Natural deduction with general elimination rules

The structure of derivations in natural deduction is analyzed through isomorphism with a suitable sequent calculus, with twelve hidden convertibilities revealed in usual natural deduction. A general formulation of conjunction and implication elimination rules is given, analogous to disjunction elimination. Normalization through permutative conversions now applies in all cases. Derivations in normal form have all major premisses of elimination rules as assumptions. Conversion in any order terminates. Through the condition that in a cut-free derivation of the sequent Γ⇒C, no inactive weakening or contraction formulas remain in Γ, a correspondence with the formal derivability relation of natural deduction is obtained: All formulas of Γ become open assumptions in natural deduction, through an inductively defined translation. Weakenings are interpreted as vacuous discharges, and contractions as multiple discharges. In the other direction, non-normal derivations translate into derivations with cuts having the cut formula principal either in both premisses or in the right premiss only. Received: 1 December 1998 / Revised version: 30 June 2000 / Published online: 18 July 2001     

A Gentzen-style axiomatization for basic predicate calculus

 We introduce a Gentzen-style sequent calculus axiomatization for Basic Predicate Calculus. Our new axiomatization is an improvement of the previous axiomatizations, in the sense that it has the subformula property. In this system the cut rule is eliminated. Received: 18 April 2000 / Published online: 2 September 2002 Mathematics Subject Classification (2000): Primary 03F05; Secondary 03F99, 03B60 Key words or phrases: Basic predicate calculus – Cut elimination – Sequent     

Contraction-free sequent calculi for geometric theories with an application to Barr's theorem

 Geometric theories are presented as contraction- and cut-free systems of sequent calculi with mathematical rules following a prescribed rule-scheme that extends the scheme given in Negri and von Plato (1998). Examples include cut-free calculi for Robinson arithmetic and real closed fields. As an immediate consequence of cut elimination, it is shown that if a geometric implication is classically derivable from a geometric theory then it is intuitionistically derivable. Received: 18 April 2001 / Published online: 10 October 2002 Mathematics Subject Classification (2000): 03F05, 18C10, 18B15 Key words or phrases: Cut elimination – Geometric theories – Barr's theorem     

Translations from natural deduction to sequent calculus

Gentzen's “Untersuchungen” [1] gave a translation from natural deduction to sequent calculus with the property that normal derivations may translate into derivations with cuts. Prawitz in [8] gave a translation that instead produced cut‐free derivations. It is shown that by writing all elimination rules in the manner of disjunction elimination, with an arbitrary consequence, an isomorphic translation between normal derivations and cut‐free derivations is achieved. The standard elimination rules do not permit a full normal form, which explains the cuts in Gentzen's translation. Likewise, it is shown that Prawitz' translation contains an implicit process of cut elimination.     

A new correctness criterion for multiplicative non-commutative proof nets

 We introduce a new correctness criterion for multiplicative non commutative proof nets which can be considered as the non-commutative counterpart to the Danos-Regnier criterion for proof nets of linear logic. The main intuition relies on the fact that any switching for a proof net (obtained by mutilating one premise of each disjunction link) can be naturally viewed as a series-parallel order variety (a cyclic relation) on the conclusions of the proof net. Received: 8 November 2000 / Revised version: 21 June 2001 / Published online: 2 September 2002 Research supported by the EU TMR Research Programme ``Linear Logic and Theoretical Computer Science'. Mathematics Subject Classification (2000): 03F03, 03F07, 03F52, 03B70 Key words or phrases: Linear and non-commutative logic – Proof nets – Series-parallel orders and order varieties     

Short proofs of normalization for the simply- typed λ-calculus,permutative conversions and Gödel's T

 Inductive characterizations of the sets of terms, the subset of strongly normalizing terms and normal forms are studied in order to reprove weak and strong normalization for the simply-typed λ-calculus and for an extension by sum types with permutative conversions. The analogous treatment of a new system with generalized applications inspired by generalized elimination rules in natural deduction, advocated by von Plato, shows the flexibility of the approach which does not use the strong computability/candidate style à la Tait and Girard. It is also shown that the extension of the system with permutative conversions by η-rules is still strongly normalizing, and likewise for an extension of the system of generalized applications by a rule of ``immediate simplification'. By introducing an infinitely branching inductive rule the method even extends to G?del's T. Received 12 October 1998 Published online: 19 December 2002 Mathematics Subject Classification (2000): 03B40 Keywords or phrases: λ-calculus – Permutative/commuting conversions – Sum types – Generalized application – G?del's T_q-ω-rule – Inductive characterization     

PSPACE complexity of modal logic <Emphasis Type="Italic">K D</Emphasis>45<Subscript>n</Subscript>

We study sequent calculus for multi-modal logic K D45_n and its complexity. We introduce a loop-check free sequent calculus. Loop-check is eliminated by using the marked modal operator □_i^•, which is used as an alternative to sequents with histories ([8], [3], [5]). All inference rules are invertible or semi-invertible. To get this, we use or branches beside common and branches. We prove the equivalence between known sequent calculus and our newly introduced efficient sequent calculus. We concentrate on the complexity analysis of the introduced sequent calculus for multi-modal logic K D45_n. We prove that the space complexity of the given calculus is polynomial (O(l ³)). We show the maximum height of the constructed derivation tree that leads to the reduction of the time and space complexity. We present a decision algorithm for multi-modal logic K D45_n and some nontrivial examples to improve the introduced loop-check free sequent calculus.     

A sequent calculus for logic of knowledge and past time: Completeness and decidability

We consider logic of knowledge and past time. This logic involves the discrete-time linear temporal operators next, until, weak yesterday, and since. In addition, it contains an indexed set of unary modal operators agent i knows.We consider the semantic constraint of the unique initial states for this logic. For the logic, we present a sequent calculus with a restricted cut rule. We prove the soundness and completeness of the sequent calculus presented. We prove the decidability of provability in the considered calculus as well. So, this calculus can be used as a basis for automated theorem proving. The proof method for the completeness can be used to construct complete sequent calculi with a restricted cut rule for this logic with other semantical constraints as well. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 427–437, July–September, 2006.     

Proof-theoretical investigation of temporal logic with time gaps

In the paper, the first-order intuitionistic temporal logic sequent calculus LBJ is considered. The invertibility of some of the LBJ rules, syntactic admissibility of the structural rules and the cut rule in LBJ, as well as Harrop and Craig's interpolation theorems for LBJ are proved. Gentzen's midsequent theorem is proved for the LBJ' calculus which is obtained from LBJ by removing the antecedent disjunction rule from it. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 255–276, July–September, 2000.     

On temporal logic S4Dbr

In this paper, we study the temporal logic S4Dbr with two temporal operators “always” and “eventually.” An equivalent sequent calculus is presented with formulae as modal clauses or modal clauses starting with operator “always.” An upper bound of deduction tree is given for propositional logic. A theorem prover for propositional logic is written in SWI-Prolog. Published in LietuvosMatematikos Rinkinys, Vol. 46, No. 2, pp. 203–214, April–June, 2006.     

The modal logic of the countable random frame

 We study the modal logic M L ^r of the countable random frame, which is contained in and `approximates' the modal logic of almost sure frame validity, i.e. the logic of those modal principles which are valid with asymptotic probability 1 in a randomly chosen finite frame. We give a sound and complete axiomatization of M L ^r and show that it is not finitely axiomatizable. Then we describe the finite frames of that logic and show that it has the finite frame property and its satisfiability problem is in EXPTIME. All these results easily extend to temporal and other multi-modal logics. Finally, we show that there are modal formulas which are almost surely valid in the finite, yet fail in the countable random frame, and hence do not follow from the extension axioms. Therefore the analog of Fagin's transfer theorem for almost sure validity in first-order logic fails for modal logic. Received: 1 May 2000 / Revised version: 29 July 2001 / Published online: 2 September 2002 Mathematics Subject Classification (2000): 03B45, 03B70, 03C99 Key words or phrases: Modal logic – Random frames – Almost sure frame validity – Countable random frame – Axiomatization – Completeness     

Saturated calculus for Horn-like sequents of a complete class of a linear temporal first order logic

A saturated calculus for the so-called Horn-like sequents of a complete class of a linear temporal logic of the first order is described. The saturated calculus contains neither induction-like postulates nor cut-like rules. Instead of induction-like postulates the saturated calculus contains a finite set of “saturated” sequents, which (1) capture and reflect the periodic structure of inductive reasoning (i.e., a reasoning which applies induction-like postulates); (2) show that “almost nothing new” can be obtained by continuing the process of derivation of a given sequent; (3) present an explicit way of generating the so-called invariant formula in induction-like rules. The saturated calculus for Horn-like sequents allows one: (1) to prove in an obvious way the completeness of a restricted linear temporal logic of the first order; (2) to construct a computer-aided proof system for this logic; (3) to prove the decidability of this logic for logically decidable Horn-like sequents. Bibliography: 15 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 220, 1995, pp. 123–144. Translated by R. Pliuškevičius.     

Efficient loop-check for KD 45 logic

We introduce a new sequent calculus for KD 45 logic. A loop-check technique is used to determine whether a sequent derivable or not. We concentrate ourselves on the efficiency of the loop-check technique used. The efficiency is obtained by making the loop-check to act locally (then we need to check only one or two current sequents), instead of a global loop-check used in known works as [3]. Moreover, the sequent calculus introduced uses a marked operator □ to integrate loop-check into an inference rule. Besides the efficient loop-check used, the sequent calculus introduced produces smaller derivation trees that reduce the derivation time. We also prove a lemma which determines the maximal number of modality rule applications in one branch of a tree. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 1, pp. 55–66, January–March, 2006.     

Two measures for proving Gentzen's Hauptsatz without mix

 This paper presents a cut-elimination procedure for classical and intuitionistic logic, in which cut is eliminated directly, without introducing the mix rule. The well-known problem of cut eliminations, when in the derivation the contractions of the cut-formulae are above the premisses of the cut, will be solved by new transformations of the derivation. Received: 1 June 2001 / Published online: 5 November 2002 Mathematics Subject Classification (2000): 03F05 Key words or phrases: Systems of sequents – Cut-elimination theorem     

Distributive-Lattice Semantics of Sequent Calculi with Structural Rules

The goal of the paper is to develop a universal semantic approach to derivable rules of propositional multiple-conclusion sequent calculi with structural rules, which explicitly involve not only atomic formulas, treated as metavariables for formulas, but also formula set variables (viz., metavariables for finite sets of formulas), upon the basis of the conception of model introduced in (Fuzzy Sets Syst 121(3):27–37, 2001). One of the main results of the paper is that any regular sequent calculus with structural rules has such class of sequent models (called its semantics) that a rule is derivable in the calculus iff it is sound with respect to each model of the semantics. We then show how semantics of admissible rules of such calculi can be found with using a method of free models. Next, our universal approach is applied to sequent calculi for many-valued logics with equality determinant. Finally, we exemplify this application by studying sequent calculi for some of such logics.      

CM-triviality and generic structures

 We show that any relational generic structure whose theory has finite closure and amalgamation over closed sets is stable CM-trivial with weak elimination of imaginaries. Received: 21 December 2001 / Published online: 5 November 2002 Mathematics Subject Classification (2000): 03C45 Key words or phrases: CM-triviality – Generic structures – Stability     

Proof-theoretical analysis of order relations

A proof-theoretical analysis of elementary theories of order relations is effected through the formulation of order axioms as mathematical rules added to contraction-free sequent calculus. Among the results obtained are proof-theoretical formulations of conservativity theorems corresponding to Szpilrajns theorem on the extension of a partial order into a linear one. Decidability of the theories of partial and linear order for quantifier-free sequents is shown by giving terminating methods of proof-search.Mathematics Subject Classification (2000): 03F05, 06A05, 06A06     

Fuzzy logics with modalities

We explore the basic fuzzy logic BL as well as propositional fuzzy logics with modalities □ and ◊ and a total accessibility relation. Formulations and proofs are given to replacement theorems for BL. A basic calculus of modal fuzzy logic is introduced. For this calculus and its extensions, we prove replacement and deduction theorems. Supported by RFBR grant No. 06-01-00358, by INTAS grant No. 04-77-7080, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-4787.2006.1. __________ Translated from Algebra i Logika, Vol. 45, No. 6, pp. 731–757, November–December, 2006.     

Cut elimination for a calculus with context-dependent rules

Context-dependent rules are an obstacle to cut elimination. Turning to a generalised sequent style formulation using deep inferences is helpful, and for the calculus presented here it is essential. Cut elimination is shown for a substructural, multiplicative, pure propositional calculus. Moreover we consider the extra problems induced by non-logical axioms and extend the results to additive connectives and quantifiers. Received: 11 April 1998 / Published online: 25 January 2001