Observational Equivalence Research Articles

Theorems for free for free: parametricity, with and without types

The polymorphic blame calculus integrates static typing, including universal types, with dynamic typing. The primary challenge with this integration is preserving parametricity: even dynamically-typed code should satisfy it once it has been cast to a universal type. Ahmed et al. (2011) employ runtime type generation in the polymorphic blame calculus to preserve parametricity, but a proof that it does so has been elusive. Matthews and Ahmed (2008) gave a proof of parametricity for a closely related system that combines ML and Scheme, but later found a flaw in their proof. In this paper we present an improved version of the polymorphic blame calculus and we prove that it satisfies relational parametricity. The proof relies on a step-indexed Kripke logical relation. The step-indexing is required to make the logical relation well-defined in the case for the dynamic type. The possible worlds include the mapping of generated type names to their types and the mapping of type names to relations. We prove the Fundamental Property of this logical relation and that it is sound with respect to contextual equivalence. To demonstrate the utility of parametricity in the polymorphic blame calculus, we derive two free theorems.

Algorithmic games for full ground references

We present a full classification of decidable and undecidable cases for contextual equivalence in a finitary ML-like language equipped with full ground storage (both integers and reference names can be stored). The simplest undecidable type is mathsf {unit}rightarrow mathsf {unit}rightarrow mathsf {unit}. At the technical level, our results marry game semantics with automata-theoretic techniques developed to handle infinite alphabets. On the automata-theoretic front, we show decidability of the emptiness problem for register pushdown automata extended with fresh-symbol generation.

A problem with observational equivalence: Disentangling the renter illusion hypothesis

The renter illusion hypothesis argues that renters are more likely to support local public expenditures than homeowners because of a misperception about the actual costs and benefits they face. Numerous empirical studies have produced mixed results which are difficult to interpret because of the lack of micro foundations for the renter illusion hypothesis. This paper helps fill that gap by introducing in a theoretical model the concept of renter illusion behaviour, which can be seen as a rational behaviour in the presence of cognitive anomalies (i.e. behavioural or bounded rationality) and asymmetric information. The resulting theoretical framework can be used to interpret and support a large share of micro- and macro-econometric findings in the previous empirical literature on renter illusion.

Culture, diffusion, and economic development: The problem of observational equivalence

This research explores the direct and barrier effects of culture on economic development. It shows both theoretically and empirically that whenever the technological frontier is at the top or bottom of the world distribution of a cultural value, there exists an observational equivalence between absolute cultural distances and cultural distances relative to the frontier, preventing the identification of its direct and barrier effects. Since the technological frontier usually has the “right” cultural values for development, it tends to be in the extremes of the distribution of cultural traits, generating observational equivalence and confounding the analysis. These results highlight the difficulty of disentangling the direct and barrier effects of culture. The empirical analysis finds suggestive evidence for direct effects of individualism and conformity with hierarchy, and barrier effects of hedonism.

The Guarded Lambda-Calculus: Programming and Reasoning with Guarded Recursion for Coinductive Types

We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive types may be transformed into coinductive types by a type-former inspired by modal logic and Atkey-McBride clock quantification, allowing the typing of acausal functions. We give a call-by-name operational semantics for the calculus, and define adequate denotational semantics in the topos of trees. The adequacy proof entails that the evaluation of a program always terminates. We introduce a program logic with L\"ob induction for reasoning about the contextual equivalence of programs. We demonstrate the expressiveness of the calculus by showing the definability of solutions to Rutten's behavioural differential equations.

Causal Inference via Algebraic Geometry: Feasibility Tests for Functional Causal Structures with Two Binary Observed Variables

AbstractWe provide a scheme for inferring causal relations from uncontrolled statistical data based on tools from computational algebraic geometry, in particular, the computation of Groebner bases. We focus on causal structures containing just two observed variables, each of which is binary. We consider the consequences of imposing different restrictions on the number and cardinality of latent variables and of assuming different functional dependences of the observed variables on the latent ones (in particular, the noise need not be additive). We provide an inductive scheme for classifying functional causal structures into distinct observational equivalence classes. For each observational equivalence class, we provide a procedure for deriving constraints on the joint distribution that are necessary and sufficient conditions for it to arise from a model in that class. We also demonstrate how this sort of approach provides a means of determining which causal parameters are identifiable and how to solve for these. Prospects for expanding the scope of our scheme, in particular to the problem of quantum causal inference, are also discussed.

Stateful applied pi calculus: Observational equivalence and labelled bisimilarity

We extend Abadi–Fournet's applied pi calculus with state cells, which are used to reason about protocols that store persistent information. Examples are protocols involving databases or hardware modules with internal state. We distinguish between private state cells, which are not available to the attacker, and public state cells, which arise when a private state cell is compromised by the attacker. For processes involving only private state cells we define observational equivalence and labelled bisimilarity in the same way as in the original applied pi calculus, and show that they coincide. Our result implies Abadi–Fournet's theorem – the coincidence of observational equivalence and labelled bisimilarity – in a revised version of the applied pi calculus. For processes involving public state cells, we can essentially keep the definition of observational equivalence, but need to strengthen the definition of labelled bisimulation in order to show that observational equivalence and labelled bisimilarity coincide in this case as well.

Publish and Perish: Creative Destruction and Macroeconomic Theory

A number of macroeconomic theories, very popular in the 1980s, seem to have completely disappeared and been replaced by the dynamic stochastic general equilibrium (DSGE) approach. We will argue that this replacement is due to a tacit agreement on a number of assumptions, previously seen as mutually exclusive, and not due to a settlement by nature. As opposed to econometrics and microeconomics and despite massive progress in the access to data and the use of statistical softwares, macroeconomic theory appears not to be a cumulative science so far. Observational equivalence of different models and the problem of identification of parameters of the models persist as will be highlighted by examining two examples: one in growth theory and a second in testing inflation persistence.

Modules, abstraction, and parametric polymorphism

Reynolds's Abstraction theorem forms the mathematical foundation for data abstraction. His setting was the polymorphic lambda calculus. Today, many modern languages, such as the ML family, employ rich module systems designed to give more expressive support for data abstraction than the polymorphic lambda calculus, but analogues of the Abstraction theorem for such module systems have lagged far behind. We give an account of the Abstraction theorem for a modern module calculus supporting generative and applicative functors, higher-order functors, sealing, and translucent signatures. The main issues to be overcome are: (1) the fact that modules combine both types and terms, so they must be treated as both simultaneously, (2) the effect discipline that models the distinction between transparent and opaque modules, and (3) a very rich language of type constructors supporting singleton kinds. We define logical equivalence for modules and show that it coincides with contextual equivalence. This substantiates the folk theorem that modules are good for data abstraction. All our proofs are formalized in Coq.

Deciding equivalence with sums and the empty type

The logical technique of focusing can be applied to the λ-calculus; in a simple type system with atomic types and negative type formers (functions, products, the unit type), its normal forms coincide with βη-normal forms. Introducing a saturation phase gives a notion of quasi-normal forms in presence of positive types (sum types and the empty type). This rich structure let us prove the decidability of βη-equivalence in presence of the empty type, the fact that it coincides with contextual equivalence, and with set-theoretic equality in all finite models.

Contextual isomorphisms

What is the right notion of "isomorphism" between types, in a simple type theory? The traditional answer is: a pair of terms that are inverse up to a specified congruence. We firstly argue that, in the presence of effects, this answer is too liberal and needs to be restricted, using Führmann's notion of thunkability in the case of value types (as in call-by-value), or using Munch-Maccagnoni's notion of linearity in the case of computation types (as in call-by-name). Yet that leaves us with different notions of isomorphism for different kinds of type. This situation is resolved by means of a new notion of "contextual" isomorphism (or morphism), analogous at the level of types to contextual equivalence of terms. A contextual morphism is a way of replacing one type with the other wherever it may occur in a judgement, in a way that is preserved by the action of any term with holes. For types of pure λ-calculus, we show that a contextual morphism corresponds to a traditional isomorphism. For value types, a contextual morphism corresponds to a thunkable isomorphism, and for computation types, to a linear isomorphism.

Characteristic bisimulation for higher-order session processes

For higher-order (process) languages, characterising contextual equivalence is a long-standing issue. In the setting of a higher-order pi -calculus with session types, we develop characteristic bisimilarity, a typed bisimilarity which fully characterises contextual equivalence. To our knowledge, ours is the first characterisation of its kind. Using simple values inhabiting (session) types, our approach distinguishes from untyped methods for characterising contextual equivalence in higher-order processes: we show that observing as inputs only a precise finite set of higher-order values suffices to reason about higher-order session processes. We demonstrate how characteristic bisimilarity can be used to justify optimisations in session protocols with mobile code communication.

Formal modelling and analysis of receipt-free auction protocols in applied pi

We formally study two privacy-type properties for e-auction protocols: bidding-price-secrecy and receipt-freeness. These properties are formalised as observational equivalences in the applied pi calculus. We analyse two receipt-free auction protocols: one proposed by Abe and Suzuki in 2002 (AS02) and the other by Howlader et al. in 2014 (HRM14). Bidding-price-secrecy of the AS02 protocol is verified using the automatic verifier ProVerif, whereas receipt-freeness of the two protocols, as well as bidding-price-secrecy of the HRM14 protocol, are proved manually.

Money in Exile: Campaign Contributions and Committee Access

Understanding how money influences the legislative process is essential for assessing American democracy, but problems of endogeneity, legality, and observational equivalence make it difficult to isolate the effect of contributions on policy. We seek to answer long-standing questions about the influence of money in Congress by exploiting a congressional procedure (committee exile) that exogenously varies a member’s influence over the policy-making process. We leverage exile as an identification strategy to show that business interests seek short-term access to influential legislators. Industries overseen by the committee decrease contributions to exiled legislators and instead direct their contributions to new committee members from the opposite party. Partisan interests, in contrast, attempt to influence electoral outcomes—boosting contributions to exiled members. Together, we provide evidence that corporations and business PACs use donations to acquire immediate access and favor—suggesting they at least anticipate that the donations will influence policy.

Automated Verification of Equivalence Properties of Cryptographic Protocols

Indistinguishability properties are essential in formal verification of cryptographic protocols. They are needed to model anonymity properties, strong versions of confidentiality, and resistance against offline guessing attacks. Indistinguishability properties can be conveniently modeled as equivalence properties. We present a novel procedure to verify equivalence properties for a bounded number of sessions of cryptographic protocols. As in the applied pi calculus, our protocol specification language is parametrized by a first-order sorted term signature and an equational theory that allows formalization of algebraic properties of cryptographic primitives. Our procedure is able to verify trace equivalence for determinate cryptographic protocols. On determinate protocols, trace equivalence coincides with observational equivalence, which can therefore be automatically verified for such processes. When protocols are not determinate, our procedure can be used for both under- and over-approximations of trace equivalence, which proved successful on examples. The procedure can handle a large set of cryptographic primitives, namely those whose equational theory is generated by an optimally reducing convergent rewrite system. The procedure is based on a fully abstract modelling of the traces of a bounded number of sessions of the protocols into first-order Horn clauses on which a dedicated resolution procedure is used to decide equivalence properties. We have shown that our procedure terminates for the class of subterm convergent equational theories. Moreover, the procedure has been implemented in a prototype tool Active Knowledge in Security Protocols and has been effectively tested on examples. Some of the examples were outside the scope of existing tools, including checking anonymity of an electronic voting protocol due to Okamoto.

Global Identification in DSGE Models Allowing for Indeterminacy

This article presents a framework for analysing global identification in log linearized Dynamic Stochastic General Equilibrium (DSGE) models that encompasses both determinacy and indeterminacy. First, it considers a frequency domain expression for the Kullback–Leibler distance between two DSGE models and shows that global identification fails if and only if the minimized distance equals 0. This result has three features: (1) it can be applied across DSGE models with different structures; (2) it permits checking whether a subset of frequencies can deliver identification; (3) it delivers parameter values that yield observational equivalence if there is identification failure. Next, the article proposes a measure for the empirical closeness between two DSGE models for a further understanding of the strength of identification. The measure gauges the feasibility of distinguishing one model from another based on a finite number of observations generated by the two models. It is shown to represent the highest possible power under Gaussianity when considering local alternatives. The above theory is illustrated using two small-scale and one medium-scale DSGE models. The results document that certain parameters can be identified under indeterminacy but not determinacy, that different monetary policy rules can be (nearly) observationally equivalent, and that identification properties can differ substantially between small and medium-scale models. For implementation, two procedures are developed and made available, both of which can be used to obtain and thus to cross validate the findings reported in the empirical applications. Although the article focuses on DSGE models, the results are also applicable to other vector linear processes with well-defined spectra, such as the (factor-augmented) vector autoregression.

Contextual equivalences in configuration structures and reversibility

Contextual equivalences equate terms that have the same observable behavior in any context. A standard contextual equivalence for CCS is the strong barbed congruence. Configuration structures are a denotational semantics for processes in which one defines equivalences that are more discriminating, i.e., that distinguish the denotation of terms equated by barbed congruence. Hereditary history preserving bisimulation (HHPB) is such a relation. We define a strong back-and-forth barbed congruence on RCCS, a reversible variant of CCS. We show that the relation induced by the back-and-forth congruence on configuration structures is equivalent to HHPB, thus providing a contextual characterization of HHPB.

On the Equivalence of Observation Structures for Petri Net Generators

Observation structures considered for Petri net generators usually assume that the firing of transitions may be observed through a static mask and that the marking of some places may be measurable. These observation structures, however, are rather limited, namely they do not cover all cases of practical interest where complex observations are possible. We consider in this paper more general ones, by correspondingly defining two new classes of Petri net generators: labeled Petri nets with outputs (LPNOs) and adaptive labeled Petri nets (ALPNs). To compare the modeling power of different Petri net generators, the notion of observation equivalence is proposed. ALPNs are shown to be the class of bounded generators possessing the highest modeling power. Looking for bridges between the different formalisms, we first present a general procedure to convert a bounded LPNO into an equivalent ALPN or even into an equivalent labeled Petri net (if any exists). Finally, we discuss the possibility of converting an unbounded LPNO into an equivalent ALPN.

Invariance, observational equivalence, and identification: Some implications for the empirical performance of affine term structure models

Abstract Scholars working on the class of affine term structure models (ATSM) use econometric theory to postulate that any two ATSMs can be connected through any combination of affine transformations (AT), leaving their implied bond prices and distributions identical. Our results, which are obtained through the derivation and testing of two hypotheses, show that, first, the application of AT introduces a lack of precision in the parameters resulting from their not being identified, and, this leads to very statistically distinct model performance. Secondly, we also demonstrate that, owing to the property of observational equivalence, AT neither improves nor worsens the empirical performance of ATSM. Results are robust to a Monte Carlo simulation experiment. Taken together, our findings show that the usage of invariant transformations introduces statistically significant differences in empirical performance that are economically unimportant. The fundamental lack of identification of the parameters introduces imprecision in ATSM when applying affine transformations, which affects the structure of the yield pricing errors and this causes statistical differences in empirical performance, while not causing substantive economic changes to it.

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