In this paper, we propose a new method to deal with continuously varying quantity in the situation calculus based on the concept of the nonstandard analysis. The essential point of the method is to devise a new model called nonstandard situation calculus, which is an interpretation of the situation calculus in the set of hyperreals. This nonstandard model allows discrete but uncountable (hyperfinite) state transition, so that we can describe and reason about the continuous (...) dynamics which are usually treated with differential equations. In this enlarged perspective of the nonstandard situation calculus, we discuss about an infinitesimal action, infinite plan and an abnormality theory for the temporal prediction based on the differentiability. (shrink)

The situation calculus is one of the most established formalisms for reasoning about action and change. In this paper we will review the basics of Reiter’s version of the situation calculus, show how knowledge and time have been addressed in this framework, and point to some of the weaknesses of the situation calculus with respect to time. We then present a modal version of the situation calculus where these problems can be overcome (...) with relative ease and without sacrificing the advantages of the original. (shrink)

This is a 1985 note aimed at regarding STRIPS as a proof strategy for an interactive theorem prover using a situation calculus formalism. It doesn't quite get there.

We investigate agents that have incomplete information and make decisions based on their beliefs expressed as situation calculus bounded action theories. Such theories have an infinite object domain, but the number of objects that belong to fluents at each time point is bounded by a given constant. Recently, it has been shown that verifying temporal properties over such theories is decidable. We take a first-person view and use the theory to capture what the agent believes about the domain (...) of interest and the actions affecting it. In this paper, we study verification of temporal properties over online executions. These are executions resulting from agents performing only actions that are feasible according to their beliefs. To do so, we first examine progression, which captures belief state update resulting from actions in the situation calculus. We show that, for bounded action theories, progression, and hence belief states, can always be represented as a bounded first-order logic theory. Then, based on this result, we prove decidability of temporal verification over online executions for bounded action theories. (shrink)

Sequential von Neumann–Morgernstern (VM) games are a very general formalism for representing multi-agent interactions and planning problems in a variety of types of environments. We show that sequential VM games with countably many actions and continuous utility functions have a sound and complete axiomatization in the situation calculus. This axiomatization allows us to represent game-theoretic reasoning and solution concepts such as Nash equilibrium. We discuss the application of various concepts from VM game theory to the theory of planning (...) and multi-agent interactions, such as representing concurrent actions and using the Baire topology to define continuous payoff functions. (shrink)

We propose a translation into Modal Logic of the ideas that formalise belief change in the Situation Calculus. This translation is extended to the case of revision. In the conclusion is presented a set of open issues.

In social interactions, it is common for individuals to possess different amounts of knowledge about a specific transaction, and those who are more knowledgeable might perform opportunistic behavior to others in their interest, which promotes their value but demotes others’ value. Such a typical social behavior is called opportunistic behavior. In this paper, we propose a formal account of opportunism based on the situation calculus. We first propose a model of opportunism that only considers a single action between (...) two agents, and then extend it to multiple actions and incorporate social context in the model. A simple example of selling a broken cup is used to illustrate our models. Through our models, we can have a thorough understanding of opportunism. (shrink)

Most agents can acquire information about their environments as they operate. A good plan for such an agent is one that not only achieves the goal, but is also executable, i.e., ensures that the agent has enough information at every step to know what to do next. In this paper, we present a formal account of what it means for an agent to know how to execute a plan and to be able to achieve a goal. Such a theory is (...) a prerequisite for producing specifications of planners for agents that can acquire information at run time. It is also essential to account for cooperation among agents. Our account is more general than previous proposals, correctly handles programs containing loops, and incorporates a solution to the frame problem. It can also be used to prove programs containing sensing actions correct. (shrink)

internal events that happen spontaneously from external events (actions). It also treats processes, e.g. a buzzer, that do not settle down. The non-monotonic reasoning is circumscription done situation by situation.

The modal object calculus is the system of logic which houses the (proper) axiomatic theory of abstract objects. The calculus has some rather interesting features in and of itself, independent of the proper theory. The most sophisticated, type-theoretic incarnation of the calculus can be used to analyze the intensional contexts of natural language and so constitutes an intensional logic. However, the simpler second-order version of the calculus couches a theory of fine-grained properties, relations and propositions and (...) serves as a framework for defining situations, possible worlds, stories, and fictional characters, among other things. In the present paper, we focus on the second-order calculus. The second-order modal object calculus is so-called to distinguish it from the second-order modal predicate calculus. Though the differences are slight, the extra expressive power of the object calculus significantly enhances its ability to resolve logical and philosophical concepts and problems. (shrink)

We consider substitutions in order sensitive situations, having in the back of our minds the case of dynamic predicate logic (DPL) with a stack semantics. We start from the semantic intuition that substitutions are move instructions on stacks: the syntactic operation [y/x] is matched by the instruction to move the value of the y-stack to the x-stack. We can describe these actions in the positive fragment of DPLE. Hence this fragment counts as a logic for DPL-substitutions. We give a (...) class='Hi'>calculus for the fragment and prove soundness and completeness. (shrink)

Context: Non-dualistic thinking is an alternative to realism and constructivism. Problem: In the absence of a distinct definition of the term “description,” the question comes up of what exactly can be included in non-dualistic descriptions, and in how far the definition of this term affects the relation between theory and empirical practice. Furthermore, this paper is concerned with the question of whether non-dualism and dualism differ in their implications. Method: I provide a wider semantic framework for the term “description” by (...) means of George Spencer Brown’s terminology in his calculus of indications as laid out in Laws of Form. The connection of descriptions and distinctions enables descriptions to comprise reflections and language as well as empirical observations. Results: Non-dualism can be thought of in different ways but still has essential elements in common with dualism. Implications: Non-dualism, as well as dualism, is an argumentation technique suitable for specific situations, but without significant differences in implications. (shrink)

This paper discusses the distinctions indicated in its title. It is argued that the distinction between syntax and semantics is much more important for the present situation in logic than other distinctions. In particular, doing formal syntax and formal semantics requires the use of an informal melanguage based on ordinary mathematics.

The intention of the presented paper is to establish within a certain modal semantic based on the situational ontology a description of the phenomenon of substantial change, which originally had been formulated within Aristotelian metaphysics – a theory based in reistic ontology. We understand substantial changesto be such changes whose subjects are primary substances (πρωται ουσι αι ) conceived as actually existing individual essences. The analysed changeability is of an existential character - it pertains to the existence of those substances. (...) The formalised description of the chosen changes, constructed in part (1) of the study, takes into consideration the necessary condition for their occurrence: the disappearing and coming into being of successively becoming substances. Part (2) presents the manner in which it is possible to reformulate the obtained characteristic of substantial change in order to create a description of a particular variety of situational changes, which belong, in turn, to the intended interpretation of modal sentential logic of change LCG - a calculus in which change is a primitive concept precised with the help of two one-argument operators C and G, read respectively as: it changes that…, a change is enabled by this, that ….. (shrink)

This paper discusses Jean van Heijenoort’s (1967) and Jaakko and Merrill B. Hintikka’s (1986, 1997) distinction between logic as auniversal language and logic as a calculus, and its applicability to Edmund Husserl’s phenomenology. Although it is argued that Husserl’s phenomenology shares characteristics with both sides, his view of logic is closer to the model-theoretical, logic-as-calculus view. However, Husserl’s philosophy as transcendental philosophy is closer to the universalist view. This paper suggests that Husserl’s position shows that holding a model-theoretical (...) view of logic does not necessarily imply a calculus view about the relations between language and the world. The situation calls for reflection about the distinction: It will be suggested that the applicability of the van Heijenoort and the Hintikkas distinction either has to be restricted to a particular philosopher’s views about logic, in which case no implications about his or her more general philosophical views should be inferred from it; or the distinction turns into a question of whether our human predicament is inescapable or whether it is possible, presumably by means of model theory, to obtain neutral answers to philosophical questions. Thus the distinction ultimately turns into a question about the correct method for doing philosophy. (shrink)

This paper gives a semantical account for the (i)ordinary propositional calculus, enriched with quantifiers binding variables standing for sentences, and with an identity-function with sentences as arguments; (ii)the ordinary theory of quantification applied to the special quantifiers; and (iii)ordinary laws of identity applied to the special function. The account includes some thoughts of Roman Suszko as well as some thoughts of Wittgenstein's Tractatus.

Robin Dunbar's hypothesis that language capacity in response to the demands of maintaining large groups suggests a more specific hypothesis that language arose from a cognitive calculus by which animals could predict their status in complex dominance situations.

Levesque et al. have defined a programming language, Golog, in order to reason about complex actions within the framework of the situation calculus. We build on previous work of ours and show how to translate Golog into linear logic, suitably augmented.

There are at least a few plausible grounds for our use of the term Beta in our title, notwithstanding that there is a key departure, in our framework, from classical Beta Existential Graphs. The situation, in brief, is as follows.The reader accustomed to Peirce’s graphical development of quantificational logic may, if desired, continue to think of formulas being written on a “sheet of assertion.” We retain the “cut” notation for negation and continue to represent conjunction simply by juxtaposition of (...) diagrams. However, on the side of departure, we dispense with the sometimes problematic “lines of identity” of classical Beta and instead represent a quantification by a.. (shrink)

The situation calculus is a logical formalism that has been extensively developed for planning. We apply the formalism in a complex multi-agent domain, modelled on the game of Clue. We find that the situation calculus, with suitable extensions, supplies a unified representation of (1) the interaction protocol, or structure of the game, (2) the dynamics of the knowledge and common knowledge of the agents, and (3) principles of strategic planning.

There are two well-developed formalizations of discrete time dynamic systems that evidently share many concerns but suffer from a lack of mutual awareness. One formalization is classical systems and automata theory. The other is the logic of actions in which the situation and event calculi are the strongest representatives. Researchers in artificial intelligence are likely to be familiar with the latter but not the former. This is unfortunate, for systems and automata theory have much to offer by way of (...) insight into problems raised in the logics of action. This paper is an outline of how the input-output view of systems and its associated solution of state realization may be applied to the formalization of dynamics that uses a situation calculus approach. In particular, because the latter usually admits incompletely specified dynamics, which induces a non-deterministic input-output system behavior, we first show that classical state realization can still be achieved if the behavior is causal. This is a novel systems-theoretic result. Then we proceed to indicate how situation calculi dynamic specifications can be understood in systems-theoretic terms, and how automata can be viewed as models of such specifications. As techniques for reasoning about automata are abundant, this will provide yet more tools for reasoning about actions. (shrink)

(Click here for selected papers published before 1996, and here for papers published after 2000.) V. Lifschitz, Foundations of logic programming ," in Principles of Knowledge Representation , CSLI Publications, 1996, pp. 69-127. E. Giunchiglia, N. Kartha and V. Lifschitz, Representing action: indeterminacy and ramifications ," Artificial Intelligence , Vol. 95, 1997, pp. 409-443. V. Lifschitz, On the logic of causal explanation ," Artificial Intelligence , Vol. 96, 1997, pp. 451-465. V. Lifschitz, Two components of an action language ," Annals (...) of Mathematics and Artificial Intelligence , Vol. 21, 1997, pp. 305-320. V. Lifschitz, Situation calculus and causal logic ," in Proceedings of the Sixth International Conference on Principles of Knowledge Representation and Reasoning, 1998, pp. 536-646. (shrink)

This paper describes an approach for reasoning in a dynamic domain with nondeterministic actions in which an agent’s beliefs correspond to the simplest, or most plausible, course of events consistent with the agent’s observations and beliefs. The account is based on an epistemic extension of the situation calculus, a first-order theory of reasoning about action that accommodates sensing actions. In particular, the account is based on a qualitative theory of nondeterminism. Our position is that for commonsense reasoning, the (...) world is most usefully regarded as deterministic, and that nondeterminism is an epistemic phenomenon, arising from an agent’s limited awareness and perception. The account offers several advantages: an agent has a set of categorical beliefs, yet can deal with equally-likely outcomes or with outcomes of differing plausibility. The agent maintains as its set of contingent beliefs the most plausible, or simplest, picture of the world, consistent with its beliefs and actions it believes it executed; yet it may modify these in light of later information. (shrink)

In this work we propose an encoding of Reiter’s Situation Calculus solution to the frame problem into the framework of a simple multimodal logic of actions. In particular we present the modal counterpart of the regression technique. This gives us a theorem proving method for a relevant fragment of our modal logic.

As a contribution to the challenge of building game-playing AI systems, we develop and analyse a formal language for representing and reasoning about strategies. Our logical language builds on the existing general Game Description Language and extends it by a standard modality for linear time along with two dual connectives to express preferences when combining strategies. The semantics of the language is provided by a standard state-transition model. As such, problems that require reasoning about games can be solved by the (...) standard methods for reasoning about actions and change. We also endow the language with a specific semantics by which strategy formulas are understood as move recommendations for a player. To illustrate how our formalism supports automated reasoning about strategies, we demonstrate two example methods of implementation: first, we formalise the semantic interpretation of our language in conjunction with game rules and strategy rules in the Situation Calculus; second, we show how the reasoning problem can be solved with Answer Set Programming. (shrink)

Possible worlds semantics for conditionals leave open the problem of how to construct models for realistic domains. In this paper, we show how to adapt logics of action and change such as John McCarthy’s Situation Calculus to conditional logics. We illustrate the idea by presenting models for conditionals whose antecedents combine a declarative condition with a hypothetical action.

Probably the most celebrated controversy in all of the history of science was that between Newton and Leibniz over the invention of the calculus. The argument ranged far beyond a mere priority dispute and took on the character of a war between two different philosophies of nature. Newton was the first to devise the methods of the calculus, but Leibniz (who independently discovered virtually identical methods) was the first to publish, in 1684. Mutual toleration passed into suspicion and, (...) at last, denunciation of each by the other as a fraud and a plagiarist. The affair became a scandal, as British mathematicians asserted Newton's claims before the public while their Continental colleagues hotly defended Leibniz's priority. Professor Hall analyzes the situation out of which the dispute arose, the circumstances that caused it to become embittered, the dispositions of the chief actors, and the shifts in their opinions of each other. (shrink)

The rational status of the Bayesian calculus for revising likelihoods is compromised by the common but still unfamiliar phenomenon of information distortion. This bias is the distortion in the evaluation of a new datum toward favoring the currently preferred option in a decision or judgment. While the Bayesian calculus requires the independent combination of the prior probability and a new datum, information distortion invalidates such independence (because the prior influences the datum). Although widespread, information distortion has not generally (...) been recognized. First, individuals are not aware when they themselves commit this bias. In addition, it is often hidden in more obvious suboptimal phenomena. Finally, the Bayesian calculus is usually explained only with undistortable data like colored balls drawn randomly. Partly because information distortion is unrecognized by the individuals exhibiting it, no way has been devised for eliminating it. Partial reduction is possible in some situation such as presenting all data simultaneously rather than sequentially with revision after each datum. The potential dangers of information distortion are illustrated for three professional revision tasks: forecasting, predicting consumer choices from internet data, and statistical inference from experimental results. The optimality of the Bayesian calculus competes with people’s natural desire that their belief systems remain coherent in the face of new data. Information distortion provides this coherence by biasing those data toward greater agreement with the currently preferred position – but at the cost of Bayesian optimality. (shrink)

In this paper, an objective conception of contexts based loosely upon situation theory is developed and formalized. Unlike subjective conceptions, which take contexts to be something like sets of beliefs, contexts on the objective conception are taken to be complex, structured pieces of the world that (in general) contain individuals, other contexts, and propositions about them. An extended first-order language for this account is developed. The language contains complex terms for propositions, and the standard predicate "ist" that expresses the (...) relation that holds between a context and a proposition just in case the latter is true in the former. The logic for the objective conception features a global classical predicate calculus, a local logic for reasoning within contexts, and axioms for propositions. The specter of paradox is banished from the logic by allowing "ist" to be nonbivalent in problematic cases: it is not in general the case, for any context c and proposition p, that either ist(c,p) or ist(c, ¬p). An important representational capability of the logic is illustrated by proving an appropriately modified version of an illustrative theorem from McCarthy's classic Blocks World example. (shrink)

The previous Part I of the paper discusses the option of the Gödel incompleteness statement (1931: whether “Satz VI” or “Satz X”) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel’s paper (1930) (and more precisely, the negation of “Satz VII”, or “the completeness theorem”) as a necessary condition for (...) granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the “completeness paper” can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russell’s logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle’s logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl’s phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel’s completeness theorem (1930: “Satz VII”) and even both and arithmetic in the sense of the “compactness theorem” (1930: “Satz X”) therefore opposing the latter to the “incompleteness paper” (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the “half” of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbert’s epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined. (shrink)