On First-Order μ-Calculus over Situation Calculus Action Theories

In this paper we study verification of situation calculus action theories against first-order mu-calculus with quantification across situations. Specifically, we consider mu-La and mu-Lp, the two variants of mu-calculus introduced in the literature for verification of data-aware processes. The former requires that quantification ranges over objects in the current active domain, while the latter additionally requires that objects assigned to variables persist across situations. Each of these two logics has a distinct corresponding notion of bisimulation. In spite of the differences we show that the two notions of bisimulation collapse for dynamic systems that are generic, which include all those systems specified through a situation calculus action theory. Then, by exploiting this result, we show that for bounded situation calculus action theories, mu-La and mu-Lp have exactly the same expressive power. Finally, we prove decidability of verification of mu-La properties over bounded action theories, using finite faithful abstractions. Differently from the mu-Lp case, these abstractions must depend on the number of quantified variables in the mu-La formula

Bounded Situation Calculus Action Theories

In this paper, we investigate bounded action theories in the situation calculus. A bounded action theory is one which entails that, in every situation, the number of object tuples in the extension of fluents is bounded by a given constant, although such extensions are in general different across the infinitely many situations. We argue that such theories are common in applications, either because facts do not persist indefinitely or because the agent eventually forgets some facts, as new ones are learnt. We discuss various classes of bounded action theories. Then we show that verification of a powerful first-order variant of the mu-calculus is decidable for such theories. Notably, this variant supports a controlled form of quantification across situations. We also show that through verification, we can actually check whether an arbitrary action theory maintains boundedness.Comment: 51 page

Global semantic typing for inductive and coinductive computing

Inductive and coinductive types are commonly construed as ontological (Church-style) types, denoting canonical data-sets such as natural numbers, lists, and streams. For various purposes, notably the study of programs in the context of global semantics, it is preferable to think of types as semantical properties (Curry-style). Intrinsic theories were introduced in the late 1990s to provide a purely logical framework for reasoning about programs and their semantic types. We extend them here to data given by any combination of inductive and coinductive definitions. This approach is of interest because it fits tightly with syntactic, semantic, and proof theoretic fundamentals of formal logic, with potential applications in implicit computational complexity as well as extraction of programs from proofs. We prove a Canonicity Theorem, showing that the global definition of program typing, via the usual (Tarskian) semantics of first-order logic, agrees with their operational semantics in the intended model. Finally, we show that every intrinsic theory is interpretable in a conservative extension of first-order arithmetic. This means that quantification over infinite data objects does not lead, on its own, to proof-theoretic strength beyond that of Peano Arithmetic. Intrinsic theories are perfectly amenable to formulas-as-types Curry-Howard morphisms, and were used to characterize major computational complexity classes Their extensions described here have similar potential which has already been applied

Embedded Finite Models beyond Restricted Quantifier Collapse

We revisit evaluation of logical formulas that allow both uninterpreted relations, constrained to be finite, as well as interpreted vocabulary over an infinite domain: denoted in the past as embedded finite model theory. We extend the analysis of "collapse results": the ability to eliminate first-order quantifiers over the infinite domain in favor of quantification over the finite structure. We investigate several weakenings of collapse, one allowing higher-order quantification over the finite structure, another allowing expansion of the theory. We also provide results comparing collapse for unary signatures with general signatures, and new analyses of collapse for natural decidable theories

On Polynomial-Time Decidability of k-Negations Fragments of FO Theories (Extended Abstract)

This paper introduces a generic framework that provides sufficient conditions for guaranteeing polynomial-time decidability of fixed-negation fragments of first-order theories that adhere to certain fixed-parameter tractability requirements. It enables deciding sentences of such theories with arbitrary existential quantification, conjunction and a fixed number of negation symbols in polynomial time. It was recently shown by Nguyen and Pak [SIAM J. Comput. 51(2): 1-31 (2022)] that an even more restricted such fragment of Presburger arithmetic (the first-order theory of the integers with addition and order) is NP-hard. In contrast, by application of our framework, we show that the fixed negation fragment of weak Presburger arithmetic, which drops the order relation from Presburger arithmetic in favour of equality, is decidable in polynomial time

MacNeille Completion and Buchholz\u27 Omega Rule for Parameter-Free Second Order Logics

Buchholz\u27 Omega-rule is a way to give a syntactic, possibly ordinal-free proof of cut elimination for various subsystems of second order arithmetic. Our goal is to understand it from an algebraic point of view. Among many proofs of cut elimination for higher order logics, Maehara and Okada\u27s algebraic proofs are of particular interest, since the essence of their arguments can be algebraically described as the (Dedekind-)MacNeille completion together with Girard\u27s reducibility candidates. Interestingly, it turns out that the Omega-rule, formulated as a rule of logical inference, finds its algebraic foundation in the MacNeille completion. In this paper, we consider a family of sequent calculi LIP = cup_{n >= -1} LIP_n for the parameter-free fragments of second order intuitionistic logic, that corresponds to the family ID_{<omega} = cup_{n <omega} ID_n of arithmetical theories of inductive definitions up to omega. In this setting, we observe a formal connection between the Omega-rule and the MacNeille completion, that leads to a way of interpreting second order quantifiers in a first order way in Heyting-valued semantics, called the Omega-interpretation. Based on this, we give a (partly) algebraic proof of cut elimination for LIP_n, in which quantification over reducibility candidates, that are genuinely second order, is replaced by the Omega-interpretation, that is essentially first order. As a consequence, our proof is locally formalizable in ID-theories

Decidable Verification of Golog Programs over Non-Local Effect Actions: Extended Version

The Golog action programming language is a powerful means to express high-level behaviours in terms of programs over actions defined in a Situation Calculus theory. In particular for physical systems, verifying that the program satisfies certain desired temporal properties is often crucial, but undecidable in general, the latter being due to the language’s high expressiveness in terms of first-order quantification and program constructs. So far, approaches to achieve decidability involved restrictions where action effects either had to be contextfree (i.e. not depend on the current state), local (i.e. only affect objects mentioned in the action’s parameters), or at least bounded (i.e. only affect a finite number of objects). In this paper, we present a new, more general class of action theories (called acyclic) that allows for context-sensitive, non-local, unbounded effects, i.e. actions that may affect an unbounded number of possibly unnamed objects in a state-dependent fashion. We contribute to the further exploration of the boundary between decidability and undecidability for Golog, showing that for acyclic theories in the two-variable fragment of first-order logic, verification of CTL properties of programs over ground actions is decidable

Information Physics: The New Frontier

At this point in time, two major areas of physics, statistical mechanics and quantum mechanics, rest on the foundations of probability and entropy. The last century saw several significant fundamental advances in our understanding of the process of inference, which make it clear that these are inferential theories. That is, rather than being a description of the behavior of the universe, these theories describe how observers can make optimal predictions about the universe. In such a picture, information plays a critical role. What is more is that little clues, such as the fact that black holes have entropy, continue to suggest that information is fundamental to physics in general. In the last decade, our fundamental understanding of probability theory has led to a Bayesian revolution. In addition, we have come to recognize that the foundations go far deeper and that Cox's approach of generalizing a Boolean algebra to a probability calculus is the first specific example of the more fundamental idea of assigning valuations to partially-ordered sets. By considering this as a natural way to introduce quantification to the more fundamental notion of ordering, one obtains an entirely new way of deriving physical laws. I will introduce this new way of thinking by demonstrating how one can quantify partially-ordered sets and, in the process, derive physical laws. The implication is that physical law does not reflect the order in the universe, instead it is derived from the order imposed by our description of the universe. Information physics, which is based on understanding the ways in which we both quantify and process information about the world around us, is a fundamentally new approach to science.Comment: 17 pages, 6 figures. Knuth K.H. 2010. Information physics: The new frontier. J.-F. Bercher, P. Bessi\`ere, and A. Mohammad-Djafari (eds.) Bayesian Inference and Maximum Entropy Methods in Science and Engineering (MaxEnt 2010), Chamonix, France, July 201

On How to Refer to Unobservable Entities

In order for us to associate a word with an object it might seem that we would need to have direct experience with both. Given the present technology, however, there are some objects with which we can have no direct experience, namely the unobservable entities postulated by scientific theories. The problem taken up here is how to refer to those entities. There are two prominent attempts to explain reference in scientific theories – the first is Ramsey and Carnap’s proposal that we exchange theoretical terms for variables and existential quantification. The second is Kripke and Evan’s causal theory of names and rigid designation. I will argue that the most plausible theory of reference to unobservables lies in between these two theories; terms that purport to refer to unobservable entities, when occurring within a theory, need to be thought of as bound variables. But when those same terms occur in sentences outside of the theory, as when spoken, for example, they occur as genuine referring expressions, which have their reference determined by a theory