Cut elimination and complexity bounds for intuitionistic epistemic logic

2020 ◽  
Vol 30 (1) ◽  
pp. 281-294
Author(s):  
Vladimir N Krupski
Keyword(s):  
Epistemic Logic ◽  
Sequent Calculus ◽  
Formal System ◽  
Point Of View ◽  
Cut Elimination ◽  
Polynomial Space ◽  
Complexity Bounds ◽  
Proof Search

Abstract The formal system of intuitionistic epistemic logic (IEL) was proposed by S. Artemov and T. Protopopescu. It provides the formal foundation for the study of knowledge from an intuitionistic point of view based on Brouwer–Heyting–Kolmogorov semantics of intuitionism. We construct a cut-free sequent calculus for IEL and establish that polynomial space is sufficient for the proof search in it. We prove that IEL is PSPACE-complete.

A multi-labelled sequent calculus for Topo-Logic

2020 ◽  
Vol 30 (2) ◽  
pp. 663-696
Author(s):  
Ian Shillito
Keyword(s):  
Structural Properties ◽  
Epistemic Logic ◽  
Search Strategy ◽  
Sequent Calculus ◽  
Search Tree ◽  
Topological Spaces ◽  
Label Technique ◽  
Proof Search ◽  
Labelled Sequent Calculus

Abstract We present a labelled sequent calculus for a trimodal epistemic logic exhibitied in Baltag et al. (2017, Logic, Rationality, and Interaction, pp. 330–346), an extension of the so called ‘Topo-Logic’. To the best of our knowledge, our calculus is the first proof-calculus for this logic. This calculus is obtained via an adaptation of the label technique by internalizing a semantics over topological spaces. This internalization leads to the generation of two kinds of labels in our calculus and the labelling of formulae by pairs of labels. These novelties give tools to provide a simple calculus that is intuitively connected to the semantics. We prove that this calculus enjoys many structural properties such as admissibility of cut, admissibility of contraction and invertibility of its rules. Finally, we exhibit a proof search strategy for our calculus that allows us to prove completeness in a direct way by the extraction of a countermodel from a failure of proof. To define this strategy, we design a tool for controlling the generation of labels in the construction of a search tree, although the termination of this strategy is still open.

scholarly journals Focused Proof-search in the Logic of Bunched Implications

2021 ◽  
pp. 247-267
Author(s):  
Alexander Gheorghiu ◽  
Sonia Marin
Keyword(s):  
Data Structure ◽  
Proof Theory ◽  
Sequent Calculus ◽  
Operational Semantics ◽  
Search Space ◽  
Difficult Problem ◽  
Cut Elimination ◽  
Proof Search ◽  
Nested Sequents

AbstractThe logic of Bunched Implications (BI) freely combines additive and multiplicative connectives, including implications; however, despite its well-studied proof theory, proof-search in BI has always been a difficult problem. The focusing principle is a restriction of the proof-search space that can capture various goal-directed proof-search procedures. In this paper we show that focused proof-search is complete for BI by first reformulating the traditional bunched sequent calculus using the simpler data-structure of nested sequents, following with a polarised and focused variant that we show is sound and complete via a cut-elimination argument. This establishes an operational semantics for focused proof-search in the logic of Bunched Implications.

VALENTINI’S CUT-ELIMINATION FOR PROVABILITY LOGIC RESOLVED

2012 ◽  
Vol 5 (2) ◽  
pp. 212-238 ◽  
Author(s):  
RAJEEV GORÉ ◽  
REVANTHA RAMANAYAKE
Keyword(s):  
Sequent Calculus ◽  
Point Of View ◽  
Cut Elimination ◽  
Provability Logic ◽  
Sequent Calculi ◽  
Original Proof ◽  
Underlying Assumption ◽  
Induction Argument ◽  
Formal Point ◽  
Explicit Rule

Valentini (1983) has presented a proof of cut-elimination for provability logic GL for a sequent calculus using sequents built from sets as opposed to multisets, thus avoiding an explicit contraction rule. From a formal point of view, it is more syntactic and satisfying to explicitly identify the applications of the contraction rule that are ‘hidden’ in proofs of cut-elimination for such sequent calculi. There is often an underlying assumption that the move to a proof of cut-elimination for sequents built from multisets is straightforward. Recently, however, it has been claimed that Valentini’s arguments to eliminate cut do not terminate when applied to a multiset formulation of the calculus with an explicit rule of contraction. The claim has led to much confusion and various authors have sought new proofs of cut-elimination for GL in a multiset setting.Here we refute this claim by placing Valentini’s arguments in a formal setting and proving cut-elimination for sequents built from multisets. The use of sequents built from multisets enables us to accurately account for the interplay between the weakening and contraction rules. Furthermore, Valentini’s original proof relies on a novel induction parameter called “width” which is computed ‘globally’. It is difficult to verify the correctness of his induction argument based on “width.” In our formulation however, verification of the induction argument is straightforward. Finally, the multiset setting also introduces a new complication in the case of contractions above cut when the cut-formula is boxed. We deal with this using a new transformation based on Valentini’s original arguments.Finally, we discuss the possibility of adapting this cut-elimination procedure to other logics axiomatizable by formulae of a syntactically similar form to the GL axiom.

scholarly journals Expansion trees with cut

2019 ◽  
Vol 29 (8) ◽  
pp. 1009-1029 ◽  
Author(s):  
Federico Aschieri ◽  
Stefan Hetzl ◽  
Daniel Weller
Keyword(s):  
Sequent Calculus ◽  
Higher Order ◽  
Point Of View ◽  
Compact Representation ◽  
Cut Elimination ◽  
Elimination Procedure ◽  
Herbrand’S Theorem ◽  
Syntactic Approach ◽  
Proof Nets ◽  
Herbrand's Theorem

AbstractHerbrand’s theorem is one of the most fundamental insights in logic. From the syntactic point of view, it suggests a compact representation of proofs in classical first- and higher-order logics by recording the information of which instances have been chosen for which quantifiers. This compact representation is known in the literature as Miller’s expansion tree proof. It is inherently analytic and hence corresponds to a cut-free sequent calculus proof. Recently several extensions of such proof representations to proofs with cuts have been proposed. These extensions are based on graphical formalisms similar to proof nets and are limited to prenex formulas.In this paper, we present a new syntactic approach that directly extends Miller’s expansion trees by cuts and also covers non-prenex formulas. We describe a cut-elimination procedure for our expansion trees with cut that is based on the natural reduction steps and shows that it is weakly normalizing.

A proof of the cut-elimination theorem in simple type theory

1973 ◽  
Vol 38 (2) ◽  
pp. 215-226
Author(s):  
Satoko Titani
Keyword(s):  
Boolean Algebra ◽  
Type Theory ◽  
Arbitrary Number ◽  
Formal System ◽  
Cut Elimination ◽  
Simple Type ◽  
Inductive Definition ◽  
Simple Type Theory ◽  
Maximal Ideals ◽  
Definition Of

In [4], I introduced a quasi-Boolean algebra, and showed that in a formal system of simple type theory, from which the cut rule is omitted, wffs form a quasi-Boolean algebra, and that the cut-elimination theorem can be formulated in algebraic language. In this paper we use the result of [4] to prove the cut-elimination theorem in simple type theory. The theorem was proved by M. Takahashi [2] in 1967 by using the concept of Schütte's semivaluation. We use maximal ideals of a quasi-Boolean algebra instead of semivaluations.The logical system we are concerned with is a modification of Schütte's formal system of simple type theory in [1] into Gentzen style.Inductive definition of types.0 and 1 are types.If τ1, …, τn are types, then (τ1, …, τn) is a type.Basic symbols.a1τ, a2τ, … for free variables of type τ.x1τ, x2τ, … for bound variables of type τ.An arbitrary number of constants of certain types.An arbitrary number of function symbols with certain argument places.

scholarly journals Possession as Linear Knowledge

10.29007/ntkm ◽  
2018 ◽  
Author(s):  
Frank Pfenning
Keyword(s):  
Distributed System ◽  
Epistemic Logic ◽  
Linear Logic ◽  
Expressive Power ◽  
Cut Elimination ◽  
Multi Agent Systems ◽  
Agent Systems ◽  
Ongoing Effort ◽  
Multi Agent ◽  
Localized Knowledge

Epistemic logic analyzes reasoning governing localized knowledge, and is thus fundamental to multi- agent systems. Linear logic treats hypotheses as consumable resources, allowing us to model evolution of state. Combining principles from these two separate traditions into a single coherent logic allows us to represent localized consumable resources and their flow in a distributed system. The slogan “possession is linear knowledge” summarizes the underlying idea. We walk through the design of a linear epistemic logic and discuss its basic metatheoretic properties such as cut elimination. We illustrate its expressive power with several examples drawn from an ongoing effort to design and implement a linear epistemic logic programming language for multi-agent distributed systems.

A constructive analysis of RM

1987 ◽  
Vol 52 (4) ◽  
pp. 939-951 ◽  
Author(s):  
Arnon Avron
Keyword(s):  
Direct Method ◽  
Hard Core ◽  
Point Of View ◽  
Cut Elimination ◽  
Famous Theorem ◽  
Direct Extension ◽  
Admissibility Of Γ ◽  
A Direct Method ◽  
Important System ◽  
Surprising Fact

The system RM is the most well-understood (and to our opinion, also the most important) system among the logics developed by the Anderson and Belnap school. In this paper we investigate RM from a constructive point of view. For example, we give a new proof of the completeness of RM relative to the Sugihara matrix (first shown by Meyer), a proof in which a p.r. procedure is presented, applying which to a sentence A in RM language yields either a proof of it in RM or a refuting valuation for it in the Sugihara matrix SZ.Two topics dealt with in this work deserve a special attention.a) The admissibility of γ. This is a famous theorem of Meyer and Dunn. In [1] Anderson and Belnap emphasize that “the Meyer-Dunn argument … guarantees the existence of a proof of B, but there is no guarantee that the proof of B is related in any sort of plausible way to the proofs of A and Ā ∨ B.” In §2 we provide such a guarantee for the RM-case. In fact, we give there a direct method of obtaining a proof of B from given proofs of A and Ā ∨ B.b) The relationships betweenRMand its full negation-implication fragment. RM is known ([1, pp. 148–149], and [3]) to be a conservative extension of (Sobociński 3-valued logic; see [4]). Anderson and Belnap admit [1, p. 149] that this fact came to them as a distinct surprise, since RM as a whole is far from being three-valued. In this paper, however, this “surprising” fact appears quite natural (see III.3). In fact, we show that , is the “hard core” of RM, since our proof of the completeness of RM is based in an essential way on the completeness of relative to the Sobociński matrix, and since the Gentzen-type calculus we develop for RM is a direct extension of a similar (but much simpler) calculus for . Because of the importance has in this work, we devote the first section to a constructive investigation of it.We note, finally, that the Gentzen-type calculus mentioned above admits cut-elimination and normal-form techniques. (Such calculi were found till now only for RM without distribution.)

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