A Complete Logical System for the Equality of Recursive Terms for Sets

A complete proof system for propositional projection temporal logic

The main goal of this paper is to provide an abstract framework for constructing proof systems for various many-valued logics. Using the framework it is possible to generate strongly complete proof systems with respect to any finitely valued deterministic and non-deterministic logic. I provide a couple of examples of proof systems for well-known many-valued logics and prove the completeness of proof systems generated by the framework.

Focused proofs are sequent calculus proofs that group inference rules into alternating positive and negative phases. These phases can then be used to define macro-level inference rules Gentzen’s original and tiny introduction and structural rules. We show here that the inference rules of labeled proof systems for modal logics can similarly be described as pairs of such phases within the LKF focused proof system for first-order classical logic. We consider the system $${ G3K} $$ of Negri for the modal logic $$K$$ and define a translation from labeled modal formulas into first-order polarized formulas and show a strict correspondence between derivations in the two systems, i.e., each rule application in $${ G3K} $$ corresponds to a bipole—a pair of a positive and a negative phases—in $${ LKF} $$ . Since geometric axioms (when properly polarized) induce bipoles, this strong correspondence holds for all modal logics whose Kripke frames are characterized by geometric properties. We extend these results to present a focused labeled proof system for this same class of modal logics and show its soundness and completeness. The resulting proof system allows one to define a rich set of normal forms of modal logic proofs.

The article proposes a new approach to reasoning about knowledge and strategies in multiagent systems. It emphasizes data, not agents, as the source of strategic knowledge. The approach brings together Armstrong's functional dependency expression from database theory, a data-informed knowledge modality based on a recent work by Baltag and van Benthem, and a newly proposed data-informed strategy modality. The main technical result is a sound and complete logical system that describes the interplay between these three logical operators.

If an agent, or a coalition of agents, has a strategy, knows that she has a strategy, and knows what the strategy is, then she has a know-how strategy. Several modal logics of coalition power for know-how strategies have been studied before.
 The contribution of the article is three-fold. First, it proposes a new class of know-how strategies that depend on the intelligence information about the opponents’ actions. Second, it shows that the coalition power modality for the proposed new class of strategies cannot be expressed through the standard know-how modality. Third, it gives a sound and complete logical system that describes the interplay between the coalition power modality with intelligence and the distributed knowledge modality in games with imperfect information.

The paper studies strategic abilities that rise from restrictions on the information sharing in multi-agent systems. The main technical result is a sound and complete logical system that describes the interplay between the knowledge and the strategic ability modalities.

The paper investigates the second-order blameworthiness or duty to warn modality "one coalition knew how another coalition could have prevented an outcome''. The main technical result is a sound and complete logical system that describes the interplay between the distributed knowledge and the duty to warn modalities.

The author introduces a temporal logic called interval division logic, IDL, based on the constructive temporal ontology. In IDL, time is regarded as a constructive object which is built from an interval by iterating the interval division every time a new temporal fact is recognized. IDL as a sound and complete logical system which is as expressive as the Buich tree automata. In order to examine how the persistence problem is treated on this ontology, the author extends a nonmonotonic version of IDL based on the model preference in which any belief is changed as late as possible in the epistemological order rather than the temporal order. The Yale shooting problem is discussed in this framework.

Sound over-approximation methods are effective for proving the absence of errors, but inevitably produce false alarms that can hamper programmers. In contrast, under-approximation methods focus on bug detection and are free from false alarms. In this work, we present two novel proof systems designed to locate the source of errors via backward under-approximation, namely Sufficient Incorrectness Logic (SIL) and its specialization for handling memory errors, called Separation SIL. The SIL proof system is minimal, sound and complete for Lisbon triples, enabling a detailed comparison of triple-based program logics across various dimensions, including negation, approximation, execution order, and analysis objectives. More importantly, SIL lays the foundation for our main technical contribution, by distilling the inference rules of Separation SIL, a sound and (relatively) complete proof system for automated backward reasoning in programs involving pointers and dynamic memory allocation. The completeness result for Separation SIL relies on a careful crafting of both the assertion language and the rules for atomic commands.

A methodology for developing sound proof systems for program verification is demonstrated by the development of a proof system for reasoning about concurrent programs based on the unity logic of K.M. Chandy and J. Misra (1988). This proof system has been validated by an automated theorem prover, a computer program that checks the correctness of proofs. The Boyer-Moore logic in which the proof system is formalized and the Boyer-Moore theorem prover which mechanizes this logic are described. The motivation behind the unity logic and the way in which it may be used to prove the correctness of concurrent programs are examined. A proof system for concurrent programs based on the unity logic system is provided as an example. An operational semantics of concurrency is formalized in Boyer-Moore logic by use of the transition system model. Unity's proof rules are proved as theorems about this operational semantics. The proofs of these theorems are mechanically checked. The entire proof system has been verified by the Boyer-Moore prover, making it possible to prove mechanically the consequence of other concurrent programs.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Theorem proving procedures for the propositional calculus have traditionally relied on syntactic manipulations of the formula to derive a proof. In particular, clausal theorem provers sometimes lose some of the obvious semantics present in the theorem, in the process of converting the theorem into an unnatural normal form. Most existing propositional theorem provers do not incorporate substitution of equals for equals as an inference rule. In this paper we develop a “natural” proof system for the propositional calculus, with the goal that most succinct mathematical proofs should be encodable as short formal proofs within the proof system. The main distinctive features of NPS are: We show that from a complexity theoretic viewpoint NPS is at least as powerful as the resolution procedure. We further demonstrate formulas on which NPS fares better than resolution. Finally, since proofs in NPS usually resemble manual proofs, we feel that NPS is easily amenable to an interactive theorem prover.

This paper presents a sound and complete proof system for the logical system whose sentences are of the form AllXareY, Some X are Y and Most X are Y, where we interpret these sentences on finite models, with the meaning of “most” being “strictly more than half.” Our proof system is syllogistic; there are no individual variables.

Starting with Hoare Logic over 50 years ago, numerous program logics have been devised to reason about the different kinds of programs encountered in the real world. This includes reasoning about computational effects, particularly those effects that cause the program execution to branch into multiple paths due to, e.g., nondeterministic or probabilistic choice. Outcome Logic reimagines Hoare Logic with branching at its core, using an algebraic representation of choice to capture programs that branch into many outcomes. In this article, we give a comprehensive account of the Outcome Logic metatheory. This includes a relatively complete proof system for Outcome Logic with the ability to reason about general purpose looping. We also show that this proof system applies to programs with various types of branching, that it subsumes some well-known logics such as Hoare Logic, and that it facilitates the reuse of proof fragments across different kinds of specifications.

Along with the convenience brought by the increasing usage of autonomous systems, unexpected accidents happened. These accidents emphasize the need for autonomous systems to possess the ability to recognize potential hazards, effectively communicate these hazards to human operators, and facilitate retrospective analyses. This ability, including recognition, prejudgment, post-analysis, and reasoning, falls within the realm of cognitive ability, which is important in improving the safety and outcomes in the decision-making process of such systems. In this paper, we present a foundational step toward addressing the safety challenges involving the cognitive ability of artificial agents. We study the interplay between knowledge and the cognitive ability of intelligent agents. The main technical result is a sound and complete bimodal logical system that describes the interplay between the knowledge and cognitive ability modalities.

Together we know how to achieve: An epistemic logic of know-how