Representation and Reasoning about Evolutions of the World in the Context of Reasoning about Actions

The first step in reasoning about actions and change involves reasoning about how the world would evolve if a certain action is executed in a certain state. Most research on this assumes the evolution to be only a single step and focus on formulating the transition function that defines changes between states due to actions. In this paper we consider cases where the evolution is more than just a single change between one state and another. This is manifested when the execution of an action may trigger execution of other actions, or when multiple agents act on the environment following certain strategies.     

Axiomatic Extensions of IMT3 Logic

In this paper we characterize, classify and axiomatize all axiomatic extensions of the IMT3 logic. This logic is the axiomatic extension of the involutive monoidal t-norm logic given by ¬φ³ ∨ φ. For our purpose we study the lattice of all subvarieties of the class IMT3, which is the variety of IMTL-algebras given by the equation ¬(x³) ∨ x ≈ ?, and it is the algebraic counterpart of IMT3 logic. Since every subvariety of IMT3 is generated by their totally ordered members, we study the structure of all IMT3-chains in order to determine the lattice of all subvarieties of IMT3. Given a family of IMT3-chains the number of elements of the largest odd finite subalgebra in the family and the number of elements of the largest even finite subalgebra in the family turns out to be a complete classifier of the variety generated. We obtain a canonical set of generators and a finite equational axiomatization for each subvariety and, for each corresponding logic, a finite set of characteristic matrices and a finite set of axioms.     

AI &; Law,Logic and Argument Schemes

This paper reviews the history of AI & Law research from the perspective of argument schemes. It starts with the observation that logic, although very well applicable to legal reasoning when there is uncertainty, vagueness and disagreement, is too abstract to give a fully satisfactory classification of legal argument types. It therefore needs to be supplemented with an argument-scheme approach, which classifies arguments not according to their logical form but according to their content, in particular, according to the roles that the various elements of an argument can play. This approach is then applied to legal reasoning, to identify some of the main legal argument schemes. It is also argued that much AI & Law research in fact employs the argument-scheme approach, although it usually is not presented as such. Finally, it is argued that the argument-scheme approach and the way it has been employed in AI & Law respects some of the main lessons to be learnt from Toulmin’s The Uses of Argument.     

Types of \mathsf{I} -Free Hereditary Right Maximal Terms

The implicational fragment of the relevance logic “ticket entailment” is closely related to the so-called hereditary right maximal terms. I prove that the terms that need to be considered as inhabitants of the types which are theorems of T_→ are in normal form and built in all but one case from and only. As a tool in the proof ordered term rewriting systems are introduced. Based on the main theorem I define FIT_→ – a Fitch-style calculus (related to FT_→) for the implicational fragment of ticket entailment.     

Number of Extensions of Non-Fregean Logics

We show that there are continuum many different extensions of SCI (the basic theory of non-Fregean propositional logic) that lie below WF (the Fregean extension) and are closed under substitution. Moreover, continuum many of them are independent from WB (the Boolean extension), continuum many lie above WB and are independent from WH (the Boolean extension with only two values for the equality relation), and only countably many lie between WH and WF.     

Conditional Probability and Defeasible Inference

We offer a probabilistic model of rational consequence relations (Lehmann and Magidor, 1990) by appealing to the extension of the classical Ramsey–Adams test proposed by Vann McGee in (McGee, 1994). Previous and influential models of non-monotonic consequence relations have been produced in terms of the dynamics of expectations (Gärdenfors and Makinson, 1994; Gärdenfors, 1993).Expectation is a term of art in these models, which should not be confused with the notion of expected utility. The expectations of an agent are some form of belief weaker than absolute certainty. Our model offers a modified and extended version of an account of qualitative belief in terms of conditional probability, first presented in (van Fraassen, 1995). We use this model to relate probabilistic and qualitative models of non-monotonic relations in terms of expectations. In doing so we propose a probabilistic model of the notion of expectation. We provide characterization results both for logically finite languages and for logically infinite, but countable, languages. The latter case shows the relevance of the axiom of countable additivity for our probability functions. We show that a rational logic defined over a logically infinite language can only be fully characterized in terms of finitely additive conditional probability. The research of both authors was supported in part by a grant from NSF, and, for Parikh, also by support from the research foundation of CUNY.     

Cooperation,Knowledge, and Time: Alternating-time Temporal Epistemic Logic and its Applications

Branching-time temporal logics have proved to be an extraordinarily successful tool in the formal specification and verification of distributed systems. Much of their success stems from the tractability of the model checking problem for the branching time logic CTL, which has made it possible to implement tools that allow designers to automatically verify that systems satisfy requirements expressed in CTL. Recently, CTL was generalised by Alur, Henzinger, and Kupferman in a logic known as Alternating-time Temporal Logic (ATL). The key insight in ATL is that the path quantifiers of CTL could be replaced by cooperation modalities, of the form , where is a set of agents. The intended interpretation of an ATL formula is that the agents can cooperate to ensure that holds (equivalently, that have a winning strategy for ). In this paper, we extend ATL with knowledge modalities, of the kind made popular in the work of Fagin, Halpern, Moses, Vardi and colleagues. Combining these knowledge modalities with ATL, it becomes possible to express such properties as group can cooperate to bring about iff it is common knowledge in that . The resulting logic — Alternating-time Temporal Epistemic Logic (ATEL) — shares the tractability of model checking with its ATL parent, and is a succinct and expressive language for reasoning about game-like multiagent systems.