Cut-Elimination and Proof Schemata

A cut elimination theorem for stationary logic

Transitive closure logic is a known extension of first-order logic obtained by introducing a transitive closure operator. While other extensions of first-order logic with inductive definitions are a priori parametrized by a set of inductive definitions, the addition of the transitive closure operator uniformly captures all finitary inductive definitions. In this paper we present an infinitary proof system for transitive closure logic which is an infinite descent-style counterpart to the existing (explicit induction) proof system for the logic. We show that, as for similar systems for first-order logic with inductive definitions, our infinitary system is complete for the standard semantics and subsumes the explicit system. Moreover, the uniformity of the transitive closure operator allows semantically meaningful complete restrictions to be defined using simple syntactic criteria. Consequently, the restriction to regular infinitary (i.e. cyclic) proofs provides the basis for an effective system for automating inductive reasoning.

I show two limitations of multiplicative (or context-independent) sequent calculi: contraction can neither be restricted to atoms nor to the bottom of a proof tree. There is a a set of rules for classical propositional logic, system SKS [1], with two interesting properties: 1. Contraction can be restricted to atoms. As a consequence, no rule in system SKS requires the duplication of formulae of unbounded size. 2. Contraction can be restricted to the bottom of a proof. As a consequence, a proof in SKS can be separated into two phases (seen bottom-up): In the first phase atoms are duplicated and in the second phase all other rules are applied and the size of the frontier of the proof (in terms of number of atoms) does not increase. System SKS is not a sequent calculus but is presented in a more general formalism, the calculus of structures [2]. While the sequent calculus restricts the application of rules to the main connective of a formula, the calculus of structures is more expressive by admitting rules that can be applied anywhere deep inside formulae. This leads to the question whether the extra expressive power of the calculus of structures is needed to observe these properties, or whether they can be observed in sequent systems as well. This question seems quickly answered by sequent system G3cp [3] in which contraction is admissible and can thus trivially be restricted to atoms or to the bottom of a proof. However, the R∧-rule in G3cp is additive (or context-sharing), so none of the above mentioned interesting consequences for system SKS follow for G3cp. Even though contraction is admissible, formulae of unbounded size still have to be duplicated in the context of an additive R∧. The separation of a proof into two phases as in system SKS also is not observable in G3cp, simply because there is no rule that just duplicates formulae. To answer the question whether similar properties to those of system SKS can be achieved in sequent systems, I thus consider systems with a multiplicative R∧-rule. For the specific case of system GS1p [3], a Gentzen-Schutte formulation of classical logic shown in Fig. 1, I now prove by counterexample that it does not admit the properties of SKS. Atoms can be either positive or negative and are denoted by a or b. The negation of an atom a is denoted by ā and is again an atom. In a proof, contraction is considered to be at the bottom if there is no application of a rule different from contraction such that one of its premises is the conclusion of a contraction. An application of the contraction rule is considered atomic if its principal formula is an atom. Theorem 1. There is a sequent that is valid but for which there is no cut-free proof in GS1p with multiplicative context treatment in which all contractions are at the bottom. Ax A, Ā Γ,A ∆, Ā Cut Γ,∆ Γ,A,B R∨ Γ,A ∨B Γ,A ∆,B R∧ Γ,∆,A ∧B Γ,A,A RC Γ,A Γ RW Γ,A Fig. 1. GS1p with multiplicative context treatment Proof. Consider the following sequent: a ∧ a, ā ∧ ā . It suffices to show that a ∧ a, . . . , a ∧ a, ā ∧ ā, . . . , ā ∧ ā (for any number of occurrences of the formulae a ∧ a and ā ∧ ā) is not provable in GS1p without contraction and cut. The connective ∨ does not occur in the conclusion. Thus, the only rules that can appear in the proof are RW ,Ax and R∧. A proof has to have all branches closed with axioms A, Ā. Since no rule that may occur introduces new formulae, the only formula that can take the place of A in an axiom is the atom a. There can be no such proof since there is always one branch in the derivation tree in which there is at most one single atom, as is shown by induction on the derivation tree: Base Case There is only one branch in a ∧ a, . . . , a ∧ a, ā ∧ ā, . . . , ā ∧ ā and it contains at most one (that is, no) single atom. Inductive Case Weakening does not increase the number of single atoms. When a R∧ rule is applied, the context is split, so the only single atom that may occur in the conclusion goes to one branch. Choose the other branch, which has at most one (that is, exactly one) atom. Theorem 2. There is a sequent that is valid but for which there is no cut-free proof in GS1p with multiplicative context treatment in which all contractions are atomic.

All the rules of the sequent calculus have the property that all the formulas that are present in the premises also occur in the conclusion. There is only one exception, the cut rule. In this chapter, it is shown using double induction that every theorem provable in Gentzen’s sequent calculi using the cut rule can also be proved without. One first proves that the mix rule is equivalent in proof-theoretic strength to the cut rule. Then one proves by induction on two complexity measures, the degree and the rank of a mix, that every application of mix can be eliminated. Several results follow from the cut-elimination theorem, including the so-called “mid-sequent” theorem and Herbrand’s theorem. In addition, one can prove the consistency of minimal, intuitionistic, and classical logic.

Cut elimination for a logic with induction and co-induction

System SKS is a set of rules for classical propositional logic presented in the calculus of structures. Like sequent systems and unlike natural deduction systems, it has an explicit cut rule, which is admissible. In contrast to sequent systems, the cut rule can easily be reduced to atomic form. This allows for a very simple cut elimination procedure based on plugging in parts of a proof, like normalisation in natural deduction and unlike cut elimination in the sequent calculus. It should thus be a good common starting point for investigations into both proof search as computation and proof normalisation as computation.

We design a proof system for propositional classical logic that integrates two languages for Boolean functions: standard conjunction-disjunction-negation and binary decision trees. We give two reasons to do so. The first is proof-theoretical naturalness: The system consists of all and only the inference rules generated by the single, simple, linear scheme of the recently introduced subatomic logic. Thanks to this regularity, cuts are eliminated via a natural construction. The second reason is that the system generates efficient proofs. Indeed, we show that a certain class of tautologies due to Statman, which cannot have better than exponential cut-free proofs in the sequent calculus, have polynomial cut-free proofs in our system. We achieve this by using the same construction that we use for cut elimination. In summary, by expanding the language of propositional logic, we make its proof theory more regular and generate more proofs, some of which are very efficient. That design is made possible by considering atoms as superpositions of their truth values, which are connected by self-dual, non-commutative connectives. A proof can then be projected via each atom into two proofs, one for each truth value, without a need for cuts. Those projections are semantically natural and are at the heart of the constructions in this article. To accommodate self-dual non-commutativity, we compose proofs in deep inference.

Probably the most significant result concerning cut-free sequent calculus proofs in linear logic is the completeness of focused proofs. This completeness theorem has a number of proof theoretic applications — e.g. in game semantics, Ludics, and proof search — and more computer science applications — e.g. logic programming, call-by-name/value evaluation. Andreoli proved this theorem for first-order linear logic 15 years ago. In the present paper, we give a new proof of the completeness of focused proofs in terms of proof transformation. The proof of this theorem is simple and modular: it is first proved for MALL and then is extended to full linear logic. Given its modular structure, we show how the proof can be extended to larger systems, such as logics with induction. Our analysis of focused proofs will employ a proof transformation method that leads us to study how focusing and cut elimination interact. A key component of our proof is the construction of a focalization graph which provides an abstraction over how focusing can be organized within a given cut-free proof. Using this graph abstraction allows us to provide a detailed study of atomic bias assignment in a way more refined that is given in Andreoli’s original proof. Permitting more flexible assignment of bias will allow this completeness theorem to help establish the completeness of a number of other automated deduction procedures. Focalization graphs can be used to justify the introduction of an inference rule for multifocus derivation: a rule that should help us better understand the relations between sequentiality and concurrency in linear logic.

Several theorems about the equivalence of familiar theories of reverse mathematics with certain well-ordering principles have been proved by recursion-theoretic and combinatorial methods (Friedman, Marcone, Montalbán et al.) and with far-reaching results by proof-theoretic technology (Afshari, Freund, Girard, Rathjen, Thomson, Valencia Vizcaíno, Weiermann et al.), employing deduction search trees and cut elimination theorems in infinitary logics with ordinal bounds in the latter case. At type level 1, the well-ordering principles are of the form $$ \text { (*) } ``\text {if}\,X \text { is well-ordered then } \, f(X) \, \text { is well-ordered''} $$where f is a standard proof-theoretic function from ordinals to ordinals (such f’s are always dilators). One aim of the paper is to present a general methodology underlying these results that enables one to construct omega-models of particular theories from \((*)\) and even \(\beta \)-models from the type 2 version of \((*)\). As \((*)\) is of complexity \(\Pi ^1_2\) such a principle cannot characterize stronger comprehensions at the level of \(\Pi ^1_1\)-comprehension. This requires a higher order version of \((*)\) that employs ideas from ordinal representation systems with collapsing functions used in impredicative proof theory. The simplest one is the Bachmann construction. Relativizing the latter construction to any dilator f and asserting that this always yields a well-ordering turns out to be equivalent to \(\Pi ^1_1\)-comprehension. This result has been conjectured and a proof strategy adumbrated roughly 10 years ago, but a detailed proof has only been worked out in recent years.KeywordsInfinite proof theoryReverse mathematicsOrdinal analysisOrdinal representation systemsCut eliminationDilators\(\omega \)-models\(\beta \)-models

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.

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.

A well established technique toward developing the proof theory of a Hilbert-style modal logic is to introduce a Gentzen-style equivalent (a Gentzenisation), then develop the proof theory of the latter, and finally transfer the metatheoretical results to the original logic (e.g., [1, 6, 8, 18, 10, 12]). In the first-order modal case, on one hand we know that the Gentzenisation of the straightforward first-order extension of GL, the logic QGL, admits no cut elimination (if the rule is included as primitive; or, if not included, then the rule is not admissible [1]). On the other hand the (cut-free) Gentzenisations of the first-order modal logics M3 and ML3 of [10, 12] do have cut as an admissible rule. The syntactic cut admissibility proof given in [18] for the Gentzenisation of the propositional provability logic GL is extremely complex, and it was the basis of the proofs of cut admissibility of the Gentzenisations of M3 and ML3, where the presence of quantifiers and quantifier rules added to the complexity and length of the proof. A recent proof of cut admissibility in a cut-free Gentzenisation of GL is given in [5] and is quite short and easy to read. We adapt it here to revisit the proofs for the cases of M3 and ML3, resulting to similarly short and easy to read proofs, only slightly complicated by the presence of quantification and its relevant rules.

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.

A cyclic proof system, called CLKID-omega, gives us another way of representing inductive definitions and efficient proof search. The 2005 paper by Brotherston showed that the provability of CLKID-omega includes the provability of LKID, first order classical logic with inductive definitions in Martin-L\"of's style, and conjectured the equivalence. The equivalence has been left an open question since 2011. This paper shows that CLKID-omega and LKID are indeed not equivalent. This paper considers a statement called 2-Hydra in these two systems with the first-order language formed by 0, the successor, the natural number predicate, and a binary predicate symbol used to express 2-Hydra. This paper shows that the 2-Hydra statement is provable in CLKID-omega, but the statement is not provable in LKID, by constructing some Henkin model where the statement is false.

In this paper we examine up to which point Modern logic can be qualified as non-Aristotelian. After clarifying the difference between logic as reasoning and logic as a theory of reasoning, we compare syllogistic with propositional and first-order logic. We touch the question of formal validity, variable and mathematization and we point out that Gentzen’s cut-elimination theorem can be seen as the rejection of the central mechanism of syllogistic – the cut-rule having been first conceived as a modus Barbara by Hertz. We then examine the non-Aristotelian aspect of some non-classical logics, in particular paraconsistent logic. We argue that a paraconsistent negation can be seen as neo-Aristotelian since it corresponds to the notion of subcontrary in Boethius’ square of opposition. We end by examining if the comparison promoted by Vasiliev between non-Aristotelian logic and non-Euclidian geometry makes sense.