Terms Of Calculus Research Articles

On the Computational Representation of Classical Logical Connectives

Many programming calculi have been designed to have a Curry-Howard correspondence with a classical logic. We investigate the effect that different choices of logical connective have on such calculi, and the resulting computational content.We identify two connectives 'if-and-only-if' and 'exclusive or' whose computational content is not well known, and whose cut elimination rules are non-trivial to define. In the case of the former, we define a term calculus and show that the computational content of several other connectives can be simulated. We show this is possible even for connectives not logically expressible with 'if-and-only-if'.

Fractional variational calculus in terms of Riesz fractional derivatives

This paper presents extensions of traditional calculus of variations for systems containing Riesz fractional derivatives (RFDs). Specifically, we present generalized Euler–Lagrange equations and the transversality conditions for fractional variational problems (FVPs) defined in terms of RFDs. We consider two problems, a simple FVP and an FVP of Lagrange. Results of the first problem are extended to problems containing multiple fractional derivatives, functions and parameters, and to unspecified boundary conditions. For the second problem, we present Lagrange-type multiplier rules. For both problems, we develop the Euler–Lagrange-type necessary conditions which must be satisfied for the given functional to be extremum. Problems are considered to demonstrate applications of the formulations. Explicitly, we introduce fractional momenta, fractional Hamiltonian, fractional Hamilton equations of motion, fractional field theory and fractional optimal control. The formulations presented and the resulting equations are similar to the formulations for FVPs given in Agrawal (2002 J. Math. Anal. Appl. 272 368, 2006 J. Phys. A: Math. Gen. 39 10375) and to those that appear in the field of classical calculus of variations. These formulations are simple and can be extended to other problems in the field of fractional calculus of variations.

A Behavioural Model for Klop's Calculus

A model characterising strong normalisation for Klop's extension of λ-calculus is presented. The main technical tools for this result are an inductive definition of strongly normalising terms of Klop's calculus and an intersection type system for terms of Klop's calculus.

Light affine lambda calculus and polynomial time strong normalization

Light Linear Logic (LLL) and Intuitionistic Light Affine Logic (ILAL) are logics that capture polynomial time computation. It is known that every polynomial time function can be represented by a proof of these logics via the proofs-as-programs correspondence. Furthermore, there is a reduction strategy which normalizes a given proof in polynomial time. Given the latter polynomial time “weak” normalization theorem, it is natural to ask whether a “strong” form of polynomial time normalization theorem holds or not. In this paper, we introduce an untyped term calculus, called Light Affine Lambda Calculus (λLA), which corresponds to ILAL. λLA is a bi-modal λ-calculus with certain constraints, endowed with very simple reduction rules. The main property of LALC is the polynomial time strong normalization: any reduction strategy normalizes a given λLA term in a polynomial number of reduction steps, and indeed in polynomial time. Since proofs of ILAL are structurally representable by terms of λLA, we conclude that the same holds for ILAL.

Resource operators for λ-calculus

We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic’s proof-nets. We show the operational behaviour of the calculus and some of its fundamental properties such as confluence, preservation of strong normalisation, strong normalisation of simply typed terms, step by step simulation of β-reduction and full composition.

Intersection types for λGtz-calculus

We introduce an intersection type assignment system for Espirito-Santo?s ?Gtz-calculus, a term calculus embodying the Curry-Howard correspondence for the intuitionistic sequent calculus. We investigate basic properties of this intersection type system. Our main result is Subject reduction property.

Parametric representation of fuzzy numbers and application to fuzzy calculus

We present several models to obtain simple parametric representations of the fuzzy numbers or intervals, based on the use of piecewise monotonic functions of different forms. The representations have the advantage of allowing flexible and easy to control shapes of the fuzzy numbers (we use the standard α - cuts setting, but also the membership functions are obtained immediately) and can be used directly to obtain error-controlled-approximations of the fuzzy calculus in terms of a finite set of parameters. The general setting is the Hermite-type interpolation, where the values and the slopes of the monotonic interpolators are given by appropriate parameters and the overall errors of the fuzzy computations can be controlled within a prefixed tolerance by eventually augmenting the total number of pieces (and of the parameters) by which the results are obtained. The representations are designed to model the lower and the upper extremal values of each α - cut (compact) intervals of the fuzzy numbers and are able to produce almost any possible configuration (differentiable, continuous or with a finite number of discontinuity points) by using parametric monotonic functions of different types. We show applications in the standard fuzzy calculus and we stress the generality and the applicability of the proposed representation to a large class of problems, including the numerical solution of fuzzy differential equations, the fuzzy linear regression and the stochastic extensions of the fuzzy mathematics. The proposed model is called the Lower–Upper representation and we denote the associated fuzzy numbers or intervals as LU-fuzzy.

Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages

This article develops, and describes how to use, results concerning disintegrations of Poisson random measures. These results are fashioned as simple tools that can be tailor-made to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The Poisson disintegration method is based on the formal statement of two results concerning a Laplace functional change of measure and a Poisson Palm/Fubini calculus in terms of random partitions of the integers {1,…,n}. The techniques are analogous to, but much more general than, techniques for the Dirichlet process and weighted gamma process developed in [Ann. Statist. 12 (1984) 351–357] and [Ann. Inst. Statist. Math. 41 (1989) 227–245]. In order to illustrate the flexibility of the approach, large classes of random probability measures and random hazards or intensities which can be expressed as functionals of Poisson random measures are described. We describe a unified posterior analysis of classes of discrete random probability which identifies and exploits features common to all these models. The analysis circumvents many of the difficult issues involved in Bayesian nonparametric calculus, including a combinatorial component. This allows one to focus on the unique features of each process which are characterized via real valued functions h. The applicability of the technique is further illustrated by obtaining explicit posterior expressions for Lévy–Cox moving average processes within the general setting of multiplicative intensity models. In addition, novel computational procedures, similar to efficient procedures developed for the Dirichlet process, are briefly discussed for these models.

Intersection Types for Light Affine Lambda Calculus

Light Affine Lambda Calculus is a term calculus for polynomial time computation ([K. Terui. Light affine lambda calculus and polytime strong normalization. In Proceedings of LICS'01, pages 209–220, 2001]). Some of the terms of Light Affine Lambda Calculus must however be regarded as errors. Intuitionistic Light Affine Logic (ILAL) types only terms without errors, but not all of them. We introduce two type assignment systems with intersection types : in the first one, typable pseudo-terms are exactly the terms without errors ; in the second one, they are exactly those that reduce to normal terms without errors.

A term calculus for (co-)recursive definitions on streamlike data structures

We introduce a system of simply typed lambda terms (with fixed point combinators) and show that a rather comprehensive class of (co-)recursion equations on streams or non-wellfounded trees can be solved in our system. Moreover certain conditions are presented which guarantee that the defined functionals are primitive recursive. As a major example we give a co-recursive treatment of Mints’ continuous cut-elimination operator.

Classical linear logic of implications

We give a simple term calculus for the multiplicative exponential fragment of Classical Linear Logic, by extending Barber and Plotkin's dual-context system for the intuitionistic case. The calculus has the non-linear and linear implications as the basic constructs, and this design choice allows a technically manageable axiomatisation without commuting conversions. Despite this simplicity, the calculus is shown to be sound and complete for category-theoretic models given by *-autonomous categories with linear exponential comonads.

A Language For Multiplicative-additive Linear Logic

A term calculus for the proofs in multiplicative-additive linear logic is introduced and motivated as a programming language for channel based concurrency. The term calculus is proved complete for a semantics in linearly distributive categories with additives. It is also shown that proof equivalence is decidable by showing that the cut elimination rewrites supply a confluent rewriting system modulo equations.

Interpretation of frequency modulation atomic force microscopy in terms of fractional calculus

It is widely recognized that small amplitude frequency modulation atomic force microscopy probes the derivative of the interaction force between tip and sample. For large amplitudes, however, such a physical connection is currently lacking, although it has been observed that the frequency shift presents a quantity intermediate to the interaction force and energy for certain force laws. Here we prove that these observations are a universal property of large amplitude frequency modulation atomic force microscopy, by establishing that the frequency shift is proportional to the half-fractional integral of the force, regardless of the force law. This finding indicates that frequency modulation atomic force microscopy can be interpreted as a fractional differential operator, where the order of the derivative∕integral is dictated by the oscillation amplitude. We also establish that the measured frequency shift varies systematically from a probe of the force gradient for small oscillation amplitudes, through to the measurement of a quantity intermediate to the force and energy (the half-fractional integral of the force) for large oscillation amplitudes. This has significant implications to measurement sensitivity, since integrating the force will smooth its behavior, while differentiating it will enhance variations. This highlights the importance in choice of oscillation amplitude when wishing to optimize the sensitivity of force spectroscopy measurements to short-range interactions and consequently imaging with the highest possible resolution.

Comparing model checking and logical reasoning for real-time systems

Abstract. We apply both model checking and logical reasoning to a real-time protocol for mutual exclusion. To this end we employ PLC-Automata, an abstract notion of programs for real-time systems. A logic-based semantics in terms of Duration Calculus is used to verify the correctness of the protocol by logical reasoning. An alternative but consistent operational semantics in terms of Timed Automata is used to verify the correctness by model checkers. Since model checking of the full model does not terminate in all cases within an acceptable time we examine abstractions and their influence on model-checking performance. We present two abstraction methods that can be applied successfully for the protocol presented.

A Calculus of Tactics and Its Operational Semantics

This paper presents work in progress on a calculus of tactics for a hypothetical interactive theorem prover based on a λ-calculus for higher-order logic (λHOL). The calculus of tactics is an extension of a calculus of open terms for λHOL. In contrast to other systems where the semantics of tactics is given by the semantics of their implementation in a general programming language (e.g. OCAML) we are able to define what a tactic does in terms of the state of the theorem prover expressed by an open term that encodes the incomplete proof created so far at that given state.We present typed operational semantics for the tactics calculus and show that it is sound and complete with respect to the calculus of open terms. The soundness theorem goes further to establish the relation between the states of the prover before and after the execution of a tactic.

A fully abstract semantics for a nondeterministic functional language with monadic types

This paper presents work in progress on a calculus of tactics for a hypothetical interactive theorem prover based on a λ-calculus for higher-order logic (λHOL). The calculus of tactics is an extension of a calculus of open terms for λHOL. In contrast to other systems where the semantics of tactics is given by the semantics of their implementation in a general programming language (e.g. OCAML) we are able to define what a tactic does in terms of the state of the theorem prover expressed by an open term that encodes the incomplete proof created so far at that given state. We present typed operational semantics for the tactics calculus and show that it is sound and complete with respect to the calculus of open terms. The soundness theorem goes further to establish the relation between the states of the prover before and after the execution of a tactic.

The honeycomb model of 𝐺𝐿_{𝑛}(ℂ) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone

The set of possible spectra ( λ , μ , ν ) (\lambda ,\mu ,\nu ) of zero-sum triples of Hermitian matrices forms a polyhedral cone, whose facets have been already studied by Knutson and Tao, Helmke and Rosenthal, Totaro, and Belkale in terms of Schubert calculus on Grassmannians. We give a complete determination of these facets; there is one for each triple of Grassmannian Schubert cycles intersecting in a unique point. In particular, the list of inequalities determined by Belkale to be sufficient is in fact minimal. We introduce puzzles, which are new combinatorial gadgets to compute Grassmannian Schubert calculus, and seem to have much interest in their own right. As the proofs herein indicate, the Hermitian sum problem is very naturally studied using puzzles directly, and their connection to Schubert calculus is quite incidental to our approach. In particular, we get new, puzzle-theoretic, proofs of the results of Horn, Klyachko, Helmke and Rosenthal, Totaro, and Belkale. Along the way we give a characterization of “rigid” puzzles, which we use to prove a conjecture of W. Fulton: “if for a triple of dominant weights λ , μ , ν \lambda ,\mu ,\nu of G L n ( C ) GL_n({\mathbb C}) the irreducible representation V ν V_\nu appears exactly once in V λ ⊗ V μ V_\lambda \otimes V_\mu , then for all N ∈ N N\in {\mathbb N} , V N λ V_{N\lambda } appears exactly once in V N λ ⊗ V N μ V_{N\lambda }\otimes V_{N\mu } .”

A comparison of two systems of ordinal notations

The standard method of generating countable ordinals from uncountable ordinals can be replaced by a use of fixed point extractors available in the term calculus of Howard’s system. This gives a notion of the intrinsic complexity of an ordinal analogous to the intrinsic complexity of a function described in Godel’s T.

Equational properties of mobile ambients

The ambient calculus is a process calculus for describing mobile computation. We develop a theory of Morris-style contextual equivalence for proving properties of mobile ambients. We prove a context lemma that allows derivation of contextual equivalences by considering contexts of a particular limited form, rather than all arbitrary contexts. We give an activity lemma that characterises the possible interactions between a process and a context. We prove several examples of contextual equivalence. The proofs depend on characterising reductions in the ambient calculus in terms of a labelled transition system.

Linearity and Passivity

A simple symmetric logic is proposed which captures both the notions of Linearity and of Passivity (Syntactic Control of Interference).Both of these systems of logic can be seen as providing varying constraints on the use of the Contraction structural rule. We show that these systems are closely related and form a natural integration.A term calculus is proposed for the logic and we establish the soundness of our proposal with respect to an operational semantics of this term calculus.