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Bounded Situation Calculus Action Theories and Decidable Verification
"... We define a notion of bounded action theory in the situation calculus, where the theory entails that in all situations, the number of ground fluent atoms is bounded by a constant. Such theories can still have an infinite domain and an infinite set of states. We argue that such theories are fairly co ..."
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We define a notion of bounded action theory in the situation calculus, where the theory entails that in all situations, the number of ground fluent atoms is bounded by a constant. Such theories can still have an infinite domain and an infinite set of states. We argue that such theories are fairly common in applications, either because facts do not persist indefinitely or because one eventually forgets some facts, as one learns new ones. We discuss various ways of obtaining bounded action theories. The main result of the paper is that verification of an expressive class of firstorder µcalculus temporal properties in such theories is in fact decidable.
Definability of Languages by Generalized FirstOrder Formulas over (N
 In 23rd Symp. on Theoretical Aspects of Comp. Sci. (STACS’06
, 2006
"... Abstract. We consider an extension of firstorder logic by modular quantifiers of a fixed modulus q. Drawing on collapse results from finite model theory and techniques of finite semigroup theory, we show that if the only available numerical predicate is addition, then sentences in this logic cannot ..."
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Abstract. We consider an extension of firstorder logic by modular quantifiers of a fixed modulus q. Drawing on collapse results from finite model theory and techniques of finite semigroup theory, we show that if the only available numerical predicate is addition, then sentences in this logic cannot define the set of bit strings in which the number of 1’s is divisible by a prime p that does not divide q. More generally, we completely characterize the regular languages definable in this logic. The corresponding statement, with addition replaced by arbitrary numerical predicates, is equivalent to the conjectured separation of the circuit complexity class ACC from NC 1. Thus our theorem can be viewed as proving a highly uniform version of the conjecture. 1
Bounded Epistemic Situation Calculus Theories
"... We define the class of ebounded theories in the epistemic situation calculus, where the number of fluent atoms that the agent thinks may be true is bounded by a constant. Such theories can still have an infinite domain and an infinite set of states. We show that for them verification of an expressi ..."
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We define the class of ebounded theories in the epistemic situation calculus, where the number of fluent atoms that the agent thinks may be true is bounded by a constant. Such theories can still have an infinite domain and an infinite set of states. We show that for them verification of an expressive class of firstorder µcalculus temporal epistemic properties is decidable. We also show that if the agent’s knowledge in the initial situation is ebounded and the objective part of an action theory maintains boundedness, then the entire epistemic theory is ebounded.
An EhrenfeuchtFraïssé Game Approach to Collapse Results in Database Theory
, 2008
"... We present a new EhrenfeuchtFraïssé game approach to collapse results in database theory. We show that, in principle, every natural generic collapse result may be proved via a translation of winning strategies for the duplicator in an EhrenfeuchtFraïssé game. Following this approach we can deal wi ..."
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We present a new EhrenfeuchtFraïssé game approach to collapse results in database theory. We show that, in principle, every natural generic collapse result may be proved via a translation of winning strategies for the duplicator in an EhrenfeuchtFraïssé game. Following this approach we can deal with certain infinite databases where previous, highly involved methods fail. We prove the natural generic collapse for Zembeddable databases over any linearly ordered context structure with arbitrary monadic predicates, and for Nembeddable databases over the context structure 〈R, <,+, MonQ, Groups〉, where Groups is the collection of all subgroups of 〈R,+ 〉 that contain the set of integers and MonQ is the collection of all subsets of a particular infinite set Q of natural numbers. This, in particular, implies the collapse for arbitrary databases over 〈N, <,+, MonQ 〉 and for Nembeddable databases over 〈R, <,+, Z, Q〉. I.e., firstorder logic with < can express the same ordergeneric queries as firstorder logic with <, +, etc. Restricting the complexity of the formulas that may be used to formulate queries to Boolean combinations of purely existential firstorder formulas, we even obtain the collapse for N
Embedded Finite Models
"... Yuri asked me, the author (A) to meet his student Quisani (Q), who often appears in public just before a new issue of the Bulletin comes out, and for whom Yuri arranges meetings with computer science logicians. As Q looked rather tired and suffering from a lack of sleep, I asked him what had caused ..."
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Yuri asked me, the author (A) to meet his student Quisani (Q), who often appears in public just before a new issue of the Bulletin comes out, and for whom Yuri arranges meetings with computer science logicians. As Q looked rather tired and suffering from a lack of sleep, I asked him what had caused it. He explained that in a recent meeting with Jan Van den Bussche, which was reported in this Column [38], he was given a chapter on embedded finite models from my book [29] as bedtime reading, but didn’t find it very easy to start reading a 14chapter book from chapter 13. So an email to Yuri followed, and a meeting with me was arranged. The following is my transcription of that meeting. A. At the very least you’re now familiar with the main definition of embedded finite models. Let’s review it first. Q. As I recall it, you start with an infinite model or structure, something like the real closed field R = 〈R, +, ·, 0, 1, <〉, and then put a finite model on it, say, a finite graph whose nodes are real numbers. A. That’s right. Formally speaking, you have two vocabularies, say Ω for an infinite structure, and σ for a finite structure, and you look at (Ω, σ)structures, where σrelations are finite.
Proceedings of the TwentyThird International Joint Conference on Artificial Intelligence Bounded Epistemic Situation Calculus Theories
"... We define the class of ebounded theories in the epistemic situation calculus, where the number of fluent atoms that the agent thinks may be true is bounded by a constant. Such theories can still have an infinite domain and an infinite set of states. We show that for them verification of an expressi ..."
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We define the class of ebounded theories in the epistemic situation calculus, where the number of fluent atoms that the agent thinks may be true is bounded by a constant. Such theories can still have an infinite domain and an infinite set of states. We show that for them verification of an expressive class of firstorder µcalculus temporal epistemic properties is decidable. We also show that if the agent’s knowledge in the initial situation is ebounded and the objective part of an action theory maintains boundedness, then the entire epistemic theory is ebounded.
Inexpressibility Results for Regular Languages in Nonregular Settings
, 2005
"... My ostensible purpose in this talk is to describe some new results (found in collaboration with Amitabha Roy) on expressibility of regular languages in certain generalizations of firstorder logic. [10]. This provides me with a good excuse for describing some the work on the algebraic theory of regu ..."
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My ostensible purpose in this talk is to describe some new results (found in collaboration with Amitabha Roy) on expressibility of regular languages in certain generalizations of firstorder logic. [10]. This provides me with a good excuse for describing some the work on the algebraic theory of regular languages in what one might call “nonregular settings”. The syntactic monoid and syntactic morphism of a regular language provide a highly effective tool for proving that a given regular language is not expressible or recognizable in certain compuational models, as long as the model is guaranteed to produce only regular languages. This includes finite automata, of course. but also formulas of propositional temporal logic, and firstorder logic, provided one is careful to restrict the expressive power of such logics. (For example, by only allowing the order relation in firstorder formulas.) Things become much harder, and quite a bit more interesting, when we drop this kind of restriction on the model. The questions that arise are important (particularly in computational complexity), and most of them are unsolved. They all point to a rich theory that extends the reach of algebraic methods beyond the domain of finite automata 1 Uniformizing Nonuniform Automata with Ramsey’s Theorem Let’s start with an especially trivial application of the syntactic monoid: Let Σ = {0, 1}, and consider the two languages