P,(not P) for any context Gamma, and
replacing the original disjunction rule with Gamma,A,B implies Gamma,(A or B).
This paper presents a classical sequent calculus which is also free of
contraction and weakening, but more symmetrically: both contraction and
weakening are absorbed into conjunction, leaving the axiom rule intact. It uses
a blended conjunction rule, combining the standard context-sharing and
context-splitting rules: Gamma,Delta,A and Gamma,Sigma,B implies
Gamma,Delta,Sigma,(A and B). We refer to this system M as minimal sequent
calculus.
We prove a minimality theorem for the propositional fragment Mp: any
propositional sequent calculus S (within a standard class of right-sided
calculi) is complete if and only if S contains Mp (that is, each rule of Mp is
derivable in S). Thus one can view M as a minimal complete core of Gentzen's
LK.Comment: To appear in Annals of Pure and Applied Logic. 15 page
Deciding regular grammar logics with converse through first-order logic
We provide a simple translation of the satisfiability problem for regular
grammar logics with converse into GF2, which is the intersection of the guarded
fragment and the 2-variable fragment of first-order logic. This translation is
theoretically interesting because it translates modal logics with certain frame
conditions into first-order logic, without explicitly expressing the frame
conditions.
A consequence of the translation is that the general satisfiability problem
for regular grammar logics with converse is in EXPTIME. This extends a previous
result of the first author for grammar logics without converse. Using the same
method, we show how some other modal logics can be naturally translated into
GF2, including nominal tense logics and intuitionistic logic.
In our view, the results in this paper show that the natural first-order
fragment corresponding to regular grammar logics is simply GF2 without extra
machinery such as fixed point-operators.Comment: 34 page
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