Likelihood,probability, and knowledge1

The modal logic LL was introduced by Halpern and Rabin as a means of doing qualitative reasoning about likelihood. Here the relationship between LL and probability theory is examined. It is shown that there is a way of translating probability assertions into LL in a sound manner, so that LL in some sense can capture the probabilistic interpretation of likelihood. However, the translation is subtle; several more obvious attempts are shown to lead to inconsistencies. We also extend LL by adding modal operators for knowledge. This allows us to reason about the interaction between knowledge and likelihood. The propositional version of the resulting logic LLK is shown to have a complete axiomatization and to be decidable in exponential time, provably the best possible.     

Weakly complete axiomatization of exogenous quantum propositional logic

A finitary axiomatization for EQPL (exogenous quantum propositional logic) is presented. The axiomatization is shown to be weakly complete relative to an oracle for analytical reasoning. The proof is carried out using a non-trivial extension of the Fagin–Halpern–Megiddo technique together with three Henkin style completions.     

A Temporal Logic for Proving Properties of Topologically General Executions

We present a generalization of the temporal propositional logic of linear time which is useful for stating and proving properties of the generic execution sequence of a parallel program or a non-deterministic program. The formal system we present is exactly that same as the third of three logics presented by Lehmann and Shelah (Information and Control53, 165–198 (1982)), but we give it a different semantics. The models are tree models of arbitrary size similar to those used in branching time temporal logic. The formulation we use allows us to state properties of the “co-meagre” family of paths, where the term “co-meagre” refers to a set whose complement is of the first category in Baire's classification looking at the set of paths in the model as a metric space. Our system is decidable, sound, and, complete for models of arbitrary size, but it has the finite model property; namely, every sentence having a model has a finite model.     

A Symmetric Lambda Calculus for Classical Program Extraction

We introduce aλ-calculus with symmetric reduction rules and “classical” types, i.e., types corresponding to formulas of classical propositional logic. The strong normalization property is proved to hold for such a calculus, as well as for its extension to a system equivalent to Peano arithmetic. A theorem on the shape of terms in normal form is also proved, making it possible to get recursive functions out of proofs ofΠ⁰₂formulas, i.e., those corresponding to program specifications.     

Sequential operators in computability logic

Computability logic (CL) is a semantical platform and research program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth which it has more traditionally been. Formulas in CL stand for (interactive) computational problems, understood as games between a machine and its environment; logical operators represent operations on such entities; and “truth” is understood as existence of an effective solution, i.e., of an algorithmic winning strategy.The formalism of CL is open-ended, and may undergo series of extensions as the study of the subject advances. The main groups of operators on which CL has been focused so far are the parallel, choice, branching, and blind operators, with the logical behaviors of the first three groups resembling those of the multiplicatives, additives and exponentials of linear logic, respectively. The present paper introduces a new important group of operators, called sequential. The latter come in the form of sequential conjunction and disjunction, sequential quantifiers, and sequential recurrences (“exponentials”). As the name may suggest, the algorithmic intuitions associated with this group are those of sequential computations, as opposed to the intuitions of parallel computations associated with the parallel group of operations. Specifically, while playing a parallel combination of games means playing all components of the combination simultaneously, playing a sequential combination means playing the components in a sequential fashion, one after one.The main technical result of the present paper is a sound and complete axiomatization of the propositional fragment of computability logic whose vocabulary, together with negation, includes all three — parallel, choice and sequential — sorts of conjunction and disjunction. An extension of this result to the first-order level is also outlined.     

Logics to formalise p-adic valued probability and their applications

Abstract

In this paper, we focus on the main results which we have developed to obtain different propositional logics for reasoning about p-adic valued probabilities. Each of these logics is a sound, complete, and decidable extension of classical propositional logic. Also, we show some applications of these logics in modelling cognitions.     

The relationship between knowledge,belief, and certainty

We consider the relation between knowledge and certainty, where a fact isknown if it is true at all worlds an agent considers possible and iscertain if it holds with probability 1. We identify certainty with belief, interpreted probabilistically. We show that if we assume one fixed probability assignment, then the logic KD45, which has been identified as perhaps the most appropriate for belief, provides a complete axiomatization for reasoning about certainty. Just as an agent may believe a fact although is false, he may be certain that a fact is true although is false. However, it is easy to see that an agent can have such false (probabilistic) beliefs only at a set of worlds of probability 0. If we restrict attention to structures where all worlds have positive probability, then S5 provides a complete axiomatization. If we consider a more general setting, where there might be a different probability assignment at each world, then by placing appropriate conditions on thesupport of the probability function (the set of worlds which have non-zero probability), we can capture many other well-known modal logics, such as T and S4. Finally, we considerMiller's principle, a well-known principle relating higher-order probabilities to lower-order probabilities, and show that in a precise sense KD45 characterizes certainty in those structures satisfying Miller's principle.     

Theory of truth degrees of propositions in the logic system Ln*

By means of infinite product of uniformly distributed probability spaces of cardinal n the concept of truth degrees of propositions in the n-valued generalized Lukasiewicz propositional logic system L _n ^* is introduced in the present paper. It is proved that the set consisting of truth degrees of all formulas is dense in [0, 1], and a general expression of truth degrees of formulas as well as a deduction rule of truth degrees is then obtained. Moreover, similarity degrees among formulas are proposed and a pseudo-metric is defined therefrom on the set of formulas, and hence a possible framework suitable for developing approximate reasoning theory in n-valued generalized Lukasiewicz propositional logic is established.     

Deterministic propositional dynamic logic: Finite models,complexity, and completeness

Let p be a formula in deterministic propositional dynamic logic. A decision procedure for the satisfiability of p is given along with a construction of a finite model for every satisfiable p. The decision procedure runs in deterministic time 2^cn and the size of the model is bounded by n² · 4ⁿ, where n is the length of p. Finally, a complete axiomatization for deterministic propositional dynamic logic is given, based on the Segerberg axoms for propositional dynamic logic.     

Exogenous Probabilistic Computation Tree Logic

We define a logic EpCTL for reasoning about the evolution of probabilistic systems. System states correspond to probability distributions over classical states and the system evolution is modelled by probabilistic Kripke structures that capture both stochastic and non–deterministic transitions. The proposed logic is a temporal enrichment of Exogenous Probabilistic Propositional Logic (EPPL). The model-checking problem for EpCTL is analysed and the logic is compared with PCTL; the semantics of the former is defined in terms of probability distributions over sets of propositional symbols, whereas the latter is designed for reasoning about distributions over paths of possible behaviour. The intended application of the logic is as a specification formalism for properties of communication protocols, and security protocols in particular; to demonstrate this, we specify relevant security properties for a classical contract signing protocol and for the so–called quantum one–time pad.     

Completeness of Kozen's Axiomatisation of the Propositional μ-Calculus

Propositional μ-calculus is an extension of the propositional modal logic with the least fixpoint operator. In the paper introducing the logic Kozen posed a question about completeness of the axiomatisation which is a small extension of the axiomatisation of the model system K. It is shown that this axiomatisation is complete.     

一类扩展的动态描述逻辑

作为描述逻辑的扩展,动态描述逻辑为语义Web服务的建模和推理提供了一种有效途径.在将语义Web服务建模为动作之后,动态描述逻辑从动作执行结果的角度提供了丰富的推理机制,但对于动作的执行过程却不能加以处理.借鉴Pratt关于命题动态逻辑的相关研究,一方面,对动态描述逻辑中动作的语义重新进行定义,将每个动作解释为由关于可能世界的序列组成的集合;另一方面,在动态描述逻辑中引入动作过程断言,用来对动作的执行过程加以刻画.在此基础上提出一类扩展的动态描述逻辑EDDL(X),其中的X表示从ALC(attributive language with complements)到SHOIN(D)等具有不同描述能力的描述逻辑.以X为描述逻辑ALCQO(attributive language with complements,qualified number restrictions and nominals)的情况为例,给出了EDDL(ALCQO)的表判定算法,并证明了算法的可终止性、可靠性和完备性.EDDL(X)可以从动作执行过程和动作执行结果两个方面对动作进行全面的刻画和推理,为语义Web服务的建模和推理提供了进一步的逻辑支持.     

Reasoning in a Restricted Temporal Logic

We consider RTL, a linear time propositional temporal logic whose only modalities are the [formula] (eventually) operator and its dual [formula] (always). Although less expressive than the full temporal logic, RTL is the fragment of temporal logic that is used most often and in many verification systems. Indeed, many properties of distributed systems discussed in the literature are RTL properties. Another advantage of RTL over the full temporal logic is in the decidability procedure; while deciding satisfiability of a formula in full temporal logic is a PSPACE complete procedure, doing so for an RTL formula is in NP. We characterize the class of ω-regular languages that are definable in RTL and show simple translations between ω-regular sets and RTL formulae that define them. We explore the applications of RTL in reasoning about communication systems. Finally, we relate variants of RTL (when interpreted over a real line segments) to several fragments of Interval Modal Logic and show that the satisfiability problem for RTL when interpreted over a real line is NP-complete.     

Developing (Meta)Theory of λ-calculus in the Theory of Contexts

We present a case study on the formal development of a non trivial (meta)theory in the Theory of Contexts using the Coq proof assistant. The methodology underlying the Theory of Contexts for reasoning on systems presented in HOAS is based on an axiomatic syntactic standpoint. We feel that one of the main advantages of this approach, is that it requires a very low logical overhead.The object system we focus on is the lazy, call-by-name λ-calculus (λ_cbn), both untyped and simply typed. We will see that the formal, fully detailed development of the theory of (λ_cbn) in the Theory of Contexts introduces a small, sustainable overhead with respect to the proofs “on the paper”. Moreover, this will allow for comparison with similar case studies developed in other approaches to the metatheoretical reasoning in higher-order abstract syntax.     

A Logic for Reasoning about Generic Judgments

This paper presents an extension of a proof system for encoding generic judgments, the logic FOλ^Δ of Miller and Tiu, with an induction principle. The logic FOλ^Δ is itself an extension of intuitionistic logic with fixed points and a “generic quantifier”, , which is used to reason about the dynamics of bindings in object systems encoded in the logic. A previous attempt to extend FOλ^Δ with an induction principle has been unsuccessful in modeling some behaviours of bindings in inductive specifications. It turns out that this problem can be solved by relaxing some restrictions on , in particular by adding the axiom B≡x.B, where x is not free in B. We show that by adopting the equivariance principle, the presentation of the extended logic can be much simplified. Cut-elimination for the extended logic is stated, and some applications in reasoning about an object logic and a simply typed λ-calculus are illustrated.     

Natural Deduction for First-Order Hybrid Logic

This is a companion paper to Braüner (2004b, Journal of Logic and Computation 14, 329–353) where a natural deduction system for propositional hybrid logic is given. In the present paper we generalize the system to the first-order case. Our natural deduction system for first-order hybrid logic can be extended with additional inference rules corresponding to conditions on the accessibility relations and the quantifier domains expressed by so-called geometric theories. We prove soundness and completeness and we prove a normalisation theorem. Moreover, we give an axiom system first-order hybrid logic.     

Probabilistic logic programming

Of all scientific investigations into reasoning with uncertainty and chance, probability theory is perhaps the best understood paradigm. Nevertheless, all studies conducted thus far into the semantics of quantitative logic programming have restricted themselves to non-probabilistic semantic characterizations. In this paper, we take a few steps towards rectifying this situation. We define a logic programming language that is syntactically similar to the annotated logics of Blair et al., 1987 and Blair and Subrahmanian, 1988, 45–73) but in which the truth values are interpreted probabilistically. A probabilistic model theory and fixpoint theory is developed for such programs. This probabilistic model theory satisfies the requirements proposed by Fenstad (in “Studies in Inductive Logic and Probabilities” (R. C. Jeffrey, Ed.), Vol. 2, pp. 251–262, Univ. of California Press, Berkeley, 1980) for a function to be called probabilistic. The logical treatment of probabilities is complicated by two facts: first, that the connectives cannot be interpreted truth-functionally when truth values are regarded as probabilities; second, that negation-free definite-clause-like sentences can be inconsistent when interpreted probabilistically. We address these issues here and propose a formalism for probabilistic reasoning in logic programming. To our knowledge, this is the first probabilistic characterization of logic programming semantics.     

Probabilistic Logic over Paths

We introduce a probabilistic modal logic PPL extending the work of [Ronald Fagin, Joseph Y. Halpern, and Nimrod Megiddo. A logic for reasoning about probabilities. Information and Computation, 87(1,2):78–128, 1990; Ronald Fagin and Joseph Y. Halpern. Reasoning about knowledge and probability. Journal of the ACM, 41(2):340–367, 1994] by allowing arbitrary nesting of a path probabilistic operator and we prove its completeness. We prove that our logic is strictly more expressive than other logics such as the logics cited above. By considering a probabilistic extension of CTL we show that this additional expressive power is really needed in some applications.