For any ordinal Λ, we can define a polymodal logic GLPΛ, with a modality [ξ] for each ξ < Λ. Thes... more For any ordinal Λ, we can define a polymodal logic GLPΛ, with a modality [ξ] for each ξ < Λ. These represent provability predicates of increasing strength. Although GLPΛ has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities, denoted GLP 0 ω. Later, Icard defined a topological model for GLP 0 ω which is very closely related to Ignatiev's. In this paper we show how to extend these constructions for arbitrary Λ. More generally, for each Θ, Λ we build a Kripke model I Θ Λ and a topological model T Θ Λ , and show that GLP 0 Λ is sound for both of these structures, as well as complete, provided Θ is large enough.
FV Time is a small-scale verification project developed in the Coq proof assistant using the Math... more FV Time is a small-scale verification project developed in the Coq proof assistant using the Mathematical Components libraries. It is a library for managing conversions between time formats (UTC and timestamps), as well as commonly used functions for time arithmetic. As a library for time conversions, its novelty is the implementation of leap seconds, which are part of the UTC standard but usually not implemented in existing libraries. Since the verified functions of FV Time are reasonably simple yet non-trivial, it nicely illustrates our methodology for verifying software with Coq. In this paper we present a description of the project, emphasizing the main problems faced while developing the library, as well as some general-purpose solutions that were produced as by-products and may be used in other verification projects. These include a refinement package between proof-oriented MathComp numbers and computation-oriented primitive numbers from the Coq standard library, as well as a set of tactics to automatically prove certain decidable statements over finite ranges through brute-force computation.
The provability logic of a theory T is the set of modal formulas, which under any arithmetical re... more The provability logic of a theory T is the set of modal formulas, which under any arithmetical realization are provable in T. We slightly modify this notion by requiring the arithmetical realizations to come from a specified set Γ. We make an analogous modification for interpretability logics. This is a paper from 2012. We first studied provability logics with restricted realizations, and show that for various natural candidates of theory T and restriction set Γ, where each sentence in Γ has a well understood (meta)-mathematical content in T , the result is the logic of linear frames. However, for the theory Primitive Recursive Arithmetic (PRA), we define a fragment that gives rise to a more interesting provability logic, by capitalizing on the well-studied relationship between PRA and IΣ 1. We then study interpretability logics, obtaining some upper bounds for IL(PRA), whose characterization remains a major open question in interpretability logic. Again this upper bound is closely relatively to linear frames. The technique is also applied to yield the non-trivial result that IL(PRA) ⊂ ILM. 2 Σ 1-sound is sufficient here. 3 There is a paper by de Jongh, Jumelet and Montagna [22] where an alternative proof of Solovay's theorem is given. In that proof, using the diagonal lemma, one finds some sentences with the required properties rather than defining the sentences and then proving the necessary properties.
The interpretability logic of a mathematical theory describes the structural behavior of interpre... more The interpretability logic of a mathematical theory describes the structural behavior of interpretations over that theory. Different theories have different logics. This paper revolves around the question what logic describes the behavior that is present in all theories with a minimum amount of arithmetic; the intersection over all such theories so to say. We denote this target logic by IL(All). In this paper we present a new principle R in IL(All). We show that R does not follow from the logic ILP0W * that contains all previously known principles. This is done by providing a modal incompleteness proof of ILP0W * : showing that R follows semantically but not syntactically from ILP0W *. Apart from giving the incompleteness proof by elementary methods, we also sketch how to work with so-called Generalized Veltman Semantics as to establish incompleteness. To this extent, a new version of this Generalized Veltman Semantics is defined and studied. Moreover, for the important principles the frame correspondences are calculated. After the modal results it is shown that the new principle R is indeed valid in any arithmetically theory. The proof employs some elementary results on definable cuts in arithmetical theories. 1 Technically speaking the property of so-called essential reflexivity is sufficient. A theory is
Abstract This paper analyses a selection of articles from European transport regulations that con... more Abstract This paper analyses a selection of articles from European transport regulations that contain algorithmic information, but may be problematic to implement. We focus on issues regarding the interpretation of tachograph data and requirements on weekly rest periods. We first show that the interpretation of data prescribed by these regulations is highly sensitive to minor variations in input, such that near-identical driving patterns may be regarded both as lawful and as unlawful. We then show that the content of the regulation may be represented in mondadic second order logic, but argue that a more computationally tame fragment would be preferrable for applications. As a case study we consider its representation in linear temporal logic, but show that a representation of the legislation requires formulas of unfeasible complexity, if at all possible.
We discuss model-checking problems as formal models of algorithmic law. Specifically, we ask for ... more We discuss model-checking problems as formal models of algorithmic law. Specifically, we ask for an algorithmically tractable general purpose model-checking problem that naturally models the European transport Regulation 561 ([49]), and discuss the reaches and limits of a version of discrete time stopwatch automata.
A fast consistency prover is a consistent poly-time axiomatized theory that has short proofs of t... more A fast consistency prover is a consistent poly-time axiomatized theory that has short proofs of the finite consistency statements of any other poly-time axiomatized theory. Krajíček and Pudlák proved in [5] that the existence of an optimal propositional proof system is equivalent to the existence of a fast consistency prover. It is an easy observation that NP = coNP implies the existence of a fast consistency prover. The reverse implication is an open question. In this paper we define the notion of an unlikely fast consistency prover and prove that its existence is equivalent to NP = coNP. Next it is proved that fast consistency provers do not exist if one considers RE axiomatized theories rather than theories with an axiom set that is recognizable in polynomial time.
In this paper we consider IL0, the closed fragment of the basic interpretability logic IL. We sho... more In this paper we consider IL0, the closed fragment of the basic interpretability logic IL. We show that we can translate GL1, the one variable fragment of Gödel-Löb's provabilty logic GL, into IL0. Invoking a result on the PSPACE completeness of GL1 we obtain the PSPACE hardness of IL0.
In this paper we introduce hyperations and cohyperations, which are forms of transfinite iteratio... more In this paper we introduce hyperations and cohyperations, which are forms of transfinite iteration of ordinal functions. Hyperations are iterations of normal functions. Unlike iteration by pointwise convergence, hyperation preserves normality. The hyperation f ξ ξ∈On of a normal function f is a sequence of normal functions so that f 0 = id, f 1 = f and for all α, β we have that f α+β = f α f β. These conditions do not determine f α uniquely; in addition, we require that f α α∈On be minimal in an appropriate sense. We study hyperations systematically and show that they are a natural refinement of Veblen progressions. Next, we define cohyperations, very similar to hyperations except that they are left-additive: given α, β, f α+β = f β f α. Cohyperations iterate initial functions which are functions that map initial segments to initial segments. We systematically study co-hyperations and see how they can be employed to define left inverses to hyperations. Hyperations provide an alternative presentation of Veblen progressions and can be useful where a more fine-grained analysis of such sequences is called for. They are very amenable to algebraic manipulation and hence are convenient to work with. Cohyperations, meanwhile, give a novel way to describe slowly increasing functions as often appear, for example, in proof theory.
In this paper we approach some questions about quantum contextuality with tools from formal logic... more In this paper we approach some questions about quantum contextuality with tools from formal logic. In particular, we consider an experiment associated with the Peres-Mermin square. The language of all possible sequences of outcomes of the experiment is classified in the Chomsky hierarchy and seen to be a regular language. Next, we make the rather evident observation that a finite set of hidden finite valued variables can never account for indeterminism in an ideally isolated repeatable experiment. We see that, when the language of possible outcomes of the experiment is regular, as is the case with the Peres-Mermin square, the amount of binary-valued hidden variables needed to de-randomize the model for all sequences of experiments up to length n grows as bad as it could be: linearly in n. We introduce a very abstract model of machine that simulates nature in a particular sense. A lower-bound on the number of memory states of such machines is proved if they were to simulate the experiment that corresponds to the Peres-Mermin square. Moreover, the proof of this lower bound is seen to scale to a certain generalization of the Peres-Mermin square. For this scaled experiment it is seen that the Holevo bound is violated and that the degree of violation increases uniformly.
This paper from 2000 is a presentation of a status quaestionis at that tiime, to wit of the probl... more This paper from 2000 is a presentation of a status quaestionis at that tiime, to wit of the problem of the interpretability logic of all reasonable arithmetical theories. We present both the arithmetical side and the modal side of the question.
This paper is the second in a series of three papers. All three papers deal with interpretability... more This paper is the second in a series of three papers. All three papers deal with interpretability logics and related matters. In the first paper a construction method was exposed to obtain models of these logics. Using this method, we obtained some completeness results, some already known, and some new. In this paper, we will set the construction method to work to obtain more results. First, the modal completeness of the logic ILM is proved using the construction method. This is not a new result, but by using our new proof we can obtain new results. Among these new results are some admissible rules for ILM and GL. Moreover, the new proof will be used to classify all the essentially ∆1 and also all the essentially Σ1 formulas of ILM. Closely related to essentially Σ1 sentences are the so-called self provers. A self-prover is a formula ϕ which implies its own provability, that is ϕ → ✷ϕ. Each formula ϕ will generate a self prover ϕ ∧ ✷ϕ. We will use the construction method to characterize those sentences of GL that generate a self prover that is trivial in the sense that it is Σ1.
In this paper we give an overview of an essential part of a Π 0 1 ordinal analysis of Peano Arith... more In this paper we give an overview of an essential part of a Π 0 1 ordinal analysis of Peano Arithmetic (PA) as presented by Beklemishev ([3]). This analysis is mainly performed within the polymodal provability logic GLPω. We reflect on ways of extending this analysis beyond PA. A main difficulty in this is to find proper generalizations of the so-called Reduction Property. The Reduction Property relates reflection principles to reflection rules. In this paper we prove a result that simplifies the reflection rules. Moreover, we see that for an ordinal analysis the full Reduction Property is not needed. This latter observation is also seen to open up ways for applications of ordinal analysis relative to some strong base theory.
Turing progressions have been often used to measure the proof-theoretic strength of mathematical ... more Turing progressions have been often used to measure the proof-theoretic strength of mathematical theories: iterate adding consistency of some weak base theory until you "hit" the target theory. Turing progressions based on n-provability give rise to a Πn+1 proof-theoretic ordinal |U | Π 0 n+1. As such, to each theory U we can assign the sequence of corresponding Πn+1 ordinals |U |n n>0. We call this sequence a Turing-Taylor expansion or spectrum of a theory. In this paper, we relate Turing-Taylor expansions of sub-theories of Peano Arithmetic to Ignatiev's universal model for the closed fragment of the polymodal provability logic GLPω. In particular, we observe that each point in the Ignatiev model can be seen as Turing-Taylor expansions of formal mathematical theories. Moreover, each sub-theory of Peano Arithmetic that allows for a Turing-Taylor expansion will define a unique point in Ignatiev's model.
This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit:
Wolfram's Principle of Computational Equivalence (PCE) implies that universal complexity abounds ... more Wolfram's Principle of Computational Equivalence (PCE) implies that universal complexity abounds in nature. This paper comprises three sections. In the first section we consider the question why there are so many universal phenomena around. So, in a sense, we seek a driving force behind the PCE if any. We postulate a principle GNS that we call the Generalized Natural Selection principle that together with the Church-Turing thesis is seen to be equivalent in a sense to a weak version of PCE. In the second section we ask the question why we do not observe any phenomena that are complex but not-universal. We choose a cognitive setting to embark on this question and make some analogies with formal logic. In the third and final section we report on a case study where we see rich structures arise everywhere.
In this paper we give an overview of an essential part of a Pi^0_1 ordinal analysis of Peano Arit... more In this paper we give an overview of an essential part of a Pi^0_1 ordinal analysis of Peano Arithmetic (PA) as presented by Beklemishev. This analysis is mainly performed within the polymodal provability logic GLP. We reflect on ways of extending this analysis beyond PA. A main difficulty in this is to find proper generalizations of the so-called Reduction Property. The Reduction Property relates reflection principles to reflection rules. In this paper we prove a result that simplifies the reflection rules. Moreover, we see that for an ordinal analysis the full Reduction Property is not needed. This latter observation is also seen to open up ways for applications of ordinal analysis relative to some strong base theory.
For any ordinal Λ, we can define a polymodal logic GLPΛ, with a modality [ξ] for each ξ < Λ. Thes... more For any ordinal Λ, we can define a polymodal logic GLPΛ, with a modality [ξ] for each ξ < Λ. These represent provability predicates of increasing strength. Although GLPΛ has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities, denoted GLP 0 ω. Later, Icard defined a topological model for GLP 0 ω which is very closely related to Ignatiev's. In this paper we show how to extend these constructions for arbitrary Λ. More generally, for each Θ, Λ we build a Kripke model I Θ Λ and a topological model T Θ Λ , and show that GLP 0 Λ is sound for both of these structures, as well as complete, provided Θ is large enough.
FV Time is a small-scale verification project developed in the Coq proof assistant using the Math... more FV Time is a small-scale verification project developed in the Coq proof assistant using the Mathematical Components libraries. It is a library for managing conversions between time formats (UTC and timestamps), as well as commonly used functions for time arithmetic. As a library for time conversions, its novelty is the implementation of leap seconds, which are part of the UTC standard but usually not implemented in existing libraries. Since the verified functions of FV Time are reasonably simple yet non-trivial, it nicely illustrates our methodology for verifying software with Coq. In this paper we present a description of the project, emphasizing the main problems faced while developing the library, as well as some general-purpose solutions that were produced as by-products and may be used in other verification projects. These include a refinement package between proof-oriented MathComp numbers and computation-oriented primitive numbers from the Coq standard library, as well as a set of tactics to automatically prove certain decidable statements over finite ranges through brute-force computation.
The provability logic of a theory T is the set of modal formulas, which under any arithmetical re... more The provability logic of a theory T is the set of modal formulas, which under any arithmetical realization are provable in T. We slightly modify this notion by requiring the arithmetical realizations to come from a specified set Γ. We make an analogous modification for interpretability logics. This is a paper from 2012. We first studied provability logics with restricted realizations, and show that for various natural candidates of theory T and restriction set Γ, where each sentence in Γ has a well understood (meta)-mathematical content in T , the result is the logic of linear frames. However, for the theory Primitive Recursive Arithmetic (PRA), we define a fragment that gives rise to a more interesting provability logic, by capitalizing on the well-studied relationship between PRA and IΣ 1. We then study interpretability logics, obtaining some upper bounds for IL(PRA), whose characterization remains a major open question in interpretability logic. Again this upper bound is closely relatively to linear frames. The technique is also applied to yield the non-trivial result that IL(PRA) ⊂ ILM. 2 Σ 1-sound is sufficient here. 3 There is a paper by de Jongh, Jumelet and Montagna [22] where an alternative proof of Solovay's theorem is given. In that proof, using the diagonal lemma, one finds some sentences with the required properties rather than defining the sentences and then proving the necessary properties.
The interpretability logic of a mathematical theory describes the structural behavior of interpre... more The interpretability logic of a mathematical theory describes the structural behavior of interpretations over that theory. Different theories have different logics. This paper revolves around the question what logic describes the behavior that is present in all theories with a minimum amount of arithmetic; the intersection over all such theories so to say. We denote this target logic by IL(All). In this paper we present a new principle R in IL(All). We show that R does not follow from the logic ILP0W * that contains all previously known principles. This is done by providing a modal incompleteness proof of ILP0W * : showing that R follows semantically but not syntactically from ILP0W *. Apart from giving the incompleteness proof by elementary methods, we also sketch how to work with so-called Generalized Veltman Semantics as to establish incompleteness. To this extent, a new version of this Generalized Veltman Semantics is defined and studied. Moreover, for the important principles the frame correspondences are calculated. After the modal results it is shown that the new principle R is indeed valid in any arithmetically theory. The proof employs some elementary results on definable cuts in arithmetical theories. 1 Technically speaking the property of so-called essential reflexivity is sufficient. A theory is
Abstract This paper analyses a selection of articles from European transport regulations that con... more Abstract This paper analyses a selection of articles from European transport regulations that contain algorithmic information, but may be problematic to implement. We focus on issues regarding the interpretation of tachograph data and requirements on weekly rest periods. We first show that the interpretation of data prescribed by these regulations is highly sensitive to minor variations in input, such that near-identical driving patterns may be regarded both as lawful and as unlawful. We then show that the content of the regulation may be represented in mondadic second order logic, but argue that a more computationally tame fragment would be preferrable for applications. As a case study we consider its representation in linear temporal logic, but show that a representation of the legislation requires formulas of unfeasible complexity, if at all possible.
We discuss model-checking problems as formal models of algorithmic law. Specifically, we ask for ... more We discuss model-checking problems as formal models of algorithmic law. Specifically, we ask for an algorithmically tractable general purpose model-checking problem that naturally models the European transport Regulation 561 ([49]), and discuss the reaches and limits of a version of discrete time stopwatch automata.
A fast consistency prover is a consistent poly-time axiomatized theory that has short proofs of t... more A fast consistency prover is a consistent poly-time axiomatized theory that has short proofs of the finite consistency statements of any other poly-time axiomatized theory. Krajíček and Pudlák proved in [5] that the existence of an optimal propositional proof system is equivalent to the existence of a fast consistency prover. It is an easy observation that NP = coNP implies the existence of a fast consistency prover. The reverse implication is an open question. In this paper we define the notion of an unlikely fast consistency prover and prove that its existence is equivalent to NP = coNP. Next it is proved that fast consistency provers do not exist if one considers RE axiomatized theories rather than theories with an axiom set that is recognizable in polynomial time.
In this paper we consider IL0, the closed fragment of the basic interpretability logic IL. We sho... more In this paper we consider IL0, the closed fragment of the basic interpretability logic IL. We show that we can translate GL1, the one variable fragment of Gödel-Löb's provabilty logic GL, into IL0. Invoking a result on the PSPACE completeness of GL1 we obtain the PSPACE hardness of IL0.
In this paper we introduce hyperations and cohyperations, which are forms of transfinite iteratio... more In this paper we introduce hyperations and cohyperations, which are forms of transfinite iteration of ordinal functions. Hyperations are iterations of normal functions. Unlike iteration by pointwise convergence, hyperation preserves normality. The hyperation f ξ ξ∈On of a normal function f is a sequence of normal functions so that f 0 = id, f 1 = f and for all α, β we have that f α+β = f α f β. These conditions do not determine f α uniquely; in addition, we require that f α α∈On be minimal in an appropriate sense. We study hyperations systematically and show that they are a natural refinement of Veblen progressions. Next, we define cohyperations, very similar to hyperations except that they are left-additive: given α, β, f α+β = f β f α. Cohyperations iterate initial functions which are functions that map initial segments to initial segments. We systematically study co-hyperations and see how they can be employed to define left inverses to hyperations. Hyperations provide an alternative presentation of Veblen progressions and can be useful where a more fine-grained analysis of such sequences is called for. They are very amenable to algebraic manipulation and hence are convenient to work with. Cohyperations, meanwhile, give a novel way to describe slowly increasing functions as often appear, for example, in proof theory.
In this paper we approach some questions about quantum contextuality with tools from formal logic... more In this paper we approach some questions about quantum contextuality with tools from formal logic. In particular, we consider an experiment associated with the Peres-Mermin square. The language of all possible sequences of outcomes of the experiment is classified in the Chomsky hierarchy and seen to be a regular language. Next, we make the rather evident observation that a finite set of hidden finite valued variables can never account for indeterminism in an ideally isolated repeatable experiment. We see that, when the language of possible outcomes of the experiment is regular, as is the case with the Peres-Mermin square, the amount of binary-valued hidden variables needed to de-randomize the model for all sequences of experiments up to length n grows as bad as it could be: linearly in n. We introduce a very abstract model of machine that simulates nature in a particular sense. A lower-bound on the number of memory states of such machines is proved if they were to simulate the experiment that corresponds to the Peres-Mermin square. Moreover, the proof of this lower bound is seen to scale to a certain generalization of the Peres-Mermin square. For this scaled experiment it is seen that the Holevo bound is violated and that the degree of violation increases uniformly.
This paper from 2000 is a presentation of a status quaestionis at that tiime, to wit of the probl... more This paper from 2000 is a presentation of a status quaestionis at that tiime, to wit of the problem of the interpretability logic of all reasonable arithmetical theories. We present both the arithmetical side and the modal side of the question.
This paper is the second in a series of three papers. All three papers deal with interpretability... more This paper is the second in a series of three papers. All three papers deal with interpretability logics and related matters. In the first paper a construction method was exposed to obtain models of these logics. Using this method, we obtained some completeness results, some already known, and some new. In this paper, we will set the construction method to work to obtain more results. First, the modal completeness of the logic ILM is proved using the construction method. This is not a new result, but by using our new proof we can obtain new results. Among these new results are some admissible rules for ILM and GL. Moreover, the new proof will be used to classify all the essentially ∆1 and also all the essentially Σ1 formulas of ILM. Closely related to essentially Σ1 sentences are the so-called self provers. A self-prover is a formula ϕ which implies its own provability, that is ϕ → ✷ϕ. Each formula ϕ will generate a self prover ϕ ∧ ✷ϕ. We will use the construction method to characterize those sentences of GL that generate a self prover that is trivial in the sense that it is Σ1.
In this paper we give an overview of an essential part of a Π 0 1 ordinal analysis of Peano Arith... more In this paper we give an overview of an essential part of a Π 0 1 ordinal analysis of Peano Arithmetic (PA) as presented by Beklemishev ([3]). This analysis is mainly performed within the polymodal provability logic GLPω. We reflect on ways of extending this analysis beyond PA. A main difficulty in this is to find proper generalizations of the so-called Reduction Property. The Reduction Property relates reflection principles to reflection rules. In this paper we prove a result that simplifies the reflection rules. Moreover, we see that for an ordinal analysis the full Reduction Property is not needed. This latter observation is also seen to open up ways for applications of ordinal analysis relative to some strong base theory.
Turing progressions have been often used to measure the proof-theoretic strength of mathematical ... more Turing progressions have been often used to measure the proof-theoretic strength of mathematical theories: iterate adding consistency of some weak base theory until you "hit" the target theory. Turing progressions based on n-provability give rise to a Πn+1 proof-theoretic ordinal |U | Π 0 n+1. As such, to each theory U we can assign the sequence of corresponding Πn+1 ordinals |U |n n>0. We call this sequence a Turing-Taylor expansion or spectrum of a theory. In this paper, we relate Turing-Taylor expansions of sub-theories of Peano Arithmetic to Ignatiev's universal model for the closed fragment of the polymodal provability logic GLPω. In particular, we observe that each point in the Ignatiev model can be seen as Turing-Taylor expansions of formal mathematical theories. Moreover, each sub-theory of Peano Arithmetic that allows for a Turing-Taylor expansion will define a unique point in Ignatiev's model.
This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit:
Wolfram's Principle of Computational Equivalence (PCE) implies that universal complexity abounds ... more Wolfram's Principle of Computational Equivalence (PCE) implies that universal complexity abounds in nature. This paper comprises three sections. In the first section we consider the question why there are so many universal phenomena around. So, in a sense, we seek a driving force behind the PCE if any. We postulate a principle GNS that we call the Generalized Natural Selection principle that together with the Church-Turing thesis is seen to be equivalent in a sense to a weak version of PCE. In the second section we ask the question why we do not observe any phenomena that are complex but not-universal. We choose a cognitive setting to embark on this question and make some analogies with formal logic. In the third and final section we report on a case study where we see rich structures arise everywhere.
In this paper we give an overview of an essential part of a Pi^0_1 ordinal analysis of Peano Arit... more In this paper we give an overview of an essential part of a Pi^0_1 ordinal analysis of Peano Arithmetic (PA) as presented by Beklemishev. This analysis is mainly performed within the polymodal provability logic GLP. We reflect on ways of extending this analysis beyond PA. A main difficulty in this is to find proper generalizations of the so-called Reduction Property. The Reduction Property relates reflection principles to reflection rules. In this paper we prove a result that simplifies the reflection rules. Moreover, we see that for an ordinal analysis the full Reduction Property is not needed. This latter observation is also seen to open up ways for applications of ordinal analysis relative to some strong base theory.
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