Whilst results from Structural Proof Theory can be couched in many formalisms, it is the sequent calculus which is the most amenable of the formalisms to metamathematical treatment. Constructive syntactic proofs are filled with bureaucratic details; rarely are all cases of a proof completed in the literature. Two intermediate results can be used to drastically reduce the amount of effort needed in proofs of Cut admissibility: Weakening and Invertibility. Indeed, whereas there are proofs of Cut admissibility which do not use Invertibility, Weakening is almost always necessary. Use of these results simply shifts the bureaucracy, however; Weakening and Invertibility, whilst more easy to prove, are still not trivial. We give a framework under which sequent calculi can be codified and analysed, which then allows us to prove various results: for a calculus to admit Weakening and for a rule to be invertible in a calculus. For the latter, even though many calculi are investigated, the general condition is simple and easily verified. The results have been applied to G3ip, G3cp, G3s, G3-LC and G4ip. Invertibility is important in another respect; that of proof-search. Should all rules in a calculus be invertible, then terminating root-first proof search gives a decision procedure for formulae without the need for back-tracking. To this end, we present some results about the manipulation of rule sets. It is shown that the transformations do not affect the expressiveness of the calculus, yet may render more rules invertible. These results can guide the design of efficient calculi. When using interactive proof assistants, every case of a proof, however complex, must be addressed and proved before one can declare the result formalised. To do this in a human readable way adds a further layer of complexity; most proof assistants give output which is only legible to a skilled user of that proof assistant. We give human-readable formalisations of Cut admissibility for G3cp and G3ip, Contraction admissibility for G4ip and Craig's Interpolation Theorem for G3i using the Isar vernacular of Isabelle. We also formalise the new invertibility results, in part using the package for reasoning about first-order languages, Nominal Isabelle. Examples are given showing the effectiveness of the formalisation. The formal proof of invertibility using the new methods is drastically shorter than the traditional, direct method.

Every compiler passes code through several stages, each a sort of mini- compiler of its own. Thus each stage may deal with the code in a different representation, which may have little to do with the source or target language. We can describe these in-memory representations as languages in their own right, which we call intermediate languages. Each intermediate language is designed to accomodate the stage of compilation that handles it. Those toward the end of the compilation pipeline, for instance, tend to have features expressing low-level details of computation. A subtler case is that of the optimization stage, whose role is to transform the program so that it runs faster, uses less memory, and so forth. The optimizer faces tradeoffs: The language should provide enough information to guide optimization algorithms, but all of this information must be kept up to date as the program is transformed. Also, establishing invariants in the language can be helpful both in implementing algorithms and in debugging the implementation, but each invariant may complicate desirable transformations or rule them out altogether. Finally, a ivlanguage where the invariants are obviously correct may have a form too awkward or otherwise unsuited to the compiler’s needs. Given the properties and invariants that we would like the language to provide, we can approach the design task in a way that gives these features without necessarily sacrificing implementability. Namely, begin with a formal language that makes the desired properties obvious, then translate it to one more suitable for implementation. We can even translate theorems about valid transformations in the formal language to derive correct algorithms in the implementation language. This dissertation explores the connections between different intermediate languages and how they can be interderived, then demonstrates how translation lead to an improvement to the Glasgow Haskell Compiler opimization engine. This dissertation includes previously published coauthored material.

Truth and falsehood, questions and answers, construction and deconstruction; most things come in dual pairs. Duality is a mirror that reveals the new from the old via opposition. This idea appears pervasively in logic, where duality inverts "true" with "false" and "and" with "or." However, even though programming languages are closely connected to logics, this kind of strong duality is not so apparent in practice. Sum types (disjoint tagged unions) and product types (structures) are dual concepts, but in the realm of programming, natural biases obscure their duality. To better understand the role of duality in programming, we shift our perspective. Our approach is based on the Curry-Howard isomorphism which says that programs following a specification are the same as proofs for mathematical theorems. This thesis explores Gentzen's sequent calculus, a logic steeped in duality, as a model for computational duality. By applying the Curry-Howard isomorphism to the sequent calculus, we get a language that combines dual programming concepts as equal opposites: data types found in functional languages are dual to co-data types (interface-based objects) found in object-oriented languages, control flow is dual to information flow, induction is dual to co-induction. This gives a duality-based semantics for reasoning about programs via orthogonality: checking safety and correctness based on a comprehensive test suite. We use the language of the sequent calculus to apply ideas from logic to issues relevant to program compilation. The idea of logical polarity reveals a symmetric basis of primitive programming constructs that can faithfully represent all user-defined data and co-data types. We reflect the lessons learned back into a core language for functional languages, at the cost of symmetry, with the relationship between the sequent calculus and natural deduction. This relationship lets us derive a pure lambda calculus with user-defined data and co-data which we further extend by bringing out the implicit control-flow in functional programs. Explicit control-flow lets us share and name control the same way we share and name data, enabling a direct representation of join points, which are essential for tractable optimization and compilation.

Programmers don't just have to write programs, they are have to reason about them. Programming languages aren't just tools for instructing computers what to do, they are tools for reasoning. And, it isn't just programmers who reason about programs: compilers and other tools reason similarly as they transform from one language into another one, or as they optimize an inefficient program into a better one. Languages, both surface languages and intermediate ones, need therefore to be both efficiently implementable and to support effective logical reasoning. However, these goals often seem to be in conflict. This dissertation studies programming language calculi inspired by the Curry-Howard correspondence, relating programming languages to proof systems. Our focus is on calculi corresponding logically to classical sequent calculus and connected computationally to abstract machines. We prove that these calculi have desirable properties to help bridge the gap between reasoning and implementation. Firstly, we explore a persistent conflict between extensionality and effects for lazy functional programs that manifests in a loss of confluence. Building on prior work, we develop a new rewriting theory for lazy functions and control which we first prove corresponds to the desired equational theory and then prove, by way of reductions into a smaller system, to be confluent. Next, we turn to the inconsistency between weak-head normalization and extensionality. Using ideas from our study of confluence, we develop a new operational semantics and series of abstract machines for head reduction which show us how to retain weak-head reduction's ease of implementation. After demonstrating the limitations of the above approach for call-by-value or types other than functions, we turn to typed calculi, showing how a type system can be used not only for mixing different kinds of data, but also different evaluation strategies in a single program. Building on variations of the reducibility candidates method such as biorthogonality and symmetric candidates, we present a uniform proof of strong normalization for our mixed-strategy system which works so long as all the strategies used satisfy criteria we isolate. This dissertation includes previously published co-authored material.

I study sequent calculus of combined logics in this thesis. Two specific logics are looked at-Logic BI that combines intuitionistic logic and multiplicative intuitionistic linear logic and Logic BBI that combines classical logic and multiplicative linear logic. A proof-theoretical study into logical combinations themsel ves then follows. To consolidate intuition about what this thesis is all about, let us suppose that we know about two different logics, Logic A developed for reasoning about Purpose A and Logic B developed for reasoning about Purpose B. Logic A serves Purpose A very well, but not Purpose B. Logic B serves Purpose B very well but not Purpose A. We wish to fulfill both Purpose A and Purpose B, but presently we can only afford to let one logic guide through our reasoning. What shall we do? One option is to be content with having Logic A with which we handle Purpose A efficiently and Purpose B rather inefficiently. Another option is to choose Logic B instead. But there is yet another option: we combine Logic A and Logic B to derive a new logic Logic C which is still one logic but which serves both Purpose A and Purpose B efficiently. The combined logic is synthetic of the strengths in more basic logics (Logic A and Logic B). As it nicely takes care of our requirements, it may be the best choice among all that have been so far considered. Yet this is not the end of the story. Depending on the manner Logic A and Logic B combine, Logic C may have extensions serving more purposes than just Purpose A and Purpose B. Ensuing is the following problem: we know about Logic A and Logic B, but we may not know about combined logics of the base logics. To understand the combined logics, we need to understand the extensions in which base logics interact each other. Analysis on the interesting parts tends to be non-trivial, however. The mentioned two specific combined logics BI and BBI do not make an exception, for which proof-theoretical development has been particularly slow. It has remained in obscurity how to properly handle base-logic interactions of the combined logics as appearing syntactically. As one objective of this thesis, I provide analysis on the syntactic phenomena of the BI and BBI base-logic interactions within sequent calculus, to augment the knowledge. For BI, I deliver, through appropriate methodologies to reason about the syntactic phenomena of the base-logic interactions, the first BI sequent calculus free of any structural rules. Given its positive consequence to efficient proof searches, this is a significant step forward in further maturity of BI proof theory. Based on the calculus, I prove decidability of a fragment of BI purely syntactically. For BBI which is closely connected to application via separation logic, I develop adequate sequent calculus conventions and consider the implication of the underlying semantics onto syntax. Sound BBI sequent calculi result with a closer syntax-semantics correspondence than previously envisaged. From them, adaptation to separation logic is also considered. To promote the knowledge of combined logics in general within computer science, it is also important that we be able to study logical combinations themselves. Towards this direction of generalisation, I present the concept of phased sequent calculus - sequent calculus which physically separates base logics, and in which a specific manner of logical combination to take place between them can be actually developed and analysed. For a demonstration, the said decidable BI fragment is formulated in phased sequent calculus, and the sense of logical combination in effect is analysed. A decision procedure is presented for the fragment.

In this thesis we examine the relationship between hypersequent and some types of labelled sequent calculi for a subset of intermediate logics—logics between intuitionistic (Int), and classical logics—that have geometric Kripke semantics, which we call Int∗/Geo. We introduce a novel calculus for a fragment of first-order classical logic, which we call partially-shielded formulae (or PSF for short), that is adequate for expressing the semantic validity of formulae in Int∗/Geo, and apply techniques from correspondence theory to provide translations of hypersequents, simply labelled sequents and relational sequents (simply labelled sequents with relational formulae) into PSF. Using these translations, we show that hypersequents and simply labelled sequents for calculi in Int∗/Geo share the same models. We also use these translations to justify various techniques that we introduce for translating simply labelled sequents into relational sequents and vice versa. In particular, we introduce a technique called "transitive unfolding" for translating relational sequents into simply labelled sequents (and by extension, hypersequents) which preserves linear models in Int∗/Geo. We introduce syntactic translations between hypersequent calculi and simply labelled sequent calculi. We apply these translations to a novel hypersequent framework HG3ipm∗ for some logics in Int∗/Geo to obtain a corresponding simply labelled sequent framework LG3ipm∗, and to an existing simply labelled calculus for Int from the literature to obtain a novel hypersequent calculus for Int. We introduce methods for translating a simply labelled sequent calculus into a cor- responding relational calculus, and apply these methods to LG3ipm∗ to obtain a novel relational framework RG3ipm∗ that bears similarities to existing calculi from the literature. We use transitive unfolding to translate proofs in RG3ipm∗ into proofs in LG3ipm∗ and HG3ipm∗ with the communication rule, which corresponds to the semantic restriction to linear models.

This thesis deals with the algorithmic undecidability (unsolvability) of provability in some non-classical logics. In fact, there are two natural variants of this problem. Fix a logic, we can study its set of theorems or its consequence relation, which is a more general problem. It is well-known that both these problems can be undecidable already for propositional logics and we provide further examples of such logics in this thesis. In particular, we study propositional substructural logics which are obtained from the sequent calculus LJ for intuitionistic logic by dropping structural rules. Our main results are the following. First, (finite) consequence relations in some basic non-associative substructural logics are shown to be undecidable. Second, we prove that a basic associative substructural logic with the contraction rule, which is notorious for being hard to handle, has an undecidable set of theorems. Since the studied logics have natural algebraic semantics, we also obtain corresponding algebraic results which are interesting in their own right.

In order to predict hydraulic jump characteristics for channel design, the jump height may be determined by calculating the subcritical sequent depth from momentum theory. In closed conduits, however, outlet submergence may fill the conduit entirely before the expected sequent depth is reached. This is called an incomplete or pressure jump (as opposed to a complete or free-surface jump), because pressure flow conditions prevail downstream. Since the momentum equation involves terms for the top width, area, and centroid of flow, the subcritical sequent depth is a function of the conduit shape in addition to the upstream depth and Froude number. This paper reviews momentum theory as applicable to closed-conduit hydraulic jumps and presents general solutions to the sequent depth problem for four commonly-shaped conduits: rectangular, circular, elliptical, and pipe arch. It also provides a numerical solution for conduits of any shape, as defined by the user. The solutions conservatively assume that the conduits are prismatic, horizontal, and frictionless within the jump length; that the pressure is hydrostatic and the velocity is uniform at each end of the jump; and that the effects of air entrainment and viscosity are negligible. The implications of these assumptions are briefly discussed. It was found that these solutions may be applied successfully to determine the subcritical sequent depth for hydraulic jumps in closed conduits of any shape or size. In practice, this may be used to quantify jump size, location, and energy dissipation.

Sequent calculi for BDI logics is a research object of the thesis. BDI logics are widely used for agent system description and implementation. Agents are autonomous systems, those acts in some environment and aspire to achieve preassigned goals. Implementation of the decision making is the main and the most complicated part in agent systems implementation. Logic calculi may be used for the decision making implementation. In this thesis, there are researched sequent calculi for BDI logics. Sequent calculi for BDI logics, like sequent calculi for other modal logics, use loop-check technique to get decidability. Inefficient loop-check takes a major part of the resources used for the derivation. For some modal logics, there are known loop-check free sequent calculi or calculi with an efficient loop-check. In this thesis, there is presented loop-check free sequent calculus for KD45 logic, which is the main fragment of the BDI logics. Introduced calculus not only eliminates loop-check, but also simplifies sequent derivation. For the branching time logic (another BDI logic fragment) there is presented sequent calculus with an efficient loop-check. Obtained results are adapted for creation sequent calculi for monoagent and multiagent BDI logics. Introduced calculi use only restricted loop-check. Moreover, loop-check is totally eliminated for some types of the loops. These results enables to create more efficient agent systems, those are based on the BDI logics. / Darbe nagrinėjami sekvenciniai skaičiavimai BDI logikoms. BDI logikos yra plačiai naudojamos agentinių sistemų aprašymui ir realizavimui. Agentai yra autonomiškos sistemos, kurios veikia kažkurioje aplinkoje ir siekia įvykdyti iš anksto apibrėžtus tikslus. Sprendimų priėmimo realizavimas yra svarbiausia ir sudėtingiausia dalis realizuojant agentines sistemas. Sprendimo priėmimo realizavimui gali būti naudojami logikos skaičiavimai. Šiame darbe ir yra nagrinėjami sekvenciniai skaičiavimai BDI logikoms. BDI logikose, kaip ir kitose modalumo logikose, yra naudojama ciklų paieška išsprendžiamumui gauti. Neefektyvi ciklų paieška užima didesnę išvedimų paieškos resursų dalį. Kai kurioms modalumo logikoms yra žinomi becikliai skaičiavimai ar skaičiavimai naudojantys efektyvią ciklų paiešką. Šiame darbe yra pateikiamas beciklis sekvencinis skaičiavimas KD45 logikai, kuri yra esminis BDI logikų fragmentas. Pateiktas skaičiavimas ne tik eliminuoja ciklų paiešką, bet ir supaprastina patį sekvencijos išvedimą. Skaidaus laiko logikai (kitam BDI logikų fragmentui) yra pateikiamas sekvencinis skaičiavimas naudojantis efektyvią ciklų paiešką. Gauti rezultatai yra pritaikyti sukuriant sekvencinius skaičiavimus vianaagentinei ir daugiaagentinei BDI logikoms. Pristatyti skaičiavimai naudoja tik apribotą ciklų paiešką. Be to, kai kurių tipų ciklus eliminuoja visiškai. Šie rezultatai įgalina kurti efektyvesnes agentines sistemas, paremtas BDI logikomis.

Darbe nagrinėjami sekvenciniai skaičiavimai BDI logikoms. BDI logikos yra plačiai naudojamos agentinių sistemų aprašymui ir realizavimui. Agentai yra autonomiškos sistemos, kurios veikia kažkurioje aplinkoje ir siekia įvykdyti iš anksto apibrėžtus tikslus. Sprendimų priėmimo realizavimas yra svarbiausia ir sudėtingiausia dalis realizuojant agentines sistemas. Sprendimo priėmimo realizavimui gali būti naudojami logikos skaičiavimai. Šiame darbe ir yra nagrinėjami sekvenciniai skaičiavimai BDI logikoms. BDI logikose, kaip ir kitose modalumo logikose, yra naudojama ciklų paieška išsprendžiamumui gauti. Neefektyvi ciklų paieška užima didesnę išvedimų paieškos resursų dalį. Kai kurioms modalumo logikoms yra žinomi becikliai skaičiavimai ar skaičiavimai naudojantys efektyvią ciklų paiešką. Šiame darbe yra pateikiamas beciklis sekvencinis skaičiavimas KD45 logikai, kuri yra esminis BDI logikų fragmentas. Pateiktas skaičiavimas ne tik eliminuoja ciklų paiešką, bet ir supaprastina patį sekvencijos išvedimą. Skaidaus laiko logikai (kitam BDI logikų fragmentui) yra pateikiamas sekvencinis skaičiavimas naudojantis efektyvią ciklų paiešką. Gauti rezultatai yra pritaikyti sukuriant sekvencinius skaičiavimus vianaagentinei ir daugiaagentinei BDI logikoms. Pristatyti skaičiavimai naudoja tik apribotą ciklų paiešką. Be to, kai kurių tipų ciklus eliminuoja visiškai. Šie rezultatai įgalina kurti efektyvesnes agentines sistemas, paremtas BDI logikomis. / Sequent calculi for BDI logics is a research object of the thesis. BDI logics are widely used for agent system description and implementation. Agents are autonomous systems, those acts in some environment and aspire to achieve preassigned goals. Implementation of the decision making is the main and the most complicated part in agent systems implementation. Logic calculi may be used for the decision making implementation. In this thesis, there are researched sequent calculi for BDI logics. Sequent calculi for BDI logics, like sequent calculi for other modal logics, use loop-check technique to get decidability. Inefficient loop-check takes a major part of the resources used for the derivation. For some modal logics, there are known loop-check free sequent calculi or calculi with an efficient loop-check. In this thesis, there is presented loop-check free sequent calculus for KD45 logic, which is the main fragment of the BDI logics. Introduced calculus not only eliminates loop-check, but also simplifies sequent derivation. For the branching time logic (another BDI logic fragment) there is presented sequent calculus with an efficient loop-check. Obtained results are adapted for creation sequent calculi for monoagent and multiagent BDI logics. Introduced calculi use only restricted loop-check. Moreover, loop-check is totally eliminated for some types of the loops. These results enables to create more efficient agent systems, those are based on the BDI logics.