Gentzen’s sequent calculus is a foundational tool for describing and investigating provability, yet its fine-grained inference rules generally do not directly support automated proof search. Incorporating synthetic inferences improves automatability, but they do not provide mechanisms for reasoning naturally about the elementary mathematical notions of equivalence and congruence. In this paper, we present a first-order framework for reasoning modulo such relations within the sequent calculus. We provide a setting in which congruences can be treated as actual equalities, mirroring the informal practice of mathematicians and eliminating the need to use the lemmas typically required for formal congruence proofs. We demonstrate that this approach remains strictly first-order, avoids the complexity of higher-order or set-theoretic constructions, and yields proof systems that retain essential meta-theoretic properties.

Abstract Treating syntactic equality as a logical connective—governed by left- and right-introduction rules within the sequent calculus—offers an elegant and powerful approach to term identity. This treatment of equality allows for the derivation of core mathematical principles, such as Peano’s axioms (excluding induction), and serves as a foundation for the Abella interactive proof assistant. However, integrating this equality into automated proof search remains challenging. We present a proof search procedure that extends unification to handle the complexities of quantifier alternation and equations that occur in both positive and negative occurrences. While established logical frameworks such as $$\lambda $$ λ Prolog and LF lack direct support for this kind of equality, our procedure enables a lightweight logical framework that addresses this gap. Our system enables unification-aware proof search across a diverse range of first-order sequent calculi that can directly use this form of equality.

Abstract We present a proof-theoretic treatment of forward chaining and saturation within a multisorted, first-order intuitionistic logic with equality. The notions of polarity and focused proofs are central to our approach since they provide a characterization of geometric implications as bipolar formulas as well as a natural setting to describe forward chaining and the concept of productive proofs . We identify conditions under which forward chaining with a given set of formulas is guaranteed to saturate in a finite number of steps. The motivation for this research stems, in part, from exploring avenues to automate the Abella theorem prover, which relies on relational specifications, and where theorems in typical proof developments are essentially bipolar formulas. We illustrate the potential benefits of automating forward chaining and saturation for Abella by presenting examples that compute congruence closure and assist in other equational and relational reasoning tasks.

Abstract Property-based testing (PBT) is a technique for validating code against an executable specification by automatically generating test-data. We present a proof-theoretical reconstruction of this style of testing for relational specifications and employ the Foundational Proof Certificate framework to describe test generators. We do this by encoding certain kinds of “proof outlines” as proof certificates that can describe various common generation strategies in the PBT literature, ranging from random to exhaustive, including their combination. We also address the shrinking of counterexamples as a first step toward their explanation. Once generation is accomplished, the testing phase is a standard logic programing search. After illustrating our techniques on simple, first-order (algebraic) data structures, we lift it to data structures containing bindings by using the $\lambda$ -tree syntax approach to encode bindings. The $\lambda$ Prolog programing language can perform both generating and checking of tests using this approach to syntax. We then further extend PBT to specifications in a fragment of linear logic.

Gentzen designed his natural deduction proof system to "come as close as possible to actual reasoning." Indeed, natural deduction proofs closely resemble the static structure of logical reasoning in mathematical arguments. However, different features of inference are compelling to capture when one wants to support the process of searching for proofs. PSF (Proof Search Framework) attempts to capture these features naturally and directly. The design and metatheory of PSF are presented, and its ability to specify a range of proof systems for classical, intuitionistic, and linear logic is illustrated.

We present a type system over language definitions that classifies parts of the operational semantics of a language in input, and models a common language design organization. The resulting typing discipline guarantees that the language at hand is automatically type sound.

When presented with a formula to prove, most theorem provers for classical first-order logic process that formula following several steps, one of which is commonly called skolemization. That process eliminates quantifier alternation within formulas by extending the language of the underlying logic with new Skolem functions and by instantiating certain quantifiers with terms built using Skolem functions. In this paper, we address the problem of checking (i.e., certifying) proof evidence that involves Skolem terms. Our goal is to do such certification without using the mathematical concepts of model-theoretic semantics (i.e., preservation of satisfiability) and choice principles (i.e., epsilon terms). Instead, our proof checking kernel is an implementation of Gentzen's sequent calculus, which directly supports quantifier alternation by using eigenvariables. We shall describe deskolemization as a mapping from client-side terms, used in proofs generated by theorem provers, into kernel-side terms, used within our proof checking kernel. This mapping which associates skolemized terms to eigenvariables relies on using outer skolemization. We also point out that the removal of Skolem terms from a proof is influenced by the polarities given to propositional connectives.

Property-based testing (PBT) is a technique for validating code against an executable specification by automatically generating test-data. We present a proof-theoretical reconstruction of this style of testing for relational specifications and employ the Foundational Proof Certificate framework to describe test generators. We do this by presenting certain kinds of "proof outlines" that can be used to describe various common generation strategies in the PBT literature, ranging from random to exhaustive, including their combination. We also address the shrinking of counterexamples as a first step towards their explanation. Once generation is accomplished, the testing phase boils down to a standard logic programming search. After illustrating our techniques on simple, first-order (algebraic) data structures, we lift it to data structures containing bindings using λ-tree syntax. The λProlog programming language is capable of performing both the generation and checking of tests. We validate this approach by tackling benchmarks in the metatheory of programming languages coming from related tools such as PLT-Redex.

We present the design of a new functional programming language, MLTS, that uses the λ-tree syntax approach to encoding bindings appearing within data structures. In this approach, bindings never become free nor escape their scope: instead, binders in data structures are permitted to move to binders within programs. The design of MLTS includes additional sites within programs that directly support this movement of bindings. In order to formally define the language's operational semantics, we present an abstract syntax for MLTS and a natural semantics for its evaluation. We shall view such natural semantics as a logical theory within a rich logic that includes both nominal abstraction and the ∇-quantifier: as a result, the natural semantics specification of MLTS can be given a succinct and elegant presentation. We present a typing discipline that naturally extends the typing of core ML programs and we illustrate the features of MLTS by presenting several examples. An on-line interpreter for MLTS is briefly described.

The logical foundation of arithmetic generally starts with a quantificational logic over relations. Of course, one often wishes to have a formal treatment of functions within this setting. Both Hilbert and Church added choice operators (such as the epsilon operator) to logic in order to coerce relations that happen to encode functions into actual functions. Others have extended the term language with confluent term rewriting in order to encode functional computation as rewriting to a normal form. We take a different approach that does not extend the underlying logic with either choice principles or with an equality theory. Instead, we use the familiar two-phase construction of focused proofs and capture functional computation entirely within one of these phases. As a result, our logic remains purely relational even when it is computing functions.

Abstract In a world where trusting software systems is increasingly important, formal methods and formal proof can help provide some basis for trust. Proof checking can help to reduce the size of the trusted base since we do not need to trust an entire theorem prover: instead, we only need to trust a (smaller and simpler) proof checker. Many approaches to building proof checkers require embedding within them a full programming language. In most modern proof checkers and theorem provers, that programming language is a functional programming language, often a variant of ML. In fact, aspects of ML (e.g., strong typing, abstract datatypes, and higher-order programming) were designed to make ML a trustworthy “meta-language” for checking proofs. While there is considerable overlap between logic programming and proof checking (e.g., both benefit from unification, backtracking search, efficient term structures, etc.), the discipline of logic programming has, in fact, played a minor role in the history of proof checking. I will argue that logic programming can have a major role in the future of this important topic.

The sequent calculus is often criticized for requiring proofs to contain large amounts of low-level syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, sequent proofs can separate closely related steps—such as instantiating a block of quantifiers—by irrelevant noise. Moreover, the sequential nature of sequent proofs forces proof steps that are syntactically non-interfering and permutable to nevertheless be written in some arbitrary order. The sequent calculus thus lacks a notion of canonicity : proofs that should be considered essentially the same may not have a common syntactic form. To fix this problem, many researchers have proposed replacing the sequent calculus with proof structures that are more parallel or geometric. Proof-nets, matings and atomic flows are examples of such revolutionary formalizms. We propose, instead, an evolutionary approach to recover canonicity within the sequent calculus, which we illustrate for classical first-order logic. The essential element of our approach is the use of a multi-focused sequent calculus as the means for abstracting away low-level details from classical cut-free sequent proofs. We show that, among the multi-focused proofs, the maximally multi-focused proofs that collect together all possible parallel foci are canonical. Moreover, if we start with a certain focused sequent proof system, such proofs are isomorphic to expansion proofs —a well known, minimalistic and parallel generalization of Herbrand disjunctions—for classical first-order logic. This technique appears to be a systematic way to recover the ‘essence of proof’ from within sequent calculus proofs.

Bisimilarity of two processes is formally established by producing a bisimulation relation that contains those two processes and obeys certain closure properties. In many situations, particularly when the underlying labeled transition system is unbounded, these bisimulation relations can be large and even infinite. The bisimulation-up-to technique has been developed to reduce the size of the relations being computed while retaining soundness, that is, the guarantee of the existence of a bisimulation. Such techniques are increasingly becoming a critical ingredient in the automated checking of bisimilarity. This paper is devoted to the formalization of the meta theory of several major bisimulation-up-to techniques for the process calculi CCS and the π-calculus (with replication). Our formalization is based on recent work on the proof theory of least and greatest fixpoints, particularly the use of relations defined (co-)inductively, and of co-inductive proofs about such relations, as implemented in the Abella theorem prover. An important feature of our formalization is that our definitions of the bisimulation-up-to relations are, in most cases, straightforward translations of published informal definitions, and our proofs clarify several technical details of the informal descriptions. Since the logic behind Abella also supports λ-tree syntax and generic reasoning using the ∇-quantifier, our treatment of the λ-calculus is both direct and natural.

In a world where trusting software systems is problematic, formal methods and formal proofs should be able to help. Proof checking can play an important role in establishing trust since such checkers can be smaller and easier to verify than, for example, entire theorem provers or model checkers. Proof checking has played an important role in the history of programming languages and, I argue, in its future. In general, proof checkers rely on programming languages which must also be trusted. In many modern proof checkers and theorem provers, that programming language is a functional programming language, often a variant of ML. In fact, parts of ML (eg., strong typing, abstract datatypes, and higher-order programming) were designed to make ML into a trustworthy metalanguage for the finding and checking proofs in LCF [2].

We show that control operators and other extensions of the Curry-Howard isomorphism can be achieved without collapsing all of intuitionistic logic into classical logic. For this purpose we introduce a unified propositional logic using polarized formulas. We define a Kripke semantics for this logic. Our proof system extends an intuitionistic system that already allows multiple conclusions. This arrangement reveals a greater range of computational possibilities, including a form of dynamic scoping. We demonstrate the utility of this logic by showing how it can improve the formulation of exception handling in programming languages, including the ability to distinguish between different kinds of exceptions and constraining when an exception can be thrown, thus providing more refined control over computation compared to classical logic. We also describe some significant fragments of this logic and discuss its extension to second-order logic.

We specify the operational semantics and bisimulation relations for the finite φ-calculus within a logic that contains the ∇ quantifier for encoding generic judgments and definitions for encoding fixed points. Since we restrict to the finite case, the ability of the logic to unfold fixed points allows this logic to be complete for both the inductive nature of operational semantics and the coinductive nature of bisimulation. The ∇ quantifier helps with the delicate issues surrounding the scope of variables within φ-calculus expressions and their executions (proofs). We illustrate several merits of the logical specifications permitted by this logic: they are natural and declarative; they contain no side-conditions concerning names of variables while maintaining a completely formal treatment of such variables; differences between late and open bisimulation relations arise from familar logic distinctions; the interplay between the three quantifiers (∀, ∃, and ∇) and their scopes can explain the differences between early and late bisimulation and between various modal operators based on bound input and output actions; and proof search involving the application of inference rules, unification, and backtracking can provide complete proof systems for one-step transitions, bisimulation, and satisfaction in modal logic. We also illustrate how one can encode the φ-calculus with replications, in an extended logic with induction and co-induction.

We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical logic, intuitionistic logic, and multiplicative-additive linear logic are derived as fragments of LKU by increasing the sensitivity of specialized structural rules to polarity information. We develop a unified, streamlined framework for proving cut-elimination in the various fragments. Furthermore, each sublogic can interact with other fragments through cut. We also consider the possibility of introducing classical-linear hybrid logics.

The linear logic exponentials !,? are not canonical: one can add to linear logic other such operators, say !l,?1, which may or may not allow contraction and weakening, and where l is from some pre-ordered set of labels. We shall call these additional operators subexponentials and use them to assign locations to multisets of formulas within a linear logic programming setting. Treating locations as subexponentials greatly increases the algorithmic expressiveness of logic. To illustrate this new expressiveness, we show that focused proof search can be precisely linked to a simple algorithmic specification language that contains while-loops, conditionals, and insertion into and deletion from multisets. We also give some general conditions for when a focused proof step can be executed in constant time. In addition, we propose a new logical connective that allows for the creation of new subexponentials, thereby further augmenting the algorithmic expressiveness of logic.

We propose a setting in which the search for a proof of B or a refutation of B (a proof of not B) can be carried out simultaneously. In contrast with the usual approach in automated deduction, we do not need to first commit to either proving B or to proving not B: instead we devise a neutral setting for attempting both a proof and a refutation. This setting is described as a two player game in which each player follows the same rules. A winning strategy translates to a proof of the formula and a winning counter-strategy translates to a refutation of the formula. The game is described for multiplicative and additive linear logic without atomic formulas. A game theoretic treatment of the multiplicative connectives is intricate and our approach to it involves two important ingredients. First, labeled graph structures are used to represent positions in a game and, second, the game playing must deal with the failure of a given player and with an appropriate resumption of play. This latter ingredient accounts for the fact that neither players might win (that is, neither B nor not B might be provable).

Many semantical aspects of programming languages, such as their operational semantics and their type assignment calculi, are specified by describing appropriate proof systems. Recent research has identified two proof-theoretic features that allow direct, logic-based reasoning about such descriptions: the treatment of atomic judgments as fixed points (recursive definitions) and an encoding of binding constructs via generic judgments. However, the logics encompassing these two features have thus far treated them orthogonally: that is, they do not provide the ability to define object-logic properties that themselves depend on an intrinsic treatment of binding. We propose a new and simple integration of these features within an intuitionistic logic enhanced with induction over natural numbers and we show that the resulting logic is consistent. The pivotal benefit of the integration is that it allows recursive definitions to not just encode simple, traditional forms of atomic judgments but also to capture generic properties pertaining to such judgments. The usefulness of this logic is illustrated by showing how it can provide elegant treatments of object-logic contexts that appear in proofs involving typing calculi and of arbitrarily cascading substitutions that play a role in reducibility arguments.

This roadmap describes ways that researchers in four areas---specification languages, program generation, correctness by construction, and programming languages---might help further the goal of verified software. It also describes what advances the "verified software" grand challenge might anticipate or demand from work in these areas. That is, the roadmap is intended to help foster collaboration between the grand challenge and these research areas.A common goal for research in these areas is to establish language designs and tool architectures that would allow multiple annotations and tools to be used on a single program. In the long term, researchers could try to unify these annotations and integrate such tools.

We consider approximating data structures with collections of the items that they contain. For examples, lists, binary trees, tuples, etc, can be approximated by sets or multisets of the items within them. Such approximations can be used to provide partial correctness properties of logic programs. For example, one might wish to specify than whenever the atom sort(t,s) is proved then the two lists t and s contain the same multiset of items (that is, s is a permutation of t). If sorting removes duplicates, then one would like to infer that the sets of items underlying t and s are the same. Such results could be useful to have if they can be determined statically and automatically. We present a scheme by which such collection analysis can be structured and automated. Central to this scheme is the use of linear logic as a computational logic underlying the logic of Horn clauses

The operational semantics of a computation system is often presented as inference rules or, equivalently, as logical theories. Specifications can be made more declarative and high level if syntactic details concerning bound variables and substitutions are encoded directly into the logic using term-level abstractions (λ-abstraction) and proof-level abstractions (eigenvariables). When one wishes to use such logical theories to support reasoning about properties of computation, the usual quantifiers and proof-level abstractions do not seem adequate: proof-level abstraction of variables with scope over sequents ( global scope) as well as over only formulas ( local scope) seem required for many examples. We will present a sequent calculus that provides this local notion of proof-level abstraction via generic judgment and a new quantifier, ∇, which explicitly manipulates such local scope. Intuitionistic logic extended with ∇ satisfies cut-elimination even when the logic is additionally strengthened with a proof theoretic notion of definitions. The resulting logic can be used to encode naturally a number of examples involving abstractions, and we illustrate the uses of ∇ with the π-calculus and an encoding of provability of an object-logic.

A powerful and declarative means of specifying computations containing abstractions involves meta-level, universally quantified generic judgments. We present a proof theory for such judgments in which signatures are associated to each sequent (used to account for eigenvariables of sequent) and to each formula in the sequent (used to account for generic variables locally scoped over the formula). A new quantifier, /spl nabla/, is introduced to explicitly manipulate the local signature. Intuitionistic logic extended with /spl nabla/ satisfies cut-elimination even when the logic is additionally strengthened with a proof theoretic notion of definitions. The resulting logic can be used to encode naturally a number of examples involving name abstractions, and we illustrate using the /spl pi/-calculus and the encoding of object-level provability.

Logical frameworks based on intuitionistic or linear logics with higher-type quantification have been successfully used to give high-level, modular, and formal specifications of many important judgments in the area of programming languages and inference systems. Given such specifications, it is natural to consider proving properties about the specified systems in the framework: for example, given the specification of evaluation for a functional programming language, prove that the language is deterministic or that evaluation preserves types. One challenge in developing a framework for such reasoning is that higher-order abstract syntax (HOAS), an elegant and declarative treatment of object-level abstraction and substitution, is difficult to treat in proofs involving induction. In this article, we present a meta-logic that can be used to reason about judgments coded using HOAS; this meta-logic is an extension of a simple intuitionistic logic that admits higher-order quantification over simply typed λ-terms (key ingredients for HOAS) as well as induction and a notion of definition . The latter concept of definition is a proof-theoretic device that allows certain theories to be treated as "closed" or as defining fixed points. We explore the difficulties of formal meta-theoretic analysis of HOAS encodings by considering encodings of intuitionistic and linear logics, and formally derive the admissibility of cut for important subsets of these logics. We then propose an approach to avoid the apparent trade-off between the benefits of higher-order abstract syntax and the ability to analyze the resulting encodings. We illustrate this approach through examples involving the simple functional and imperative programming languages PCF and PCF := . We formally derive such properties as unicity of typing, subject reduction, determinacy of evaluation, and the equivalence of transition semantics and natural semantics presentations of evaluation.

Logical frameworks based on intuitionistic or linear logics with higher-type quantification have been successfully used to give high-level, modular, and formal specifications of many important judgments in the area of programming languages and inference systems. Given such specifications, it is natural to consider proving properties about the specified systems in the framework: for example, given the specification of evaluation for a functional programming language, prove that the language is deterministic or that the subject-reduction theorem holds. One challenge in developing a framework for such reasoning is that higher-order abstract syntax (HOAS), an elegant and declarative treatment of object-level abstraction and substitution, is difficult to treat in proofs involving induction. In this paper we present a meta-logic that can be used to reason about judgments coded using HOAS; this meta-logic is an extension of a simple intuitionistic logic that admits higher-order quantification over simply typed /spl lambda/-terms (key ingredients for HOAS) as well as induction and a notion of definition. The latter concept of a definition is a proof-theoretic device that allows certain theories to be treated as "closed" or as defining fixed points. The resulting meta-logic can specify various logical frameworks and a large range of judgments regarding programming languages and inference systems. We illustrate this point through examples, including the admissibility of cut for a simple logic and subject reduction, determinacy of evaluation, and the equivalence of SOS and natural semantics presentations of evaluation for a simple functional programming language.

The theory of cut-free sequent proofs has been used to motivate and justify the design of a number of logic programming languages. Two such languages, λProlog and its linear logic refinement, Lolli [15], provide for various forms of abstraction (modules, abstract data types, and higher-order programming) but lack primitives for concurrency. The logic programming language, LO (Linear Objects) [2] provides some primitives for concurrency but lacks abstraction mechanisms. In this paper we present Forum, a logic programming presentation of all of linear logic that modularly extends λProlog, Lolli, and LO. Forum, therefore, allows specifications to incorporate both abstractions and concurrency. To illustrate the new expressive strengths of Forum, we specify in it a sequent calculus proof system and the operational semantics of a programming language that incorporates references and concurrency. We also show that the meta theory of linear logic can be used to prove properties of the object-languages specified in Forum.

The theory of cut-free sequent proofs has been used to motivate and justify the design of a number of logic programming languages. Two such languages, /spl lambda/Prolog and its linear logic refinement, Lolli (J. Hodas and D. Miller, 1994), provide for various forms of abstraction (modules, abstract data types, higher-order programming) but lack primitives for concurrency. The logic programming language, LO (Linear Objects) (J. Andreoli and R. Pareschi, 1991) provides for concurrency but lacks abstraction mechanisms. We present Forum, a logic programming presentation of all of linear logic that modularly extends the languages /spl lambda/Prolog, Lolli, and LO. Forum, therefore, allows specifications to incorporate both abstractions and concurrency. As a meta-language, Forum greatly extends the expressiveness of these other logic programming languages. To illustrate its expressive strength, we specify in Forum a sequent calculus proof system and the operational semantics of a functional programming language that incorporates such nonfunctional features as counters and references.>

When logic programming is based on the proof theory of intuitionistic logic, it is natural to allow implications in goals and in the bodies of clauses. Attempting to prove a goal of the form D ⊃ G from the context (set of formulas) Γ leads to an attempt to prove the goal G in the extended context Γ ∪ {D}. Thus contexts, represented as the left-hand side of intuitionistic sequents, grow as stacks during the bottom-up search for a cut-free proof. While such an intuitionistic notion of context provides for elegant specifications of many computations, contexts can be made more expressive and flexible if they are based on linear logic. After presenting two equivalent formulations of a fragment of linear logic, we show that the fragment has a goal-directed interpretation, thereby partially justifying calling it a logic program-ming language. Logic programs based on the intuitionistic theory of hereditary Harrop formulas can be modularly embedded into this linear logic setting. Programming examples taken from theorem proving, natural language parsing, and data base programming are presented: each example requires a linear, rather than intuitionistic, notion of context to be modeled adequately. An interpreter for this logic programming language must address the problem of splitting contexts; that is, in the attempt to prove a multiplicative conjunction (tensor), say G1 ⊗ G2, from the context Δ the latter must be split into disjoint contexts Δ1 and Δ2 for which G1 follows from Δ1 and G2 follows from Δ2. Since there is an exponential number of such splits, it is important to delay the choice of a split as much as possible. A mechanism for the lazy splitting of contexts is presented based on viewing proof search as a process that takes a context, consumes part of it, and returns the rest (to be consumed elsewhere). In addition, we use collections of Kripke interpretations indexed by a commutative monoid to provide models for this logic programming language and show that logic programs admit canonical models.

An important activity in collaborative writing is communicating about changes to texts,, This paper reports on a software system, ji'exible cliff, that finds and reports differences ("cliffs") between versions of texts.The system is flexible, allowing users to control several aspects of its operation including what changes are reported and how they are shown when they are reported.We argue that such flexibility is necessary to support users' different social and cognitive needs.

We consider the problem of mechanically constructing abstract machines from operational semantics, producing intermediate-level specifications of evaluators guaranteed to be correct with respect to the operational semantics. We construct these machines by repeatedly applying correctness-preserving transformations to operational semantics until the resulting specifications have the form of abstract machines. Though not automatable in general, this approach to constructing machine implementations can be mechanized, providing machine-verified correctness proofs. As examples, we present the transformation of specifications for both call-by-name and call-by-value evaluation of the untyped λ-calculus into abstract machines that implement such evaluation strategies. We also present extensions to the call-by-value machine for a language containing constructs for recursion, conditionals, concrete data types, and built-in functions. In all cases, the correctness of the derived abstract machines follows from the (generally transparent) correctness of the initial operational semantic specification and the correctness of the transformations applied.

The intuitionistic notion of context is refined by using a fragment of J.-Y. Girard's (Theor. Comput. Sci., vol.50, p.1-102, 1987) linear logic that includes additive and multiplicative conjunction, linear implication, universal quantification, the of course exponential, and the constants for the empty context and for the erasing contexts. It is shown that the logic has a goal-directed interpretation. It is also shown that the nondeterminism that results from the need to split contexts in order to prove a multiplicative conjunction can be handled by viewing proof search as a process that takes a context, consumes part of it, and returns the rest (to be consumed elsewhere). Examples taken from theorem proving, natural language parsing, and database programming are presented: each example requires a linear, rather than intuitionistic, notion of context to be modeled adequately.>

Miller, D., G. Nadathur, F. Pfenning and A. Scedrov, Uniform proofs as a foundation for logic programming, Annals of Pure and Applied Logic 51 (1991) 125–157. A proof-theoretic characterization of logical languages that form suitable bases for Prolog-like programming languages is provided. This characterization is based on the principle that the declarative meaning of a logic program, provided by provability in a logical system, should coincide with its operational meaning, provided by interpreting logical connectives as simple and fixed search instructions. The operational semantics is formalized by the identification of a class of cut-free sequent proofs called uniform proofs. A uniform proof is one that can be found by a goal-directed search that respects the interpretation of the logical connectives as search instructions. The concept of a uniform proof is used to define the notion of an abstract logic programming language, and it is shown that first-order and higher-order Horn clauses with classical provability are examples of such a language. Horn clauses are then generalized to hereditary Harrop formulas and it is shown that first-order and higher-order versions of this new class of formulas are also abstract logic programming languages if the inference rules are those of either intuitionistic or minimal logic. The programming language significance of the various generalizations to first-order Horn clauses is briefly discussed.

It has been argued elsewhere that a logic programming language with function variables and λ-abstractions within terms makes a good meta-programming language, especially when an object-language contains notions of bound variables and scope. The λProlog logic programming language and the related Elf and Isabelle systems provide meta-programs with both function variables and λ-abstractions by containing implementations of higher order unification. This paper presents a logic programming language, called Lλ, that also contains both function variables and λ-abstractions, although certain restrictions are placed on occurrences of function variables. As a result of these restrictions, an implementation of Lλdoes not need to implement full higher-order unification. Instead, an extension to first-order unification that respects bound variable names and scopes is all that is required. Such unification problems are shown to be decidable and to possess most general unifiers when unifiers exist. A unification algorithm and logic programming interpreter are described and proved correct. Several examples of using Lλ as a meta-programming language are presented.

A generalization of Horn clauses to a higher-order logic is described and examined as a basis for logic programming. In qualitative terms, these higher-order Horn clauses are obtained from the first-order ones by replacing first-order terms with simply typed λ-terms and by permitting quantification over all occurrences of function symbols and some occurrences of predicate symbols. Several proof-theoretic results concerning these extended clauses are presented. One result shows that although the substitutions for predicate variables can be quite complex in general, the substitutions necessary in the context of higher-order Horn clauses are tightly constrained. This observation is used to show that these higher-order formulas can specify computations in a fashion similar to first-order Horn clauses. A complete theorem-proving procedure is also described for the extension. This procedure is obtained by interweaving higher-order unification with backchaining and goal reductions, and constitutes a higher-order generalization of SLD-resolution. These results have a practical realization in the higher-order logic programming language called λProlog.

Consideration of the question of meaning in the framework of linguistics often requires an allusion to sets and other higher-order notions. The traditional approach to representing and reasoning about meaning in a computational setting has been to use knowledge representation systems that are either based on first-order logic or that use mechanisms whose formal justifications are to be provided after the fact. In this paper we shall consider the use of a higher-order logic for this task. We first present a version of definite clauses (positive Horn clauses) that is based on this logic. Predicate and function variables may occur in such clauses and the terms in the language are the typed λ-terms. Such term structures have a richness that may be exploited in representing meanings. We also describe a higher-order logic programming language, called λProlog, which represents programs as higher-order definite clauses and interprets them using a depth-first interpreter. A virtue of this language is that it is possible to write programs in it that integrate syntactic and semantic analyses into one computational paradigm. This is to be contrasted with the more common practice of using two entirely different computation paradigms, such as DCGs or ATNs for parsing and frames or semantic nets for semantic processing. We illustrate such an integration in this language by considering a simple example, and we claim that its use makes the task of providing formal justifications for the computations specified much more direct.