scholarly journals A Tableau Proof System with Names for Modal Mu-calculus

10.29007/lwqm ◽  
2018 ◽  
Author(s):  
Colin Stirling
Keyword(s):  
Modal Logic ◽  
Expressive Power ◽  
Proof System ◽  
Temporal Logics ◽  
General Issue ◽  
Proof Systems

Howard Barringer was a pioneer in the study of temporal logics withfixpoints. Their addition adds considerable expressive power.One general issue is how to defineproof systems for such logics. Here we examine proof systems formodal logic with fixpoints. We present a tableau proof systemfor checking validity of formulaswhich uses names to keep track of unfoldings of fixpoint variables.

scholarly journals QMA variants with polynomially many provers

2013 ◽  
Vol 13 (1&2) ◽  
pp. 135-157
Author(s):  
Sevag Gharibian ◽  
Jamie Sikora ◽  
Sarvagya Upadhyay
Keyword(s):  
Bounded Function ◽  
Expressive Power ◽  
Proof System ◽  
Alternate Proof ◽  
Cone Programming ◽  
Parallel Repetition ◽  
Proof Systems ◽  
Classical Proof

We study three variants of multi-prover quantum Merlin-Arthur proof systems. We first show that the class of problems that can be efficiently verified using polynomially many quantum proofs, each of logarithmic-size, is exactly \class{MQA} (also known as QCMA), the class of problems which can be efficiently verified via a classical proof and a quantum verifier. We then study the class $\class{BellQMA}(\poly)$, characterized by a verifier who first applies unentangled, nonadaptive measurements to each of the polynomially many proofs, followed by an arbitrary but efficient quantum verification circuit on the resulting measurement outcomes. We show that if the number of outcomes per nonadaptive measurement is a polynomially-bounded function, then the expressive power of the proof system is exactly \class{QMA}. Finally, we study a class equivalent to \class{QMA}($m$), denoted $\class{SepQMA}(m)$, where the verifier's measurement operator corresponding to outcome {\it accept} is a fully separable operator across the $m$ quantum proofs. Using cone programming duality, we give an alternate proof of a result of Harrow and Montanaro [FOCS, pp. 633--642 (2010)] that shows a perfect parallel repetition theorem for $\class{SepQMA}(m)$ for any $m$.

LINEAR TIME IN HYPERSEQUENT FRAMEWORK

2016 ◽  
Vol 22 (1) ◽  
pp. 121-144 ◽  
Author(s):  
ANDRZEJ INDRZEJCZAK
Keyword(s):  
Proof Theory ◽  
Linear Time ◽  
Expressive Power ◽  
Sequent Calculi ◽  
Temporal Logics ◽  
Structural Rules ◽  
Proof Systems ◽  
Nonclassical Logics ◽  
Time Flow ◽  
Basic Calculus

AbstractHypersequent calculus (HC), developed by A. Avron, is one of the most interesting proof systems suitable for nonclassical logics. Although HC has rather simple form, it increases significantly the expressive power of standard sequent calculi (SC). In particular, HC proved to be very useful in the field of proof theory of various nonclassical logics. It may seem surprising that it was not applied to temporal logics so far. In what follows, we discuss different approaches to formalization of logics of linear frames and provide a cut-free HC formalization ofKt4.3, the minimal temporal logic of linear frames, and some of its extensions. The novelty of our approach is that hypersequents are defined not as finite (multi)sets but as finite lists of ordinary sequents. Such a solution allows both linearity of time flow, and symmetry of past and future, to be incorporated by means of six temporal rules (three for future-necessity and three dual rules for past-necessity). Extensions of the basic calculus with simple structural rules cover logics of serial and dense frames. Completeness is proved by Schütte/Hintikka-style argument using models built from saturated hypersequents.

scholarly journals Constructing weak simulations from linear implications for processes with private names

2019 ◽  
Vol 29 (8) ◽  
pp. 1275-1308 ◽  
Author(s):  
Ross Horne ◽  
Alwen Tiu
Keyword(s):  
Expressive Power ◽  
Dual Pair ◽  
Proof System ◽  
Branching Time ◽  
First Order ◽  
Order Extension

AbstractThis paper clarifies that linear implication defines a branching-time preorder, preserved in all contexts, when used to compare embeddings of process in non-commutative logic. The logic considered is a first-order extension of the proof system BV featuring a de Morgan dual pair of nominal quantifiers, called BV1. An embedding of π-calculus processes as formulae in BV1 is defined, and the soundness of linear implication in BV1 with respect to a notion of weak simulation in the π -calculus is established. A novel contribution of this work is that we generalise the notion of a ‘left proof’ to a class of formulae sufficiently large to compare embeddings of processes, from which simulating execution steps are extracted. We illustrate the expressive power of BV1 by demonstrating that results extend to the internal π -calculus, where privacy of inputs is guaranteed. We also remark that linear implication is strictly finer than any interleaving preorder.

PROOF SYSTEMS FOR VARIOUS FDE-BASED MODAL LOGICS

2019 ◽  
Vol 13 (4) ◽  
pp. 720-747
Author(s):  
SERGEY DROBYSHEVICH ◽  
HEINRICH WANSING
Keyword(s):  
Modal Logic ◽  
Sequent Calculus ◽  
Axiom System ◽  
Modal Logics ◽  
Cut Elimination ◽  
Proof Systems ◽  
Image Position ◽  
Axiomatic Extension

AbstractWe present novel proof systems for various FDE-based modal logics. Among the systems considered are a number of Belnapian modal logics introduced in Odintsov & Wansing (2010) and Odintsov & Wansing (2017), as well as the modal logic KN4 with strong implication introduced in Goble (2006). In particular, we provide a Hilbert-style axiom system for the logic $BK^{\square - } $ and characterize the logic BK as an axiomatic extension of the system $BK^{FS} $. For KN4 we provide both an FDE-style axiom system and a decidable sequent calculus for which a contraction elimination and a cut elimination result are shown.

scholarly journals A duality between proof systems for cyclic term graphs

2007 ◽  
Vol 17 (3) ◽  
pp. 439-484 ◽  
Author(s):  
CLEMENS GRABMAYER
Keyword(s):  
Close Connection ◽  
Proof System ◽  
Consistency Checking ◽  
Duality Result ◽  
First Order ◽  
Proof Systems ◽  
Similar Proof ◽  
Equational Specifications ◽  
Cyclic Term

This paper presents a proof-theoretic observation about two kinds of proof systems for bisimilarity between cyclic term graphs.First we consider proof systems for demonstrating that μ term specifications of cyclic term graphs have the same tree unwinding. We establish a close connection between adaptations for μ terms over a general first-order signature of the coinductive axiomatisation of recursive type equivalence by Brandt and Henglein (Brandt and Henglein 1998) and of a proof system by Ariola and Klop (Ariola and Klop 1995) for consistency checking. We show that there exists a simple duality by mirroring between derivations in the former system and formalised consistency checks, which are called ‘consistency unfoldings', in the latter. This result sheds additional light on the axiomatisation of Brandt and Henglein: it provides an alternative soundness proof for the adaptation considered here.We then outline an analogous duality result that holds for a pair of similar proof systems for proving that equational specifications of cyclic term graphs are bisimilar.

scholarly journals Universal Hash Proofs and a Paradigm for Adaptive Chosen Ciphertext Secure Public-Key Encryption

2001 ◽  
Vol 8 (37) ◽  
Author(s):  
Ronald Cramer ◽  
Victor Shoup
Keyword(s):  
Special Kind ◽  
Public Key ◽  
Proof System ◽  
Reasonable Assumption ◽  
Encryption Scheme ◽  
Public Key Encryption ◽  
Proof Systems ◽  
Encryption Schemes ◽  
Hash Proof System ◽  
Chosen Ciphertext Attack

We present several new and fairly practical public-key encryption schemes and prove them secure against adaptive chosen ciphertext attack. One scheme is based on Paillier's Decision Composite Residuosity (DCR) assumption, while another is based in the classical Quadratic Residuosity (QR) assumption. The analysis is in the standard cryptographic model, i.e., the security of our schemes does not rely on the Random Oracle model.<br /> <br />We also introduce the notion of a universal hash proof system. Essentially, this is a special kind of non-interactive zero-knowledge proof system for an NP language. We do not show that universal hash proof systems exist for all NP languages, but we do show how to construct very efficient universal hash proof systems for a general class of group-theoretic language membership problems.<br /> <br />Given an efficient universal hash proof system for a language with certain natural cryptographic indistinguishability properties, we show how to construct an efficient public-key encryption schemes secure against adaptive chosen ciphertext attack in the standard model. Our construction only uses the universal hash proof system as a primitive: no other primitives are required, although even more efficient encryption schemes can be obtained by using hash functions with appropriate collision-resistance properties. We show how to construct efficient universal hash proof systems for languages related to the DCR and QR assumptions. From these we get corresponding public-key encryption schemes that are secure under these assumptions. We also show that the Cramer-Shoup encryption scheme (which up until now was the only practical encryption scheme that could be proved secure against adaptive chosen ciphertext attack under a reasonable assumption, namely, the Decision Diffie-Hellman assumption) is also a special case of our general theory.

scholarly journals The Glory of the Past and Geometrical Concurrency

10.29007/klcl ◽  
2018 ◽  
Author(s):  
Cristian Prisacariu
Keyword(s):  
Modal Logic ◽  
Geometric Model ◽  
Expressive Power ◽  
Geometrical Model ◽  
Geometrical Aspect ◽  
Fine Grained ◽  
The Past ◽  
Action Refinement ◽  
Higher Dimensional ◽  
Natural Way

This paper contributes to the general understanding of the "geometrical model of concurrency" that was named higher dimensional automata (HDAs) by Pratt and van Glabbeek. In particular we provide some understanding of the modal logics for such models and their expressive power in terms of the bisimulation that can be captured.The geometric model of concurrency is interesting from two main reasons: its generality and expressiveness, and the natural way in which autoconcurrency and action refinement are captured.Logics for this model, though, are not well investigated, where a simple, yet adequate, modal logic over HDAs was only recently introduced.As this modal logic, with two existential modalities, "during" and "after", captures only split bisimulation, which is rather low in the spectrum of van Glabbeek and Vaandrager, the immediate question was what small extension of this logic could capture the more fine-grained hereditary history preserving bisimulation (hh)?In response, the work in this paper provides several insights. One is the fact that the geometrical aspect of HDAs makes it possible to use for capturing the hh-bisimulation, a standard modal logic that does not employ event variables, opposed to the two logics (over less expressive models) that we compare with. The logic that we investigate here uses standard backward-looking modalities (i.e., past modalities) and extends the previously introduced logic (called HDML) that had only forward, action-labelled, modalities.Since the direct proofs are rather intricate, we try to understand better the above issues by introducing a related model that we call ST-configuration structures, which extend the configuration structures of van Glabbeek and Plotkin. We relate this model to HDAs, and redefine and prove the earlier results in the light of this new model. These offer a different view on why the past modalities and geometrical concurrency capture the hereditary history preserving bisimulation.Additional correlating insights are also gained.

scholarly journals Temporalising Separation Logic for Planning with Search Control Knowledge

2017 ◽  
Author(s):  
Xu Lu ◽  
Cong Tian ◽  
Zhenhua Duan
Keyword(s):  
Temporal Logic ◽  
Expressive Power ◽  
Separation Logic ◽  
Temporal Constraints ◽  
Ai Planning ◽  
Temporal Logics ◽  
Spatial Constraints ◽  
Spatio Temporal ◽  
Control Knowledge ◽  
Search Control

Temporal logics are widely adopted in Artificial Intelligence (AI) planning for specifying Search Control Knowledge (SCK). However, traditional temporal logics are limited in expressive power since they are unable to express spatial constraints which are as important as temporal ones in many planning domains. To this end, we propose a two-dimensional (spatial and temporal) logic namely PPTL^SL by temporalising separation logic with Propositional Projection Temporal Logic (PPTL). The new logic is well-suited for specifying SCK containing both spatial and temporal constraints which are useful in AI planning. We show that PPTL^SL is decidable and present a decision procedure. With this basis, a planner namely S-TSolver for computing plans based on the spatio-temporal SCK expressed in PPTL^SL formulas is developed. Evaluation on some selected benchmark domains shows the effectiveness of S-TSolver.

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