Composition-Nominative Logics in Rigorous Development of Software Systems

We propose methods for solving the satisfiability and validity problems in many-sorted composition-nominative pure predicate logics (without functions and with equality). These logics are algebra-based logics of many-sorted partial predicates constructed in a semantic-syntactic style on the methodological basis that is common with programming; they can be considered as generalizations of traditional many-sorted logics on classes of partial predicates that do not have fixed arity. We show the reduction of the satisfiability problem to the same problem for many-sorted classical first-order pure predicate logic with equality. As validity is dual to satisfiability, the method proposed can be adopted to the validity problem. This enables us to use existent satisfiability and validity checking procedures developed for classical logic also for solving these problems in composition-nominative pure predicate logics with equality.

In their recent article “A Hierarchy of Classical and Paraconsistent Logics”, Eduardo Barrio, Federico Pailos and Damien Szmuc (BPS hereafter) present novel and striking results about meta-inferential validity in various three valued logics. In the process, they have thrown open the door to a hitherto unrecognized domain of non-classical logics with surprising intrinsic properties, as well as subtle and interesting relations to various familiar logics, including classical logic. One such result is that, for each natural number n, there is a logic which agrees with classical logic on tautologies, inferences, meta-inferences, meta-meta-inferences, meta-meta-...(n - 3 times)-meta-inferences, but that disagrees with classical logic on n + 1-meta-inferences. They suggest that this shows that classical logic can only be characterized by defining its valid inferences at all orders. In this article, I invoke some simple symmetric generalizations of BPS’s results to show that the problem is worse than they suggest, since in fact there are logics that agree with classical logic on inferential validity to all orders but still intuitively differ from it. I then discuss the relevance of these results for truth theory and the classification problem.

We discuss a scheme for defining and reasoning about partial recursive functions within a classical two-valued logic in which all terms denote. We show how a total extension of the partial function introduced by a recursive declaration may be axiomatized within a classical logic, and illustrate by an example the kind of reasoning that our scheme supports. By presenting a naive set-theoretic semantics, we show that the system we propose is logically consistent. Our work is motivated largely by the pragmatic issues arising from mechanical theorem proving – we discuss some of the practical benefits and limitations of our scheme for mechanical verification of software and hardware systems.

In this paper we investigate the use of Artificial Immune Systems' principles to cope with the satisfiability problem. We describe ClonSAT, a new iterative approach for solving the well known Maximum Satisfiability (Max-SAT) problem. This latter has been shown to be NP-hard if the number of variables per clause is greater than 3. The underlying idea is to harness the optimization capabilities of artificial clonal selection algorithm to achieve good quality solutions for MaxSAT problem. To foster the process, a local search has been used. The obtained results are very encouraging and show the feasibility and effectiveness of the proposed hybrid approach.

Boolean Satisfiability (SAT), the first problem proven to be NP-complete, is intractable on digital computers based on the von Neumann architecture. An efficient SAT solver can benefit many applications such as artificial intelligence, circuit design, and functional verification. Recently, a SAT solver approach based on a deterministic, continuous-time dynamical system (CTDS) was introduced [1]. This approach shows polynomial analog time-complexity on even the hardest k-SAT (k ≥ 3) problem instances, but at an energy cost dependent on exponentially growing auxiliary variables. This paper reports a novel analog hardware SAT solver, AC-SAT, implementing the CTDS via incorporating novel, analog circuit design ideas. AC-SAT is intended to be used as a co-processor and is programmable for handling different problem specifications. Furthermore, with its modular design, AC-SAT can be readily extended to solve larger size problems. SPICE simulation results show that AC-SAT can indeed solve the SAT problems, and it has speedup factors of ~10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> on even the hardest 3-SAT problems, when compared with a state-of-the-art SAT solver on digital computers.

The formulation of a Propositional Satisfiability (SAT) problem instance is vital to efficient solving. This has motivated research on preprocessing, and inprocessing techniques where reformulation of a SAT instance is interleaved with solving. Preprocessing and inprocessing are highly effective in extending the reach of SAT solvers, however they necessarily operate on the lowest level representation of the problem, the raw SAT clauses, where higher-level patterns are difficult and/or costly to identify. Our approach is different: rather than reformulate the SAT representation directly, we apply automated reformulations to a higher level representation (a constraint model) of the original problem. Common Subexpression Elimination (CSE) is a family of techniques to improve automatically the formulation of constraint satisfaction problems, which are often highly beneficial when using a conventional constraint solver. In this work we demonstrate that CSE has similar benefits when the reformulated constraint model is encoded to SAT and solved using a state-of-the-art SAT solver. In some cases we observe speed improvements of over 100 times.KeywordsConstraint ModelNegation Normal FormImplied ConstraintCommon Subexpression EliminationGlobal Cardinality ConstraintThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Linear search algorithms have been shown to be particularly effective for solving partial Maximum Satisfiability (MaxSAT) problem instances. These algorithms start by adding a new relaxation variable to each soft clause and solving the resulting formula with a SAT solver. Whenever a Model is found, a new constraint on the relaxation variables is added such that Models with a greater or equal value are excluded. However, if the problem instance has a large number of relaxation variables, then adding a new constraint over these variables can lead to the exploration of a much larger search space. This article proposes new algorithms to improve the performance of linear search algorithms for MaxSAT by using Models found by the SAT solver to partition the relaxation variables. These algorithms add a new constraint on a subset of relaxation variables, thus intensifying the search on that subspace. In addition, the proposed algorithms are further enhanced by using incremental approaches where learned clauses are kept between calls to the SAT solver. Experimental results show that Model-based algorithms can outperform a classic linear search algorithm in several problem instances. Moreover, incremental approaches further improve the performance of linear search and Model-based algorithms for MaxSAT. Overall, Model-based algorithms are competitive with state-of-the-art MaxSAT solvers, and can improve their performance when solving problem instances with a large number of soft clauses.

In this thesis, we introduce and examine four new temporal logic formalisms that can be used as specification languages for the automated verification of the reliability of hardware and software designs with respect to a desired behavior. The work is organized in two parts. In the first one, we reason about two logics for computations, GCTL* and MCTL*, which are useful to describe a correct execution of monolithic closed systems. In the second one, instead, we focus on two logics for strategies, SL and mATL*, which are useful to formalize several interesting properties about interactive plays in multi-entities systems modeled as multi-agent games. In the “Logics for Computations” part, we first study the immersion of the idea of graded quantifications into the temporal-logic framework. In first order logic, existential and universal quantifiers express the concepts of the existence of at least one individual object satisfying a formula, or that all individual objects satisfy a formula. In other logics, these quantifiers have been generalized to express that, for a given non-negative integer n, at least n or all but n individuals satisfy a particular formula. Here, we consider GCTL, a temporal logic with graded path quantifiers, which allows to describe properties like “there exist at least n different classes of computational fluxes in which a system reaches a predetermined state”, where the classes over paths are computed by means of a predetermined equivalence relation. More precisely, we uniformly extend the classic concept of graded quantifiers from states to paths, through the use of a concept of path equivalence with respect to a given path formula. About this logic, in particular, we study the expressiveness and succinctness relationships with respect to GµCalculus and the complexity of the satisfiability problem, which results to be ExpTime-Complete. This research is partially based on the works [BMM09] “Graded Computation Tree Logic” and [BMM10] “Graded Computation Tree Logic with Binary Coding” published, respectively, in the proceedings of the “IEEE Symposium on Logic in Computer Science, 2009” and “EACSL Annual Conference on Computer Science Logic, 2010”. Preliminary results can be also found in [Mog07]. Furthermore, we consider special quantifiers over substructures, which allow to select, using parametric criteria, small critical parts of a system to be successively verified. In literature, there are some attempts to define a logic that allows to modify the underlying structure under exam and then to verify on it some assigned property. However, as far as we know, none of them is able to select minimal submodels of a given property describing the criteria on which then execute the verification process. Here, we base our work on the search of a new operator that merges the concept of quantifiers on structures with that one derived by a generalization of the concept of pruning. The results of this work, is a class of three different extensions of CTL* with minimal model quantifiers, which we name MCTL*. Regarding these logics, we study several reductions among them, as well as the satisfiability problem that we prove to be highly undecidability, i.e., Σ_1^1-Hard, for two out of the three cases. This research is partially based on the work [MM09] “Branching-Time Temporal Logics with Minimal Model Quantifiers” published in the proceedings of the “International Conference on Developments in Language Theory, 2009”. In the “Logics for Strategies” part, we first study the problem of defining a new specification language through which it is possible to express several important properties of multi-entities systems that are neither expressible using classical monolithic temporal logics, such as CTL*, nor using two-agent-teams temporal logics, such as ATL*. In literature, we can found some proposal of logics that try to achieve this goal, but unfortunately, none of them succeeds completely on all the aspects. Among them, one of the most important attempts is CHP-SL, a logics in which one can use variables over strategies. However, this logic has a deep weakness, since it does not allow to describe games with more than two players and even two-players concurrent games. Here, we introduce SL, a logic with a syntax similar in some aspects to the first order logic, in which the strategies of the agent building the game are treated as first order objects on which we can quantify. This logic generalizes CHP-SL, by allowing the specification of the correct behavior of multi-agent concurrent games. In SL, for example, we are able to express very complex but useful Nash equilibria that are not expressible with CHP-SL. We enlighten that Nash equilibrium is one of the most important concepts in game theory. For the introduced logic, we solve two problems left open in the work on CHP-SL. Precisely, we show that the related model-checking problem is 2ExpTime-Complete, thus not harder of the same problem for several subsumed logics, while we prove that its satisfiability problem is highly undecidable, i.e., Σ_1^1-Hard. This research is partially based on the work [MMV10a] “Reasoning About Strategies” published in the proceedings of the “IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, 2010”. Finally, we consider the concept of relentful strategic reasoning, i.e., a formalism that expresses the ability of a strategy to be used not only to achieve a first given goal, but also to change its final goal in dependence of the history of the play. In the context of planning, memoryful quantification, i.e., quantification over computations that does not lose information about the past along the time, is one of the principal way to express the fact that a system is able to achieve a desired result, and shift to a different goal if some event happens. However, this kind of quantification was not considered before in the context of multi-agent planning. Here, we introduce mATL*, a fusion of the classic alternating temporal logic ATL* with memoryful quantification, with the aim of covering the previous idea. About this logics, we prove that, although it is equivalent to ATL*, it is exponentially more succinct. Nevertheless, we prove that both the model-checking and the satisfiability problems remain 2ExpTime-Complete, as for ATL*. This research is partially based on the work [MMV10b] “Relentful Strategic Reasoning in Alternating-Time Temporal Logic” published in the proceedings of the “International Conference on Logic for Programming Artificial Intelligence and Reasoning, 2010”.

We describe a method for introducing “partial functions” into ACL2, that is, functions not defined everywhere. The function “definitions” are actually admitted via the encapsulation principle: the new function symbol is constrained to satisfy the appropriate equation. This is permitted only when a witness function can be exhibited, establishing that the constraint is satisfiable. Of particular interest is the observation that every tail recursive definition can be witnessed in ACL2. We describe a macro that allows the convenient introduction of arbitrary tail recursive functions, and we discuss how such functions can be used to prove theorems about state machine models without reasoning about “clocks” or counting the number of steps until termination. Our macro for introducing “partial functions” also permits a variety of other recursive schemes, and we briefly illustrate some of them.

On the interpretability of arithmetic in temporal logic

The Tarskian classical relevant logic \(\mathsf {TR}\) arises from Tarski’s work on the foundations of the calculus of relations and on first-order logic restricted to finitely many variables, presented by Tarski and Givant their book, A Formalization of Set Theory without Variables, and summarized in the first nine sections. \(\mathsf {TR}\) is closely related to the well-known logic \(\mathsf {KR}\). Every formula of relevance logic has a corresponding sentence in Tarski’s extended first-order logic of binary relations with operators on the relation symbols. A formula is in \(\mathsf {TR}\) (by definition), or in \(\mathsf {KR}\) (by a theorem), if and only if its corresponding sentence can be proved in first-order logic, using at most four variables, from the assumptions that all binary relations are dense and, for \(\mathsf {TR}\), commute under composition, or, for \(\mathsf {KR}\), are symmetric. The vocabulary of \(\mathsf {TR}\) is the same as the classical relevant logic \(\mathsf {CR^*}\) proposed by Meyer and Routley but \(\mathsf {TR}\) properly contains \(\mathsf {CR^*}\). The frames characteristic for \(\mathsf {TR}\) are the ones that are characteristic for \(\mathsf {CR^*}\) and satisfy an extra frame condition. There are formulas in \(\mathsf {TR}\) (but not in \(\mathsf {CR^*}\)) that correspond to this frame condition and provide a counterexample to a theorem of T. Kowalski. The frames characteristic for \(\mathsf {TR}\), or \(\mathsf {KR}\), are the ones whose complex algebras are integral dense relation algebras that are commutative, or symmetric, respectively. For both classes, the number of isomorphism types grows like the number of isomorphism types of ternary relations. Asymptotic formulas are obtained for both classes. Similar results apply to a hierarchy of logics defined by the number of variables used in the first-order proofs of their corresponding sentences.

Modern satisfiability (SAT) solvers provide an efficient implementation of classical propositional logic. Their input language, however, is based on the conjunctive normal form (CNF) of propositional formulas. To use SAT solver technology in practice, a user must create the input clauses in one way or another. A typical approach is to write a procedural program that generates formulas on the basis of some input data relevant for the problem domain and translates them into CNF. In this paper, we propose a declarative approach where the intended clauses are specified in terms of rules in analogy to answer set programming (ASP). This allows the user to write first-order specifications for intended clauses in a schematic way by exploiting term variables. We develop a formal framework required to define the semantics of such specifications. Moreover, we provide an implementation harnessing state-of-the-art ASP grounders to accomplish the grounding step of clauses. As a result, we obtain a general-purpose clause-level grounding approach for SAT solvers. Finally, we illustrate the capabilities of our specification methodology in terms of combinatorial and application problems.

Despite the superior performance of large language models to generate natural language texts, it is hard to generate texts with correct logic according to a given task, due to the difficulties for neural models to capture strict logic from free-form texts. In this paper, we propose a novel graph-based language model, Logical-GLM, to extract strict logic from free-form texts and then infuse into language models. Specifically, we first capture information from natural language instructions and construct logical probability graphs that generally describe domains. Next, we generate logical skeletons to guide language model training, infusing domain knowledge into language models. At last, we alternately optimize the searching policy of graphs and language models until convergence. The experimental results show that Logical-GLM is both effective and efficient compared with traditional language models, despite using smaller-scale training data and fewer parameters. Our approach can generate instructional texts with more correct logic owing to the internalized domain knowledge. Moreover, the search of logical graphs reflects the inner mechanism of the language models, which improves the interpretability of black-box models.

We classify the computational complexity of the satisfiability, validity, and model-checking problems for propositional independence, inclusion, and team logic. Our main result shows that the satisfiability and validity problems for propositional team logic are complete for alternating exponential-time with polynomially many alternations.

In this paper, we present a Satisfiability Modulo Theory based (SMT-based) bounded model checking (BMC) method for Timed Real-Weighted Interpreted Systems and for the existential fragment of the Weighted Epistemic Computation Tree Logic. SMT-based bounded model checking consists in translating the existential model checking problem for a modal logic and for a model to the satisfiability problem of a quantifier-free first-order formula. We have implemented the SMT-BMC method and performed the BMC algorithm on Timed Weighted Generic Pipeline Paradigm benchmark. The preliminary experimental results demonstrate the feasibility of the method. To perform the experiments, we used the state of the art SMT-solver Z3.