Logic programming and theorem proving

Logic programming is programming in which a program is a set of logical sentences, usually facts and if–then rules, and running the program means proving a query from them. Automated theorem proving is the wider task of making a computer find, or check, proofs in a formal logic. Both rest on one rule: resolution with unification.

This page is one of the family pages of Symbolic AI techniques, the index of every major symbolic method. For the field as a whole see What is symbolic AI?, and for how these ideas fit the wider story see the history of symbolic AI.

1. Logic as a tool: propositional and first-order logic

Propositional logic

What it is. Propositional logic builds formulas from atomic statements ($p$, $q$, …) with the connectives $\neg$, $\wedge$, $\vee$ and $\to$. A formula is satisfiable if some assignment of true and false to its atoms makes it true, and valid if every assignment does. Validity is decidable: in the worst case, check every row of the truth table.

How it is used. Almost every automated reasoner first puts formulas in clause form: a conjunction of clauses, each a disjunction of literals. The implication $p\to q$ becomes the clause $\neg p\vee q$. The practical descendants of propositional reasoning, SAT and SMT solvers, have their own page: constraint satisfaction, SAT and SMT.

Limits. Deciding satisfiability is NP-complete (Cook, 1971), and propositional logic cannot talk about objects or say “for all”.

First-order logic

What it is. First-order logic adds terms that name objects (constants such as $\mathrm{ann}$, variables such as $x$, function symbols such as $f\left(x\right)$), predicates over them, and the quantifiers $\forall$ and $\exists$. It is expressive enough for most of mathematics and for most knowledge a program needs to state.

What it guarantees, and what it does not. Gödel’s completeness theorem (1930) says the valid first-order formulas are exactly the provable ones, so a program that enumerates proofs will eventually confirm any valid formula [1]. Church and Turing showed in 1936 that no program can also always confirm the invalid ones [2] [3]. First-order validity is therefore semi-decidable: a prover can succeed on every true theorem, but on a non-theorem it may run forever. Every system on this page lives with that fact, either by restricting the logic (Datalog, ASP) or by accepting that search can fail to terminate (Prolog, first-order provers).

Herbrand’s theorem

Who and when. Jacques Herbrand, in his 1930 doctoral thesis at the University of Paris [4]. The same thesis already sketched the idea of unification.

Why it matters. The theorem reduces a first-order question to a sequence of propositional ones: generate ground instances, test them, repeat. Early provers did exactly that. Davis and Putnam’s 1960 procedure [5] was built on it, and its propositional core, refined in 1962 by Davis, Logemann and Loveland, still underlies modern SAT solvers. The weakness was blind instantiation: there are infinitely many ground terms and no good way to guess the right ones. Resolution fixed that.

2. Resolution and unification

Resolution

Who and when. John Alan Robinson, “A Machine-Oriented Logic Based on the Resolution Principle”, Journal of the ACM, 1965 [6]. Robinson called it machine-oriented because it has one inference rule, easy for a computer to apply many times, rather than the many rules of a logic designed for people.

How it works. Take two clauses containing complementary literals, unify those literals, and combine the rest:

To prove a goal $G$ from premises $\Gamma$, add the clauses of $\neg G$ and resolve until the empty clause $\mathrm{\square }$ appears. The empty clause is a contradiction, so $\Gamma \cup \{\neg G\}$ is unsatisfiable and $\Gamma \models G$. This is proof by refutation.

Worked example: a resolution refutation. Premises: whoever is a parent of a parent is a grandparent; Ann is Bill’s parent; Bill is Carl’s parent. Goal: Ann is Carl’s grandparent. In clause form, with the goal negated:

Where it is used today. Resolution, and its equality-aware successor, the superposition calculus, is the core of first-order provers such as Vampire and E (§4). Its restriction to Horn clauses is the execution model of Prolog (§3).

Limits. Unrestricted resolution generates clauses far faster than it finds the empty one, so practical provers depend on refinements (ordering restrictions, subsumption) and clause-selection heuristics. Plain resolution also handles equality badly.

Unification

What it is. Unification finds a substitution that makes two terms identical. Robinson’s 1965 paper gave the first general algorithm and proved that when two terms can be unified at all, there is a most general unifier (mgu) from which every other unifier can be obtained by further substitution [6].

Worked example. Unify $p(X,f(Y\left)\right)$ with $p(a,f(X\left)\right)$. Matching argument by argument: $X=a$ gives $\{X\mapsto a\}$; applying it, the second arguments become $f\left(Y\right)$ and $f\left(a\right)$, so $Y\mapsto a$. The mgu is

Unification fails on a clash of function symbols ($f\left(X\right)$ against $g\left(Y\right)$) and on the occurs check: $X$ cannot be unified with $f\left(X\right)$, because no finite term equals a term containing itself.

Efficiency. Robinson’s algorithm can take exponential time and space on adversarial inputs. Paterson and Wegman (1976, journal version 1978) gave a linear-time algorithm [7], and Martelli and Montanari (1982) an efficient one presented as rewriting a set of equations [8].

Where it is used today. Every Prolog system, every resolution prover, type inference in languages of the ML family, and pattern matching in rewriting engines. A known shortcut: most Prolog systems omit the occurs check by default for speed, which in rare programs lets them build cyclic terms and draw unsound conclusions. ISO Prolog offers unify_with_occurs_check/2 for when it matters.

3. Logic programming: SLD resolution, Prolog, Datalog, ASP

SLD resolution

Who and when. Robert Kowalski’s “Predicate Logic as Programming Language” (IFIP Congress, Stockholm, 1974) set out the procedural reading of Horn clauses: a rule A :- B, C. can be read as “to solve A, solve B and then C” [9]. Van Emden and Kowalski (1976) gave logic programs a matching declarative semantics: the least Herbrand model, reached as the least fixpoint of a one-step consequence operator [10]. The inference rule got its name, SLD resolution (Selective Linear Definite clause resolution), from Maarten van Emden; Apt and van Emden (1982) proved it sound and complete for definite programs [11].

How it works. A goal is a list of atoms to prove. At each step select one atom (Prolog takes the leftmost), find a program clause whose head unifies with it, replace the atom by that clause’s body, and apply the unifier to the whole goal. Success is the empty goal; the composed substitutions are the answer. Because each step resolves the current goal against an input clause, the derivation is linear, which is what makes it cheap to run.

Worked example: a Prolog derivation. The program and the query:

The same program answers questions with variables: ?- ancestor(ann, W). returns W = bill and, on backtracking, W = carl. Each answer is a substitution read off a successful branch.

Limits. SLD resolution is complete in the sense that every answer lies on some branch of the tree, but Prolog’s depth-first search can fall down an infinite branch before reaching it. Write the recursive clause as ancestor(X, Y) :- ancestor(X, Z), parent(Z, Y). and the query above never returns: the leftmost goal calls itself before consuming any data. The logic is identical; the search is not.

Prolog

Who and when. Alain Colmerauer and Philippe Roussel, with Robert Kowalski’s collaboration, in the artificial intelligence group at Aix-Marseille University. A preliminary version ran at the end of 1971 and a more definitive one at the end of 1972; Roussel named it, from PROgrammation en LOGique [12]. David H. D. Warren’s compiler for DEC-10 Prolog made it fast, and his 1983 abstract instruction set, the Warren Abstract Machine, became the standard way to implement Prolog [13]. The ISO standard is ISO/IEC 13211-1:1995 [14].

How it works. A Prolog program is a set of Horn clauses; execution is SLD resolution with the leftmost-goal selection rule, clauses tried in textual order, and chronological backtracking, as in the example above. Prolog adds what pure logic lacks for programming: arithmetic, input and output, the cut (!) for pruning the search, and negation as failure, \+ G, which succeeds when G cannot be proved. Clark (1978) explained negation as failure as reasoning over the completion of a program, in which each predicate’s clauses are read as “if and only if” [15]. Its role in default reasoning is covered on non-monotonic reasoning.

Where it is used today. SWI-Prolog, SICStus and GNU Prolog are maintained systems. Prolog is used for rule-heavy applications, parsing, teaching and research; IBM’s Watson question-answering system used it for pattern matching over parse trees. Japan’s Fifth Generation Computer Systems project built on logic programming; see the history page.

Limits. The procedural parts (cut, side effects, clause order) mean a Prolog program’s behaviour is not fully described by its logic. Negation as failure is sound only under a closed-world reading: what cannot be proved is treated as false, which is a strong assumption for incomplete knowledge.

Datalog

What it is. Datalog is logic programming restricted for databases: no function symbols, so the set of possible facts is finite, and every variable in a rule’s head must appear in its body. The field of logic and databases took shape around a 1977 workshop organised by Hervé Gallaire and Jack Minker; the name Datalog is credited to David Maier. The standard survey is Ceri, Gottlob and Tanca (1989) [16].

How it works. Datalog is evaluated bottom-up. Start from the stored facts and apply all rules to derive new ones until nothing changes. The result is the least fixpoint of the program’s immediate-consequence operator $T_P$, which is also the least Herbrand model of van Emden and Kowalski [10]:

For the ancestor program: round 1 derives ancestor(ann,bill) and ancestor(bill,carl) from the parent facts; round 2 derives ancestor(ann,carl); round 3 derives nothing new, so evaluation stops. Semi-naive evaluation joins only the facts that are new since the previous round, which avoids re-deriving the same facts. Bottom-up evaluation terminates on the left-recursive version that loops in Prolog.

Where it is used today. Program analysis is the largest modern use. Soufflé (Jordan, Scholz and Subotić, 2016) compiles Datalog to parallel C++ and runs points-to analyses such as Doop [17]; Semmle’s query language, now GitHub’s CodeQL, is an object-oriented Datalog variant. Datomic uses Datalog as its query language, and SQL:1999 added recursive queries in the same spirit.

Limits. Datalog is deliberately not Turing-complete. Evaluating a fixed program is polynomial in the data (P-complete); with the program also part of the input the problem is EXPTIME-complete. Negation and aggregation need restrictions such as stratification to keep a unique meaning.

Answer set programming

Who and when. Michael Gelfond and Vladimir Lifschitz defined stable models in 1988 [18]. In 1999 Marek and Truszczyński, and Niemelä, independently proposed using logic programs under this semantics as a paradigm for solving search problems [19] [20]; the name answer set programming (ASP) dates from the same year.

How it works. Given a candidate set of atoms $M$, form the reduct $P^M$: delete every rule with a negative literal $\mathrm{not}b$ where $b\in M$, and delete the remaining negative literals from the rest. The reduct has no negation, so it has a least model $\mathrm{Cn}\left(P^M\right)$.

Worked example. The program p :- not q. q :- not p. has two stable models. Try $M=\left\{p\right\}$: the second rule is deleted because $p$ is in $M$, the first becomes the fact p., its least model is $\left\{p\right\}$, which equals $M$. By symmetry $\left\{q\right\}$ is stable too; $\varnothing$ and $\{p,q\}$ are not. The single rule p :- not p. has no stable model at all. Each stable model is one solution. That is how ASP encodes a search problem: a choice rule generates candidates and constraints discard bad ones. Graph 3-colouring in the input language of the clingo system:

Where it is used today. Configuration, scheduling, planning, bioinformatics and diagnosis. The main solvers are clingo from the Potassco project at the University of Potsdam, whose search borrows conflict-driven learning from SAT solvers [21], and DLV.

Limits. ASP programs are first grounded, instantiated over every combination of constants, and grounding can blow up on large domains. Deciding whether a normal program has a stable model is NP-complete, so hard instances stay hard.

4. Automated theorem provers

Automated theorem provers: Otter, Vampire, E

What they are. Programs that take a set of first-order axioms and a conjecture and search, without human help, for a proof, almost always a refutation in the style of §2. The first program usually called a theorem prover is the Logic Theorist of Newell, Shaw and Simon (1956), which proved 38 of the first 52 theorems of chapter 2 of Principia Mathematica.

Who and when. Otter and its successor Prover9 were written by William McCune at Argonne National Laboratory. McCune’s related prover EQP settled the Robbins problem in 1996, showing that every Robbins algebra is a Boolean algebra, a long-standing open question; the search took about eight days [22]. Vampire, begun by Andrei Voronkov at the University of Manchester and now developed by an international team, has won the main first-order division of the annual CADE ATP System Competition (CASC) many times [23]. E, by Stephan Schulz, began at the Technical University of Munich and is built on the equational superposition calculus [24]. Provers are compared on the TPTP problem library.

How they work. Modern provers use the given-clause loop: keep a set of processed clauses and a queue of unprocessed ones; repeatedly pick the most promising unprocessed clause, draw all inferences between it and the processed set, simplify, and discard redundant results. Most of a prover’s strength is in the clause-selection heuristics, term indexing and redundancy elimination, not in the calculus.

Where they are used today. As back-ends for proof assistants (Isabelle’s Sledgehammer sends goals to provers such as E and Vampire and replays their proofs), and in hardware verification: after the 1994 Pentium division bug, AMD, Intel and others adopted theorem proving to check floating-point operations.

Limits. Semi-decidability means a prover that has not found a proof has proved nothing; time-outs are routine. Success depends on stating the problem so the search space stays small, and on axiom selection when the library is large.

5. Interactive proof assistants

Proof assistants: LCF, HOL, Isabelle, Rocq (Coq), Lean

What they are. A proof assistant is a system in which a person writes definitions, statements and proofs, and the machine checks every step. Automation fills in routine steps; the human supplies the ideas. The result is a proof certified by a small program rather than by a referee.

Who and when. Robin Milner’s LCF (Stanford, 1972; Edinburgh LCF, 1979) introduced the design most systems still use [25]. Theorems are values of an abstract type that only the inference rules can create, so any tactic, however complicated, can at worst fail; it cannot produce a false theorem. LCF also introduced the language ML to write those tactics. The HOL family and Isabelle, begun by Lawrence Paulson in 1986 [26], descend from it. Coq, based on the Calculus of Constructions of Thierry Coquand and Gérard Huet [27], was first released in 1989 by INRIA; it received the ACM Software System Award in 2013 and was renamed the Rocq Prover with version 9.0 in March 2025. Lean was started by Leonardo de Moura at Microsoft Research in 2013 [28]; Lean 4, a reimplementation that is also a programming language, appeared in 2021 [29].

How they work. Rocq and Lean are based on dependent type theory and use the Curry–Howard correspondence: a proposition is a type, and a proof is a program of that type, so checking a proof is type checking. A one-line Lean 4 proof that addition of natural numbers is commutative, by citing a library lemma:

Isabelle/HOL and HOL Light use classical higher-order logic with an LCF-style kernel; the principle is the same.

Where they are used today. Large mathematics: Georges Gonthier and Benjamin Werner formalised the four colour theorem in Coq in 2005 [30]; a Coq proof of the Feit–Thompson odd order theorem was completed in 2012 [31]; the Flyspeck project finished a formal proof of the Kepler conjecture in HOL Light and Isabelle in 2014 [32]; Lean’s community library mathlib holds a large and growing share of undergraduate and research mathematics. Verified software: the CompCert C compiler is proved correct in Rocq and the seL4 kernel in Isabelle/HOL, both covered on formal verification and program synthesis. And AI: in 2024 Google DeepMind’s AlphaProof produced Lean proofs of International Mathematical Olympiad problems at silver-medal standard, with Lean’s kernel, not the neural network, certifying each proof.

Limits. Formal proof is expensive: large developments take person-years, and the proof is only as useful as the statement. A correct proof of the wrong theorem is still wrong. Trust also rests on the kernel, the logic’s consistency and the hardware, which is why kernels are kept small and some systems have independent proof checkers.

6. Timeline

7. What these techniques cannot do

• Decide everything. First-order provers are semi-decision procedures. The decidable fragments (propositional logic, Datalog, ASP over finite domains, description logics) buy termination by giving up expressiveness.
• Escape combinatorial explosion. Resolution, SLD search, grounding and SAT all face exponential worst cases. Heuristics make typical problems tractable; they do not change the worst case.
• Write their own knowledge. A logic program knows exactly what its rules and facts say. Authoring and maintaining those rules is the knowledge-acquisition bottleneck that limited expert systems (see expert systems).
• Check the specification. A proof shows that a statement follows from axioms. Whether the statement captures what anyone wanted is outside the logic.
• Ground their symbols. parent(ann, bill) is true because it is written there; see what is a symbolic system?.

Many current systems pair these techniques with learned models: a neural network proposes proof steps, lemmas or program candidates, and a symbolic checker accepts or rejects them. That pattern is described on neuro-symbolic AI.

8. How this connects to fail-safe models

The proof assistant design is the clearest precedent for how a fail-safe model separates proposing from admitting. In LCF and its descendants, tactics can be clever, heuristic, even written by a neural network; none of that matters to soundness, because only a small kernel can mint a theorem. AlphaProof works the same way: the network searches, Lean decides. A fail-safe model applies that separation to facts and actions: a model may propose; only the floor admits a fact.

Logic programming also shows where the analogy has to be careful. Prolog’s negation as failure turns “not provable” into “false”. A fail-safe model must not make that move: when the evidence for a claim is missing, the safe answer is unknown or an abstention, not a confident negative. Closed-world reasoning is correct for a database that is complete by construction; it is a failure mode when the knowledge is partial.

Peel, by Perslis Research, is to our knowledge the first fail-safe model. Its knowledge is typed, sourced cards, its learning is readable counts, and there is no neural network in the loop that decides. It is a research prototype, not a certified safety system. How Perslis uses symbolic methods more broadly is on symbolic AI at Perslis, and our paper Traversing Data in Symbolic Systems treats typed-relation traversal, a close relative of the Datalog fixpoint above, as a retrieval primitive [33].

9. Questions

What is logic programming?

Logic programming is a style of programming in which a program is a set of logical sentences, usually facts and if-then rules, and running it means proving a query from them. Prolog, Datalog and answer set programming are the main logic programming languages. The answer to a query is the substitution, or the set of facts, that makes the query true.

What is resolution in artificial intelligence?

Resolution is a single inference rule for clause-form logic, introduced by J. A. Robinson in 1965. From two clauses containing complementary literals it derives a new clause combining the rest, after unifying the literals. A set of clauses is unsatisfiable exactly when resolution can derive the empty clause, so theorems are proved by refuting their negation.

What is unification?

Unification is the process of finding a substitution for variables that makes two terms identical. For example, p(X, f(Y)) and p(a, f(X)) unify with X = a and Y = a. When two terms are unifiable there is a most general unifier. Unification is the matching step inside resolution, Prolog and type inference.

Is Prolog still used?

Yes. SWI-Prolog, SICStus Prolog and GNU Prolog are actively maintained, Prolog has an ISO standard, and it is used for rule-based systems, parsing, teaching and research. IBM Watson used Prolog for pattern matching over parse trees. Its ideas also live on in Datalog engines used for program analysis.

What is the difference between Prolog and Datalog?

Datalog is a restricted form of logic programming with no function symbols, evaluated bottom-up from the facts to a fixpoint. Every Datalog program terminates and its result does not depend on rule order. Prolog allows function symbols, is Turing-complete and runs top-down with depth-first search, so a logically correct Prolog program can still loop.

What is answer set programming?

Answer set programming is a form of declarative programming for search problems, based on the stable model semantics of Gelfond and Lifschitz (1988). A program describes candidate solutions with choice rules and rules out bad ones with constraints; a solver such as clingo returns the stable models, each of which is one solution.

What is the difference between an automated theorem prover and a proof assistant?

An automated theorem prover such as Vampire or E searches for a proof on its own and either finds one or gives up. A proof assistant such as Rocq, Isabelle or Lean lets a person write the proof interactively while a small trusted kernel checks every step. Proof assistants often call automated provers for routine subgoals.

Can a computer decide whether any first-order statement is true?

No. Church and Turing proved in 1936 that first-order validity is undecidable. It is semi-decidable: a prover can eventually confirm every valid statement, but on a statement that is not valid it may run forever. Decidable fragments such as propositional logic and Datalog trade expressiveness for guaranteed termination.

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