The calculus of structures is a proof theoretical formalism which generalizes the sequent calculus with the feature of deep inference: In contrast to the sequent calculus, inference rules can be applied at any depth inside a formula, bringing shorter proofs than any other formalisms supporting analytical proofs. However, deep applicability of the inference rules causes greater nondeterminism th...

The calculus of structures is a proof theoretical formalism which generalizes the sequent calculus with the feature of deep inference: in contrast to the sequent calculus, inference rules can be applied at any depth inside a formula, bringing shorter proofs than all other formalisms supporting analytical proofs. However, deep applicability of inference rules causes greater nondeterminism than i...

Deep inference is a proof theoretical methodology that generalises the traditional notion of inference of the sequent calculus. Deep inference provides more freedom in design of deductive systems for different logics and a rich combinatoric analysis of proofs. In particular, construction of exponentially shorter analytic proofs becomes possible, but with the cost of a greater nondeterminism tha...

Cut elimination is shown, in a constructive way, to hold in sequent calculi labelled with truth values for a wide class of normal modal logics, supporting global and local reasoning and allowing a general frame semantics. The complexity of cut elimination is studied in terms of the increase of logical depth of the derivations. A hyperexponential worst case bound is established. The subformula p...

In this work we show how some useful reductions known from ordinary intuitionistic propositional calculus can be modiied for intuitionistic linear logic (without modalities). The main reductions we consider are: (1) reduction of the depth of formulas in the sequents by addition of new variables, and (2)elimination of linear disjunction, tensor and constant F. Both transformations preserve deduc...

This paper studies the complexity of constant depth propositional proofs in the cedent and sequent calculus. We discuss the relationships between the size of tree-like proofs, the size of dag-like proofs, and the heights of proofs. The main result is to correct a proof construction in an earlier paper about transformations from proofs with polylogarithmic height and constantly many formulas per...

We prove a quadratic upper bound for the depth of cut free proofs in propositional intuitionistic logic formalized with Gentzen’s sequent calculus. We discuss bounds on the necessary number of reuses of left implication rules. We exhibit an example showing that this quadratic bound is optimal. As a corollary, this gives a new proof that propositional validity for intuitionistic logic is in PSPACE.

We see a cut-free infinitary sequent system for common knowledge. Its sequents are essentially trees and the inference rules apply deeply inside of these trees. This allows to give a syntactic cut-elimination procedure which yields an upper bound of φ20 on the depth of proofs, where φ is the Veblen function.

The worst-case complexity of cut elimination in sequent calculi for first order based logics is investigated in terms of the increase in logical depth of the deduction. It is shown that given a calculus satisfying a general collection of sufficient conditions for cut elimination and given a deduction with cuts, there exists a cut free deduction with a logical depth, in the worst case, hyper-exp...

Deep inference is a proof theoretical methodology that generalizes the traditional notion of inference in the sequent calculus: in contrast to the sequent calculus, the deductive systems with deep inference do not rely on the notion of main connective, and permit the application of the inference rules at any depth inside logical expressions, in a way which resembles the application of term rewr...