Proof Search Research Articles

"My Person" or "A Person": CarePartner and Practitioner Perceptions Caring for Persons in Disordered Consciousness

<h3>Research Objectives</h3> Compare perceptions of carepartners (CP) and rehabilitation practitioners (RP) caring for a person in disordered consciousness (DoC) following brain injury. <h3>Design</h3> Qualitative, exploratory design. <h3>Setting</h3> Post-acute care (PAC) and community. <h3>Participants</h3> 16 CPs for persons in DoC. 21 RPs working in PAC. <h3>Interventions</h3> N/A. <h3>Main Outcome Measures</h3> N/A. <h3>Results</h3> CP and RP perceptions of personhood in persons in DoC are compared. The CP searching for "my person" through the recovery process and the RP searching for "a person". CPs described this through subthemes of looking for their old person ("Just like the old Katie",), searching for evidence their person is "in there" ("I am in here why can't I get to you") and the emergence of a "new person" ("he does the loudest laugh, I guess this is the new person"). In contrast RPs described their search for "a person". RPs search for scientific proof of "a person". They described this through subthemes of searching for consistent and purposeful behavior ("Is she truly consistent? Like 100%?"), cognitive dissonance associated with interpretation of observed behaviors ("Did I just see that?"), and the search for signs of consciousness ("it's not a black and white thing"). <h3>Conclusions</h3> CPs and RPs for persons in DoC have different world views through which they see the person. These worldviews can lead to challenges with perceptions of recovery. A shared understanding of "my person" and "a person" is essential to ensure CPs and RPs exchange information supportive to everyone's needs. <h3>Author(s) Disclosures</h3> There are no disclosures for any of the authors on this poster.

Concise outlines for a complex logic: a proof outline checker for TaDA

Modern separation logics allow one to prove rich properties of intricate code, e.g., functional correctness and linearizability of non-blocking concurrent code. However, this expressiveness leads to a complexity that makes these logics difficult to apply. Manual proofs or proofs in interactive theorem provers consist of a large number of steps, often with subtle side conditions. On the other hand, automation with dedicated verifiers typically requires sophisticated proof search algorithms that are specific to the given program logic, resulting in limited tool support that makes it difficult to experiment with program logics, e.g., when learning, improving, or comparing them. Proof outline checkers fill this gap. Their input is a program annotated with the most essential proof steps, just like the proof outlines typically presented in papers. The tool then checks automatically that this outline represents a valid proof in the program logic. In this paper, we systematically develop a proof outline checker for the TaDA logic, which reduces the checking to a simpler verification problem, for which automated tools exist. Our approach leads to proof outline checkers that provide substantially more automation than interactive provers, but are much simpler to develop than custom automatic verifiers.

Intelligent Tutoring Systems Authoring Tools For Nonprogrammer Authors: A Systematic Review

Many studies have been shown that Intelligent Tutoring Systems (ITSs) contribute to a remarkable improvement in the educational outcomes of students. Several authoring tools have been developed to contribute to making ITSs easier to build and widespread. However, there is still a lacks of current understanding in the ITSs community on how authoring tools could support nonprogrammer authors to build ITSs. This study uses a systematic literature review (SLR) method to identify primary empirical studies for nonprogrammers regarding the usage of ITSs authoring tools. The concern of this study is summarizing and analyzing the state of the art of the development of authoring tools. This study identifies the ITSs components and types that can be authored, the technologies used, and also search for proof of the advantages of ITSs authoring tools as well.

Proof Theory of Skew Non-Commutative MILL

Monoidal closed categories naturally model NMILL, non-commutative multiplicative intuitionistic linear logic: the monoidal unit and tensor interpret the multiplicative verum and conjunction; the internal hom interprets linear implication. In recent years, the weaker notion of (left) skew monoidal closed category has been proposed by Ross Street, where the three structural laws of left and right unitality and associativity are not required to be invertible, they are merely natural transformations with a specific orientation. A question arises: is it possible to find a logic which is naturally modelled by skew monoidal closed categories? We answer positively by introducing a cut-free sequent calculus for a skew version of NMILL that is a presentation of the free skew monoidal closed category. We study the proof-theoretic semantics of the sequent calculus by identifying a calculus of derivations in normal form, obtained from an adaptation of Andreoli's focusing technique to the skew setting. The resulting focused sequent calculus peculiarly employs a system of tags for keeping track of new formulae appearing in the antecedent and appropriately reducing non-deterministic choices in proof search. Focusing solves the coherence problem for skew monoidal closed categories by exhibiting an effective procedure for deciding equality of maps in the free such category.

A Bi-Directional Extensible Interface Between Lean and Mathematica

We implement a user-extensible ad hoc connection between the Lean proof assistant and the computer algebra system Mathematica. By reflecting the syntax of each system in the other and providing a flexible interface for extending translation, our connection allows for the exchange of arbitrary information between the two systems. We show how to make use of the Lean metaprogramming framework to verify certain Mathematica computations, so that the rigor of the proof assistant is not compromised. We also use Mathematica as an untrusted oracle to guide proof search in the proof assistant and interact with a Mathematica notebook from within a Lean session. In the other direction, we import and process Lean declarations from within Mathematica. The proof assistant library serves as a database of mathematical knowledge that the CAS can display and explore.

Calculi, countermodel generation and theorem prover for strong logics of counterfactual reasoning

AbstractWe present hypersequent calculi for the strongest logics in Lewis’ family of conditional systems, characterized by uniformity and total reflexivity. We first present a non-standard hypersequent calculus, which allows a syntactic proof of cut elimination. We then introduce standard hypersequent calculi, in which sequents are enriched by additional structures to encode plausibility formulas and diamond formulas. Proof search using these calculi is terminating, and the completeness proof shows how a countermodel can be constructed from a branch of a failed proof search. We then describe tuCLEVER, a theorem prover that implements the standard hypersequent calculi. The prover provides a decision procedure for the logics, and it produces a countermodel in case of proof search failure. The prover tuCLEVER is inspired by the methodology of leanTAP and it is implemented in Prolog. Preliminary experimental results show that the performances of tuCLEVER are promising.1

Certifying derivation of state machines from coroutines

One of the biggest implementation challenges in security-critical network protocols is nested state machines. In practice today, state machines are either implemented manually at a low level, risking bugs easily missed in audits; or are written using higher-level abstractions like threads, depending on runtime systems that may sacrifice performance or compatibility with the ABIs of important platforms (e.g., resource-constrained IoT systems). We present a compiler-based technique allowing the best of both worlds, coding protocols in a natural high-level form, using freer monads to represent nested coroutines , which are then compiled automatically to lower-level code with explicit state. In fact, our compiler is implemented as a tactic in the Coq proof assistant, structuring compilation as search for an equivalence proof for source and target programs. As such, it is straightforwardly (and soundly) extensible with new hints, for instance regarding new data structures that may be used for efficient lookup of coroutines. As a case study, we implemented a core of TLS sufficient for use with popular Web browsers, and our experiments show that the extracted Haskell code achieves reasonable performance.

Spreads and Packings of PG(3,2), Formally!

We study how to formalize in the Coq proof assistant the smallest projective space PG(3,2). We then describe formally the spreads and packings of PG(3,2), as well as some of their properties. The formalization is rather straightforward, however as the number of objects at stake increases rapidly, we need to exploit some symmetry arguments as well as smart proof techniques to make proof search and verification faster and thus tractable using the Coq proof assistant. This work can be viewed as a first step towards formalizing projective spaces of higher dimension, e.g. PG(4,2), or larger order, e.g. PG(3,3).

Editorial introduction to J.UCS special issue Challenges for Smart Environments – Human-Centered Computing, Data Science, and Ambient Intelligence I

Editorial introduction to J.UCS special issue Challenges for Smart Environments – Human-Centered Computing, Data Science, and Ambient Intelligence I

INFORMATION IN PROPOSITIONAL PROOFS AND ALGORITHMIC PROOF SEARCH

AbstractWe study from the proof complexity perspective the (informal) proof search problem (cf. [17, Sections 1.5 and 21.5]): •Is there an optimal way to search for propositional proofs?We note that, as a consequence of Levin’s universal search, for any fixed proof system there exists a time-optimal proof search algorithm. Using classical proof complexity results about reflection principles we prove that a time-optimal proof search algorithm exists without restricting proof systems iff a p-optimal proof system exists.To characterize precisely the time proof search algorithms need for individual formulas we introduce a new proof complexity measure based on algorithmic information concepts. In particular, to a proof system P we attach information-efficiency function $i_P(\tau )$ assigning to a tautology a natural number, and we show that: • $i_P(\tau )$ characterizes time any P-proof search algorithm has to use on $\tau $ ,•for a fixed P there is such an information-optimal algorithm (informally: it finds proofs of minimal information content),•a proof system is information-efficiency optimal (its information-efficiency function is minimal up to a multiplicative constant) iff it is p-optimal,•for non-automatizable systems P there are formulas $\tau $ with short proofs but having large information measure $i_P(\tau )$ .We isolate and motivate the problem to establish unconditional super-logarithmic lower bounds for $i_P(\tau )$ where no super-polynomial size lower bounds are known. We also point out connections of the new measure with some topics in proof complexity other than proof search.

Human-Centered Automated Proof Search

Human-Centered Automated Proof Search

A coinductive approach to proof search through typed lambda-calculi

In reductive proof search, proofs are naturally generalized by solutions, comprising all (possibly infinite) structures generated by locally correct, bottom-up application of inference rules. We propose a rather natural extension of the Curry-Howard paradigm of representation, from proofs to solutions: to represent solutions by (possibly infinite) terms of the coinductive variant of the typed lambda-calculus that represents proofs. We take this as a starting point for a new, comprehensive approach to proof search; our case study is proof search in the sequent calculus LJT for intuitionistic implication logic. A second, finitary representation is proposed, comprising a syntax of lambda-terms extended with a formal greatest fixed point, and a type system that can be seen as a logic of coinductive proofs. In the finitary system, fixed-point variables enjoy a relaxed form of binding that allows the detection of cycles through the type system. Formal sums are used in both representations to express alternatives in the search process, so that not only individual solutions but actually solution spaces are expressed. Moreover, formal sums are used in the coinductive syntax to define “decontraction” (contraction bottom-up)—an operation whose theory we initiate in this paper. A semantics is defined assigning a coinductive lambda-term to each finitary term, making use of decontraction as a semantical match to the relaxed form of binding of fixed-point variables present in the finitary system. The main result is the existence of an equivalent finitary representation for any full solution space expressed coinductively. This result is the main ingredient in the proof that our logic of coinductive proofs is sound and complete with respect to the coinductive semantics. These results are the foundation for an original approach to proof search, where the search builds the finitary representation of the full solution space, and the a posteriori analysis typically consisting in applying a syntax-directed procedure or function. The paper illustrates the potential of the methodology to the study of proof search and inhabitation problems in the simply-typed lambda-calculus, reviewing results detailed elsewhere, and including new results that obtain extensive generalizations of the so-called monatomic theorem.

Neighbourhood semantics and labelled calculus for intuitionistic infinitary logic

Abstract Neighbourhood semantics for intuitionistic logic extended with countable conjunctions and disjunctions is introduced and shown equivalent to topological semantics, with an indirect completeness proof as payoff. The new semantics is used to obtain a labelled sequent calculus with good structural properties. In particular, admissibility of weakening and contraction, invertibility with preservation of height for each rule and cut elimination are shown. Finally, a direct Tait–Schütte–Takeuti style form of completeness via the extraction of a countermodel from a failed proof search is proved.

The undecidability of proof search when equality is a logical connective

One proof-theoretic approach to equality in quantificational logic treats equality as a logical connective: in particular, term equality can be given both left and right introduction rules in a sequent calculus proof system. We present a particular example of this approach to equality in a first-order logic setting in which there are no predicate symbols (apart from equality). After we illustrate some interesting applications of this logic, we show that provability in this logic is undecidable.

Ceptre: A Language for Modeling Generative Interactive Systems

We present a rule specification language called Ceptre,intended to enable rapid prototyping for experimental game mechanics, especially in domains that depend on procedural generation and multi-agent simulation. Ceptre can be viewed as an explication of a new methodology for understanding games based on linear logic, a formal logic concerned with resource usage. We present a correspondence between gameplay and proof search in linear logic, building on prior work on generating narratives. In Ceptre, we introduce the ability to add interactivity selectively into a generative model, enabling inspection of intermediate states for debugging and exploration as well as a means of play. We claim that this methodology can support game designers and researchers in designing, anaylzing, and debugging the core systems of their work in generative, multi-agent gameplay. To support this claim, we provide two case studies implemented in Ceptre, one from interactive narrative and one from a strategy-like domain.

A Deep Reinforcement Learning Approach to First-Order Logic Theorem Proving

Automated theorem provers have traditionally relied on manually tuned heuristics to guide how they perform proof search. Deep reinforcement learning has been proposed as a way to obviate the need for such heuristics, however, its deployment in automated theorem proving remains a challenge. In this paper we introduce TRAIL, a system that applies deep reinforcement learning to saturation-based theorem proving. TRAIL leverages (a) a novel neural representation of the state of a theorem prover and (b) a novel characterization of the inference selection process in terms of an attention-based action policy. We show through systematic analysis that these mechanisms allow TRAIL to significantly outperform previous reinforcement-learning-based theorem provers on two benchmark datasets for first-order logic automated theorem proving (proving around 15% more theorems).

Interpolating bit-vector formulas using uninterpreted predicates and Presburger arithmetic

The inference of program invariants over machine arithmetic, commonly called bit-vector arithmetic, is an important problem in verification. Techniques that have been successful for unbounded arithmetic, in particular Craig interpolation, have turned out to be difficult to generalise to machine arithmetic: existing bit-vector interpolation approaches are based either on eager translation from bit-vectors to unbounded arithmetic, resulting in complicated constraints that are hard to solve and interpolate, or on bit-blasting to propositional logic, in the process losing all arithmetic structure. We present a new approach to bit-vector interpolation, as well as bit-vector quantifier elimination (QE), that works by lazy translation of bit-vector constraints to unbounded arithmetic. Laziness enables us to fully utilise the information available during proof search (implied by decisions and propagation) in the encoding, and this way produce constraints that can be handled relatively easily by existing interpolation and QE procedures for Presburger arithmetic. The lazy encoding is complemented with a set of native proof rules for bit-vector equations and non-linear (polynomial) constraints, this way minimising the number of cases a solver has to consider. We also incorporate a method for handling concatenations and extractions of bit-vector efficiently.

Interpolation for intermediate logics via injective nested sequents

AbstractWe introduce a novel, semantically inspired method of constructing nested sequent calculi for propositional intermediate logics. Applying recently developed methods for proving Craig interpolation to these nested sequent calculi, we obtain constructive proofs of the interpolation property for most non-trivial interpolable intermediate logics, as well as Lyndon interpolation for Gödel logic. Finally, we provide a prototype implementation combining proof search and countermodel construction.

Uniform labelled calculi for preferential conditional logics based on neighbourhood semantics

Abstract The preferential conditional logic $ \mathbb{PCL} $, introduced by Burgess, and its extensions are studied. First, a natural semantics based on neighbourhood models, which generalizes Lewis’ sphere models for counterfactual logics, is proposed. Soundness and completeness of $ \mathbb{PCL} $ and its extensions with respect to this class of models are proved directly. Labelled sequent calculi for all logics of the family are then introduced. The calculi are modular and have standard proof-theoretical properties, the most important of which is admissibility of cut that entails a syntactic proof of completeness of the calculi. By adopting a general strategy, root-first proof search terminates, thereby providing a decision procedure for $ \mathbb{PCL} $ and its extensions. Finally, semantic completeness of the calculi is established: from a finite branch in a failed proof attempt it is possible to extract a finite countermodel of the root sequent. The latter result gives a constructive proof of the finite model property of all the logics considered.

Extensional Higher-Order Paramodulation in Leo-III

Leo-III is an automated theorem prover for extensional type theory with Henkin semantics and choice. Reasoning with primitive equality is enabled by adapting paramodulation-based proof search to higher-order logic. The prover may cooperate with multiple external specialist reasoning systems such as first-order provers and SMT solvers. Leo-III is compatible with the TPTP/TSTP framework for input formats, reporting results and proofs, and standardized communication between reasoning systems, enabling e.g. proof reconstruction from within proof assistants such as Isabelle/HOL. Leo-III supports reasoning in polymorphic first-order and higher-order logic, in all normal quantified modal logics, as well as in different deontic logics. Its development had initiated the ongoing extension of the TPTP infrastructure to reasoning within non-classical logics.