Proof-graphs: a Thorough Cycle Treatment, Normalization and Subformula Property

A normalization procedure is given for classical natural deduction with the standard rule of indirect proof applied to arbitrary formulas. For normal derivability and the subformula property, it is sufficient to permute down instances of indirect proof whenever they have been used for concluding a major premiss of an elimination rule. The result applies even to natural deduction for classical modal logic.

We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof.A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by “nearly linear” is meant the ratio of proof lengths is O(α(n)) where α is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O(n . α(n)). Also we show that a Frege proof of n lines can be transformed into a tree-like Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O(n log n).

Full classical S5 in natural deduction with weak normalization

This paper studies the relationship between Argumentation Logic (AL), a recently defined logic based on the study of argumentation in AI, and classical Propositional Logic (PL). In particular, it shows that AL and PL are logically equivalent in that they have the same entailment relation from any given classically consistent theory. This equivalence follows from a correspondence between the non-acceptability of (arguments for) sentences in AL and Natural Deduction (ND) proofs of the complement of these sentences. The proof of this equivalence uses a restricted form of ND proofs, where hypotheses in the application of the Reductio of Absurdum inference rule are required to be “relevant” to the absurdity derived in the rule. The paper also discusses how the argumentative re-interpretation of PL could help control the application of ex-falso quodlibet in the presence of inconsistencies.

Gentzen's systems of natural deduction and sequent calculus were byproducts in his program of proving the consistency of arithmetic and analysis. It is suggested that the central component in his results on logical calculi was the use of a tree form for derivations. It allows the composition of derivations and the permutation of the order of application of rules, with a full control over the structure of derivations as a result. Recently found documents shed new light on the discovery of these calculi. In particular, Gentzen set up five different forms of natural calculi and gave a detailed proof of normalization for intuitionistic natural deduction. An early handwritten manuscript of his thesis shows that a direct translation from natural deduction to the axiomatic logic of Hilbert and Ackermann was, in addition to the influence of Paul Hertz, the second component in the discovery of sequent calculus. A system intermediate between the sequent calculus LI and axiomatic logic, denoted LIG in unpublished sources, is implicit in Gentzen's published thesis of 1934–35. The calculus has half rules, half “groundsequents,” and does not allow full cut elimination. Nevertheless, a translation from LI to LIG in the published thesis gives a subformula property for a complete class of derivations in LIG. After the thesis, Gentzen continued to work on variants of sequent calculi for ten more years, in the hope to find a consistency proof for arithmetic within an intuitionistic calculus.

Constructive linear-time temporal logic: Proof systems and Kripke semantics

We present a Natural Deduction proof system for the propositional modal µ-calculus, and its formalization in the Calculus of Inductive Constructions. We address several problematic issues, such as the use of higher-order abstract syntax in inductive sets in presence of recursive constructors, the encoding of modal (sequent-style) rules and of context sensitive grammars. The formalization can be used in the system Coq, providing an experimental computer-aided proof environment for the interactive development of error-free proofs in the µ-calculus. The techniques we adopt can be readily ported to other languages and proof systems featuring similar problematic issues.

Intermediate verification languages like Why3 and Boogie have made it much easier to build program verifiers, transforming the process into a logic compilation problem rather than a proof automation one. Why3 in particular implements a rich logic for program specification with polymorphism, algebraic data types, recursive functions and predicates, and inductive predicates; it translates this logic to over a dozen solvers and proof assistants. Accordingly, it serves as a backend for many tools, including Frama-C, EasyCrypt, and GNATProve for Ada SPARK. But how can we be sure that these tools are correct? The alternate foundational approach, taken by tools like VST and CakeML, provides strong guarantees by implementing the entire toolchain in a proof assistant, but these tools are harder to build and cannot directly take advantage of SMT solver automation. As a first step toward enabling automated tools with similar foundational guarantees, we give a formal semantics in Coq for the logic fragment of Why3. We show that our semantics are useful by giving a correct-by-construction natural deduction proof system for this logic, using this proof system to verify parts of Why3’s standard library, and proving sound two of Why3’s transformations used to convert terms and formulas into the simpler logics supported by the backend solvers.

We describe a natural deduction system NDIL for the second order intuitionistic linear logic which admits normalization and has a subformula property. NDIL is an extension of the system for !-free multiplicative linear logic constructed by the author and elaborated by A. Babaev. Main new feature here is the treatment of the modality !. It uses a device inspired by D. Prawitz' treatment of S4 combined with a construction \( \) introduced by the author to avoid cut-like constructions used in \(\otimes\)-elimination and global restrictions employed by Prawitz. Normal form for natural deduction is obtained by Prawitz translation of cut-free sequent derivations.

Cut-free proofs in Herbelin’s sequent calculus are in 1-1 correspondence with normal natural deduction proofs. For this reason Herbelin’s sequent calculus has been considered a privileged middle-point between L-systems and natural deduction. However, this bijection does not extend to proofs containing cuts and Herbelin observed that his cutelimination procedure is not isomorphic to β-reduction.In this paper we equip Herbelin’s system with rewrite rules which, at the same time: (1) complete in a sense the cut elimination procedure firstly proposed by Herbelin; and (2) perform the intuitionistic “fragment” of the tq-protocol - a cut-elimination procedure for classical logic defined by Danos, Joinet and Schellinx. Moreover we identify the subcalculus of our system which is isomorphic to natural deduction, the isomorphism being with respect not only to proofs but also to normalisation.Our results show, for the implicational fragment of intuitionistic logic, how to embed natural deduction in the much wider world of sequent calculus and what a particular cut-elimination procedure normalisation is.KeywordsClassical LogicIntuitionistic LogicReduction RuleNatural DeductionSequent CalculusThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study how to postpone the application of the reductio ad absurdum rule ( $$\mathsf {raa}$$ ) in classical natural deduction. This technique is connected with two normalization strategies for classical logic, due to Prawitz and Seldin, respectively. We introduce a variant of Seldin’s strategy for the postponement of $$\mathsf {raa}$$ , which induces a negative translation (a variant of Kuroda’s one) from classical to intuitionistic and minimal logic. Through this translation, Glivenko’s theorem from classical to intuitionistic and minimal logic is proven.

Functional coroutines are a restricted form of control mechanism, where each coroutine is represented with both a continuation and an environment. This restriction was originally obtained by considering a constructive version of Parigot's classical natural deduction which is sound and complete for the Constant Domain logic. In this article, we present a refinement of de Groote's abstract machine for functional coroutines and we prove its correctness. Therefore, this abstract machine also provides a direct computational interpretation of the Constant Domain logic.

A modification of natural deduction system for intuitionistic predicate logic, the extension of natural deduction system, will be defined. The main result of such modification will be that each conversion from the set of conversions of a cut-elimination procedure in the system of sequents has the corresponding conversion in the set of conversions of a normalization procedure in that extension of natural deduction.

Multi-focused Proofs with Different Polarity Assignments

Direct consistency proof of Gentzen's system of natural deduction.