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A Machine-Oriented Logic based on the Resolution Principle.
... Inference plays an important role in the context of Max-SAT solving (Li et al., 2007;Narodytska & Bacchus, 2014;Abramé & Habet, 2014) and this has led to an increasing interest in studying proof systems for Max-SAT in the literature (Larrosa & Heras, 2005;Bonet, Levy, & Manyà, 2006Larrosa & Rollon, 2020a, 2020bBonet & Levy, 2020;Filmus, Mahajan, Sood, & Vinyals, 2020;Cherif, Habet, & Py, 2022). In particular, Max-SAT resolution (Larrosa & Heras, 2005;Bonet et al., 2006Bonet et al., , 2007 is one of the first known complete systems for Max-SAT and is a natural extension of the resolution rule (Robinson, 1965) used in the context of SAT. Max-SAT resolution proofs are more constrained than their SAT counterparts as the premise clauses are replaced by the conclusions when applying Max-SAT resolution. ...
... On the other hand, to prove that a CNF formula is unsatisfiable, we need to refute the existence of a model. A well-known SAT refutation system is based on an inference rule for SAT called resolution (Robinson, 1965). The resolution rule, defined below, deduces a clause called resolvent which can be added to the formula. ...
... Definition 1 (Resolution (Robinson, 1965)). Given two clauses c 1 = (x ∨ A) and c 2 = (x ∨ B), the resolution rule deduces a third additional clause as follows: ...
Current Max-SAT solvers are able to efficiently compute the optimal value of an input instance but they do not provide any certificate of its validity. In this paper, we present a tool, called MS-Builder, which generates certificates for the Max-SAT problem in the particular form of a sequence of equivalence-preserving transformations. To generate a certificate, MS-Builder iteratively calls a SAT oracle to get a SAT resolution refutation which is handled and adapted into a sound refutation for Max-SAT. In particular, we prove that the size of the computed Max-SAT refutation is linear with respect to the size of the initial refutation if it is semi-read-once, tree-like regular, tree-like or semi-tree-like. Additionally, we propose an extendable tool, called MS-Checker, able to verify the validity of any Max-SAT certificate using Max-SAT inference rules. Both tools are evaluated on the unweighted and weighted benchmark instances of the 2020 Max-SAT Evaluation.
... The laws, the axioms, and, the processes, of such an Algebra will be identical in their whole extent with the laws, the axioms, and, the processes of an Algebra of Logic. [7,[37][38]. ...
... A clausal sentence is either a literal, an atomic sentence or a negation of an atomic sentence, or a disjunction of literals, e.g., p, ¬p. If p and q are logical constants, then the following three are clausal sentences: p, ¬p, ¬p ∨ q (for a complete discussion, see [37]). ...
... First-order logic programming [Apt, 1990, Lloyd, 1994] is a declarative paradigm based on formal logic, that views computing as a procedure to find a proof for a logical theory, i.e: a set of logical sentences. A program encodes this logical theory and the proof is typically produced -if finding one is possible -using the resolution principle [Robinson, 1965], which provides a single rule of deductive inference that is sound and complete for proving statements constructed using the syntax of logic programs. The most CHAPTER 2. BACKGROUND NOTIONS widespread syntax for logic programming is known as Horn logic [Horn, 1951], a Turing complete [Tärnlund, 1977] restricted (in the form of rules it allows) version of first-order predicate logic. ...
... Of course, generating T ∞ P is computationally expensive and not the only method to determine if P |= A. Many strategies and algorithms have been developed in the literature, most notably following the resolution principle [Robinson, 1965]. The most popular variation of the resolution principle is called selective linear definite (SLD) resolution [Kowalski and Kuehner, 1971] and is the one used by Prolog. ...
Machine ethics is an uprising sub-field of artificial intelligence fueled by the interest and concerns about the deployment of automated agents in our everyday life. As these agents gain independence from human intervention and make decisions with possible impact on human welfare, real concerns are rising across domains.Due to those reasons, various approaches have been proposed to imbue automated agents with ethical considerations. Several research currents have developed models stemming from psychology and philosophy in an effort to adapt decision-making algorithms to consider ethical values so that the impact of agents on people is bounded and guided by these notions.Most of these approaches consist of either reasoning and applying a set of well-known ethical restrictions, also known as principles (top-down), or inferring them based on carefully crafted datasets through learning algorithms (bottom-up).In this thesis, we look at the problem of implementing these ethical principles in the context of tasks involving sequences of interdependent decisions, i.e: automated planning. We show how certain notions can be modeled using preference-based frameworks, as in top-down approaches, and how these preferences can be inferred from a corpus of data like bottom-up methodologies, to develop a hybrid approach that can be applied to planning problems. An implementation for each facet of our approach is provided in order to test our ideas in practical scenarios.
... The notions of read-once, tree-like and dag-like refutations also apply to CNF formulas [4]. In case of CNF formulas refutations are typically studied in the resolution refutation system [22]. It is important to note that both tree-like and dag-like refutations are complete in that every unsatisfiable CNF formula and linear system has a tree-like refutation and a dag-like refutation. ...
... 19: Φ ← Make-2CNF (U, d). 20: if (Φ is satisfiable) then 21: v ← satisfying assignment to Φ. 22: else Φ has a resolution refutation R. 23: return (The integer refutation corresponding to R (see Sect. 6)) 24: for i = 1 . . . n do 25: if d i ∈ Z then 26: y i ← d i . ...
This paper is concerned with the design and analysis of a bit-scaling based algorithm for the problem of checking integer feasibility in a system of unit two variable per inequality (UTVPI) constraints (IF). The insights developed during algorithm design result in new techniques for extracting refutations of integer feasibility in such systems. Recall that a UTVPI constraint is a linear constraint of the form: \(a_i\cdot x_i+a_j \cdot x_j \le b_{ij}\), where \(a_i, a_j \in \{0,1,-1\}\) and \(b_{ij} \in {\mathbb {Z}}\). These constraints arise in a number of application domains including but not limited to program verification (array bounds checking and abstract interpretation), operations research (packing and covering), and logic programming. Over the years, several algorithms have been proposed for the IF problem. Most of these algorithms are based on two inference rules, viz. the transitive rule and the tightening rule. None of these algorithms are bit-scaling. In other words, the running times of these algorithms are parameterized only by the number of variables and the number of constraints in the UTVPI constraint system (UCS) and not by the sizes of input constants. We introduce a novel algorithm for the IF problem, which is based on a collection of new insights. These insights are used to design a new bit-scaling algorithm for IF that runs in \(O(\sqrt{n}\cdot m \cdot \log _2 C)\) time, where n denotes the number of variables, m denotes the number of constraints, and C denotes the absolute value of the most negative constant defining the UCS. An interesting consequence of our research is the development of techniques for extracting refutations of integer infeasibility in UCSs that are linearly feasible. If the UCS is linearly feasible, then our algorithm creates a 2CNF formula. The UCS has an integer refutation (i.e., does not have a lattice point) if and only if the created 2CNF formula has a resolution refutation (i.e., is unsatisfiable).
... Figure 13 presents a unification algorithm U , which takes a constraint and produces a substitution θ and a pattern π . The algorithm can be understood as extending Robinson's unification algorithm (Robinson, 1965) to handle variational types and dynamic types and to support error tolerance. To support error tolerance, the unification not only returns a substitution but also a typing pattern. ...
... Int. Case (e) unifies two static types and is delegated to Robinson's unification algorithm (Robinson, 1965). Case (f) unifies two function types by unifying their respective argument and return types. ...
Gradual typing allows programs to enjoy the benefits of both static typing and dynamic typing. While it is often desirable to migrate a program from more dynamically typed to more statically typed or vice versa, gradual typing itself does not provide a way to facilitate this migration. This places the burden on programmers who have to manually add or remove type annotations. Besides the general challenge of adding type annotations to dynamically typed code, there are subtle interactions between these annotations in gradually typed code that exacerbate the situation. For example, to migrate a program to be as static as possible, in general, all possible combinations of adding or removing type annotations from parameters must be tried out and compared. In this paper, we address this problem by developing migrational typing , which efficiently types all possible ways of replacing dynamic types with fully static types for a gradually typed program. The typing result supports automatically migrating a program to be as static as possible or introducing the least number of dynamic types necessary to remove a type error. The approach can be extended to support user-defined criteria about which annotations to modify. We have implemented migrational typing and evaluated it on large programs. The results show that migrational typing scales linearly with the size of the program and takes only 2–4 times longer than plain gradual typing.
... The following proposition is known as the resolution technique in the literature, which was first proved in (Robinson 1965), and then used in many SAT algorithms. Definition 1. (Resolution on a variable) Let F be a CNFformula containing a variable x. ...
... We may always use F \x to denote the CNF-formula after resolving a variable x in F . Proposition 1. (Robinson 1965) Let F be a CNF-formula containing a variable x and F \x be the CNF-formula after resolving on variable x. Then F has a satisfying assignment if and only if F \x does. ...
We show that the CNF satisfiability problem can be solved O^*(1.2226^m) time, where m is the number of clauses in the formula, improving the known upper bounds O^*(1.234^m) given by Yamamoto 15 years ago and O^*(1.239^m) given by Hirsch 22 years ago. By using an amortized technique and careful case analysis, we successfully avoid the bottlenecks in previous algorithms and get the improvement.
... The Davis-Putnam algorithm [Davis et al. (1960)] and DPLL backtracking algorithm [Davis et al. (1962)] were among the earliest effective algorithms for propositional knowledge bases. Thereafter Robinson developed the full resolution rule [Robinson (1965)]. Due to the close relation between propositional inference and the satisfiability problem (SAT), all the algorithms developed for SAT are actually working for propositional inference either [Gu et al. (1997)]. ...
... Yani, tümevarım çıkarım kuralları tümdengelim çıkarım kurallarının tersi olarak elde edilebilir. Bu düşünceyle, (Muggleton ve Buntine, 1988) ters çözümlemeyi, (Robinson, 1965) tarafından bulunan bir tümdengelimli çıkarım kuralı olan çözümlemenin (resolotion) tersi olarak sunmuşlardır. Ters çözümleme, günümüzde aşağıdan yukarıya TMP"de öne çıkan bir genelleştirme operatörüdür. ...
Word Sense Disambiguation (WSD) is one of the important processes needed
for natural language processing applications and is defined as determining the sense of a
multi-sense word in a given context. The aim of this thesis is to report on the
performance results achieved by applying some WSD approaches to Turkish texts and
present the evaluations made using results.
In the study, firstly an unsupervised corpus based WSD application developed
with collocation knowledge is presented and then its results are evaluated. Afterwards,
an alternative WSD application developed considering the insufficiency of in the
performance results achieved in that application is accounted for in detail. This latter
application rests on Inductive Logic Programming (ILP), which is a method that
circumvents the incapability of traditional machine learning approaches in employing
background knowledge and making deductive inferences. To this effect, the topic of ILP
is given a detailed explanation and its applicability to WSD is demonstrated with
empirical results obtained using Turkish data.
... The origin of unification theory is usually attributed to Julia Robinson [50]. The classical syntactic unification problem is as follows: given two term s, t (built from function symbols and variables), find a unifier for them, that is, a uniform replacement of the variables occurring in s and t by other terms that makes s and t identical. ...
In this paper, we study projective algebras in varieties of (bounded) commutative integral residuated lattices. We make use of a well-established construction in residuated lattices, the ordinal sum, and the order property of divisibility. Via the connection between projective and splitting algebras, we show that the only finite projective algebra in \(\mathsf {{FL}_{ew}}\) is the two-element Boolean algebra. Moreover, we show that several interesting varieties have the property that every finitely presented algebra is projective, such as locally finite varieties of hoops. Furthermore, we show characterization results for finite projective Heyting algebras, and finitely generated projective algebras in locally finite varieties of bounded hoops and BL-algebras. Finally, we connect our results with the algebraic theory of unification.
... Resolution, which is a way to tell whether a propositional formula is satisfiable and to prove that a first-order formula is unsatisfiable, is a rule of inference leading to a refutation theorem-proving technique for sentences in CPL and FOL. It can be traced back to (Davis & Putnam, 1960), and then improved in 1965 by John Alan Robinson (Robinson, 1965). ...
Computation tree logic (CTL) is an essential specification language in the field of formal verification. In systems design and verification, it is often important to update existing knowledge with new attributes and subtract the irrelevant content while preserving the given properties on a known set of atoms. Under the scenario, given a specification, the weakest sufficient condition (WSC) and the strongest necessary condition (SNC) are dual concepts and very informative in formal verification. In this article, we generalize our previous results (i.e., the decomposition, homogeneity properties, and the representation theorem) on forgetting in bounded CTL to the unbounded one. The cost we pay is that, unlike the bounded case, the result of forgetting in CTL may no longer exist. However, SNC and WSC can be obtained by the new forgetting machinery we are presenting. Furthermore, we complement our model-theoretic approach with a resolution-based method to compute forgetting results in CTL. This method is currently the only way to compute forgetting results for CTL and temporal logic. The method always terminates and is sound. That way, we set up the resolution-based approach for computing WSC and SNC in CTL.
... The challenge of converting data into simple and humanly comprehensible logic has been addressed in many areas of previous research. In the early 1960s, work on formal logic led to the inception of logical programming and rule-learning algorithms [1][2][3][4][5][6][7][8]. The latter-including algorithms such as Corels [9], Slipper [10], Skope-Rules [11], RuleFit [12], LRI [13], MLRules [14], and more-often rely on greedy approaches to extract short Boolean expressions from more complex models (such as large decision trees). ...
Artificial intelligence and machine learning have demonstrated remarkable results in science and applied work. However, present AI models, developed to be run on computers but used in human-driven applications, create a visible disconnect between AI forms of processing and human ways of discovering and using knowledge. In this work, we introduce a new concept of “Human Knowledge Models” (HKMs), designed to reproduce human computational abilities. Departing from a vast body of cognitive research, we formalized the definition of HKMs into a new form of machine learning. Then, by training the models with human processing capabilities, we learned human-like knowledge, that humans can not only understand, but also compute, modify, and apply. We used several datasets from different applied fields to demonstrate the advantages of HKMs, including their high predictive power and resistance to noise and overfitting. Our results proved that HKMs can efficiently mine knowledge directly from the data and can compete with complex AI models in explaining the main data patterns. As a result, our study reveals the great potential of HKMs, particularly in the decision-making applications where “black box” models cannot be accepted. Moreover, this improves our understanding of how well human decision-making, modeled by HKMs, can approach the ideal solutions in real-life problems.
... As a consequence, new complete inference systems for MaxSAT have had to be defined [20]. They are MaxSAT extensions of either the resolution rule [33] or semantic tableaux [11,15,34]. ...
We define a new MaxSAT tableau calculus based on resolution. Given a multiset of propositional clauses ϕ, we prove that the calculus is sound in the sense that if the minimum number of contradictions derived among the branches of a completed tableau for ϕ is m, then the minimum number of unsatisfied clauses in ϕ is m. We also prove that it is complete in the sense that if the minimum number of unsatisfied clauses in ϕ is m, then the minimum number of contradictions among the branches of any completed tableau for ϕ is m. Moreover, we describe how to extend the proposed calculus to solve Weighted Partial MaxSAT.
... The most widely used propositional proof system is RES (Robinson 1965). A proof in RES is a sequence of clauses C 1 , . . . ...
Modern complete SAT solvers almost uniformly implement variations of the clause learning framework introduced by Grasp and Chaff. The success of these solvers has been theoretically explained by showing that the clause learning framework is an implementation of a proof system which is as poweful as resolution. However, exponential lower bounds are known for resolution, which suggests that significant advances in SAT solving must come from implementations of more powerful proof systems. We present a clause learning SAT solver that uses extended resolution. It is based on a restriction of the application of the extension rule. This solver outperforms existing solvers on application instances from recent SAT competitions as well as on instances that are provably hard for resolution.
... This iteration continues until a clause is generated (typically the empty clause for refutation theorem proving) that signals a proof has been found, the set of clauses is saturated, or a timeout is reached. For more details on saturation (Robinson 1965) and saturation-calculi, we refer the reader to (Bachmair and Ganzinger 1998). ...
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).
... Clausal programs are sets of clauses. Robinson (1965) showed that a single rule of inference (the resolution principle) is both sound and refutation complete for clausal logic. However, reasoning about full clausal logic is computationally expensive (Nienhuys-Cheng & Wolf, 1997). ...
Inductive logic programming (ILP) is a form of machine learning. The goal of ILP is to induce a hypothesis (a set of logical rules) that generalises training examples. As ILP turns 30, we provide a new introduction to the field. We introduce the necessary logical notation and the main learning settings; describe the building blocks of an ILP system; compare several systems on several dimensions; describe four systems (Aleph, TILDE, ASPAL, and Metagol); highlight key application areas; and, finally, summarise current limitations and directions for future research.
... Robinson created the original unification algorithm [12]. The significance of this algorithm was described by [8] with: ...
The Robinson unification algorithm has exponential worst case behavior. This triggered the development of (semi-)linear versions around 1976 by Martelli and Montanari as well as by Paterson and Wegman (J Comput Syst Sci 16(2):158–167, 1978, https://doi.org/10.1016/0022-0000(78)90043-0). Another version emerged by Baader and Snyder around 2001. While these versions are distinctly faster on larger input pairs, the Robinson version still does better than them on small-sized inputs. This paper describes yet another (semi-)linear version that is faster and challenges also the Robinson version on small-sized inputs. All versions have been implemented and compared against each other on different types and sizes of input pairs.
... The main challenge of using either the chase algorithm [2,67], or resolution [81] or semantic tableau [50] to develop satisfiability checking procedures is to ensure that the procedure terminates [21,35,36,58,59,62,63,70]. In developing resolution-based decision procedure the key issue is to ensure a finitely bounded search space. This can be achieved by defining a finitely bounded clausal class to which any BCQ problem can be mapped to and all inferences produce conclusions that belong to this class. ...
Answering Boolean conjunctive query over logical constraints is an essential problem in knowledge representation. Other problems in computer science such as constraint satisfaction and homomorphism problems can also be seen as Boolean conjunctive query answering problems. This paper develops saturation-based Boolean conjunctive query answering and rewriting procedures for the guarded, the loosely guarded and the clique guarded fragments. We improve existing resolution-based decision procedures for the guarded and the loosely guarded fragments, and devise a saturation-based approach deciding Boolean conjunctive query answering problems for the guarded, the loosely guarded and the clique guarded fragments. Based on the saturation-based query answering procedure, we introduce a novel saturation-based query rewriting setting that aims to back-translate the saturated clausal set derived from saturation-based query answering procedures, to a (Skolem-symbol-free) first-order formula, and devise a saturation-based query rewriting procedures for all these guarded fragments. Unlike mainstream query answering and rewriting approaches, our procedures derive a compact saturation that is reusable even if the data changes. This paper lays the theoretical foundations for the first practical Boolean conjunctive query answering and the first saturation-based Boolean conjunctive query rewriting procedures for the guarded, the loosely guarded and the clique guarded fragments.
... Unification is a key mechanism in resolution [41] and paramodulation-based [36] theorem proving. Since Plotkin's work [40] on equational unification, i.e., E-unification modulo an equational theory E, it is widely used for increased effectiveness. ...
Equational unification and matching are fundamental mechanisms in many automated deduction applications. Supporting them efficiently for as wide as possible a class of equational theories, and in a typed manner supporting type hierarchies, benefits many applications; but this is both challenging and nontrivial. We present Maude 3.2’s efficient support of these features as well as of symbolic reachability analysis of infinite-state concurrent systems based on them.
Data warehouses have demonstrated their applicability in numerous application fields such as agriculture, the environment and health. This paper proposes a general framework for defining a data warehouse and its aggregations using logic programming. The objective is to show that data managers can easily express, in Prolog, traditional data warehouse queries and combine data aggregation operations with other advanced Prolog features. It is shown that this language provides advanced features to aggregate information in an in-memory database. This paper targets data managers; it shows them the direct writing of data warehouse queries in Prolog using an easily understandable syntax. The queries are not necessarily in an optimal form from a processing point of view, but a data manager can easily use or write them.
The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: weather it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic. The main argument consists in the contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. Thus, the pair of arithmetic and set are similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate: correspondingly, by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. The axiom of choice transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. The Gödel incompleteness statement relies on the contradiction of the axioma of induction and infinity.
The semantic foundations for logic programming are usually separated into two different approaches. The operational semantics, which uses SLD-resolution, the proof method that computes answers in logic programming, and the declarative semantics, which sees logic programs as formulas and its semantics as models. Here, we define a new operational semantics called TSLD-resolution, which stands for Typed SLD-resolution, where we include a value “wrong”, that corresponds to the detection of a type error at run-time. For this we define a new typed unification algorithm. Finally we prove the correctness of TSLD-resolution with respect to a typed declarative semantics.KeywordsLogic programmingOperational semanticsTypes
The use of the Ethereum blockchain platform [17] has experienced an enormous growth since its very first transaction back in 2015 and, along with it, the verification and optimization of the programs executed in the blockchain (known as Ethereum smart contracts) have raised considerable interest within the research community.
A clause C is syntactically relevant in some clause set N , if it occurs in every refutation of N . A clause C is syntactically semi-relevant, if it occurs in some refutation of N . While syntactic relevance coincides with satisfiability (if C is syntactically relevant then $$N\setminus \{C\}$$ N \ { C } is satisfiable), the semantic counterpart for syntactic semi-relevance was not known so far. Using the new notion of a conflict literal we show that for independent clause sets N a clause C is syntactically semi-relevant in the clause set N if and only if it adds to the number of conflict literals in N . A clause set is independent, if no clause out of the clause set is the consequence of different clauses from the clause set.
Furthermore, we relate the notion of relevance to that of a minimally unsatisfiable subset (MUS) of some independent clause set N . In propositional logic, a clause C is relevant if it occurs in all MUSes of some clause set N and semi-relevant if it occurs in some MUS. For first-order logic the characterization needs to be refined with respect to ground instances of N and C .
A four-valued semantics for the modal logic K is introduced. Possible worlds are replaced by a hierarchy of four-valued valuations, where the valuations of the first level correspond to valuations that are legal w.r.t. a basic non-deterministic matrix, and each level further restricts its set of valuations. The semantics is proven to be effective, and to precisely capture derivations in a sequent calculus for K of a certain form. Similar results are then obtained for the modal logic KT, by simply deleting one of the truth values.
Proof production for SMT solvers is paramount to ensure their correctness independently from implementations, which are often prohibitively difficult to verify. Historically, however, SMT proof production has struggled with performance and coverage issues, resulting in the disabling of many crucial solving techniques and in coarse-grained (and thus hard to check) proofs. We present a flexible proof-production architecture designed to handle the complexity of versatile, industrial-strength SMT solvers and show how we leverage it to produce detailed proofs, including for components previously unsupported by any solver. The architecture allows proofs to be produced modularly, lazily, and with numerous safeguards for correctness. This architecture has been implemented in the state-of-the-art SMT solver cvc5. We evaluate its proofs for SMT-LIB benchmarks and show that the new architecture produces better coverage than previous approaches, has acceptable performance overhead, and supports detailed proofs for most solving components.
The study of clause redundancy in Boolean satisfiability (SAT) has proven significant in various terms, from fundamental insights into preprocessing and inprocessing to the development of practical proof checkers and new types of strong proof systems. We study liftings of the recently-proposed notion of propagation redundancy—based on a semantic implication relationship between formulas—in the context of maximum satisfiability (MaxSAT), where of interest are reasoning techniques that preserve optimal cost (in contrast to preserving satisfiability in the realm of SAT). We establish that the strongest MaxSAT-lifting of propagation redundancy allows for changing in a controlled way the set of minimal correction sets in MaxSAT. This ability is key in succinctly expressing MaxSAT reasoning techniques and allows for obtaining correctness proofs in a uniform way for MaxSAT reasoning techniques very generally. Bridging theory to practice, we also provide a new MaxSAT preprocessor incorporating such extended techniques, and show through experiments its wide applicability in improving the performance of modern MaxSAT solvers.
Choice logics constitute a family of propositional logics and are used for the representation of preferences, with especially qualitative choice logic (QCL) being an established formalism with numerous applications in artificial intelligence. While computational properties and applications of choice logics have been studied in the literature, only few results are known about the proof-theoretic aspects of their use. We propose a sound and complete sequent calculus for preferred model entailment in QCL, where a formula F is entailed by a QCL-theory T if F is true in all preferred models of T. The calculus is based on labeled sequent and refutation calculi, and can be easily adapted for different purposes. For instance, using the calculus as a cornerstone, calculi for other choice logics such as conjunctive choice logic (CCL) can be obtained in a straightforward way.
We describe Goéland, an automated theorem prover for first-order logic that relies on a concurrent search procedure to find tableau proofs, with concurrent processes corresponding to individual branches of the tableau. Since branch closure may require instantiating free variables shared across branches, processes communicate via channels to exchange information about substitutions used for closure. We present the proof search procedure and its implementation, as well as experimental results obtained on problems from the TPTP library.
We present a decision procedure for intermediate logics relying on a modular extension of the SAT-based prover $$\texttt {intuitR} $$ intuitR for $$\mathrm {IPL} $$ IPL (Intuitionistic Propositional Logic). Given an intermediate logic L and a formula $$\alpha $$ α , the procedure outputs either a Kripke countermodel for $$\alpha $$ α or the instances of the characteristic axioms of L that must be added to $$\mathrm {IPL} $$ IPL in order to prove $$\alpha $$ α . The procedure exploits an incremental SAT-solver; during the computation, new clauses are learned and added to the solver.
We consider a logic used to describe sets of configurations of distributed systems, whose network topologies can be changed at runtime, by reconfiguration programs. The logic uses inductive definitions to describe networks with an unbounded number of components and interactions, written using a multiplicative conjunction, reminiscent of Bunched Implications [37] and Separation Logic [39]. We study the complexity of the satisfiability and entailment problems for the configuration logic under consideration. Additionally, we consider the robustness property of degree boundedness (is every component involved in a bounded number of interactions?), an ingredient for decidability of entailments.
Problems in many theories axiomatised by unit equalities (UEQ), such as groups, loops, lattices, and other algebraic structures, are notoriously difficult for automated theorem provers to solve. Consequently, there has been considerable effort over decades in developing techniques to handle these theories, notably in the context of Knuth-Bendix completion and derivatives. The superposition calculus is a generalisation of completion to full first-order logic; however it does not carry over all the refinements that were developed for it, and is therefore not a strict generalisation. This means that (i) as of today, even state of the art provers for first-order logic based on the superposition calculus, while more general, are outperformed in UEQ by provers based on completion, and (ii) the sophisticated techniques developed for completion are not available in any problem which is not in UEQ. In particular, this includes key simplifications such as ground joinability, which have been known for more than 30 years. In fact, all previous completeness proofs for ground joinability rely on proof orderings and proof reductions, which are not easily extensible to general clauses together with redundancy elimination. In this paper we address this limitation and extend superposition with ground joinability, and show that under an adapted notion of redundancy, simplifications based on ground joinability preserve completeness. Another recently explored simplification in completion is connectedness. We extend this notion to “ground connectedness” and show superposition is complete with both connectedness and ground connectedness. We implemented ground joinability and connectedness in a theorem prover, iProver, the former using a novel algorithm which we also present in this paper, and evaluated over the TPTP library with encouraging results.
A code X is not primitivity preserving if there is a primitive list $$\mathbf {w}\in \texttt {lists}\, X$$ w ∈ lists X whose concatenation is imprimitive. We formalize a full characterization of such codes in the binary case in the proof assistant Isabelle/HOL. Part of the formalization, interesting on its own, is a description of $$\{x,y\}$$ { x , y } -interpretations of the square xx if $$\left| y \right| \le \left| x \right| $$ y ≤ x . We also provide a formalized parametric solution of the related equation $$x^jy^k = z^\ell $$ x j y k = z ℓ .
The modal logic \({\mathsf {K}}\) is commonly used to represent and reason about necessity and possibility and its extensions with combinations of additional axioms are used to represent knowledge, belief, desires and intentions. Here we present local reductions of all propositional modal logics in the so-called modal cube, that is, extensions of \({\mathsf {K}}\) with arbitrary combinations of the axioms \({\mathsf {B}}\), \({\mathsf {D}}\), \({\mathsf {T}}\), \({\mathsf {4}}\) and \({\mathsf {5}}\) to a normal form comprising a formula and the set of modal levels it occurs at. Using these reductions we can carry out reasoning for all these logics with the theorem prover
. We define benchmarks for these logics and experiment with the reduction approach as compared to an existing resolution calculus with specialised inference rules for the various logics.
Lash is a higher-order automated theorem prover created as a fork of the theorem prover Satallax. The basic underlying calculus of Satallax is a ground tableau calculus whose rules only use shallow information about the terms and formulas taking part in the rule. Lash uses new, efficient C representations of vital structures and operations. Most importantly, Lash uses a C representation of (normal) terms with perfect sharing along with a C implementation of normalizing substitutions. We describe the ways in which Lash differs from Satallax and the performance improvement of Lash over Satallax when used with analogous flag settings. With a 10 s timeout Lash outperforms Satallax on a collection TH0 problems from the TPTP. We conclude with ideas for continuing the development of Lash.
We propose a cut-free cyclic system for Transitive Closure Logic (TCL) based on a form of hypersequents, suitable for automated reasoning via proof search. We show that previously proposed sequent systems are cut-free incomplete for basic validities from Kleene Algebra (KA) and Propositional Dynamic Logic (PDL), over standard translations. On the other hand, our system faithfully simulates known cyclic systems for KA and PDL, thereby inheriting their completeness results. A peculiarity of our system is its richer correctness criterion, exhibiting ‘alternating traces’ and necessitating a more intricate soundness argument than for traditional cyclic proofs.
We present new results on the application of semantic- and knowledge-based reasoning techniques to the analysis of cloud deployments. In particular, to the security of Infrastructure as Code configuration files, encoded as description logic knowledge bases. We introduce an action language to model mutating actions; that is, actions that change the structural configuration of a given deployment by adding, modifying, or deleting resources. We mainly focus on two problems: the problem of determining whether the execution of an action, no matter the parameters passed to it, will not cause the violation of some security requirement (static verification), and the problem of finding sequences of actions that would lead the deployment to a state where (un)desirable properties are (not) satisfied (plan existence and plan synthesis). For all these problems, we provide definitions, complexity results, and decision procedures.
The importance of subsumption testing for redundancy elimination in first-order logic automatic reasoning is well-known. Although the problem is already NP-complete for first-order clauses, the meanwhile developed test pipelines efficiently decide subsumption in almost all practical cases. We consider subsumption between first-oder clauses of the Bernays-Schönfinkel fragment over linear real arithmetic constraints: BS(LRA). The bottleneck in this setup is deciding implication between the LRA constraints of two clauses. Our new sample point heuristic preempts expensive implication decisions in about 94% of all cases in benchmarks. Combined with filtering techniques for the first-order BS part of clauses, it results again in an efficient subsumption test pipeline for BS(LRA) clauses.
The cvc5 SMT solver solves quantifier-free nonlinear real arithmetic problems by combining the cylindrical algebraic coverings method with incremental linearization in an abstraction-refinement loop. The result is a complete algebraic decision procedure that leverages efficient heuristics for refining candidate models. Furthermore, it can be used with quantifiers, integer variables, and in combination with other theories. We describe the overall framework, individual solving techniques, and a number of implementation details. We demonstrate its effectiveness with an evaluation on the SMT-LIB benchmarks.
I introduce renaming-enriched sets (rensets for short), which are algebraic structures axiomatizing fundamental properties of renaming (also known as variable-for-variable substitution) on syntax with bindings. Rensets compare favorably in some respects with the well-known foundation based on nominal sets. In particular, renaming is a more fundamental operator than the nominal swapping operator and enjoys a simpler, equationally expressed relationship with the variable-freshness predicate. Together with some natural axioms matching properties of the syntactic constructors, rensets yield a truly minimalistic characterization of λ-calculus terms as an abstract datatype – one involving an infinite set of unconditional equations, referring only to the most fundamental term operators: the constructors and renaming. This characterization yields a recursion principle, which (similarly to the case of nominal sets) can be improved by incorporating Barendregt’s variable convention. When interpreting syntax in semantic domains, my renaming-based recursor is easier to deploy than the nominal recursor. My results have been validated with the proof assistant Isabelle/HOL.
The characterizing properties of a proof-theoretical presentation of a given logic may hang on the choice of proof formalism, on the shape of the logical rules and of the sequents manipulated by a given proof system, on the underlying notion of consequence, and even on the expressiveness of its linguistic resources and on the logical framework into which it is embedded. Standard (one-dimensional) logics determined by (non-deterministic) logical matrices are known to be axiomatizable by analytic and possibly finite proof systems as soon as they turn out to satisfy a certain constraint of sufficient expressiveness. In this paper we introduce a recipe for cooking up a two-dimensional logical matrix (or B-matrix) by the combination of two (possibly partial) non-deterministic logical matrices. We will show that such a combination may result in B-matrices satisfying the property of sufficient expressiveness, even when the input matrices are not sufficiently expressive in isolation, and we will use this result to show that one-dimensional logics that are not finitely axiomatizable may inhabit finitely axiomatizable two-dimensional logics, becoming, thus, finitely axiomatizable by the addition of an extra dimension. We will illustrate the said construction using a well-known logic of formal inconsistency called mCi. We will first prove that this logic is not finitely axiomatizable by a one-dimensional (generalized) Hilbert-style system. Then, taking advantage of a known 5-valued non-deterministic logical matrix for this logic, we will combine it with another one, conveniently chosen so as to give rise to a B-matrix that is axiomatized by a two-dimensional Hilbert-style system that is both finite and analytic.
The analysis of complex dynamic systems is a core research topic in formal methods and AI, and combined modelling of systems with data has gained increasing importance in applications such as business process management. In addition, process mining techniques are nowadays used to automatically mine process models from event data, often without correctness guarantees. Thus verification techniques for linear and branching time properties are needed to ensure desired behavior.
Here we consider data-aware dynamic systems with arithmetic (DDSAs), which constitute a concise but expressive formalism of transition systems with linear arithmetic guards. We present a CTL $$^*$$ ∗ model checking procedure for DDSAs that addresses a generalization of the classical verification problem, namely to compute conditions on the initial state, called witness maps , under which the desired property holds. Linear-time verification was shown to be decidable for specific classes of DDSAs where the constraint language or the control flow are suitably confined. We investigate several of these restrictions for the case of CTL $$^*$$ ∗ , with both positive and negative results: witness maps can always be found for monotonicity and integer periodicity constraint systems, but verification of bounded lookback systems is undecidable. To demonstrate the feasibility of our approach, we implemented it in an SMT-based prototype, showing that many practical business process models can be effectively analyzed.
The ontology of Leśniewski is commonly regarded as the most comprehensive calculus of names and the theoretical basis of mereology. However, ontology was not examined by means of proof-theoretic methods so far. In the paper we provide a characterization of elementary ontology as a sequent calculus satisfying desiderata usually formulated for rules in well-behaved systems in modern structural proof theory. In particular, the cut elimination theorem is proved and the version of subformula property holds for the cut-free version.
Automated theorem provers (ATPs) are today used to attack open problems in several areas of mathematics. An ongoing project by Kinyon and Veroff uses Prover9 to search for the proof of the Abelian Inner Mapping (AIM) Conjecture, one of the top open conjectures in quasigroup theory. In this work, we improve Prover9 on a benchmark of AIM problems by neural synthesis of useful alternative formulations of the goal. In particular, we design the 3SIL (stratified shortest solution imitation learning) method. 3SIL trains a neural predictor through a reinforcement learning (RL) loop to propose correct rewrites of the conjecture that guide the search.
3SIL is first developed on a simpler, Robinson arithmetic rewriting task for which the reward structure is similar to theorem proving. There we show that 3SIL outperforms other RL methods. Next we train 3SIL on the AIM benchmark and show that the final trained network, deciding what actions to take within the equational rewriting environment, proves 70.2% of problems, outperforming Waldmeister (65.5%). When we combine the rewrites suggested by the network with Prover9, we prove 8.3% more theorems than Prover9 in the same time, bringing the performance of the combined system to 90%.
The development of computerized proof systems, such as Coq, Matita, Agda, Lean, HOL 4, HOL Light, Isabelle/HOL, Mizar, etc. is a major step forward in the never ending quest of mathematical rigor.
Definition packages in theorem provers provide users with means of defining and organizing concepts of interest. This system description presents a new definition package for the hybrid systems theorem prover KeYmaera X based on differential dynamic logic (dL). The package adds KeYmaera X support for user-defined smooth functions whose graphs can be implicitly characterized by dL formulas. Notably, this makes it possible to implicitly characterize functions, such as the exponential and trigonometric functions, as solutions of differential equations and then prove properties of those functions using dL’s differential equation reasoning principles. Trustworthiness of the package is achieved by minimally extending KeYmaera X ’s soundness-critical kernel with a single axiom scheme that expands function occurrences with their implicit characterization. Users are provided with a high-level interface for defining functions and non-soundness-critical tactics that automate low-level reasoning over implicit characterizations in hybrid system proofs.
Treating a saturation-based automatic theorem prover (ATP) as a Las Vegas randomized algorithm is a way to illuminate the chaotic nature of proof search and make it amenable to study by probabilistic tools. On a series of experiments with the ATP Vampire, the paper showcases some implications of this perspective for prover evaluation.
There exist several results on deciding termination and computing runtime bounds for triangular weakly non-linear loops (twn-loops). We show how to use results on such subclasses of programs where complexity bounds are computable within incomplete approaches for complexity analysis of full integer programs. To this end, we present a novel modular approach which computes local runtime bounds for subprograms which can be transformed into twn-loops. These local runtime bounds are then lifted to global runtime bounds for the whole program. The power of our approach is shown by our implementation in the tool $$\textsf {KoAT}$$ KoAT which analyzes complexity of programs where all other state-of-the-art tools fail.
Explanations for description logic (DL) entailments provide important support for the maintenance of large ontologies. The “justifications” usually employed for this purpose in ontology editors pinpoint the parts of the ontology responsible for a given entailment. Proofs for entailments make the intermediate reasoning steps explicit, and thus explain how a consequence can actually be derived. We present an interactive system for exploring description logic proofs, called Evonne , which visualizes proofs of consequences for ontologies written in expressive DLs. We describe the methods used for computing those proofs, together with a feature called signature-based proof condensation . Moreover, we evaluate the quality of generated proofs using real ontologies.
Our goal is to develop a logic-based component for hybrid – machine learning plus logic – commonsense question answering systems. The paper presents an implementation GK of default logic for handling rules with exceptions in unrestricted first order knowledge bases. GK is built on top of our existing automated reasoning system with confidence calculation capabilities. To overcome the problem of undecidability of checking potential exceptions, GK performs delayed recursive checks with diminishing time limits. These are combined with the taxonomy-based priorities for defaults and numerical confidences.
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