Proof Rules Research Articles

The complexity of disjunction in intuitionistic logic

AbstractWe study procedures for the derivability problem of fragments of intuitionistic logic. Intuitionistic logic is known to be PSPACE-complete, with implication being one of the main contributors to this complexity. In fact, with just implication alone, we still have a PSPACE-complete logic. We study fragments of intuitionistic logic with restricted implication and develop algorithms for these fragments which are based on the proof rules. We identify a core fragment whose derivability is solvable in linear time. Adding disjunction elimination to this core gives a logic which is solvable in co-NP. These sub-procedures are applicable to a wide variety of logics with rules of a similar flavour. We also show that we cannot do better than co-NP whenever disjunction elimination interacts with other rules.

The future is ours: prophecy variables in separation logic

Early in the development of Hoare logic, Owicki and Gries introduced auxiliary variables as a way of encoding information about the history of a program’s execution that is useful for verifying its correctness. Over a decade later, Abadi and Lamport observed that it is sometimes also necessary to know in advance what a program will do in the future . To address this need, they proposed prophecy variables , originally as a proof technique for refinement mappings between state machines. However, despite the fact that prophecy variables are a clearly useful reasoning mechanism, there is (surprisingly) almost no work that attempts to integrate them into Hoare logic. In this paper, we present the first account of prophecy variables in a Hoare-style program logic that is flexible enough to verify logical atomicity (a relative of linearizability) for classic examples from the concurrency literature like RDCSS and the Herlihy-Wing queue. Our account is formalized in the Iris framework for separation logic in Coq. It makes essential use of ownership to encode the exclusive right to resolve a prophecy, which in turn enables us to enforce soundness of prophecies with a very simple set of proof rules.

Automatically Verifying Temporal Properties of Pointer Programs with Cyclic Proof

In this article, we investigate the automated verification of temporal properties of heap-aware programs. We propose a deductive reasoning approach based on cyclic proof. Judgements in our proof system assert that a program has a certain temporal property over memory state assertions, written in separation logic with user-defined inductive predicates, while the proof rules of the system unfold temporal modalities and predicate definitions as well as symbolically executing programs. Cyclic proofs in our system are, as usual, finite proof graphs subject to a natural, decidable soundness condition, encoding a form of proof by infinite descent. We present a proof system tailored to proving CTL properties of nondeterministic pointer programs, and then adapt this system to handle fair execution conditions. We show both versions of the system to be sound, and provide an implementation of each in the Cyclist theorem prover, yielding an automated tool that is capable of automatically discovering proofs of (fair) temporal properties of pointer programs. Experimental evaluation of our tool indicates that our approach is viable, and offers an interesting alternative to traditional model checking techniques.

Axiomatization and computability of a variant of iteration-free PDL with fork

We devote this paper to the axiomatization and the computability of PDL0Δ, a variant of iteration-free PDL with fork. Concerning the axiomatization, our results are based on the following: although the program operation of fork is not modally definable in the ordinary language of PDL, it becomes definable in a modal language strengthened by the introduction of propositional quantifiers. Instead of using axioms to define the program operation of fork in the language of PDL enlarged with propositional quantifiers, we add an unorthodox rule of proof that makes the canonical model standard for the program operation of fork and we use large programs for the proof of the Truth Lemma. Concerning the computability, we prove by a selection procedure that PDL0Δ has a strong finite property, hence is decidable.

An intuitionistic formula hierarchy based on high‐school identities

AbstractWe revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal) propositional sequent calculi are formula (i.e., sequent) isomorphisms corresponding to the high‐school identities, we show that one can obtain a more compact variant of a proof system, consisting of non‐invertible proof rules only, and where the invertible proof rules have been replaced by a formula normalization procedure. Moreover, for certain proof systems such as the G4ip sequent calculus of Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the non‐invertible proof rules as strict inequalities between exponential polynomials; a careful combinatorial treatment is given in order to establish this fact. Finally, we extend the exponential polynomial analogy to the first‐order quantifiers, showing that it gives rise to an intuitionistic hierarchy of formulas, resembling the classical arithmetical hierarchy, and the first one that classifies formulas while preserving isomorphism.

Electronic civil lawsuit proceedings comparative study

The electronic development has imposed itself on all areas of life, including the legal field, as the technological developments have greatly affected the legislations issued by the legislator, as the legislator tries hard to keep pace with technological developments in various fields. In the field of litigation, the courts seek to make the most of modern technologies And the courts and the law behind them to resort to dealing with this type of courts only serve the parties (the citizen) and the courts also, the advantages of resorting to this type of courts are numerous, they are easy too much of the right of litigation, as the parties to the case they can raise their claim electronically, as well as these courts save time effort and reduce the expenses spent by the parties to the suit, complete confidentiality of the plaintiff and provide, as the litigation remotely provides complete secrecy. The electronic filing system has many advantages. It provides a kind of technology that allows lawyers and litigants to submit legal documents in an electronic manner according to an integrated system. This system also reduces the cost of excessive litigation fees and eliminates the huge quantities of documents and documents related to the case that are filled Which allows the courts to function more effectively. The system also provides the possibility of receiving documents at any time daily, even on holidays and public holidays, for 24 hours. Therefore, jurisprudence tried to apply the new concepts of electronic evidence to the existing rules of evidence. Faced with this situation, jurisprudence began to question the degree of legal validity that the law of evidence could give to electronic signatures, and whether the traditional rules of the law of evidence could confer the power of electronic documents to the same degree as the paper documents. Therefore, the men of jurisprudence and law began to find solutions by distinguishing between two cases. The first is the case of a prior agreement between the parties that regulates the validity of the electronic signature; the second is the validity of the electronic signature in the absence of an agreement regulating its validity. The establishment of electronic proceedings is also subject to many legal difficulties, since electronic props that are stored on electronic media such as CD, discs and flash, and the methods of storing, storing and retrieving them are not easily adapted to traditional rules of proof

Combining logical and distributional methods in type-logical grammars

We propose a low-level way of combining distributional and logical ideas into a single formal system. This will be an instantiation of a more general system, adding weights to proof rules. These weights will not measure some sort of "confidence the proof is valid", but rather act as a way to prefer some proofs over others, where preference can mean "easier to process (for humans)" or "more coherent (combining words that make sense together)". The resulting system of weighted theorem proving can be implemented either as a best-first proof search strategy or as a polynomial-time approximation of proof search for NP-complete parsing problems.

From mathematical axioms to mathematical rules of proof: recent developments in proof analysis.

A short text in the hand of David Hilbert, discovered in Göttingen a century after it was written, shows that Hilbert had considered adding a 24th problem to his famous list of mathematical problems of the year 1900. The problem he had in mind was to find criteria for the simplicity of proofs and to develop a general theory of methods of proof in mathematics. In this paper, it is discussed to what extent proof theory has achieved the second of these aims. This article is part of the theme issue 'The notion of 'simple proof' - Hilbert's 24th problem'.

LoET-E: A Refined Theory for Proving Security Properties of Cryptographic Protocols

Nowadays, more and more new cryptographic protocols are emerging, and the security analysis of emerging cryptographic protocols is increasingly important. The logic of events is an axiomatic method based on theorem proving, designed around message automation with actions for possible protocol steps; it figured out types of information transmitted in the protocols and also presented novel proof rules and mechanism. However, with the emergence of various cryptographic protocols, the logic of events lacks corresponding axioms and rules in the process of proving certain cryptographic protocols, so it needs a further extension. Based on the logical framework of protocol composition logic, this paper presents a refined theory of the logic of events called LoET-E, in which the novel rules about the freshness of nonces, the event attributes of messages, and the states of the predicate is presented; the concepts of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Fresh$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Gen$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$FirstSend$ </tex-math></inline-formula> is introduced; and the definition of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$has$ </tex-math></inline-formula> and the honesty axiom of LoET is extended. The refined theory can guarantee the correctness, integrity, and validity of the original axioms, ensure the consistency of event classes and basic sequences in the proof process, reduce the complexity and redundancy in the protocol analysis process, and most importantly, extend the provable range of cryptographic protocols.

The Evolution of the Liability of Credit Rating Agencies in the United States and in the European Union: Regulation after the Crisis

Credit rating agencies have assessed the creditworthiness of issuers and debt instruments for over a century. Nevertheless, in the United States and in the European Union a first regulation of rating services was passed only at the beginning of the twenty-first century, respectively in 2006 and 2009. Statutory liability rules were later adopted in the United States with the Dodd-Frank Act of 2010 and in the European Union with Regulation (EU) no 462/2013. Despite some similarities between the American and European existing discipline, significant differences still exist and pave the way for regulatory arbitrage opportunities in the ratings market. The reasons for divergence are clearly historically based and derive, in part, from the different traditions of the two legal systems. Accordingly, this article compares the evolution of the US regulatory framework and case law on the liability of rating organizations towards investors with the uniform rules adopted by the European Union since 2009, absent a comparable case law at the European level. It is argued that, in both systems, while it is easier to establish liability in case of intent, burden of proof rules generally place a significant, if not insurmountable, obstacle to damage compensation for investors.

증명책임의 원칙, 발견과 해설

사후 사실심 단계에서 사실을 재현하고 완전히 유죄를 입증하는 것은 불가능하며, 항상 불확실성이라는 문제가 따르게 된다. 미국의 경우 사실확정이 불가능한 경우, 증거법과 헌법에 따라 사실심 판단자가 이용할 수 있는 증거판단에 관한 몇 가지 법칙이 존재한다. 그 몇 가지 법칙이란 세가지 수준의 확실성에 상응하는 바, (i) 아마도 발생하였을 것이다.(ii) 발생했을 고도의 개연성이 존재한다.(iii) 발생하였음이 거의 확실하다.” 등이 그것이다. 입증책임 법칙에 의존하는 것을 정당화하는 공통적 사법정책은 기능적으로 (i) 사실판단에 있어 오류발생가능성에 대하여 총체적으로, 또는 특정유형 오류발생에 대하여 대응하기 위한 것으로, 남은 오류위험에 대하여 특정 당사자에게 위험을 적절하게 배분하기 위한 것이다. 현재 적용되고 해석상 다투어지는 보통법상의 입증책임 원칙은 정확성과 공정성이라는 정책적 가치를 지지, 표준화, 형평화할 수 있는 사실확정을 확실하게 담보하지 못하고 있다. 이러한 부분은 생명과 자유가 심각하게 문제되는 형사소송 또는 민사상 사기로 인한 소송에서 이용되는 중간적 입증책임 원칙의 경우 특히 부작용을 잉태하고 있다. 이러한 문제에 대한 마이클 파르도의 이론은 매우 설득력이 있는데, 그의 이론에 따르면 확률에 기초한 사실심 법관의 판단은 2차원 설명법칙에 따라 행하여 져야 한다는 것으로 동 문제에 대처함에 있어 효과적이다. 2차원 설명법칙에 따르면 연역적 과정이 뒤바뀌게 되는데, 사실 판단자는 만약 사실을 설명하는 가정적 제언이 사실이라면 여러 제언 중 어떠한 제언이 현출된 증거에 가장 부합하는가에 기초하여 사실을 확정하게 된다.

Weakest Precondition Reasoning for Expected Runtimes of Randomized Algorithms

This article presents a wp--style calculus for obtaining bounds on the expected runtime of randomized algorithms. Its application includes determining the (possibly infinite) expected termination time of a randomized algorithm and proving positive almost--sure termination—does a program terminate with probability one in finite expected time? We provide several proof rules for bounding the runtime of loops, and prove the soundness of the approach with respect to a simple operational model. We show that our approach is a conservative extension of Nielson’s approach for reasoning about the runtime of deterministic programs. We analyze the expected runtime of some example programs including the coupon collector’s problem, a one--dimensional random walk and a randomized binary search.

A UTP approach for rTiMo

AbstractrTiMo is a real-time version of TiMo (Timed Mobility), which is a process algebra for mobile distributed systems. In this paper, we investigate the denotational semantics for rTiMo. A trace variabletris introduced to record the communications among processes as well as the location where the communication action takes place. Based on the formalized model, we study a set of algebraic laws, especially the laws about the migration and communication with real-time constraints. In order to facilitate the algebraic reasoning about the parallel expansion laws, we enrich rTiMo with a form ofguardedchoice. This can enable us to convert every parallel program to the guarded choice form. Moreover, we also provide a set of proof rules, which can be used to verify the correctness and real-time properties of programs.

Finite-trace linear temporal logic: coinductive completeness

Linear temporal logic (LTL) is suitable not only for infinite-trace systems, but also for finite-trace systems. Indeed, LTL is frequently used as a trace specification formalism in runtime verification. The completeness of LTL with only infinite or with both infinite and finite traces has been extensively studied, but similar direct results for LTL with only finite traces are missing. This paper proposes a sound and complete proof system for finite-trace LTL. The axioms and proof rules are natural and expected, except for one rule of coinductive nature, reminiscent of the Godel-Lob axiom. A direct decision procedure for finite-trace LTL satisfiability, a PSPACE-complete problem, is also obtained as a corollary.

A Logical Analysis of Framing for Specifications with Pure Method Calls

For specifying and reasoning about object-based programs, it is often attractive for contracts to be expressed using calls to pure methods. It is useful for pure methods to have contracts, including read effects, to support local reasoning based on frame conditions. This leads to puzzles such as the use of a pure method in its own contract. These ideas have been explored in connection with verification tools based on axiomatic semantics, guided by the need to avoid logical inconsistency, and focusing on encodings that cater for first-order automated provers. This article adds pure methods and read effects to region logic, a first-order program logic that features frame-based local reasoning and provides modular reasoning principles for end-to-end correctness. Modular reasoning is embodied in a proof rule for linking a module’s method implementations with a client that relies on the method contracts. Soundness is proved with respect to conventional operational semantics and uses an extensional (i.e, relational) interpretation of read effects. Applicability to tools based on SMT solvers is demonstrated through machine-checked verification of examples. The developments in this article can guide the implementations of linking as used in modular verifiers and serve as a basis for studying observationally pure methods and encapsulation.

Uma concepção minimalista e garantista da presunção de inocência

O artigo visa abordar as múltiplas facetas que a presunção de inocência incorpora nos modernos sistemas jurídicos desde uma perspectiva crítica. Neste sentido, a partir de uma metodologia analítica busca-se demonstrar que algumas dessas facetas se sobrepõem a outros direitos e institutos jurídicos o que, finalmente, longe de aumentar as garantias dos cidadãos, chega a produzir confusão e falta de controlabilidade das decisões judiciais. Assim, defende-se a conveniência conceitual e prática de se pensar a presunção de inocência evitando sobreposições com outros direitos ou conceitos jurídicos; como standards de prova ou ônus da prova. Daí a referência a uma concepção minimalista e garantista da presunção de inocência e, como será possível ver, a defesa da presunção de inocência como uma regra de segunda ordem cuja aplicação cobraria sentido em contextos de incerteza sobre a satisfação do standard de prova.

The Sequent Calculus Trainer with Automated Reasoning - Helping Students to Find Proofs

The sequent calculus is a formalism for proving validity of statements formulated in First-Order Logic. It is routinely used in computer science modules on mathematical logic. Formal proofs in the sequent calculus are finite trees obtained by successively applying proof rules to formulas, thus simplifying them step-by-step. Students often struggle with the mathematical formalities and the level of abstraction that topics like formal logic and formal proofs involve. The difficulties can be categorised as syntactic or semantic. On the syntactic level, students need to understand what a correctly formed proof is, how rules can be applied (on paper for instance) without leaving the mathematical framework of the sequent calculus, and so on. Beyond this, on the semantic level, students need to acquire strategies that let them find the right proof. The Sequent Calculus Trainer is a tool that is designed to aid students in learning the techniques of proving given statements formally. In this paper we describe the didactical motivation behind the tool and the techniques used to address issues on the syntactic as well as on the semantic level.

„Polecam!” – naukowiec / „Polecam!” – konsument. Znaczenie obecności społecznych lub naukowych dowodów słuszności dla oceny wiarygodności reklam w kontekście refleksyjności poznawczej, maksymalizacji i światopoglądu scjentystycznego

The article examines the relationship between the perceived reliability of advertisements using the rule of social proof and the rule of authority in the context of cognitive reflection, maximisation and the scientific worldview. Results collected in a sample of 110 participants showed that cognitive reflection was not associated with the assessment of the credibility of advertisings embedded in various contexts. Furthermore, maximisation correlated positively with the credibility of the advertisements embedded in social and scientific contexts. As the scientific worldview increased, so too did the perceived level of credibility of advertisements containing scientific references. Hierarchical regression analysis was used to identify the space of predictors of the attitude towards differentiated advertising messages. The structural equations modelling (with bootstrap) revealed more complex connections between maximisation and the scientific worldview in the range of effects on the perceived reliability of advertisements containing scientific arguments.

T-Barrier Certificates: A Continuous Analogy to k-Induction

Safety proofs of discrete and continuous systems often use related proof approaches, and insight can be obtained by comparing reasoning methods across domains. For example, proofs using inductive invariants in discrete systems are analogous to barrier certificate methods in continuous systems. In this paper, we present and prove the soundness of continuous and hybrid analogs to the k-induction proof rule, which we call t-barrier certificates.The method combines symbolic reasoning and time-bounded reachability along the barrier in order to prove system safety. Compared with traditional barrier certificates, a larger class of functions can be shown to be t-barrier certificates, so we expect them to be easier to find. Compared with traditional reachability analysis, t-barrier certificates can be computationally tractable in nonlinear settings despite large initial sets, and they prove time-unbounded safety. We demonstrate the feasibility of the approach with a nonlinear harmonic oscillator example, using sympy and Z3 for symbolic reasoning and Flow⁎ for reachability analysis.

A new proof rule for almost-sure termination

We present a new proof rule for proving almost-sure termination of probabilistic programs, including those that contain demonic non-determinism. An important question for a probabilistic program is whether the probability mass of all its diverging runs is zero, that is that it terminates "almost surely". Proving that can be hard, and this paper presents a new method for doing so. It applies directly to the program's source code, even if the program contains demonic choice. Like others, we use variant functions (a.k.a. "super-martingales") that are real-valued and decrease randomly on each loop iteration; but our key innovation is that the amount as well as the probability of the decrease are parametric. We prove the soundness of the new rule, indicate where its applicability goes beyond existing rules, and explain its connection to classical results on denumerable (non-demonic) Markov chains.