Definite Formulae, Negation-as-Failure, and the Base-extension Semantics of Intuitionistic Propositional Logic

Proof-theoretic semantics (P-tS) is the paradigm of semantics in which meaning in logic is based on proof (as opposed to truth). A particular instance of P-tS for intuitionistic propositional logic (IPL) is its base-extension semantics (B-eS). This semantics is given by a relation called support, explaining the meaning of the logical constants, which is parameterized by systems of rules called bases that provide the semantics of atomic propositions. In this paper, we interpret bases as collections of definite formulae and use the operational view of the latter as provided by uniform proof-search -- the proof-theoretic foundation of logic programming (LP) -- to establish the completeness of IPL for the B-eS. This perspective allows negation, a subtle issue in P-tS, to be understood in terms of the negation-as-failure protocol in LP. Specifically, while the denial of a proposition is traditionally understood as the assertion of its negation, in B-eS we may understand the denial of a proposition as the failure to find a proof of it. In this way, assertion and denial are both prime concepts in P-tS.Comment: submitte

From Proof-theoretic Validity to Base-extension Semantics for Intuitionistic Propositional Logic

Proof-theoretic semantics (P-tS) is the approach to meaning in logic based on \emph{proof} (as opposed to truth). There are two major approaches to P-tS: proof-theoretic validity (P-tV) and base-extension semantics (B-eS). The former is a semantics of arguments, and the latter is a semantics of logical constants in a logic. This paper demonstrates that the B-eS for intuitionistic propositional logic (IPL) encapsulates the declarative content of a basic version of P-tV. Such relationships have been considered before yielding incompleteness results. This paper diverges from these approaches by accounting for the constructive, hypothetical setup of P-tV. It explicates how the B-eS for IPL works

Proof-theoretic Semantics and Tactical Proof

The use of logical systems for problem-solving may be as diverse as in proving theorems in mathematics or in figuring out how to meet up with a friend. In either case, the problem solving activity is captured by the search for an \emph{argument}, broadly conceived as a certificate for a solution to the problem. Crucially, for such a certificate to be a solution, it has be \emph{valid}, and what makes it valid is that they are well-constructed according to a notion of inference for the underlying logical system. We provide a general framework uniformly describing the use of logic as a mathematics of reasoning in the above sense. We use proof-theoretic validity in the Dummett-Prawitz tradition to define validity of arguments, and use the theory of tactical proof to relate arguments, inference, and search.Comment: submitte

Defining Logical Systems via Algebraic Constraints on Proofs

We comprehensively present a program of decomposition of proof systems for non-classical logics into proof systems for other logics, especially classical logic, using an algebra of constraints. That is, one recovers a proof system for a target logic by enriching a proof system for another, typically simpler, logic with an algebra of constraints that act as correctness conditions on the latter to capture the former; for example, one may use Boolean algebra to give constraints in a sequent calculus for classical propositional logic to produce a sequent calculus for intuitionistic propositional logic. The idea behind such forms of reduction is to obtain a tool for uniform and modular treatment of proof theory and provide a bridge between semantics logics and their proof theory. The article discusses the theoretical background of the project and provides several illustrations of its work in the field of intuitionistic and modal logics. The results include the following: a uniform treatment of modular and cut-free proof systems for a large class of propositional logics; a general criterion for a novel approach to soundness and completeness of a logic with respect to a model-theoretic semantics; and a case study deriving a model-theoretic semantics from a proof-theoretic specification of a logic.Comment: submitte

Simultaneous increasing of thermal conversion efficiency and BMEP while reducing emissions

Paper presented at the 9th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Malta, 16-18 July, 2012.The Downsizing of internal combustion engines (ICE) is al- ready recognized as a very suitable method for the concurrent enhancement of the indicated fuel conversion efficiency (IFCE) and the break mean effective pressure (BMEP) while also decreasing the CO2 and NOx emissions [1], [2]. The Ultra-Downsizing concept was introduced in [3] as a still higher development stage of ICE and implemented by means of real Atkinson cycles, using asymmetrical crank mechanisms, combined with a very intensive multistage high- pressure turbocharging with intensive intercooling. This allows an increase of ICE performance while keeping the thermal and mechanical strain strength of engine components within the current usual limits. The investigations from [3] were carried out using the simu- lation tool BOOST (AVL Co). The three-stage turbocharging with intensive intercooling used in this process and the release of heat during combustion are controlled by numerous parame- ters. As a consequence, it is very difficult to harmonize them in order to optimize concurrently IFCE, BMEP and emissions. For this reason, the ideal V,p,T-model presented in [4] has been revised, improved and adapted to better meet the BOOST simu- lations. With help of this new, ideal V,p,T-model, it is possible to evaluate adequately the potential for improving the perfor- mance of Ultra-Downsizing.dc201

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist's B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established.Comment: 27 page

Sized Types for low-level Quantum Metaprogramming

One of the most fundamental aspects of quantum circuit design is the concept of families of circuits parametrized by an instance size. As in classical programming, metaprogramming allows the programmer to write entire families of circuits simultaneously, an ability which is of particular importance in the context of quantum computing as algorithms frequently use arithmetic over non-standard word lengths. In this work, we introduce metaQASM, a typed extension of the openQASM language supporting the metaprogramming of circuit families. Our language and type system, built around a lightweight implementation of sized types, supports subtyping over register sizes and is moreover type-safe. In particular, we prove that our system is strongly normalizing, and as such any well-typed metaQASM program can be statically unrolled into a finite circuit.Comment: Presented at Reversible Computation 2019. Final authenticated publication is available online at https://doi.org/10.1007/978-3-030-21500-2_

Line orientation adaptation: local or global?

Prolonged exposure to an oriented line shifts the perceived orientation of a subsequently observed line in the opposite direction, a phenomenon known as the tilt aftereffect (TAE). Here we consider whether the TAE for line stimuli is mediated by a mechanism that integrates the local parts of the line into a single global entity prior to the site of adaptation, or the result of the sum of local TAEs acting separately on the parts of the line. To test between these two alternatives we used the fact the TAE transfers almost completely across luminance contrast polarity [1]. We measured the TAE using adaptor and test lines that (1) either alternated in luminance polarity or were of a single polarity, and (2) either alternated in local orientation or were of a single orientation. We reasoned that if the TAE was agnostic to luminance polarity and was parts-based, we should obtain large TAEs using alternating-polarity adaptors with single-polarity tests. However we found that (i) TAEs using one-alternating-polarity adaptors with all-white tests were relatively small, increased slightly for two-alternating-polarity adaptors, and were largest with all-white or all-black adaptors. (ii) however TAEs were relatively large when the test was one-alternating polarity, irrespective of the adaptor type. (iii) The results with orientation closely mirrored those obtained with polarity with the difference that the TAE transfer across orthogonal orientations was weak. Taken together, our results demonstrate that the TAE for lines is mediated by a global shape mechanism that integrates the parts of lines into whole prior to the site of orientation adaptation. The asymmetry in the magnitude of TAE depending on whether the alternating-polarity lines was the adaptor or test can be explained by an imbalance in the population of neurons sensitive to 1st-and 2nd-order lines, with the 2nd-order lines being encoded by a subset of the mechanisms sensitive to 1st-order lines