166 research outputs found
Complete Sequent Calculi for Induction and Infinite Descent
This paper formalises and compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system, LKID, supports traditional proof by induction, with induction rules formulated as rules for introducing inductively defined predicates on the left of sequents. We show LKID to be cut-free complete with respect to a natural class of Henkin models; the eliminability of cut follows as a corollary. The second system, LKID ω, uses infinite (non-well-founded) proofs to represent arguments by infinite descent. In this system, the left-introduction rules for inductively defined predicates are simple case-split rules, and an infinitary, global condition on proof trees is required in order to ensure soundness. We show LKID ω to be cut-free complete with respect to standard models, and again infer the eliminability of cut. The infinitary system LKID ω is unsuitable for formal reasoning. However, it has a natural restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs, which is so suited. We demonstrate that this restricted “cyclic ” proof system, CLKID ω, subsumes LKID, and conjecture that CLKID ω and LKID are in fact equivalent, i.e., that proof by induction is equivalent to regular proof by infinite descent.
The Lambek calculus with iteration: two variants
Formulae of the Lambek calculus are constructed using three binary
connectives, multiplication and two divisions. We extend it using a unary
connective, positive Kleene iteration. For this new operation, following its
natural interpretation, we present two lines of calculi. The first one is a
fragment of infinitary action logic and includes an omega-rule for introducing
iteration to the antecedent. We also consider a version with infinite (but
finitely branching) derivations and prove equivalence of these two versions. In
Kleene algebras, this line of calculi corresponds to the *-continuous case. For
the second line, we restrict our infinite derivations to cyclic (regular) ones.
We show that this system is equivalent to a variant of action logic that
corresponds to general residuated Kleene algebras, not necessarily
*-continuous. Finally, we show that, in contrast with the case without division
operations (considered by Kozen), the first system is strictly stronger than
the second one. To prove this, we use a complexity argument. Namely, we show,
using methods of Buszkowski and Palka, that the first system is -hard,
and therefore is not recursively enumerable and cannot be described by a
calculus with finite derivations
Towards Automated Reasoning in Herbrand Structures
Herbrand structures have the advantage, computationally speaking, of being guided by the definability of all elements in them. A salient feature of the logics induced by them is that they internally
exhibit the induction scheme, thus providing a congenial, computationally-oriented framework for
formal inductive reasoning. Nonetheless, their enhanced expressivity renders any effective proof
system for them incomplete. Furthermore, the fact that they are not compact poses yet another prooftheoretic challenge. This paper offers several layers for coping with the inherent incompleteness and
non-compactness of these logics. First, two types of infinitary proof system are introduced—one
of infinite width and one of infinite height—which manipulate infinite sequents and are sound and
complete for the intended semantics. The restriction of these systems to finite sequents induces a
completeness result for finite entailments. Then, in search of effectiveness, two finite approximations
of these systems are presented and explored. Interestingly, the approximation of the infinite-width
system via an explicit induction scheme turns out to be weaker than the effective cyclic fragment of the
infinite-height system
Integrating a Global Induction Mechanism into a Sequent Calculus
Most interesting proofs in mathematics contain an inductive argument which
requires an extension of the LK-calculus to formalize. The most commonly used
calculi for induction contain a separate rule or axiom which reduces the valid
proof theoretic properties of the calculus. To the best of our knowledge, there
are no such calculi which allow cut-elimination to a normal form with the
subformula property, i.e. every formula occurring in the proof is a subformula
of the end sequent. Proof schemata are a variant of LK-proofs able to simulate
induction by linking proofs together. There exists a schematic normal form
which has comparable proof theoretic behaviour to normal forms with the
subformula property. However, a calculus for the construction of proof schemata
does not exist. In this paper, we introduce a calculus for proof schemata and
prove soundness and completeness with respect to a fragment of the inductive
arguments formalizable in Peano arithmetic.Comment: 16 page
Intuitionistic G\"odel-L\"ob logic, \`a la Simpson: labelled systems and birelational semantics
We derive an intuitionistic version of G\"odel-L\"ob modal logic ()
in the style of Simpson, via proof theoretic techniques. We recover a labelled
system, , by restricting a non-wellfounded labelled system for
to have only one formula on the right. The latter is obtained using
techniques from cyclic proof theory, sidestepping the barrier that 's
usual frame condition (converse well-foundedness) is not first-order definable.
While existing intuitionistic versions of are typically defined over
only the box (and not the diamond), our presentation includes both modalities.
Our main result is that coincides with a corresponding
semantic condition in birelational semantics: the composition of the modal
relation and the intuitionistic relation is conversely well-founded. We call
the resulting logic . While the soundness direction is proved using
standard ideas, the completeness direction is more complex and necessitates a
detour through several intermediate characterisations of .Comment: 25 pages including 8 pages appendix, 4 figure
- …